Online Quantile Regression
Yinan Shen^1, Dong Xia^1, Wen-Xin Zhou^2

^1Department of Mathematics, Hong Kong University of Science and Technology

^2Department of Information and Decision Sciences, University of Illinois at Chicago

[1]Dong Xia’s research was partially supported by Hong Kong RGC Grant GRF 16302323.

This paper addresses the challenge of integrating sequentially arriving data within the quantile regression framework, where the number of features is allowed to grow with the number of observations, the horizon is unknown, and memory is limited. In contrast to least squares and robust regression methods, quantile regression models different segments of the distribution, accommodating variations in the relationship between predictors and the response, thereby providing a more comprehensive view of the entire conditional distribution. We employ stochastic sub-gradient descent to minimize the empirical check loss and study its statistical properties and regret performance. In our analysis, we unveil the delicate interplay between updating iterates based on individual observations versus batches of observations, revealing distinct regularity properties in each scenario. Our method ensures long-term optimal estimation irrespective of the chosen update strategy. Importantly, our contributions go beyond prior works by achieving exponential-type concentration inequalities and attaining optimal regret and error rates that exhibit only  sensitivity to initial errors. A key insight from our study is the delicate statistical analyses and the revelation that appropriate stepsize schemes significantly mitigate the impact of initial errors on subsequent errors and regrets. This underscores the robustness of stochastic sub-gradient descent in handling initial uncertainties, emphasizing its efficacy in scenarios where the sequential arrival of data introduces uncertainties regarding both the horizon and the total number of observations. Additionally, when the initial error rate is well-controlled, there is a trade-off between short-term error rate and long-term optimality. Due to the lack of delicate statistical analysis for squared loss, we also briefly discuss its properties and proper schemes. Extensive simulations support our theoretical findings.

§ INTRODUCTION

Online learning aims to effectively incorporate sequentially arriving data and make timely predictions. In contrast to offline learning, where all data is stored on a server or machine prior to the analysis, online learning does not exploit the entire sample pool simultaneously, thereby alleviating both storage and computation pressure. Consequently, online approaches are more applicable to large-scale datasets [LeCun et al., 1989, Rajalakshmi et al., 2019, Finn et al., 2019], streaming asset pricing data [Soleymani and Paquet, 2020], and find applications in the increasingly popular field of reinforcement learning [Gao et al., 2019, Han et al., 2022, Ren and Zhou, 2023]. Further applications of online learning can be found in [Bottou, 1998, Hazan, 2016, Orabona, 2019, Cesa-Bianchi and Lugosi, 2006, Hoffman et al., 2010].

In classical offline linear regression, estimation and inference are based on a pre-collected sample of independent observations $\{(\mathbf{X}_i,Y_i)\}_{i=1}^n$ satisfying $Y_i=\mathbf{X}_i^{\top}\boldsymbol{\beta}^* +\xi_i$, where $\boldsymbol{\beta}^* \in\mathbb{R}^d$ is the unknown parameter of interest, and $\xi_i$ denotes the random noise variable, satisfying $\mathbb{E}(\xi_i|\mathbf{X}_i) = 0$. It is well-known that the ordinary least squares estimator $\hat{\boldsymbol{\beta}}$ achieves the minimax rate of convergence, i.e., $\mathbb{E} \| \hat{\boldsymbol{\beta}} - \boldsymbol{\beta}^* \|_2^2=O(\sigma^2 d /n)$, where $\sigma^2$ is such that $\mathbb{E}(\xi_i^2|\mathbf{X}_i) \leq \sigma^2$ almost surely. In contrast, the entire sample is inaccessible in the online setting, and the total number of observations may even be unknown. Specifically, at time $t$, only $n_t$ pairs of observations can be used:

\begin{align}
Y_i^{(t)}=\X_i^{(t)\top}\Bbeta^*+\xi_i^{(t)},\quad i=1,\dots,n_t. \label{eq:model}
\end{align}

While the data are received sequentially, online learning refrains from repeatedly using the data for model refitting. Instead, it sequentially updates the estimator. In contrast, offline methods leverage the entire sample, and based on the available data, the offline method [He and Shao, 2000, Koenker, 2005] leads to a statistically optimal estimator with mean squared error of order $O(\sigma^2 \cdot d/\sum_{t=0}^{T}n_t)$, where $T$ is called the horizon.
This prompts the question of whether online methods can achieve error rates comparable to their offline counterparts. Must we compromise accuracy for more efficient computation and reduced storage in online learning? Furthermore, with the accumulation of more information over time, the theoretical offline error rates decline. Is it feasible to sustain this downward trend in optimal error rates over time? Particularly when the horizon (maximum time $T$) is unknown, isolated error rates at specific times lack persuasiveness.

Often, an online learner must make timely predictions as data arrive sequentially, either one at a time or in batches.
The concept of Regret, originating from [Savage, 1951] and prominently featured in various online studies such as [Goldenshluger and Zeevi, 2013, Ren and Zhou, 2023, Han et al., 2020], serves as a reflection of prediction accuracy throughout the learning procedure. Regret at the horizon $T$ is defined as

\begin{align}
\textsf{Regret}_T:=\max_{\Bbeta}\EE \Bigg\{ \sum_{t=0}^{T} f_{t}(\Bbeta_{t})-f_t(\Bbeta) \Bigg\} ,
\label{eq:regret}
\end{align}

where $f_t(\cdot)$ and $\Bbeta_{t}$ denote the loss function and estimator at time $t$, respectively. The expectation is taken over the data from $t=0$ to $T$. Regret serves as a metric to evaluate the efficacy of a sequential estimation scheme. Optimal regret occurs when error rates exhibit a scaling behavior of $1/t$ as time $t$ becomes sufficiently large [Hazan, 2016]. In contrast, [Cai et al., 2023] assumes a known horizon and focuses more on the ultimate accuracy of estimation. However, this setting bears resemblance to the sample splitting technique [Xia and Yuan, 2021, Zou et al., 2020].

Online estimation with the squared loss in equation (<ref>) has been extensively studied over the last two decades. See, for example, [Bottou, 1998], [Zinkevich, 2003], [Hazan et al., 2007] and [Langford et al., 2009], among others.
Stochastic gradient descent stands out as a natural and elegant methodology for handling sequentially arriving data, integrating streaming information, and minimizing the regret in (<ref>). [Bottou, 1998] laid the foundation for the general theoretical framework of online learning, demonstrating the asymptotic convergence of gradient descent toward extremal points. Under certain conditions, the seminal work of [Zinkevich, 2003] establishes the iterative convergence of gradient descent with stepsize $O(t^{-1/2})$. [Hazan et al., 2007] first demonstrated that employing a stepsize of $O(t^{-1})$ results in optimal $O(\log T)$ regrets. The pioneering online work by [Duchi and Singer, 2009] examines the performance of iterative sub-gradient descent when applied to a non-differentiable loss function. The authors provide insights into convergence dynamics and establish bounds on regrets for various stepsize schemes.
We refer to [Hazan et al., 2007], [Langford et al., 2009], [Streeter and McMahan, 2010], [McMahan and Streeter, 2010], [Duchi et al., 2011], and references in [Orabona, 2019] for more gradient descent-based algorithms in online learning. In the context of offline learning, stochastic (sub-)gradient descent offers a means to alleviate the computational burden [Zhang, 2004]. This method involves updating the current iterate using only a single randomly selected observation pair, as opposed to incorporating the entire sample.
[Rakhlin et al., 2011] introduces a stepsize scheme of $O(1/t)$, resulting in a sub-linear convergence rate equivalent to that of the online counterpart for a strongly convex objective. In offline stochastic optimization, the total sample size is known, enabling the incorporation of this information into stepsize schemes. As a result, the statistical optimality remains constant over time.
We refer to [Duchi and Singer, 2009], [Cai et al., 2023], and [Puchkin et al., 2023] for examples of constant stepsize schemes that take into account the horizon.
[Delyon and Juditsky, 1993], [Johnson and Zhang, 2013], and [Roux et al., 2012] focus on improving the sub-linear convergence rate of stochastic descent. While offline stochastic gradient descent and online learning share the ability to alleviate computational pressure, they exhibit notable distinctions. In online learning, the ultimate horizon and the number of observations remain unknown and may be infinite. Therefore, in online learning, the pursuit is directed toward consistently achieving dynamic, statistically optimal error rates. Moreover, online learner often needs to predict the response for an incoming covariate-only input. The prediction accuracy of an online learning algorithm is characterized by regret.

The delicate statistical analysis of squared loss remains a crucial area of study. Although online learning and (stochastic) gradient descent techniques have been extensively explored from an optimization perspective for both squared and non-differentiable loss functions, the literature has largely assumed the empirical loss function to have some particular regularity properties, such as strong convexity and/or smoothness. Additionally, the overall domain has often been restricted to a bounded convex region [Hazan et al., 2007, Duchi and Singer, 2009, Langford et al., 2009, Rakhlin et al., 2011]. A comprehensive discussion of online learning algorithms from an optimization perspective can be found in the review papers by [Hazan, 2016] and [Orabona, 2019]. However, the empirical strong convexity does not necessarily hold even when the squared loss is used if the storage size is relatively small. A delicate analysis from the statistical perspective is necessary to overcome these challenges. See, for instance, [Chen et al., 2019], [Han et al., 2023] and references therein.  [Fan et al., 2018] shows that the expected error rate of order $O(\log (T)/T)$ is attainable for sparse online regression under the $O(1/t)$ stepsize scheme, leading to an optimal $O(\log T)$ regret. [Cai et al., 2023] studies online tensor learning and proposes a multi-phase online gradient descent algorithm.
The step size is locally constant within each phase and of order $O(t^{-1})$ in the long term, resulting in the desired decreasing error rates of $O(1/t)$ at the end of each phase. Their approach achieves a linear convergence during each phase and eventually delivers an estimator that is statistically optimal with respect to the total sample size and achieves a desired $O(\log T)$ regret. However, in their schemes, the initial error has a lasting influence on the regrets, and the error rate is not scalable to the noise.

The preceding statistical online frameworks reveal shortcomings when addressing quantile regression, leading to suboptimal error rates and weakened success probability guarantees, especially under heavy-tailed noise. This paper aims to amend these limitations in quantile regression. Section <ref> offers a complementary discussion on online least squares and its stepsize scheme. To fit conditional quantile models sequentially,  we employ a straightforward yet effective sub-gradient descent method. Furthermore, we provide delicate statistical analyses to design a stepsize sequence that demonstrates geometric decay, remains constant, or decays at a rate of $1/t$, depending on the proximity of the iterate to the oracle. This approach deviates from the scheme proposed by [Duchi and Singer, 2009], and a thorough comparison between the two is elaborated in Section <ref>, underlining the advantages of the proposed statistical scheme. In the context of heavy-tailed noise, we establish exponential-type tail bounds for the online estimators introduced in this paper, thereby surpassing the polynomial bounds obtained in [Han et al., 2022].  This improvement in tail bounds attests to the robustness and reliability of the proposed online estimators when employed with heavy-tailed data.

In this work, we refer to online learning as the scenario in which data arrives sequentially. The complete dataset and the total number of observations are never available simultaneously, regardless of how many observations can be used each time. Under the linear model (<ref>), we discuss three settings: when the newly arriving dataset $\calD_t := \{\X_i^{(t)}, Y_i^{(t)}\}_{i=1}^{n_t}$ contains only one observation pair (requiring $d$ units of storage), when it contains at least $O(d)$ samples, and when the server can store an infinite number of samples, respectively. The first setting has been extensively studied [Duchi and Singer, 2009, Hazan et al., 2007, Hazan, 2016, Cai et al., 2023, Langford et al., 2009, Mhammedi et al., 2019, Han et al., 2022] and still necessitates careful statistical analysis, as mentioned earlier. In addition, [Do et al., 2009] discusses the arrival of a batch of data each time. However, there is no significant difference between a single sample arriving and a batch arriving from an optimization standpoint, as regularity properties are inherently assumed in most existing literature.  From a statistical perspective, we will demonstrate that these two settings exhibit significantly different properties. More specifically, this paper aims to answer the following series of questions: Is it possible to achieve a continuously decreasing error rate as new data arrive under constrained storage? Furthermore, is it possible to achieve optimality along the way? How much information is lost compared with offline approaches? In online learning, failure probabilities accumulate across iterations, influencing the maximum horizon for guaranteed convergence and underscoring the importance of the success probability for each iteration. The combined presence of heavy-tailed noise and the arrival of a single data at each instance highlights the challenges in achieving a strong success probability guarantee. Moreover, what is the role of initialization? Does the initial error rate necessarily have a long-term effect on the error rate or regrets? Is it possible to allow arbitrary initialization?
We summarize our contributions toward addressing these questions as follows.

* This paper introduces a statistical analysis framework for online quantile regression, addressing the challenge posed by the empirical loss function lacking strong convexity and smoothness.  To attain long-term statistical optimality under heavy-tailed noise, the optimal stepsize scheme should be guided by the statistical regularity properties of quantile loss. In contrast to prior works, our established error rates scale proportionally with the noise level. Moreover, the online approach strategically compromises a slight reduction in statistical accuracy for enhanced computational efficiency and reduced storage requirements compared to offline works. The negligible reduction in accuracy becomes inconsequential when the time horizon is sufficiently large, as demonstrated in Section <ref>.

* Our algorithm achieves statistically optimal rates even in the presence of heavy-tailed noise, and admits a high probability guarantee where the failure probability decays exponentially fast in the dimension. This implies the validity of our theory even when the unknown horizon grows exponentially fast. For example, in the scenario where only one datum arrives each time, the failure probability of our established error rate is $O\big(\exp(-c_0d)\big)$, where $c_0>0$ is a universal constant. This allows for an unknown horizon $T$ as large as $T=O(\exp(c_0d))$. In sharp contrast, the error rates established by [Cai et al., 2023] and [Han et al., 2022] hold with probability $1-d^{-O(1)}$ under sub-Gaussian noise, implying that their allowed horizon should not exceed $d^{O(1)}$.

* Our analysis reveals that the initial error has a transient impact on both estimation accuracy and regret. Consequently, the statistical error bounds obtained are applicable to arbitrary initializations, eliminating the necessity of confining them to a compact region. This is achieved through careful statistical analyses and the introduction of a newly proposed stepsize scheme. Stochastic (sub-)gradient descent methods for both quantile loss and squared loss benefit significantly from a stepsize scheme designed based on statistical properties. For instance, in the scenario where $\xi_i^{(t)} \sim N(0, \sigma^2)$ and $n_t=1$ in (<ref>), we establish a regret bound of $\sum_{t=0}^T\EE\|\Bbeta_t-\Bbeta^*\|^2\leq O(\|\Bbeta_{0}-\Bbeta^*\|_2^2 + \sigma^2\log T)$. In contrast, [Cai et al., 2023] demonstrates a regret of $O(\lambda^2 \log T)$, where $\lambda>\sigma\vee \ltwo{\Bbeta_0-\Bbeta^*}$ denotes the signal strength. Notably, this implies a substantially larger regret than the one obtained in this work.

The rest of this paper is organized as follows.  Section <ref> introduces the quantile/check loss function, alongside the online sub-gradient descent algorithm. It further elaborates on the convergence dynamics and the regret bound, delving into detailed discussions of the three different settings. In Section <ref>, we conduct extensive numerical experiments that include evaluating the stepsize scheme, comparing accuracy with offline regressions, and analyzing convergence dynamics. A brief discussion of online least squares from a statistical perspective is also presented. The proof of Theorem <ref> is provided in Section <ref>, while the proofs of the remaining theorems are deferred to the supplementary material.

§ ONLINE QUANTILE REGRESSION VIA ADAPTIVE SUB-GRADIENT DESCENT

Throughout this section, we consider the conditional quantile model (<ref>), that is, the conditional $\tau$-th quantile of $\xi_i^{(t)}$ given $\X_i^{(t)}$ is zero for some predetermined $\tau \in (0, 1)$. Quantile regression (QR) plays a crucial role in unraveling pathways of dependence between the outcome and a collection of features, which remain elusive through conditional mean regression analysis, such as the least squares estimation. Since its introduction by [Koenker and Bassett Jr, 1978], QR as undergone extensive study, encompassing theoretical understanding, methodological development for various models and data types, and practical applications in a wide range of fields. The main focus of existing literature centers around the formulation of methodologies and theories for QR utilizing static data, where a complete dataset is available. This is referred to as the offline setting in online learning literature. The most fundamental and well-studied method for estimating $\Bbeta^*$ involves empirical risk minimization, or statistical $M$-estimation. The associated loss function $\rho_{Q,\tau}(x):=\tau x\mathbbm{1}(x\geq 0)+(\tau-1)x\mathbbm{1}(x<0)$ is known as the check loss or quantile loss. We refer to [Koenker, 2005] and [31] for a comprehensive exploration of offline quantile regression, covering statistical theory, computational methods, as well as diverse extensions and applications.

In contrast to the least squares method, the non-smooth nature of the quantile loss introduces additional challenges to QR, particularly in the era of big data. Established algorithms employed for this purpose include the simplex algorithm [Koenker and d'Orey, 1987], interior point methods with preprocessing [Portnoy and Koenker, 1997], alternating direct method of multipliers, and proximal methods [Parikh et al., 2014]. More recently, [Fernandes et al., 2021] and [He et al., 2023] have demonstrated that convolution-smoothed quantile regression attains desired asymptotic and non-asymptotic properties, provided that the smoothing parameter (bandwidth) is appropriately selected as a function of the sample size and dimensionality.
When addressing low-rank matrix regression/completion problems using either absolute loss or quantile loss, recent studies by [Cambier and Absil, 2016], [Li et al., 2020], [Charisopoulos et al., 2021], [Tong et al., 2021], [Ma and Fattahi, 2023]  and [Shen et al., 2023] have extensively examined the sub-gradient descent method from both computational and statistical standpoints. The offline methods mentioned above crucially hinge on specific regularity properties inherent in the empirical loss function. However, in the online setting, characterized by sequential data arrival and an unknown total number of observations, the assurance of offline regularity properties becomes untenable with only a limited number of available samples. Consequently, neither the offline results nor the corresponding proof techniques are applicable in this online context.
As an illustration, in an offline setting based on model (<ref>), [Shen et al., 2023] demonstrated the existence of certain parameters $\mu_1, \mu_2>0$ such that with high probability,

\frac{1}{n}\sum_{i=1}^n ( \vert Y_i-\inp{\X_i}{\Bbeta} \vert- \vert Y_i-\inp{\X_i}{\Bbeta^*} \vert )\geq \max\{\mu_1\ltwo{\Bbeta-\Bbeta^*}-\gamma, \mu_2\ltwo{\Bbeta-\Bbeta^*}^2\}

holds for all $\Bbeta$ under the sample size requirement $n\geq Cd$, $\gamma=\EE|\xi|$ is the first absolute moment of the noise variable. However, in the context of online learning, where the stored data is limited, such as when $n=1$, the above inequality does not hold in general.

The foundational contributions to online learning by [Duchi and Singer, 2009], [Duchi et al., 2011] and [Johnson and Zhang, 2013] employ sub-gradient descent to address scenarios characterized by the lack of differentiability in the loss function at specific points. These studies establish the properties of iterates under various stepsize schemes, with a primary emphasis on the excess risk, leaving the analysis of estimation error  $\ltwo{\Bbeta_t-\Bbeta^*}$ unaddressed. In the absence of strong convexity and/or certain smoothness, the upper bounds on the excess risk cannot be straightforwardly extended to those on the estimation error.
From the statistical viewpoint, [Jiang and Yu, 2022] and [Wang et al., 2022] consider online quantile regression within the batch learning framework in which the data arrive in batches, and establish their asymptotic normality as the batch size goes to infinity. It is worth noticing that the requirement for diverging batch sizes renders these methods impractical in scenarios where only one observation or a few observations arrive at a time.

This paper is dedicated to the exploration of online quantile regression with unknown horizon and sample sizes (batch or total). We aim to provide a non-asymptotic analysis of online sub-gradient descent. By using a customized stepsize scheme with explicit dependencies on dimensionality, sample size, and noise scale, we establish optimal rates of convergence for online QR estimators. Additionally, we seek to demonstrate that near-optimal regret performance can be achieved. Let $\calS_t$ be the set of observations acquired at time $t$, which will be used to update the current iterate $\Bbeta_{t}$. Consequently, the loss function at $t$ is given by

f_t(\Bbeta):=\sum_{(\X_i^{(t)},Y_i^{(t)})\in\calS_t}\rho_{Q,\tau}(Y_i^{(t)}- \langle \X_i^{(t)}, \Bbeta \rangle).

We will explore three settings: (i) $\calS_t=\calD_t$, containing only one data vector, referred to as the online learning setting, (ii) $\calS_t=\calD_t$, comprising a set of observations with a size of at least $O(d)$, termed as the online batch learning setting, and (iii) $\calS_t$ containing all the data vectors accumulated up to time $t$, expressed as $\calS_t=\cup_{l=0}^t\calD_l$, recognized as the sequential learning with infinite storage setting.

Online sub-gradient descent. Initiated at some $\Bbeta_0$, the online QR estimates are iteratively defined via sub-gradient descent as

\Bbeta_{t+1}=\Bbeta_{t}-\eta_t\g_t, \quad t= 0, 1, \ldots,

where $\g_t\in\partial f_t(\Bbeta_{t})$ is the sub-gradient of $f_t$ at $\Bbeta_{t}$, and $\{ \eta_t \}_{t=0, 1, \ldots }$ constitutes a sequence of stepsize parameters (learning rates). The loss function varies across different settings, exhibiting distinct regularity properties. Consequently, tailored stepsize schemes are imperative in the three settings to attain the desired convergence properties. We will show that these customized stepsize schemes yield statistically optimal estimators and achieve near-optimal regret performances. Importantly, these schemes effectively adapt to varying dimensions and unknown horizons.

Prior to presenting the theoretical guarantees and convergence dynamics, we outline the essential assumptions concerning the model.

[Random covariate]
The covariate vector $\X\in\RR^d$ follows the Gaussian distribution $N(\boldsymbol{0},\bSigma)$, where $\bSigma$ is symmetric and positive definite. There exist absolute constants $\kl,\ku>0$ such that $\kl\leq\lambda_{\min}(\bSigma)\leq\lambda_{\max}(\bSigma)\leq\ku$.

[Noise distribution]
Given $\X \in \RR^d$, the noise variable $\xi$ has the conditional density function $h_{\xi|\X}(\cdot)$ and distribution function $H_{\xi|\X}(\cdot)$, respectively. The conditional $\tau$-th quantile of $\xi$ given $\X$ is zero, i.e., $H_{\xi|\X}(0)=\tau$.
There exists some $\varepsilon>0$ such that $\EE |\xi|^{1+\varepsilon}<+\infty$. Let $\gamma=\EE|\xi|$.  There exist constants $b_0, b_1>0$ (possibly depending on $\gamma$) such that

\begin{align*}
b_0^{-1} \leq \inf_{|x|\leq 8(\ku/\kl)^{1/2}\gamma} h_{\xi|\X}(x) \leq  \sup_{x\in \RR} h_{\xi|\X}(x)\leq b_1^{-1}.
\end{align*}

The finite $(1+\eps)$-th moment condition on $\xi$ is primarily needed to ensure that $\lim_{x\to +\infty}x \{ 1-H_{\xi|\X}(x) \}=0$. Both local lower bounds and global upper bounds on the (conditional) density $h_{\xi|\X}$ are commonly imposed in the QR literature, especially in non-asymptotic settings (e.g., [Belloni and Chernozhukov, 2011]).

§.§ Online Learning

The server receives a single new data point $(\X_t,Y_t)$ following the linear model (<ref>). The online sub-gradient descent algorithm calculates the corresponding sub-gradient $\g_t \in \partial f_t(\Bbeta_t)$ and subsequently updates the current estimate ($\Bbeta_t$) in the direction of the negative sub-gradient, all performed without storing this new observation. Here $f_t(\Bbeta)=\rho_{Q,\tau}(Y_t-\X_t^{\top}\Bbeta)$ denotes the loss function at time $t$.

The expected excess risk $\EE[f_t(\Bbeta)-f_t(\Bbeta^*) ]$, as a function of $\Bbeta$, demonstrates a two-phase regularity property, visually depicted in Figure <ref>. When $\Bbeta$ is distant from the population risk minimizer $\Bbeta^{\ast}$, the expected excess risk exhibits a first-order lower bound, decaying linearly in $\| \Bbeta-\Bbeta^{\ast} \|_2$. However, as $\Bbeta$ approaches proximity to $\Bbeta^{\ast}$, a quadratic lower bound emerges concerning $\|\Bbeta-\Bbeta^{\ast}\|_2$. Importantly, it is noteworthy that, in contrast to offline works such as [Tong et al., 2021, Shen et al., 2023], this regularity property pertains to the expectation, given that the empirical loss is based on only one single observation.
The empirical loss lacks a guaranteed high probability concentration property; for instance, the variance of $f_t(\Bbeta)$ may significantly overshadow its expectation. As a result, the commonly observed monotone-decreasing trend in the estimation error, such as $\ltwo{\Bbeta_{t+1}-\Bbeta^* }\leq \ltwo{\Bbeta_{t}-\Bbeta^* }$ in offline learning algorithms, does not occur. In fact, a more nuanced analysis is essential to understand the convergence dynamics of online sub-gradient algorithms for quantile regression.

Expected excess risk of the objective function.
$Y$-axis: lower bound of $\EE[f_t(\Bbeta)-f_t(\Bbeta^*) ]$; $X$-axis: the value of $\|\Bbeta-\Bbeta^{\ast}\|_2$. It shows that the lower bound varies from a linear to quadratic dependence on $\|\Bbeta-\Bbeta^{\ast}\|_2$ as $\Bbeta$ gets closer to $\Bbeta^*$.

Interestingly, the following theorem illustrates that, despite the online sub-gradient descent algorithm's inability to ensure error contraction as $\ltwo{\Bbeta_{t+1}-\Bbeta^* }<\ltwo{\Bbeta_{t}-\Bbeta^* }$, the error rates $\|\Bbeta_t-\Bbeta^{\ast}\|_2$ can be bounded by a monotone-decreasing sequence with high probability. The convergence of the online sub-gradient descent algorithm unfolds in two distinct phases, as suggested by the two-phase regularity properties depicted in Figure <ref>.

Under Assumptions <ref> and <ref>, there exist universal positive constants $c_0,c_1,c_2,c_3,C_0$ such that, for an arbitrary initialization $\Bbeta_{0}$, the sequence $\{\Bbeta_t\}_{t\geq  1}$ generated by the online sub-gradient descent algorithm follows the dynamics outlined below:

* in phase one, i.e., when $\ltwo{\Bbeta_{t}-\Bbeta^*}\geq 8 \kl^{-1/2}\gamma$, by selecting a stepsize $\eta_t\in [c_1,c_2]\cdot  \frac{\sqrt{C_l}}{C_u \max(\tau, 1-\tau)^2 } \frac{1}{d} \ltwo{\Bbeta_{t}-\Bbeta^*} $, it holds with probability at least  $1-\exp(-c_0d)$ that

\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq C_0\left(1-c_3\frac{\kl}{\ku}\frac{1}{\max(\tau , 1-\tau)^2 d}\right)^{t+1}\cdot\ltwo{\Bbeta_{0}-\Bbeta^*}^2;

the conclusion of phase one occurs after $t_1=O(d\log(\ltwo{\Bbeta_{0}-\Bbeta^*}/\gamma))$ iterations.

* in phase two, i.e., when $\ltwo{\Bbeta_{t}-\Bbeta^*}< 8\sqrt{\kl^{-1}}\gamma$, by choosing the stepsize $\eta_t=(b_0/\kl)\cdot(\ca/(t-t_1+\cb d))$ with certain constants $\cb>\ca>12$, it holds with probability exceeding $1-\exp(-c_0d)$ that

\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq C^*\frac{\ku}{\kl^2}\frac{d}{t+1-t_1+\cb d} b_0^2,

where the constant $C^*$ depends on $\ca,\cb$.

In accordance with Theorem <ref>, the algorithm exhibits linear convergence during the first phase, concluding after $t_1$ iterations. This phase requires a total of $t_1$ observations and employs an approximately geometrically decaying stepsize scheme. In the second phase, the algorithm converges sub-linearly, employing a  $O(1/(t+d))$ decaying stepsize scheme. The failure probability in both phases decays exponentially in $d$.
To our knowledge, our result stands as the first instance in online learning literature to provide an exponential-type high probability bound. Previous works employing the squared loss generally guarantee a polynomial-type tail probability, even under sub-Gaussian noise [Cai et al., 2023, Han et al., 2022, Bastani and Bayati, 2020].  This enhancement holds significant implications for practical applications. It suggests that our algorithm remains applicable even when the ultimate horizon $T$ is as large as  $e^{O(d)}$. Theorem <ref> asserts that, provided $d\ll T\ll e^{c_0d}$, the final estimator $\Bbeta_T$, with high probability, attains an error rate that is minimax optimal under offline settings. Online methods can adapt to the unknown horizon, and offer substantial reductions in computation costs, as well as savings in memory and storage.

The final error rate achieved after the second phase iterations is proportional to the noise level, independent of the initialization error. In existing works, the initialization error often exerts a lasting impact on the final error rate, even when using the squared loss. For instance, the online algorithm [Fan et al., 2018] obtains an expected error rate $\EE\ltwo{\Bbeta_T-\Bbeta^*}^2\leq d\ltwo{\Bbeta_0-\Bbeta^*}^2\log (T)/T$. Moreover, the error rate  established by [Cai et al., 2023] is $\ltwo{\Bbeta_T-\Bbeta^*}^2\leq d\ltwo{\Bbeta_0-\Bbeta^*}^2/T$, which holds with high probability as long as $T\ll {\rm poly}(d)$. Theorem <ref> reveals that the initialization error only has a short-time effect in the online sub-gradient descent algorithm, dissipating after the second phase iterations begin.  This two-phase convergence phenomenon is elucidated through a meticulous analysis of the algorithm's dynamics. It is worth noting that the two-phase convergence and the transient impact of initialization error are not exclusive to quantile regression; they also manifest in least squares, as per our analysis framework. A comprehensive discussion of these properties for online least squares will be presented in Section <ref>. Numerical simulations in that section demonstrate that a two-phase stepsize scheme yields significantly improved accuracy.

Assume that the same conditions as in Theorem <ref> hold, and let the horizon satisfy $T\geq C^*d \max\{1, \log(\sqrt{\kl}\ltwo{\Bbeta_{0}-\Bbeta^*}/\gamma)\}$ with a constant $C^{\ast}$ depending on $(\kl,\ku,\ca,\cb)$. Then, with probability at least $1-T\exp(-c_0d)$, the online sub-gradient descent algorithm produces a final estimate with an error rate

\ltwo{\Bbeta_T-\Bbeta^*}^2\leq C_3\frac{d}{T}b_0^2,

where $C_3>0$ is a constant.

Corollary <ref> demonstrates that the online sub-gradient descent algorithm attains the minimax optimal error rate, provided the horizon is not too small. This makes it a favorable choice over the offline approach, considering the benefits of reduced computation and storage costs. However, it is advisable to opt for offline approaches in cases where the horizon is small. A comprehensive examination of their numerical performances is conducted in Section <ref>.

Suppose the initialization is already situated in the second phase region, i.e., $\ltwo{\Bbeta_{0}-\Bbeta^*}< 8 \kl^{-1/2}\gamma$. It is advisable to initiate the phase two iterations directly ($t_1=0$), where the choice of the parameter $C_a$ plays a pivotal role in determining short-term accuracy and long-term optimality. Selecting $C_a>12$, according to Theorem <ref>, the updated estimates achieve an error rate of $\ltwo{\Bbeta_{t}-\Bbeta^*}^2\leq C^*\frac{\ku}{\kl^2}\frac{d}{t+\cb d} b_0^2$ with high probability. This upper bound may exceed the initialization error when $t$ is small, indicating potential accuracy fluctuations in the early stage. Nevertheless, the algorithm eventually converges and yields a statistically optimal estimator.
On the contrary, opting for a sufficiently small parameter $\ca<\min\{12, (\kl/\sqrt{\ku})\ltwo{\Bbeta_{0}-\Bbeta^*}/b_0\}$ results in a more stable convergence in the early stage. However, it comes at the cost of sacrificing long-term accuracy, as the updated estimates possess an error bound

\ltwo{\Bbeta_{t}-\Bbeta^*}^2\leq \bar{C}^*\left(\frac{d}{t+\cb d}\right)^{\frac{\ca}{12}}\cdot \ltwo{\Bbeta_{0}-\Bbeta^*}^2,

achieving a sub-optimal error rate in the end.
This phenomenon is illustrated in Figure <ref>. Specifically, in Figure <ref>, when the initialization satisfies $\ltwo{\Bbeta_0-\Bbeta^*}\leq 0.2\EE|\xi|$, it demonstrates that a small $C_a$ ensures short-term accuracy but compromises long-term optimality. On the other hand, a large $C_a$ guarantees long-term optimality at the expense of short-term accuracy. Figure <ref> presents even more pronounced differences, where the initialization satisfies $\ltwo{\Bbeta_0-\Bbeta^*}\leq 0.02\EE|\xi|$.

Trade-off between short-term accuracy and long-term optimality. The initialization $\Bbeta_0$ is already in proximity to  $\Bbeta^{\ast}$, enabling the online sub-gradient descent algorithm to bypass the first phase and start the second phase iterations immediately. A small stepsize can ensure short-term accuracy but may compromise losing long-term optimality, whereas a large stepsize can guarantee long-term optimality but at the expense of short-term accuracy. Here, the dimension is set to $d=100$, and the noise follows a Student $t_{\nu}$ distribution with $\nu=1.1$.

Under the same conditions and stepsize scheme as in Theorem <ref>, the online sub-gradient descent algorithm achieves an upper bound on regret given by

\begin{align*}
\textsf{Regret}_T\leq C_1(\ku/\kl)\sqrt{\ku} \, d\ltwo{\Bbeta_{0}-\Bbeta^*}+C_2\ca\frac{\ku^2}{\kl^2}\frac{b_0^2}{b_1}d\log\bigg(1+ \frac{T }{\cb d}\bigg),
\end{align*}

for any $T$.

The regret upper bound exhibits logarithmic growth with respect to the horizon $T$, aligning with the well-established optimal regret bounds in online optimization [Hazan et al., 2007, Orabona, 2019]. The other term in the regret upper bound is solely dependent on the initialization error and remains unaffected by the horizon $T$. This indicates that the initialization error does not exert a persistent influence over the regret. In contrast, regret upper bounds established by [Hazan et al., 2007, Orabona, 2019, Cai et al., 2023, Han et al., 2022] all include the multiplicative term $\ltwo{\Bbeta_{0}-\Bbeta^*}^2\log T$, suggesting that the regret could become considerably large as a consequence of bad initialization.

§.§ Batch Learning

Suppose the server receives a batch of data points $\calD_t=\{(\X_i^{(t)},Y_i^{(t)})\}_{i=1}^{n_t}$ at each time $t$, comprising $n_t$ pairs of i.i.d. observations that follow the linear model (<ref>). The  loss function at $t$ is

\begin{align}
f_t(\Bbeta)=\frac{1}{n_t}\sum_{i=1}^{n_t} \rho_{Q,\tau}(Y_i^{(t)}-\X_i^{(t)\top}\Bbeta),
\label{eq:loss-batch}
\end{align}

which is a sample average based on $n_t$ observations. The special case of a single data point studied in Section <ref> corresponds to $n_t=1$. As $n_t$ increases, the objective function (<ref>) becomes more amenable to analysis. Unlike our focus, [Jiang and Yu, 2022] and [Wang et al., 2022] study asymptotic batch learning for quantile regression as the batch size $n_t$ goes to infinity. Moreover, [Do et al., 2009] concentrates on the optimization perspective of batch learning. More recently, the advantages of batch learning have been extensively explored in bandit algorithms [Ren and Zhou, 2023, Gao et al., 2019, Han et al., 2020]. The computational and statistical aspects of batched quantile regression are largely underexplored. In comparison to Section <ref>, having more available data points brings technical convenience in analyzing the non-smooth objective function when the iterate is not close to the ground truth, and it also yields different properties.

The excess risk and sub-gradient of objective function play central roles in characterizing the convergence dynamic of (sub-)gradient descent algorithms. While the expected excess risk properties illustrated in Figure <ref> also hold for the objective function (<ref>), the sub-gradient of the objective function $f_t(\Bbeta)$, denoted by $\g_t\in \partial f_t(\Bbeta)$, behaves distinctly from that in Section <ref>. Three phases of expected properties of the sub-gradient length are illustrated in Figure <ref>.  Essentially, when $\Bbeta$ is sufficiently far away from the ground truth, the expected norm of sub-gradient (i.e., $\EE\|\g\|_2^2$) has a constant upper bound, which is independent of the distance $\|\Bbeta-\Bbeta^{\ast}\|_2$. If $\Bbeta$ is closer but not too close to $\Bbeta^{\ast}$, the expected norm of sub-gradient is upper bounded by the distance $O(\|\Bbeta-\Bbeta^{\ast}\|_2)$ without dependence on the batch size. Finally, if $\Bbeta$ is very close to the ground truth, the expected norm of the sub-gradient only depends on the batch size, dimension, and noise parameters.

Expected length of the sub-gradient for the empirical quantile loss. $Y$-axis: upper bound of $\EE[\ltwo{\g}^2|\Bbeta]$, where $\g\in\partial \frac{1}{n}\sum_{i=1}^{n} \rho_{Q,\tau}(Y_i-\X_i^{\top}\Bbeta)$; $X$-axis: $\|\Bbeta-\Bbeta^{\ast}\|_2$. It reveals three phases of properties associated with the sub-gradient, depending on the closeness between $\Bbeta$ and $\Bbeta^{\ast}$.

Hence, the online sub-gradient descent algorithm developed for (<ref>) demonstrates a three-phase convergence, provided that the stepsize scheme is carefully designed in accordance with the discussed statistical properties. Write $\bar \tau = \max\{ \tau, 1-\tau \}$.

Suppose Assumptions <ref> and <ref> hold. Let $\Bbeta_0$ be an arbitrary initialization satisfying $\ltwo{\Bbeta_0-\Bbeta^*}\leq D_0$ for some $D_0>0$. There exist universal constants $\{c_i\}_{i=0}^5,C_0,C_1,C_2>0$ such that if the batch size $n_t\geq n\geq C_0(\ku/\kl) \bar \tau^2 d$ for all $t$, then the sequence $\{\Bbeta_t\}_{t\geq 1}$ generated by the online sub-gradient descent algorithm exhibits the following dynamics:

* during phase one, where $\ltwo{\Bbeta_{t}-\Bbeta^*}\geq 8 \kl^{-1/2} \gamma$, by choosing a stepsize $\eta_t\in[c_1,c_2]\sqrt{\kl}/\ku\cdot\ltwo{\Bbeta_{t}-\Bbeta^*}$, it holds with probability exceeding $1-\exp(-c_0d)-\exp\big(-\sqrt{n_t/\log(n_t)}\big)$ that

\ltwo{\Bbeta_{t+1}-\Bbeta^*}\leq\left(1-\frac{1}{100}\frac{\kl}{\ku}\right)\ltwo{\Bbeta_{t}-\Bbeta^*};

the conclusion of phase one is reached after $t_1=O(\log(\ltwo{\Bbeta_{0}-\Bbeta^*}/\gamma))$ iterations;

* in phase two, characterized by $C_1 C_u^{1/2} C_l^{-1} \bar \tau  \sqrt{ d/n}\cdot b_0\leq\ltwo{\Bbeta_{t}-\Bbeta^*}<8\kl^{-1/2}\gamma$, by selecting a constant stepsize $\eta_t=\eta\in(\kl/\ku^2)(b_1^2/b_0)\cdot[c_1,c_2]$, it holds with probability at least  $1-3\exp(-c_0d)-\exp(-\sqrt{n_t/\log n_t})$ that

\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq\left(1-0.0005\frac{b_1^2}{b_0^2}\frac{\kl^2}{\ku^2}\right)\cdot\ltwo{\Bbeta_{t}-\Bbeta^*}^2;

phase two requires $O\big(\log((n/d)(\gamma/b_0))\big)$ iterations to conclude, providing an estimate $\Bbeta_{t_2}$ that satisfies  $\ltwo{\Bbeta_{t_2}-\Bbeta^*}\asymp  C_u^{1/2} C_l^{-1}  \bar \tau \sqrt{ d/n}\cdot b_0$ under the specified conditions;

* in phase three, characterized by $\ltwo{\Bbeta_{t}-\Bbeta^*}\leq C_1 C_u^{1/2}C_l^{-1}  \bar \tau \sqrt{ d/n}\cdot b_0$, by choosing a stepsize $\eta_t:=  \frac{\ca}{\kl}\frac{b_0}{t+\cb - t_2}$ with arbitrary constants $\cb>\ca\geq 12$, it holds with probability exceeding $1-c_2\exp(-c_1d)-\exp(-\sqrt{n_t/\log n_t})$ that

$$\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq C^*\frac{\ku^2b_0^2}{\kl^2b_1^2} \frac{\bar \tau^2 }{t+\cb+1-t_2}\frac{d}{n}\cdot \frac{\ku}{\kl^2} b_0^2,$$

where the constant $C^*$ depends solely on $\ca,\cb$, and $t_2$ denotes the conclusion time of the phase two convergence.

According to Theorem <ref>, the online sub-gradient descent algorithm exhibits fast linear convergence during phases one and two. The stepsize schemes used in these phases mirror those observed in the two-phase convergence of offline quantile regression [Shen et al., 2023]. Specifically, the first phase requires $t_1$ iterations and uses a total of $O(d\log(\ltwo{\Bbeta_{0}-\Bbeta^*}/\gamma))$ data vectors. This aligns with the data scale required in phase one of the online learning setting, as outlined in Theorem <ref>. In the third phase, a stepsize of $O(1/t)$ is essential for attaining long-term statistical optimality. The final error rate exhibits a continual decrease as additional data becomes available, demonstrating the efficacy of the proposed stepsize scheme in seamlessly integrating sequentially arriving data with an unknown horizon. Furthermore, at each iteration, the failure probability diminishes exponentially with respect to the dimension.

Under the conditions of Theorem <ref>, and if the horizon $T\geq C_0(\log(\ltwo{\Bbeta_{0}-\Bbeta^*}/b_0) + \log(n/d))$, then,   with a probability exceeding $1-c_1T\exp(-c_0d)-T\exp(-\sqrt{n/\log n})$, the online sub-gradient descent algorithm produces a final estimate with an error rate given by

\ltwo{\Bbeta_T-\Bbeta^*}^2\leq C^{\ast}\frac{\ku^2b_0^2}{\kl^2b_1^2} \frac{\ku \bar \tau }{\kl^2} \cdot \frac{d}{nT} \cdot b_0^2,

where the constant $C^{\ast}$ depends only on $C_a$ and $C_b$, and $\bar \tau = \max\{ \tau, 1-\tau\}$.

If all batch sizes $n_t\asymp n$, the total number of consumed data points amounts to $O(Tn)$. In this case, the error rate achieved in Corollary <ref> matches the minimax optimal rate in offline learning [Sun et al., 2020, He et al., 2023]. Notably, this error rate is immune to initialization errors, analogous to the rate achieved in online learning as detailed in Corollary <ref>. When the initialization is sufficiently close to $\Bbeta^{\ast}$, a trade-off exists between short-term accuracy and long-term optimality, as observed in Remark <ref>.

Under the same conditions and stepsize scheme as in Theorem <ref>, the online sub-gradient descent algorithm attains a regret upper bound that for any $T\geq 1$,

\begin{align*}
\textsf{Regret}_T\leq C_1\frac{\ku^3}{\kl^3}\frac{b_0^2}{b_1^2}\max\{\sqrt{\ku}\ltwo{\Bbeta_{0}-\Bbeta^*},\gamma^2/b_0\}+C_2\frac{\ku^3}{\kl^3}\frac{b_0^3}{b_1^3}\frac{d}{n}b_0\log\left(\frac{T-t_1+\cb}{\cb} \right).
\end{align*}

The regret upper bound comprises two terms: one depending on initialization error but independent of $T$, and the other exhibiting logarithmic growth in $T$ yet independent of initialization error. When consuming an equivalent number of total data points (denoted by $N$ and assuming $N\geq C_1\max\{ n\log(\ltwo{\Bbeta_{0}-\Bbeta^* }/b_0), n\log(n/d)\}$), both online learning and batch learning yield estimators with an optimal error rate of $O(b_0^2\,d/N)$. Nevertheless, batch learning achieves a significantly smaller regret, benefiting from faster learning in each iteration by leveraging a larger pool of data points. Specifically, as demonstrated in Theorem <ref>, online learning achieves a regret upper bound of $O(d\ltwo{\Bbeta_{0}-\Bbeta^*}+d\log(N/d))$, while batch learning attains a much smaller regret upper bound of $O(\ltwo{\Bbeta_{0}-\Bbeta^*}+(d/n)\log(N/n))$. However, it is noteworthy that batch learning necessitates a substantially larger storage size.

§.§ Sequential Learning with Infinite Storage

Consider the scenario where the $\Bbeta_{t+1}$ server possesses unlimited storage, storing all historical data and updating the estimate solely when a new sample arrives. Additionally, the server updates the estimate by incorporating all observations received up to or including time $t$. Let $\{(Y_i^{(t)},\X_i^{(t)})\}_{i=1}^{n_t}$ denote the $n_t$ observations arriving at time $t$. Consequently, the loss function at time $t$ is defined as

\begin{align}\label{eq:infinite-loss}
f_t(\Bbeta):=\frac{1}{\sum_{l=0}^t n_l}\sum_{l=0}^{t}\sum_{i=1}^{n_l}\rho_{Q,\tau}(Y_i^{(l)}-\X_i^{(l)\top}\Bbeta),
\end{align}

The online sub-gradient descent algorithm updates the iterate through $\Bbeta_{t+1}=\Bbeta_{t}-\eta_t\cdot \g_t$, where $\eta_t$ is the stepsize and $\g_t\in\partial f_t(\Bbeta_t)$ is a sub-gradient. Despite the algorithm requiring a pass through all accumulated data points at each iteration, it remains computationally more efficient compared to the offline approach [Shen et al., 2023].

The objective function (<ref>) maintains its inherent two-phase property, as depicted in Figure <ref>. Notably, the considerable storage and computation costs associated with this formulation contribute to an expeditious convergence of the algorithm, as stated in the following theorem.

Suppose Assumptions <ref> and  <ref> hold, and let $\Bbeta_0$ be an arbitrary initialization satisfying   $\ltwo{\Bbeta_0-\Bbeta^*}\leq D_0$ for some $D_0>0$. There exist absolute positive constants $c_0,c_1,c_2,C_1,C_2$ such that, if the initial storage $n_0\geq C_1d$, then the sequence $\{\Bbeta_t\}_{t\geq 1}$ generated by the online sub-gradient descent algorithm has the following dynamics:

* During phase one, when $\ltwo{\Bbeta_{t}-\Bbeta^*}\geq 8\gamma$, by selecting a step size $\eta_t:= \frac{\sqrt{\kl}}{8\ku} \left(1-\frac{1}{100}\frac{\kl}{\ku}\right)^{t}\cdot D_0$, it holds with probability at least $1-\exp(-c_0d)-\exp(-\sqrt{\sum_{l=0}^{t} n_l/\log \sum_{l=0}^{t} n_l})$ that

\ltwo{\Bbeta_{t+1}-\Bbeta^*}\leq\left(1-\frac{1}{100}\frac{\kl}{\ku}\right)^{t+1}\cdot D_0 .

Phase one concludes after  $t_1=O\big(\log(\|\Bbeta_0-\Bbeta^{\ast}\|_2/\gamma)\big)$ iterations.

* In phase two, when $\ltwo{\Bbeta_{t}-\Bbeta^*}< 8\gamma$, by selecting a stepsize $\eta_t:=\eta\asymp \frac{\kl}{\ku^2}\cdot\frac{b_1^2}{b_0}$, it holds with probability at least $1-\exp(-c_0d)-\exp(-\sqrt{\sum_{l=0}^{t} n_l/\log \sum_{l=0}^{t} n_l})$ that

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq \left(1-c_1\frac{b_1^2}{b_0^2}\cdot\frac{\kl^2}{\ku^2}\right)\cdot\ltwo{\Bbeta_{t}-\Bbeta^*}^2 + c_2\frac{d}{\ku\sum_{l=0}^t n_l}\cdot b_1^2.
\end{align*}

As shown in Theorem <ref>,  the initial phase necessitates a sequence of stepsizes characterized by geometric decay, while the subsequent phase demands a constant stepsize. Both phases are proven to exhibit linear convergence. For the sake of simplicity, assume $n_l=n$ holds for all $l\geq 1$. The error rate during the second phase (for $t > t_1$) can be expressed as

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq \left(1-c_1\frac{b_1^2}{b_0^2}\cdot\frac{\kl^2}{\ku^2}\right)^{t+1-t_1}\cdot\ltwo{\Bbeta_{t_1}-\Bbeta^*}^2 +C_2 \frac{d}{n_0+tn}\cdot\frac{\ku}{\kl^2}\cdot b_0^2\cdot\left(\frac{b_0}{b_1}\cdot\frac{\ku}{\kl}\right)^2,
\end{align*}

resulting in the subsequent corollary.

Assume the same conditions as in Theorem <ref> hold, and that $n_l=n$ for all $l\geq 1$. If the ultimate horizon $T\geq C^*\ltwo{\Bbeta_{0}-\Bbeta^*}/\gamma$ for a constant $C^*$ that may depend on $(\kl,\ku,n_0, n)$, then with probability exceeding $1-T\exp(-c_0d)\\-\sum_{t=0}^{T}\exp(-\sqrt{ (n_0+tn)/\log  (n_0+tn)}) $,

$$\ltwo{\Bbeta_{T}-\Bbeta^*}\leq C\left(\frac{b_0}{b_1}\cdot\frac{\ku}{\kl}\right)^2\cdot \frac{\ku}{\kl^2}\cdot \frac{d}{n_0+Tn} b_0^2.$$

The corollary above illustrates that despite the server updating only once upon the arrival of a new sample group, it consistently attains statistical optimality in the presence of an unknown horizon.

§ NUMERICAL SIMULATIONS

In this section, we numerically examine the performance of the proposed online QR algorithms. We first demonstrate the effectiveness of our stepsize scheme, and then compare the final accuracy of different online estimators with that of the offline estimate. We further illustrate the advantage of online quantile regression over online least squares regression. We use the relative error $\ltwo{\Bbeta_t-\Bbeta^*}/\ltwo{\Bbeta^*}$ as the main metric. For simplicity, the initialization $\Bbeta_0$ is set to be $\boldsymbol{0}$ throughout our numerical studies.

Stepsize Scheme. We begin by demonstrating the effectiveness of our stepsize scheme, as outlined in Theorem <ref>, which is theoretically guided by statistical regularities. The selection of parameters in the proposed stepsize is quite flexible, both theoretically (as outlined in Theorem <ref>) and in practice. For instance, concerning the parameter $C_a$ required by the stepsize schedule in the second phase of iterations, we observe that our algorithm always performs well as long as it is not excessively small.

In all the experiments, the stepsize for the first phase is scheduled as $\eta_t=(1-0.5/d)^t\eta_0$, where $\eta_0$ is the initial stepsize. The choice of stepsize in second phase involves specific parameters where we set $\ca = 20$ and $\cb=30$. We fix the dimension at $d=100$, the unknown horizon  $T=10^5$, and sample $t_{\nu}$-distributed noise with a degree of freedom $\nu=1.1$. The performance of proposed stepsize scheme is compared with those of two existing alternative stepsize scheme: the $\eta_t=O(1/t)$ decaying scheme and a constant stepsize scheme $\eta_t\equiv \emph{const}$. For detailed discussions on these stepsize schemes, we refer to [Duchi and Singer, 2009] and [Zhang, 2004].

The convergence performances of online sub-gradient descent under the aforementioned three stepsize schemes are presented in Figure <ref>. For both the moderate and strong SNR cases, our proposed stepsize scheme can ensure a linear convergence of the online sub-gradient descent algorithm in the first phase. The linear convergence behavior stops once the algorithm reaches at a sufficiently accurate estimate, i.e., when the error rate is dominated by the noise scale $\EE |\xi|$. Our proposed  scheme then resets the stepsize and the algorithm enters second phase of iterations. The error rate continues to decrease in the second phase, which is sub-linear and exhibits an $O(1/t)$ convergence rate. The two-phase convergence phenomenon is consistent with the theoretical discoveries. In contrast, the constant stepsize scheme can also achieve an error rate the same as the one our algorithm achieves at the end of first phase iterations. However, it cannot further improve the estimate or may converge too slowly resulting into a statistically sub-optimal estimate. It is also observed that employing a relatively large constant stepsize can facilitate faster convergence in the initial stage, which was claimed by the theoretical results in [Cai et al., 2023]. The performance under the $O(1/t)$ stepsize scheme is considerably inferior to those under the other two stepsize schemes.

Relative error versus time/iterations in online (one-sample) learning ($n_t\equiv 1$). The dimension $d=100$, unknown horizon $T=10^5$, and quantile loss parameter $\tau=1/2$. The convergence performances of online sub-gradient descent are examined under three stepsize schemes: Statistical stands for our stepsize scheme guided by Theorem <ref>, Constant stands for the stepsize scheme $\eta_t\equiv \emph{const}$ [Zhang, 2004], $O(1/t)$ means the decaying stepsize scheme $\eta_t=O(1/t)$ [Duchi and Singer, 2009]. Left(a): moderate SNR; right(b): strong SNR.

Statistical Accuracy Comparisons. We now evaluate the statistical accuracy of the final estimator output by the online sub-gradient descent algorithm. The error rate achieved by offline quantile regression is used as the benchmark. The dimension $d$ and noise distribution are set the same as those in the previous set of simulations. Both the online one-sample learning and batch learning are studied in the experiment. The total sample size is set at $n=20,000$. Figure <ref> displays a box plot of error rates based on 50 replications. For each simulation, the online batch learning algorithm and one-sample learning algorithm take approximately 2 and 10 seconds, respectively, on a MacBook Pro 2020. The offline learning, implemented using the   package, takes more than $2$ minutes. Figure <ref> shows that offline learning achieves the best statistical accuracy when the total sample size is relatively small, while online one-sample learning and batch learning achieve comparable statistical accuracy.
These are consistent with our theoretical findings. Simulation results for the large sample size $n=50,000$ are displayed in Figure <ref>, in which case the offline learning achieves only slightly better accuracy than its online counterparts. However, online algorithms enjoy much higher computational efficiencies. On the same Mac Pro, the online one-sample, batch, and offline learning methods take approximately 30, 5, and 520 seconds, respectively.

Error rates of offline and online regression using quantile loss $\rho_{Q,\tau}(\cdot)$. Online One Sample refers to the online learning algorithm studied in Section <ref>. The dimension $d=100$, total sample size $n=20,000$, the batch size $n_t\equiv 100$, and noise has a $t_{1.1}$ distribution. Box-plots are drawn based on 50 independent simulations.

Error rates of offline and online regression using quantile loss $\rho_{Q,\tau}(\cdot)$. Online One Sample refers to the online learning algorithm studied in Section <ref>. The dimension $d=100$, total sample size $n=50,000$, the batch size $n_t\equiv 200$, and noise has a $t_{1.1}$ distribution. Box-plots are drawn based on 50 independent simulations.

Convergence Dynamics Comparisons. While online QR is motivated primarily for treating heavy-tailed noise, it is still of interest to examine its performance under Gaussian noise. Towards that end, we compare the performance of our proposed online QR algorithms with that of classical online least squares algorithm [Zhang, 2004]. We will show that the proposed online QR algorithms are not only robust in the presence of heavy-tailed noise or responses, but also are as efficient as the classical online least squares algorithm if the noise is Gaussian.

While online learning with square loss has been extensively studied [Orabona, 2019, Hazan, 2016], its stepsize scheme guided by statistical properties remains relatively under-explored. Here we briefly explain the appropriate stepsize scheme to achieve long-term statistical optimality for online least squares algorithm with a focus on sub-Gaussian noise and undetermined horizon.  Considering the square loss function at time $t$ as $f_t(\Bbeta):=(Y_t-\X_t^{\top}\Bbeta)^2$, the expected length of gradient $\g_t$ satisfies

\begin{align*}
\EE\left[\|\g_t\|_2^2\big|\Bbeta_t\right]\leq 4d\ku(\EE\xi^2+\ku\ltwo{\Bbeta_t-\Bbeta^*}^2).
\end{align*}

The gradient length is decided primarily by $\ltwo{\Bbeta_t-\Bbeta^* }^2$ when it is large. Conversely, when $\Bbeta_t$ is sufficiently close to the oracle, the gradient length is determined by $\EE\xi^2$. This implies an appropriate stepsize scheme for online least squares algorithm should also consists of two phases: a constant stepsize $\eta_t\equiv \eta=O(d^{-1})$ in the first phase; a decaying stepsize schedule $\eta_t=O\big((t+d)^{-1}\big)$ in the second phase. The detailed proof of the convergence dynamics under this stepsize scheme is almost identical to that of Theorem <ref> and thus omitted.

We compare the convergence dynamics of online one-sample QR algorithm and online least squares algorithm (equipped with the aforementioned two-phase stepsize scheme). The dimension $d=100$, quantile loss parameter $\tau=0.5$, and horizon is unknown. The results under Gaussian noise and $t_{1.1}$ noise are presented in Figure <ref>. The convergence dynamics of the two algorithms are comparable under Gaussian noise, both showing a linear convergence in the first phase and an $O(1/t)$ decaying rate afterwards. They achieve almost the same statistical accuracy in the end. Under $t_{1.1}$-distributed noise, online least squares algorithm does not converge. In contrast, the proposed online QR algorithm ensures stable convergence and achieves error rates comparable to those under Gaussian noise.

$\frac{\ltwo{\Bbeta^*}}{\EE|\xi|}=20$, Gaussian noise

$\frac{\ltwo{\Bbeta^*}}{\EE|\xi|}=200$, Gaussian noise

$\frac{\ltwo{\Bbeta^*}}{\EE|\xi|}=20$, $t_{1.1}$ noise

$\frac{\ltwo{\Bbeta^*}}{\EE|\xi|}=200$,  $t_{1.1}$ noise

Relative error versus time/iterations for online QR and least squares algorithms. The dimension $d=100$, unknown horizon $T=10^5$, and quantile loss parameter $\tau=1/2$. The proposed online QR algorithm is robust to heavy-tailed noise and performs as good as online least squares under Gaussian noise.

§ PROOF OF THEOREM <REF>

This section presents the proof of Theorem <ref>. The update procedure $\Bbeta_{t+1}=\Bbeta_t-\eta_t\g_t$ yields

\begin{align}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&=\ltwo{\Bbeta_{t}-\Bbeta^*}^2-2\eta_t\inp{\Bbeta_{t}-\Bbeta^*}{\g_t}+\eta_t^2\ltwo{\g_t}^2 ,
\label{eq:update one sample}
\end{align}

where the sub-gradient $\g_t$ is of the form
$\g_t=\X_t \{ -\tau\cdot1_{(Y_t>\X_t^{\top}\Bbeta_{t})}+(1-\tau)\cdot 1_{(Y_t<\X_t^{\top}\Bbeta_{t})} +\delta \cdot1_{(Y_t=\X_t^{\top}\Bbeta_{t})} \}$, and $\delta$ can take any value in the range  $[-\tau,1-\tau]$. It thus follows that

\begin{align*}
\ltwo{\g_t}^2&\geq\min\{\tau, 1-\tau \}^2 \ltwo{\X_t\cdot 1_{(Y_t\neq \X_t\Bbeta_t ) }}^2+\delta^2 \ltwo{\X_t\cdot 1_{ ( Y_t= \X_t\Bbeta_t ) }}^2,\\ \ltwo{\g_t}^2&\leq\max\{\tau , 1-\tau \}^2 \ltwo{\X_t\cdot 1_{(Y_t\neq \X_t\Bbeta_t)}}^2+\delta^2 \ltwo{\X_t\cdot 1_{(Y_t= \X_t\Bbeta_t) }}^2.
\end{align*}

By Lemma <ref>, $\ltwo{\X_t}^2\leq 2\ku d$ holds with probability at least $1-\exp(-cd)$, which further implies  $\ltwo{\g_t}^2\leq 2\max\{\tau , 1-\tau \}^2 \ku   d$ for all $t\geq 0$.

First Phase Analysis.
We aim to establish the convergence dynamics $\ltwo{\Bbeta_{t}-\Bbeta^*}\leq C^* (1-c_5)^{t}\ltwo{\Bbeta_{0}-\Bbeta^*}$ through induction, where the constant $C^*>1$ remains fixed throughout the proof and $0<c_5<1$ is some sufficiently small constant which will be specified later.

Initially, it trivially holds that $\ltwo{\Bbeta_{0}-\Bbeta^*}\leq C^* \ltwo{\Bbeta_{0}-\Bbeta^*}$. For ease of notation, let $D_l:=(1-c_5)^{l}\ltwo{\Bbeta_{0}-\Bbeta^*}$. Assuming the established bound $\ltwo{\Bbeta_{l}-\Bbeta^*}\leq C^* D_l$ holds for $l=0,1,\dots,t$, our aim is to demonstrate the validity of $\ltwo{\Bbeta_{t+1}-\Bbeta^*}\leq C^*  D_{t+1}$.

By the property of sub-gradient, $\inp{\Bbeta_{t}-\Bbeta^*}{\g_t}\geq f_t(\Bbeta_{t})-f_t(\Bbeta^*)$. Substituting this into equation (<ref>) yields

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2 &\leq \ltwo{\Bbeta_{t}-\Bbeta^*}^2-2\eta_t \{ (f_t(\Bbeta_{t}) -f_t(\Bbeta^*) \} + 2\eta_t^2 \ku\bar{\tau}^2d \\
& = \ltwo{\Bbeta_{t}-\Bbeta^*}^2-2\eta_t\EE \{ f_t(\Bbeta_{t}) -f_t(\Bbeta^*) | \Bbeta_{t} \} + 2 \eta_t^2 \ku\bar{\tau}^2d\\
&~~~~  -2\eta_t \big[ f_t(\Bbeta_{t}) -f_t(\Bbeta^*)-\EE\{ f_t(\Bbeta_{t}) -f_t(\Bbeta^*) | \Bbeta_{t} \}  \big] ,
\end{align*}

where $\bar{\tau}=\max\{\tau,1-\tau\}$.
The convexity of the check loss implies $\rho_{Q,\tau}(x_1+x_2)\leq \rho_{Q,\tau}(x_1)+\rho_{Q,\tau}(x_2)$. This, combined with Lemma <ref>, implies

\begin{align*}
\EE \{ f_t(\Bbeta_{t}) -f_t(\Bbeta^*) | \Bbeta_{t}  \}  &\geq \EE \{ \rho_{Q,\tau}(-\inp{\X_t}{\Bbeta_{t}-\Bbeta^*}) -\rho_{Q,\tau}(\xi_t)-\rho_{Q,\tau}(-\xi_t) | \Bbeta_{t} \} \\
&\geq\sqrt{\frac{\kl}{2\pi}} \ltwo{\Bbeta_{t}-\Bbeta^*}-\gamma.
\end{align*}

During phase one in which $\ltwo{\Bbeta_{t}-\Bbeta^*}\geq 8 \kl^{-1/2} \gamma$, it follows from the above upper and lower bounds that

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq \ltwo{\Bbeta_{t}-\Bbeta^*}^2-\sqrt{\frac{2\kl}{\pi}}\eta_t \ltwo{\Bbeta_{t}-\Bbeta^*} +2\eta_t\gamma+ 2\eta_t^2 \ku\bar{\tau}^2d\\
&~~~~ -2\eta_t  \big[ f_t(\Bbeta_{t}) -f_t(\Bbeta^*)-\EE\{ f_t(\Bbeta_{t}) -f_t(\Bbeta^*) | \Bbeta_{t} \}  \big]\\
&\leq \ltwo{\Bbeta_{t}-\Bbeta^*}^2-\frac{1}{3}\eta_t\sqrt{\kl} \ltwo{\Bbeta_{t}-\Bbeta^*} + 2\eta_t^2 \ku\bar{\tau}^2d\\
&~~~~  -2\eta_t  \big[ f_t(\Bbeta_{t}) -f_t(\Bbeta^*)-\EE\{ f_t(\Bbeta_{t}) -f_t(\Bbeta^*) | \Bbeta_{t} \}  \big] .
\end{align*}

By choosing the stepsize $\eta_t= c\frac{\sqrt{\kl}}{\ku}\frac{\ltwo{\Bbeta_t-\Bbeta^*}}{\bar{\tau}^2d} $ with $c\in[1/150,1/100]$, we have

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq \left(1-\frac{1}{500}\frac{\kl}{\ku}\frac{1}{\bar{\tau}^2d}\right)\ltwo{\Bbeta_{t}-\Bbeta^*}^2\\
&~~~~ -2c\frac{\sqrt{\kl}}{\ku}\frac{1}{\bar{\tau}^2d}\cdot\ltwo{\Bbeta_t-\Bbeta^*}\big[  f_t(\Bbeta_{t}) -f_t(\Bbeta^*)-\EE \{ f_t(\Bbeta_{t}) -f_t(\Bbeta^*)\big| \Bbeta_{t} \}   \big].
\end{align*}

Applying this iterative bound repeatedly from $t=0$, we have

\begin{align*}
&\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq \left(1-\frac{1}{500}\frac{\kl}{\ku}\frac{1}{\bar{\tau}^2d}\right)^{t+1}\ltwo{\Bbeta_{0}-\Bbeta^*}^2\\
&-2c\frac{\sqrt{\kl}}{\ku}\frac{1}{\bar{\tau}^2d} \sum_{l=0}^{t} \left(1-\frac{1}{500}\frac{\kl}{\ku}\frac{1}{\bar{\tau}^2d}\right)^{t-l}\ltwo{\Bbeta_l-\Bbeta^*}\big[ f_l(\Bbeta_{l}) -f_l(\Bbeta^*)-\EE\{ f_l(\Bbeta_{l}) -f_l(\Bbeta^*) | \Bbeta_{l}\} \big].
\end{align*}

Under the event of $\cup_{l=0}^t\{\ltwo{\Bbeta_{l}-\Bbeta^*}^2\leq C^*D_l^2\}$, we have
$$\|f_l(\Bbeta_{l}) -f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_{l}) -f_l(\Bbeta^*)\big| \Bbeta_{l}\right] \|_{\Psi_2}\leq C_1\sqrt{C^*\ku}\bar{\tau}D_l$$
for each $l=0,\dots,t$, where $\|\cdot\|_{\Psi_2}$ denotes the Orlicz norm with Orlicz function $\Psi(x)=\exp(x^2)-1$. Then, according to a variant of Azuma's inequality from [Shamir, 2011], it holds with probability exceeding $1-\exp(-c_1d)$ that

\begin{align*}
&~~~~~\left|\sum_{l=0}^{t} \left(1-\frac{1}{500}\frac{\kl}{\ku}\frac{1}{\bar{\tau}^2d}\right)^{t-l}\ltwo{\Bbeta_l-\Bbeta^*}\big[f_l(\Bbeta_{l}) -f_l(\Bbeta^*)-\EE \{ f_l(\Bbeta_{l}) -f_l(\Bbeta^*) | \Bbeta_{l}\} \big]\right|\\
&\leq \bar{\tau} C^{*} \sqrt{\ku d\sum_{l=0}^{t} \left(1-\frac{1}{500}\frac{\kl}{\ku}\frac{1}{\bar{\tau}^2d}\right)^{2t-2l}D_l^4}\\
&= \bar{\tau}C^*\sqrt{ \ku d\sum_{l=0}^{t} \left(1-\frac{1}{500}\frac{\kl}{\ku}\frac{1}{\bar{\tau}^2d}\right)^{2t-2l} \left(1-c_5\frac{\kl}{\ku}\frac{1}{\bar{\tau}^2d}\right)^{4l}}\times D_0^2\\
&\leq \bar{\tau}C^*\sqrt{\ku d\sum_{l=0}^{t} \left(1-c_5\frac{\kl}{\ku}\frac{1}{\bar{\tau}^2d}\right)^{6t-2l}}\times D_0^2\\
&\leq \bar{\tau}^2C^*d\times\left(1-c_5\frac{\kl}{\ku}\frac{1}{\bar{\tau}^2d}\right)^{2t} D_{0}^2\times \sqrt{c_5^{-1}\frac{\ku^2}{\kl}},
\end{align*}

where the third equation uses the inequality $1-3a\leq (1-a)^3$ with $0<a<1$, and $1/500=3c_5$. Putting together the pieces, we conclude that

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq \left(1-\frac{1}{500}\frac{\kl}{\ku}\frac{1}{\bar{\tau}^2d}\right)^{t+1}\ltwo{\Bbeta_{0}-\Bbeta^*}^2\\&~~~~~~~~~~~~~~~~~~~+0.02\times\sqrt{1500}C^*\left(1-c_5\frac{\kl}{\ku}\frac{1}{\bar{\tau}^2d}\right)^{2t} D_{0}^2\\
&\leq C^*\left(1-c_5\frac{\kl}{\ku}\frac{1}{\bar{\tau}^2d}\right)^{2t+2} D_0^2,
\end{align*}

which completes the proof of phase one convergence.

Second Phase Analysis.
The verification of the second phase is also conducted through induction. The proof at $t_1+1$ is omitted due to its simplicity, as it can be derived analogously to the subsequent proof. We assume the event $\cup_{l=t_1}^{t}{\ltwo{\Bbeta_{l}-\Bbeta^*}^2\leq\frac{C^*d}{l-t_1+\cb d}\frac{\ku}{\kl^2}b_0^2}$ holds and proceed to establish the dynamics at $t+1$. Within the second phase, in accordance with Lemma <ref>, the lower bound for the expectation of the loss function is of second order:

\EE\left[f_t(\Bbeta_{t}) -f_t(\Bbeta^*)\big|\Bbeta_{t}\right]\geq\frac{\kl}{12b_0}\ltwo{\Bbeta_t-\Bbeta^*}^2.

Subsequently, equation (<ref>) can be upper bounded as

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq \ltwo{\Bbeta_{t}-\Bbeta^*}^2-\frac{1}{6b_0}\eta_t\kl \ltwo{\Bbeta_{t}-\Bbeta^*}^2 + 2\eta_t^2 \ku\bar{\tau}^2d\\
&~~~~~~~~~~~~~~~~~~~~~~-2\eta_t\left[f_t(\Bbeta_{t}) -f_t(\Bbeta^*)-\EE\left[f_t(\Bbeta_{t}) -f_t(\Bbeta^*)\big| \Bbeta_{t}\right] \right].
\end{align*}

Incorporating the chosen stepsize $\eta_t=\frac{\ca}{t-t_1+\cb d}\frac{b_0}{\kl}$ into the aforementioned equation, we obtain

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq\left(1-\frac{\ca}{12(t-t_1+\cb d)}\right)\ltwo{\Bbeta_{t}-\Bbeta^*}^2+2b_0^2\bar{\tau}^2\frac{\ku}{\kl^2}\frac{\ca^2}{(t-t_1+\cb d)^2}d\\
&~~~~~~~~~~~~~-2\frac{\ca}{t-t_1+\cb d}\frac{b_0}{\kl}\left[f_t(\Bbeta_{t}) -f_t(\Bbeta^*)-\EE\left[f_t(\Bbeta_{t}) -f_t(\Bbeta^*)\big| \Bbeta_{t}\right] \right].
\end{align*}

Summing the above equation over the range from $t_1$, we arrive at

\begin{equation}
\begin{split}
&\leq\prod_{l=t_1}^{t}\left(1-\frac{\ca}{12(l-t_1+\cb d)}\right)\ltwo{\Bbeta_{t_1}-\Bbeta^*}^2\\
&~~~+2b_0^2\bar{\tau}^2\frac{\ku}{\kl^2}d\sum_{l=t_1}^{t}\left(1-\frac{\ca}{12(l+1-t_1+\cb d)}\right)\cdots \left(1-\frac{\ca}{12(t-t_1+\cb d)}\right)\frac{\ca^2}{(l-t_1+\cb d)^2}\\
&~~~-2\frac{b_0}{\kl} \sum_{l=t_1}^{t}\left(1-\frac{\ca}{12(l+1-t_1+\cb d)}\right)\cdots \left(1-\frac{\ca}{12(t-t_1+\cb d)}\right) \frac{\ca}{l-t_1+\cb d}\\&~~~~~~~~~~\times\left[f_l(\Bbeta_{l}) -f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_{l}) -f_l(\Bbeta^*)\big| \Bbeta_{l}\right] \right].
\end{split}
\label{eq2}
\end{equation}

A sharp bound for the above equation is the key point of the proof. We shall analyze each of the three terms on the right-hand side respectively. Note that with $\ca>12$, we have $\left(1-\frac{\ca}{12(l-t_1+\cb d)}\right)\leq \frac{l-t_1+\cb d}{l+1-t_1+\cb d}$. Then, the first term on the right-hand side satisfies

$$\prod_{l=t_1}^{t}\left(1-\frac{\ca}{12(l-t_1+\cb d)}\right)\ltwo{\Bbeta_{t_1}-\Bbeta^*}^2\leq\frac{\cb d}{t+1-t_1+\cb d}\ltwo{\Bbeta_{t_1}-\Bbeta^*}^2. $$

Consider the product sequence, which can be bounded as

\begin{equation}
\begin{split}
&~~~\left(1-\frac{\ca}{12(l+1-t_1+\cb d)}\right)\cdots \left(1-\frac{\ca}{12(t-t_1+\cb d)}\right)\\&=\exp\left(\sum_{k=l+1}^{t}\log\left(1-\frac{\ca}{12(k-t_1+\cb d)}\right)\right)\\
&\leq \exp\left(-\sum_{k=l+1}^{t}\frac{\ca}{12(k-t_1+\cb d)}\right)\leq \exp\left(-\frac{\ca}{12}\int_{l+1-t_1}^{t+1-t_1} \frac{1}{x+\cb d}\; dx\right)\\
&\leq\exp\left(-\frac{\ca}{12}\log\left(\frac{t+1-t_1+\cb d}{l+1-t_1+\cb d}\right)\right)=\left(\frac{l+1-t_1+\cb d}{t+1-t_1+\cb d}\right)^{\frac{\ca}{12}},
\label{eq:1-1}
\end{split}
\end{equation}

where the first inequality uses the property $\log(1+x)\leq x$, and the third line leverages the connections between integrals and sums of sequences. Consequently, the second term in equation <ref> can be bounded as follows:

\begin{align*}
&~~~\sum_{l=t_1}^{t}\left(1-\frac{\ca}{12(l+1-t_1+\cb d)}\right)\cdots \left(1-\frac{\ca}{12(t-t_1+\cb d)}\right)\frac{\ca^2}{(l-t_1+\cb d)^2}\\
&\leq \sum_{l=t_1}^{t}\left(\frac{l+1-t_1+\cb d}{t+1-t_1+\cb d}\right)^{\frac{\ca}{12}}\frac{\ca^2}{(l-t_1+\cb d)^2}\\
&\leq \left(\frac{1+\cb d}{\cb d}\right)^{\frac{\ca}{12}}\frac{\ca^2}{( t+1-t_1+\cb d)^{\frac{\ca}{12}}}\sum_{l=t_1}^{t}(l-t_1+\cb d)^{\frac{\ca}{12}-2}.
\end{align*}

With $\ca>12$, we have $\sum_{l=t_1}^{t}(l-t_1+\cb d)^{\frac{\ca}{12}-2}\leq\int_{0}^{t-t_1+1}(x+\cb d)^{\frac{\ca}{12}-2}\; dx=\frac{1}{\frac{\ca}{12}-1}(t-t_1+1+\cb d )^{\frac{\ca}{12}-1}$. In this way, all together, we obtain the upper bound of the second term in equation <ref>

\begin{align*}
&~~~\sum_{l=t_1}^{t}\left(1-\frac{\ca}{12(l+1-t_1+\cb d)}\right)\cdots \left(1-\frac{\ca}{12(t-t_1+\cb d)}\right)\frac{\ca^2}{(l-t_1+\cb d)^2}\\
&\leq \left(\frac{1+\cb d}{\cb d}\right)^{\frac{\ca}{12}}\frac{\ca^2}{ t+1-t_1+\cb d}.
\end{align*}

Under the event of $\cup_{l=t_1}^t\{\ltwo{\Bbeta_{l}-\Bbeta^*}^2\leq\frac{C^*d}{l-t_1+\cb d}\frac{\ku}{\kl^2}b_0^2\}$, we have

\begin{align*}
\| f_l(\Bbeta_{l}) -f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_{l}) -f_l(\Bbeta^*)\big| \Bbeta_{l}\right]\|_{\Psi_2}\leq C\bar{\tau}\sqrt{\frac{C^*d}{l-t_1+\cb d}\frac{\ku^2}{\kl^2} }b_0.
\end{align*}

By Theorem 2 in [Shamir, 2011] and equation (<ref>), with probability exceeding $1-c\exp(-cd)$, we have the upper bound of the third term in equation (<ref>),

\begin{align*}
&~~~\frac{b_0}{\kl}\sum_{l=t_1}^{t}\left(1-\frac{\ca}{12(l+1-t_1+\cb d)}\right)\cdots \left(1-\frac{\ca}{12(t-t_1+\cb d)}\right) \frac{\ca}{l-t_1+\cb d}\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\times\left[f_l(\Bbeta_{l}) -f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_{l}) -f_l(\Bbeta^*)\big| \Bbeta_{l}\right] \right]\\
&\leq \ca\frac{\ku}{\kl^2}\sqrt{\sum_{l=t_1}^{t}\left(\frac{l+1-t_1+\cb d}{t+1-t_1+\cb d} \right)^{\frac{\ca}{6}}\frac{1}{(l-t_1+\cb d)^2} \frac{C^*d^2}{l-t_1+\cb d}}b_0^2\\
&=\ca\frac{\ku}{\kl^2}b_0^2\frac{\sqrt{C^*}d}{(t+1-t_1+\cb d)^{\frac{\ca}{12}}}\sqrt{\left(\frac{1+\cb d}{\cb d}\right)^{\frac{\ca}{6}}\sum_{l=t_1}^t (l-t_1+\cb d)^{\frac{\ca}{6}-3}}\\
&\leq \ca\frac{\ku}{\kl^2}b_0^2\frac{\sqrt{C^*}d}{t+1-t_1+\cb d}\times \left(\frac{1+\cb d}{\cb d}\right)^{\frac{\ca}{12}}.
\end{align*}

Finally, combine the upper bound of the three terms and then we have

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq \frac{\cb d}{t+1-t_1+\cb d}\ltwo{\Bbeta_{t_1}-\Bbeta^*}^2+2\left(\frac{1+\cb d}{\cb d}\right)^{\frac{\ca}{12}}\frac{\ca^2}{ t+1-t_1+\cb d}\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\ca\frac{\ku}{\kl^2}b_0^2\frac{\sqrt{C^*}d}{t+1-t_1+\cb d}\times \left(\frac{1+\cb d}{\cb d}\right)^{\frac{\ca}{12}}\\
&\leq C^*\frac{\ku}{\kl^2}b_0^2\frac{d}{t+1-t_1+\cb d},
\end{align*}

where $\left(\frac{1+\cb d}{\cb d}\right)^{\frac{\ca}{12}}<2^{\ca/12}$ is independent of $d$ and $C^*$ is some sufficiently large constant, possibly dependent on $\ca,\cb$. This completes the proof of Theorem <ref>.

§ DISCUSSIONS

This paper addresses online quantile regression in low dimensions, and its analytical framework has the potential for extension to the examination of the stochastic sub-gradient method in low-rank regression under quantile loss. Let $\mathbf{M}^* \in \mathbb{R}^{d_1 \times d_2}$ be the true low-rank matrix with $\operatorname{rank}(\mathbf{M}^*) = r$. At time $t$, the observed data is denoted as $(Y_t, \mathbf{X}_t)$, where $Y_t = \inp{\mathbf{X}_t}{\mathbf{M}^*} + \xi_t$, and $\mathbf{X}_t \in \mathbb{R}^{d_1 \times d_2}$ is the sensing matrix. The loss function at time $t$ is defined as $f_t(\mathbf{M}) = \rho_{Q,\tau}(Y_t - \inp{\mathbf{X}_t}{\mathbf{M}})$.
Maintaining the low-rank structure of the iterate over time is crucial. Specifically, the online Riemannian optimization updates the iterate as

\M_{t+1}=\operatorname{SVD}_r(\M_t-\eta_t\calP_{\TT_t}(\G_t)),

where $\G_t\in\partial f_t(\M_t)$ is the sub-gradient.
We refer to [Vandereycken, 2013] and [Mishra et al., 2014] for an in-depth exploration of the closed-form of $\calP_{\TT_t}(\mathbf{G}_t)$ and further discussions on Riemannian optimization. In each iteration, emphasis is placed on retaining only the leading $r$ components, raising the conceptual question of whether effective information has been preserved. We defer the investigation of online low-rank regression under quantile loss to future studies.

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Supplementary Material to “Online Quantile Regression"

§ PROOF OF MAIN RESULTS

This section presents proofs of the main results for each setting.

§.§ Proof of Online Learning

In this subsection, we shall first prove Remark <ref>, and then prove the regret bound, which is a subsequent result of the expected estimation error rates.

§.§.§ Proof of Remark <ref>: Online Learning with Good Initialization

We are going to prove the convergence dynamics when $\ltwo{\Bbeta_{0}-\Bbeta^*}< 8\sqrt{\kl^{-1}}\gamma$. According to proof of Theorem <ref>, we have

\begin{equation}
\begin{split}
&~~~\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq\prod_{l=0}^{t}\left(1-\frac{\ca}{12(l+\cb d)}\right)\ltwo{\Bbeta_{0}-\Bbeta^*}^2\\
&~~~+2b_0^2\bar{\tau}^2\frac{\ku}{\kl^2}d\sum_{l=0}^{t}\left(1-\frac{\ca}{12(l+1+\cb d)}\right)\cdots \left(1-\frac{\ca}{12(t+\cb d)}\right)\frac{\ca^2}{(l+\cb d)^2}\\
&~~~-2\frac{b_0}{\kl} \sum_{l=0}^{t}\left(1-\frac{\ca}{12(l+1+\cb d)}\right)\cdots \left(1-\frac{\ca}{12(t+\cb d)}\right) \frac{\ca}{l+\cb d}\\&~~~~~~~~~~\times\left[f_l(\Bbeta_{l}) -f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_{l}) -f_l(\Bbeta^*)\big| \Bbeta_{l}\right] \right],
\end{split}
\label{eq1}
\end{equation}

also, the following upper bound of the product series holds

\begin{align*}
\left(1-\frac{\ca}{12(l+\cb d)}\right)\cdots \left(1-\frac{\ca}{12(t+\cb d)}\right)
\leq\left(\frac{l+1+\cb d}{t+1+\cb d}\right)^{\frac{\ca}{12}}.
\end{align*}

Case One: large $\ca$ If $\ca>12$, the analyses are exactly the ones in Theorem <ref> and the upper bound is

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq C^*\frac{\ku}{\kl^2}\frac{d}{t+1-t_1+\cb d} b_0^2.
\end{align*}

Case One: small $\ca$ If $\ca<12$, the first term of equation (<ref>) is bounded with

\begin{align*}
\prod_{l=0}^{t}\left(1-\frac{\ca}{12(l+\cb d)}\right)\ltwo{\Bbeta_{0}-\Bbeta^*}^2\leq \left(\frac{1+\cb d}{t+1+\cb}\right)^{\frac{\ca}{12}}.
\end{align*}

In addition, the second term of equation (<ref>) has the following upper bound,

\begin{align*}
&~~~~b_0^2\bar{\tau}^2\frac{\ku}{\kl^2}d\sum_{l=0}^{t}\left(1-\frac{\ca}{12(l+1+\cb d)}\right)\cdots \left(1-\frac{\ca}{12(t+\cb d)}\right)\frac{\ca^2}{(l+\cb d)^2}\\
&\leq b_0^2\bar{\tau}^2\frac{\ku}{\kl^2}d\sum_{l=0}^{t} \left(\frac{l+1+\cb d}{t+1+\cb d}\right)^{\frac{\ca}{12}}\frac{\ca^2}{(l+\cb d)^2}\\
&\leq  b_0^2\bar{\tau}^2\frac{\ku}{\kl^2}d\frac{\ca^2}{(t+1+\cb d)^{\frac{\ca}{12}}}\sum_{l=0}^{t}\frac{1}{(l+1+\cb d)^{2-\frac{\ca}{12}}}\\
&\leq 4 b_0^2\bar{\tau}^2\frac{\ku}{\kl^2}\frac{ d^{\frac{\ca}{12}}\ca^2}{(t+1+\cb d)^{\frac{\ca}{12}}},
\end{align*}

where the last line uses $\sum_{l=0}^{t}\frac{1}{(l+1+\cb d)^{2-\frac{\ca}{12}}}\leq\int_{0}^{t+1}\frac{1}{(x+ \cb d)^{2-\frac{\ca}{12}} }\; dx\leq\frac{1}{1-\ca/12}\frac{1}{(\cb d)^{1-\frac{\ca}{12}}}$ and $\cb\geq 1$. It is worth noting that, under the event of $\cup_{l=0}^t \{\ltwo{\Bbeta_{l}-\Bbeta^*}^2\leq C^*(\frac{d}{l+\cb d})^{\frac{\ca}{12}} \ltwo{\Bbeta_{0}-\Bbeta^*}^2\}$, it has

\begin{align*}
\|f_l(\Bbeta_{l}) -f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_{l}) -f_l(\Bbeta^*)\big| \Bbeta_{l}\right] \|_{\psi_2} \leq \bar{\tau}\sqrt{\ku C^*\left(\frac{d}{l+\cb d}\right)^{\frac{\ca}{12}} \ltwo{\Bbeta_{0}-\Bbeta^*}^2}.
\end{align*}

Thus according to Theorem 2 in [Shamir, 2011], the upper bound of the last term for equation (<ref>) is obtained with probability exceeding $1-\exp(-cd)$,

\begin{align*}
&~~~\frac{b_0}{\kl} \sum_{l=0}^{t}\left(1-\frac{\ca}{12(l+1+\cb d)}\right)\cdots \left(1-\frac{\ca}{12(t+\cb d)}\right) \frac{\ca}{l+\cb d}\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\times\left[f_l(\Bbeta_{l}) -f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_{l}) -f_l(\Bbeta^*)\big| \Bbeta_{l}\right] \right]\\
&\leq \bar{\tau}\ca\frac{b_0}{\kl}\sqrt{C^*\ku d \sum_{l=0}^t \left(\frac{l+1+\cb d}{t+1+\cb d}\right)^{\frac{\ca}{6} } \frac{1}{(l+\cb d)^2} \left(\frac{d}{l+\cb d}\right)^{\frac{\ca}{12}} \ltwo{\Bbeta_{0}-\Bbeta^*}^2}\\
&\leq \bar{\tau}\sqrt{C^*}\sqrt{\ku}\frac{1}{\kl}\frac{d^{\frac{\ca }{12}}}{(t+1+\cb d)^{\frac{\ca}{12}}}\cdot\ca b_0\ltwo{\Bbeta_{0}-\Bbeta^*}.
\end{align*}

Thus altogether, by having $\ca<\min\{12, (\kl/\sqrt{\ku})\ltwo{\Bbeta_{0}-\Bbeta^*}/b_0\}$, we obtain the following bound for equation (<ref>) with some sufficient large constant $C^*$, where $C^*$ may depend on the constants $\ca,\cb$,

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq C^*\left(\frac{d}{t+1+\cb d}\right)^{\frac{\ca}{12}} \ltwo{\Bbeta_{0}-\Bbeta^*}^2.
\end{align*}

§.§.§ Proof of Theorem <ref>

Before proving the regret bound, we first prove the following lemma, which establishes the expected error rate dynamics. It is worth noting that it is not a consequence of the empirical dynamics Theorem <ref>.

Assume the same conditions and stepsizes as Theorem <ref>. Then in the first phase we have $\EE\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq (1-c_1(\kl/\ku)/(\max\{\tau^2,(1-\tau)^2\} d))^{t+1}\ltwo{\Bbeta_{0}-\Bbeta^*}^2$ and in the second phase we have $\EE\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq C\frac{\ku}{\kl^2}\frac{d}{t+1-t_1+\cb d} b_0^2$.

Note that the expectation of $\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\right]$ is

\begin{align*}
\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\right]&=\EE\left[\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\big| \Bbeta_{t}\right]\right]\\
&=\EE\left[\EE\left[\ltwo{\Bbeta_{t}-\Bbeta^*}^2-2\eta_t\inp{\Bbeta_{t}-\Bbeta^*}{\g_t}+\eta_t^2\ltwo{\g_t}^2\big| \Bbeta_{t}\right]\right].
\end{align*}

First, we would consider the inner conditional expectation. According to the sub-gradient definition, and the independence between $\eta_t$ and $(\X_{t},Y_{t})$, we have

\begin{align*}
&~~~\EE\left[\ltwo{\Bbeta_{t}-\Bbeta^*}^2-2\eta_t\inp{\Bbeta_{t}-\Bbeta^*}{\g_t}+\eta_t^2\ltwo{\g_t}^2\big| \Bbeta_{t}\right]\leq\ltwo{\Bbeta_{t}-\Bbeta^*}^2-2\eta_t\EE\left[f_t(\Bbeta_t)-f_t(\Bbeta^*)\big|\Bbeta_{t}\right]+\eta_t^2 \EE\left[\ltwo{\g_t}^2\big| \Bbeta_{t}\right].
\end{align*}

First Phase Analysis.
With triangle inequality and Gaussian expectation calculations Lemma <ref>, we have $\EE\left[f_t(\Bbeta_t)-f_t(\Bbeta^*)\big|\Bbeta_{t}\right]\geq \sqrt{\kl}\sqrt{\frac{1}{2\pi}}\ltwo{\Bbeta_{t}-\Bbeta^*}-\gamma$. And the sub-gradient term could be bounded with $\EE\left[\ltwo{\g_t}^2\big| \Bbeta_{t}\right]\leq \max\{\tau^2,(1-\tau)^2\}\EE\ltwo{\X_{t}}^2\leq \ku \max\{\tau^2,(1-\tau)^2\} d$. Thus, in all we have

\begin{align*}
&~~~~\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\big| \Bbeta_{t}\right]\\
&\leq \ltwo{\Bbeta_{t}-\Bbeta^*}^2-\frac{3}{4}\eta_t\sqrt{\kl}\ltwo{\Bbeta_{t}-\Bbeta^*}+2\eta_t\gamma+\eta_t^2\ku\max\{\tau^2,(1-\tau)^2\}d\\
&\leq \ltwo{\Bbeta_{t}-\Bbeta^*}^2-\frac{1}{2}\eta_t\sqrt{\kl}\ltwo{\Bbeta_{t}-\Bbeta^*}+\eta_t^2\ku\max\{\tau^2,(1-\tau)^2\}d,
\end{align*}

which uses the region restrictions of the first region $\ltwo{\Bbeta_{t}-\Bbeta^*}\geq 8\sqrt{\kl^{-1}}\gamma$.
Inserting the stepsize, which is characterized in Theorem <ref>, yields

\begin{align*}
\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\big| \Bbeta_{t}\right]\leq \left(1-\frac{1}{350}\frac{\kl}{\ku}\frac{1}{\bar{\tau}^2d}\right)\ltwo{\Bbeta_{t}-\Bbeta^*}^2.
\end{align*}

Take expectation over $\Bbeta_{t}$ and then the above equation is

\begin{align*}
\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\right]\leq\left(1-\frac{1}{350}\frac{\kl}{\ku}\frac{1}{\bar{\tau}^2d}\right)\EE\left[\ltwo{\Bbeta_{t}-\Bbeta^*}^2\right],
\end{align*}

which completes the convergence proof of the first phase.

Second Phase Analysis.
Lemma <ref> shows that in the second phase, the lower bound of the expected excess risk is a second-order one, $\EE[f_t(\Bbeta_{t}) -f_t(\Bbeta^*)|\Bbeta_{t}]\geq \frac{1}{12b_0}\kl\ltwo{\Bbeta_{t}-\Bbeta^*}^2$. Thus we have

\begin{align*}
\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\big| \Bbeta_{t}\right]&\leq \ltwo{\Bbeta_{t}-\Bbeta^*}^2-\frac{1}{6b_0}\eta_t\kl\ltwo{\Bbeta_{t}-\Bbeta^*}^2+\eta_t^2\ku\max\{\tau^2,(1-\tau)^2\}d.
\end{align*}

By specifying the stepsize $\eta_t=\frac{\ca}{t-t_1+\cb d}\frac{b_0}{\kl}$, we obtain

\begin{align*}
\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\big| \Bbeta_{t}\right]&\leq\left(1-\frac{\ca}{6(t-t_1+\cb d)}\right)\ltwo{\Bbeta_{t}-\Bbeta^*}^2+\frac{\ku}{\kl^2}\frac{\ca^2}{(t-t_1+\cb d)^2}db_0^2.
\end{align*}

Take expectation over $\Bbeta_{t}$ on each side of the equation and insert the upper bound at $t$, $\EE\left[\ltwo{\Bbeta_{t}-\Bbeta^*}^2\right]\leq C\frac{\ku}{\kl^2}\frac{\ca d}{t-t_1+\cb d}b_0^2$, into the above equation,

\begin{align*}
\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\right]&\leq C\left(1-\frac{\ca}{6(t-t_1+\cb d)}\right)\frac{\ku}{\kl^2}\frac{\ca d}{t-t_1+\cb d}b_0^2+\frac{\ku}{\kl^2}\frac{\ca^2}{(t-t_1+\cb d)^2}db_0^2\\
&=C\left(1-\frac{\ca/2}{6(t-t_1+\cb d)}\right)\frac{\ku}{\kl^2}\frac{\ca d}{t-t_1+\cb d}b_0^2.
\end{align*}

Note that $1-\frac{\ca/2}{6(t-t_1+\cb d)}\leq1-\frac{1}{t-t_1+\cb d}\leq \frac{t-t_1+\cb d}{t+1-t_1+\cb d}$ and then we finally obtain $\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\right]\leq C\frac{\ku}{\kl^2}\frac{\ca d}{t+1-t_1+\cb d}b_0^2$, which completes proofs of the second phase.

Then we are ready to prove the regret in Theorem <ref>. According to the expected excess risk bound in Lemma <ref>, we have $\Bbeta^*=\arg\max_{\Bbeta}\EE\sum_{t=0}^{T} f_{t}(\Bbeta_{t})-f_t(\Bbeta)$.  In the first phase, it has

\begin{align*}
\end{align*}

where the second line uses $f_t(\cdot)$ is Lipschitz and the last line follows from Lemma <ref>. In the second phase, it arrives at

\begin{align*}
\EE\left[f_t(\Bbeta_{t})-f_t(\Bbeta^*)\right]&=\EE\left[\EE\left[f_t(\Bbeta_{t})-f_t(\Bbeta^*)\big|\Bbeta_t\right]\right]\leq\frac{\ku}{b_1}\EE\ltwo{\Bbeta_{t}-\Bbeta^*}^2\\
&\leq 2\frac{\ku^2}{\kl^2}\frac{d}{t-t_1+\cb d}b_0^2,
\end{align*}

whose first inequality follows from Lemma <ref> and the last line is based on Lemma <ref>. Thus altogether, we have

\begin{align*}
&\leq\sum_{t=0}^{t_1}\sqrt{\ku}\left(1-c(\kl/\ku)/(\max\{\tau^2,(1-\tau)^2\}d)\right)^t\ltwo{\Bbeta_{0}-\Bbeta^*}+Cdb_0^2\frac{\ku^2}{\kl^2}\sum_{t=t_1}^{T}\frac{1}{t-t_1+\cb d}\\
&\leq C_1(\ku/\kl)\sqrt{\ku}d\ltwo{\Bbeta_{0}-\Bbeta^*}+C_2 (\ku^2/\kl^2)db_0^2\log\left(\frac{T-t_1+\cb d}{\cb d}\right),
\end{align*}

which completes the proof.

§.§ Proof of Batch Learning

In this subsection, we will establish the batch learning setting in a sequential manner, starting with the proof of Theorem <ref>, followed by an examination of convergence under well-controlled initialization conditions, and concluding with the demonstration of Theorem <ref>.

§.§.§ Proof of Theorem <ref>

The update $\Bbeta_{t+1}=\Bbeta_{t}-\eta_t\g_t$ guarantees that

\begin{align}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2
&= \ltwo{\Bbeta_t-\Bbeta^*}^2-2\eta_t\inp{\Bbeta_t-\Bbeta^*}{\g_t}+\eta_t^2\ltwo{\g_t}^2.
\label{eq:batch update}
\end{align}

The analyses of the terms $\inp{\Bbeta_t-\Bbeta^*}{\g_t}$ and $\ltwo{\g_t}^2$ exhibit significant variations depending on the distance between the current iterate $\Bbeta_t$ and the oracle $\Bbeta^*$. Consequently, we will examine their respective properties in accordance with this divergence.

First Phase Analysis.
We shall prove the convergence dynamics by induction. Initially, it holds obviously for $\Bbeta_0$. We continue to assume for the iteration of $t$, it has $\ltwo{\Bbeta_{t}-\Bbeta^*}\leq \left(1-\frac{1}{100}\frac{\kl}{\ku}\right)^{t}\cdot D_0$ and we are going to prove the convergence dynamics at $t+1$. For convenience, we denote $D_t:=\left(1-\frac{1}{100}\frac{\kl}{\ku}\right)^{t}\cdot D_0$ and $D_{t+1}:=\left(1-\frac{1}{100}\frac{\kl}{\ku}\right)^{t+1}\cdot D_0$. We proceed to assume the event

\begin{align*}
\bcalE_t&:=\left\{\sup_{\Bbeta_1,\Bbeta_2\in\RR^d} \left|f_t(\Bbeta_1)-f_t(\Bbeta_2)-\EE\left[ f_t(\Bbeta_1)-f_t(\Bbeta_2)\right]\right|\cdot\ltwo{\Bbeta_1-\Bbeta_2}^{-1}\leq C_1\max\{\tau,1-\tau\}\sqrt{\ku d/n_t}\right\}
\end{align*}
holds. Specifically, Proposition <ref> shows that $\PP(\bcalE_t)\geq1-\exp(-C_2d)-\exp(-\sqrt{n_t/\log n_t})$. Then the intermediate term of (<ref>) could be lower bounded with
$$\inp{\Bbeta_t-\Bbeta^*}{\g_t}\geq f_t(\Bbeta_t)-f_t(\Bbeta^*)\geq \EE\left[f_t(\Bbeta_t)-f_t(\Bbeta^*)\big|\Bbeta_t \right]-C_1\max\{\tau,1-\tau\}\sqrt{\ku\frac{d}{n_t}}\ltwo{\Bbeta_t-\Bbeta^*},$$

which is due to the sub-gradient definition and uses the event $\bcalE_t$. Also the convexity of quantile loss $\rho_{Q,\tau}(x_1+x_2)\leq \rho_{Q,\tau}(x_1)+\rho_{Q,\tau}(x_2)$ and Lemma <ref> lead to

\begin{align*}
\EE\left[f_t(\Bbeta_t)-f_t(\Bbeta^*)\big|\Bbeta_t \right]&\geq \frac{1}{n_t}\sum_{i=1}^{n_t}\EE\left[\rho_{Q,\tau}(\inp{\X_i^{(t)}}{\Bbeta^*-\Bbeta_t})-\rho_{Q,\tau}(-\xi_i)-\rho_{Q,\tau}(\xi_i)\big| \Bbeta_t\right]\\
&\geq \frac{1}{\sqrt{2\pi}} \sqrt{\kl}\ltwo{\Bbeta_t-\Bbeta^*}-\gamma.
\end{align*}

Thus the batch size $n_t\geq C(\ku/\kl)\max\{\tau^2,(1-\tau)^2\} d$ and the phase region $\ltwo{\Bbeta_t-\Bbeta^*}\geq8\sqrt{\kl^{-1}}\gamma$ guarantee that

\begin{align*}
\inp{\Bbeta_t-\Bbeta^*}{\g_t}\geq \frac{1}{6} \sqrt{\kl}\ltwo{\Bbeta_t-\Bbeta^*}.
\end{align*}

Besides, Lemma <ref> proves that under event $\bcalE_t$, $\ltwo{\g_t}\leq\sqrt{\ku}$. In this way, equation (<ref>) could be upper bounded with

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq \ltwo{\Bbeta_{t}-\Bbeta^*}^2- \frac{1}{3}\eta_t\sqrt{\kl}\ltwo{\Bbeta_t-\Bbeta^*}+ \eta_t^2\ku.
\end{align*}

We need to regard the right hand side of the above equation as a quadratic function of $\ltwo{\Bbeta_t-\Bbeta^*}$ and by induction $\ltwo{\Bbeta_t-\Bbeta^*}\leq D_{t}$, we have

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq D_t^2- \frac{1}{3}\eta_t\sqrt{\kl}D_t+ \eta_t^2\ku.
\end{align*}

The stepsize value $\eta_t=(1-\frac{1}{100}\frac{\kl}{\ku})^{t}\eta_0\in \frac{\sqrt{\kl}}{\ku}D_t\cdot [1/8,5/24]$ infers that

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq \left(1-\frac{1}{50}\frac{\kl}{\ku}\right)D_t^2,
\end{align*}

which shows $\ltwo{\Bbeta_{t+1}-\Bbeta^*}\leq (1-\frac{1}{100}\frac{\kl}{\ku})^{t+1}D_0=D_{t+1}$.

Second Phase Analysis.
In the second phase, we shall still assume event $\bcalE_t$ holds, which is defined and discussed in the first phase analyses. Note that according to Lemma <ref>, in this case, where $\ltwo{\Bbeta_t-\Bbeta^*}\leq 8\sqrt{\kl^{-1}}\gamma$, the expectation of excess risk has a quadratic lower bound,

\begin{align*}
\EE\left[f_t(\Bbeta_t)-f_t(\Bbeta^*)\big|\Bbeta_t \right]
&\geq \frac{1}{12b_0}\kl\ltwo{\Bbeta_t-\Bbeta^*}^2.
\end{align*}

The second phase region has the constraints $\ltwo{\Bbeta_{t}-\Bbeta^*}\geq C\max\{\tau,1-\tau\} \sqrt{(\ku/\kl^2)d/n}b_0$. Then under event $\bcalE_t$, we have

\begin{align*}
f_t(\Bbeta_t)-f_t(\Bbeta^*)&\geq \frac{1}{12b_0}\kl\ltwo{\Bbeta_t-\Bbeta^*}^2-C_1\max\{\tau,1-\tau\}\sqrt{\ku d/n_t}\ltwo{\Bbeta_t-\Bbeta^*}\\
&\geq \frac{1}{24b_0}\kl\ltwo{\Bbeta_t-\Bbeta^*}^2.
\end{align*}

In addition, Lemma <ref> proves that under the event $\bcalE_t$, the following holds in the second phase region:

$$\ltwo{\g_t}^2\leq 4\frac{1}{b_1^2}\ku^2\ltwo{\Bbeta-\Bbeta^*}^2.$$

Then together with the sub-gradient definition $\inp{\g_t}{\Bbeta_t-\Bbeta^*}\geq f_t(\Bbeta_t)-f_t(\Bbeta^*)$, equation (<ref>) can be upper bounded as

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq\ltwo{\Bbeta_{t}-\Bbeta^*}^2-\eta_t\frac{1}{12b_0}\kl\ltwo{\Bbeta_{t}-\Bbeta^*}^2+\eta_t^2 \frac{4}{b_1^2}\ku^2\ltwo{\Bbeta-\Bbeta^*}^2.
\end{align*}

By having stepsize $\eta_t\in \frac{\kl}{\ku^2}\frac{b_1^2}{b_0}\cdot [c_1,c_2]$, we obtain

Third Phase Analysis. We shall prove the convergence dynamics by induction. The proof at $t_2$ is trivial, which can be obtained similarly to the following analyses, and hence it is skipped. Then we are going to prove $\ltwo{\Bbeta_{t+1}-\Bbeta^*}$ when the desired inequality holds for $t_2,\dots,t$. It is worth noting that Lemma <ref> and Proposition <ref> show: in the region of $\ltwo{\Bbeta_t-\Bbeta^*}\leq C_1\max\{\tau,1-\tau\}\sqrt{(\ku/\kl^2)d/n}\cdot b_0$,  with probability exceeding $1-\exp(-C_1d)-\exp(-\sqrt{n_t/\log n_t})$, it has
$$\ltwo{\g_t}^2\leq C_3^2\bar{\tau}^2\ku\frac{d}{n}\frac{\ku^2b_0^2}{\kl^2b_1^2},$$

where we denote $\bar{\tau}:=\max\{\tau,1-\tau\}$, for convenience. According to the loss function expectation calculations in Lemma <ref>, we have

\begin{align*}
\EE\left[f_t(\Bbeta_{t})-f_t(\Bbeta^*)\big| \Bbeta_{t}\right]&\geq \frac{1}{24b_0}\ltwo{\Bbeta_{t}-\Bbeta^*}^2.
\end{align*}

Then together with the sub-gradient definition, we have

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq \ltwo{\Bbeta_{t}-\Bbeta^*}^2-\eta_t\frac{1}{12b_0}\kl\ltwo{\Bbeta_{t}-\Bbeta^*}^2+C_3^2\eta_t^2\bar{\tau}^2\ku\frac{d}{n}\frac{\ku^2b_0^2}{\kl^2b_1^2}\\
&~~~~~~~~~~~~~~~~~~~+2\eta_t\left[f_t(\Bbeta_t)-f_t(\Bbeta^*)-\EE\left[f_t(\Bbeta_t)-f_t(\Bbeta^*)\big|\Bbeta_{t} \right]\right].
\end{align*}

Insert the stepsize $\eta_t=\frac{\ca}{t-t_2+\cb}\frac{b_0}{\kl}$  into the above equation and then it arrives at

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq \left(1-\frac{\ca}{12(t-t_2+\cb)}\right)\ltwo{\Bbeta_{t}-\Bbeta^*}^2+C_3^2\bar{\tau}^2\frac{\ca^2}{(t-t_2+\cb)^2}\frac{\ku}{\kl^2}\frac{d}{n}\frac{\ku^2b_0^2}{\kl^2b_1^2}b_0^2\\
&~~~~~~~~~~~~~~~~~~~+2\frac{b_0}{\kl}\frac{\ca}{t-t_2+\cb}\left[f_t(\Bbeta_t)-f_t(\Bbeta^*)-\EE\left[f_t(\Bbeta_t)-f_t(\Bbeta^*)\big|\Bbeta_{t} \right]\right].
\end{align*}

Applying the above bound repeatedly from $t_2$, we have

\begin{equation}
\begin{split}
&+C_3^2\bar{\tau}^2\frac{\ku}{\kl^2}\frac{\ku^2b_0^2}{\kl^2b_1^2}\frac{d}{n}b_0^2\sum_{l=t_2}^{t}\left(1-\frac{\ca}{12(l+1-t_2+\cb)}\right)\cdots\left(1-\frac{\ca}{12(t-t_2+\cb)}\right) \frac{\ca^2}{(l-t_2+\cb)^2}\\
&+\frac{b_0}{\kl}\sum_{l=t_2}^{t}\left(1-\frac{\ca}{12(l+1-t_2+\cb)}\right)\cdots\left(1-\frac{\ca}{12(t-t_2+\cb)}\right)\frac{\ca}{l-t_2+\cb}\\&~~~~~~~~~~~~~~~~~~~~~~~~~~ \times\left[f_l(\Bbeta_l)-f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_l)-f_l(\Bbeta^*)\big|\Bbeta_{l} \right]\right].
\end{split}
\label{eq3}
\end{equation}

A sharp analysis of equation (<ref>) is key to the proof. Notice that with $\ca\geq 12$, we have $1-\frac{\ca}{12(l-t_2+\cb)}\leq1-\frac{1}{l-t_2+\cb}\leq \frac{l-t_2+\cb}{l+1-t_2+\cb}$. In this way, the first term on the right hand side could be bounded with

\begin{align*}
\prod_{l=t_2}^{t}\left(1-\frac{\ca}{12(l-t_2+\cb)}\right)\ltwo{\Bbeta_{t_2}-\Bbeta^*}^2\leq\frac{\cb}{t+1-t_2+\cb}\ltwo{\Bbeta_{t_2}-\Bbeta^*}^2.
\end{align*}

The product sequence satisfies, for each $l=t_2,\dots,t$, that

\begin{equation}
\begin{split}
&=\exp\left(\sum_{k=l+1}^{t}\log\left(1- \frac{\ca}{12(k-t_2+\cb)}\right)\right)\leq\exp\left(-\sum_{k=l+1}^{t} \frac{\ca}{12(k-t_2+\cb)}\right)\\
&\leq\exp\left(- \frac{\ca}{12}\int_{l+1-t_2}^{t+1-t_2}\frac{1}{x+\cb}\; dx\right)= \exp\left(- \frac{\ca}{12} \log\left(\frac{t+1-t_2+\cb}{l+1-t_2+\cb}\right)\right)\\
\end{split}
\label{eq:2-1}
\end{equation}

The second term of equation (<ref>) thus has the following upper bound:

\begin{align*}
&~~~\sum_{l=t_2}^{t}\left(1-\frac{\ca}{12(l+1-t_2+\cb)}\right)\cdots\left(1-\frac{\ca}{12(t-t_2+\cb)}\right) \frac{\ca^2}{(l-t_2+\cb)^2}\\
&\leq\left(\frac{\cb+1}{\cb}\right)^{\frac{\ca}{12}}\frac{\ca^2}{\left(t+1-t_2+\cb \right)^{\frac{\ca}{12}}}\sum_{l=t_2}^{t} \left(l-t_2+\cb\right)^{\frac{\ca}{12}-2}.
\end{align*}

With $\ca>12$, we have $\sum_{l=t_2}^{t} \left(l-t_2+\cb\right)^{\frac{\ca}{12}-2}\leq\int_{0}^{t-t_2+1}(x+\cb)^{\frac{\ca}{12}-2}\; dx= \frac{1}{\frac{\ca}{12}-1}(t-t_2+1+\cb)^{\frac{\ca}{12}-1}$.
Hence, we have the upper bound of the second term

\begin{align*}
&~~~~\sum_{l=t_2}^{t}\left(1-\frac{\ca}{12(l+1-t_2+\cb)}\right)\cdots\left(1-\frac{\ca}{12(t-t_2+\cb)}\right) \frac{\ca^2}{(l-t_2+\cb)^2}\\
&\leq\frac{1}{\frac{\ca}{12}-1}\left(\frac{\cb+1}{\cb}\right)^{\frac{\ca}{12}} \frac{1}{t+1+\cb-t_2}.
\end{align*}

Then, it suffices to bound the last term of equation (<ref>). It is worth noting that under the event of $\cup_{l=t_2}^{t}\left\{ \ltwo{\Bbeta_l-\Bbeta^*}^2\leq C^*\frac{\ku}{\kl^2}\frac{\ku^2b_0^2}{\kl^2b_1^2}\frac{1}{l-t_2+\cb}\frac{d}{n}b_0^2\right\}$, we have

$$\|f_l(\Bbeta_l)-f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_l)-f_l(\Bbeta^*)\big|\Bbeta_{l} \right]\|_{\Psi_2}\leq\sqrt{C^*\frac{\ku^2}{\kl^2}\frac{\ku^2b_0^2}{\kl^2b_1^2}\frac{1}{l-t_2+\cb}\frac{d}{n^2} }b_0.$$

Furthermore, invoking Azuma's sub-Gaussian inequality (e.g., Theorem 2 in [Shamir, 2011]) in conjunction with equation (<ref>), it holds with probability exceeding $1-\exp(-cd)$ that

\begin{align*}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \times\left[f_l(\Bbeta_l)-f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_l)-f_l(\Bbeta^*)\big|\Bbeta_{l} \right]\right]\\
&\leq \frac{b_0}{\kl}\left(d\sum_{l=t_2}^{t}\prod_{k=l+1}^{t}\left(1-\frac{\ca}{12(k-t_2+\cb)}\right)^2\frac{\ca^2}{(l-t_2+\cb)^2}\right.\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\left.\times \left\|f_l(\Bbeta_l)-f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_l)-f_l(\Bbeta^*)\big|\Bbeta_{l} \right]\right\|_{\Psi_2}^2\right)^{1/2}\\
&\leq \bar{\tau}^2\frac{\ku}{\kl^2}\frac{\ku b_0}{\kl b_1}\frac{d}{n}b_0^2\frac{C\ca\sqrt{C^*}}{(t+1-t_2+\cb )^{\frac{ \ca }{12}}}\sqrt{ \sum_{l=t_2}^{t} (l+1-t_2+\cb)^{\frac{\ca}{6}-3}}\\
&\leq \bar{\tau}^2\frac{\ku}{\kl^2}\frac{\ku b_0}{\kl b_1}\frac{d}{n}b_0^2\frac{C\ca\sqrt{C^*}}{t+1-t_2+\cb }.
\end{align*}

Therefore, combining the pieces, equation (<ref>) can be bounded from above as

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq C^*\bar{\tau}^2\frac{\ku}{\kl^2}\frac{d}{n}\frac{\ku^2b_0^2}{\kl^2b_1^2}\frac{1}{t+1-t_2+\cb }b_0^2,
\end{align*}

where $C^*>\cb^2$ is some sufficiently large constant.

§.§.§ Analysis of Batch Learning with Well-controlled Initial Errors

We will elucidate the convergence dynamics under the condition of well-controlled initial error rates, specifically when $\ltwo{\Bbeta_{0}-\Bbeta^*}^2\leq C_1^2\max{\tau^2 ,(1-\tau)^2}(\ku/\kl^2)(d/n)\cdot b_0^2$. Employing a similar analysis as applied in the third phase of Theorem <ref>, we obtain

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq \ltwo{\Bbeta_{t}-\Bbeta^*}^2-\eta_t\frac{1}{12b_0}\kl\ltwo{\Bbeta_{t}-\Bbeta^*}^2+C_3^2\eta_t^2\bar{\tau}^2\ku\frac{d}{n}\frac{\ku^2b_0^2}{\kl^2b_1^2}\\
&~~~~~~~~~~~~~~~~~~~-2\eta_t\left[f_t(\Bbeta_t)-f_t(\Bbeta^*)-\EE\left[f_t(\Bbeta_t)-f_t(\Bbeta^*)\big|\Bbeta_{t} \right]\right].
\end{align*}

Inserting the stepsize $\eta_t=\frac{\ca}{t+\cb}\frac{b_0}{\kl}$ into the above equation results in

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq\left(1-\frac{\ca}{12(t+\cb )}\right)\ltwo{\Bbeta_{t}-\Bbeta^*}^2+C_3^2\bar{\tau}^2\frac{\ku}{\kl^2}\frac{\ca^2}{(t+\cb)^2}\frac{d}{n}b_0^2\\
&~~~~~~~~~~~~~~~-2\frac{\ca}{t+\cb }\frac{b_0}{\kl} \left[f_t(\Bbeta_t)-f_t(\Bbeta^*)-\EE\left[f_t(\Bbeta_t)-f_t(\Bbeta^*)\big|\Bbeta_{t} \right]\right].
\end{align*}

Moreover, the preceding equation implies

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq\prod_{l=0}^{t}\left(1-\frac{\ca}{12(l+\cb )}\right)\ltwo{\Bbeta_{0}-\Bbeta^*}^2\\
&~~~+C_3^2\bar{\tau}^2\frac{\ku^2b_0^2}{\kl^2b_1^2}\frac{\ku}{\kl^2}\sum_{l=0}^t \left(1-\frac{\ca}{12(l+1+\cb )}\right)\cdots\left(1-\frac{\ca}{12(t+\cb )}\right)\frac{\ca^2}{(l+\cb )^2}\frac{d}{n}b_0^2\\
&~~~-\frac{b_0}{\kl}\sum_{l=0}^t \left(1-\frac{\ca}{12(l+1+\cb )}\right)\cdots\left(1-\frac{\ca}{12(t+\cb )}\right)\frac{\ca}{l+\cb }\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~\times\left[f_l(\Bbeta_l)-f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_l)-f_t(\Bbeta^*)\big|\Bbeta_{l} \right]\right].
\end{align*}

Case One: large $\ca$. With $\ca>12$, the first term on the right hand side can be bounded as

\begin{align*}
\prod_{l=0}^{t}\left(1-\frac{\ca}{12(l+\cb )}\right)\ltwo{\Bbeta_{0}-\Bbeta^*}^2&\leq\prod_{l=0}^{t}\left(1-\frac{1}{l+\cb }\right)\ltwo{\Bbeta_{0}-\Bbeta^*}^2\leq\frac{\cb}{t+1+\cb}\ltwo{\Bbeta_{0}-\Bbeta^*}^2.
\end{align*}

The product sequence is bounded by

\begin{align*}
\left(1-\frac{\ca}{12(l+1+\cb )}\right)\cdots\left(1-\frac{\ca}{12(t+\cb )}\right)&\leq \exp\left(-\frac{\ca}{12}\sum_{k=l+1}^{t} \frac{1}{k+\cb}\right)\\
&\leq\exp\left(-\frac{\ca}{12}\int_{l+1}^{t+1}\frac{1}{x+\cb}\; dx\right)\\
\end{align*}

Subsequently, the second term can be bounded as

\begin{align*}
&~~~\sum_{l=0}^t \left(1-\frac{\ca}{12(l+1+\cb )}\right)\cdots\left(1-\frac{\ca}{12(t+\cb )}\right)\frac{\ca^2}{(l+\cb )^2}\\
&\leq\sum_{l=0}^t \left(\frac{l+\cb+1}{t+\cb+1}\right)^{\frac{\ca}{12}}\frac{\ca^2}{(l+\cb )^2}\leq \left(\frac{\cb+1}{\cb}\right)^{\frac{\ca}{12}}\frac{\ca^2}{t+\cb+1}.
\end{align*}

Next, consider the last term. It is noteworthy that under the event $\cup_{l=0}^t{\ltwo{\Bbeta_{l}-\Bbeta^*}^2\leq C^*\bar{\tau}^2\frac{\ku}{\kl^2}\frac{1}{l+\cb} \frac{d}{n} b_0^2}$, we have

\begin{align*}
\|f_l(\Bbeta_l)-f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_l)-f_t(\Bbeta^*)\big|\Bbeta_{l} \right] \|_{\Psi_2}^2\leq C^*\bar{\tau}^4\frac{d}{n^2}\frac{\ku^2}{\kl^2}\frac{1}{l+\cb}b_0^2.
\end{align*}

Therefore, in accordance with Theorem 2 in [Shamir, 2011], with probability at least $1-\exp(-cd)$, the last term satisfies

\begin{align*}
&~~~\frac{b_0}{\kl}\sum_{l=0}^t \left(1-\frac{\ca}{12(l+1+\cb )}\right)\cdots\left(1-\frac{\ca}{12(t+\cb )}\right)\frac{\ca}{l+\cb }\\
&~~~~~~~~~~~~~~~~~~~~~~~~~\times\left[f_l(\Bbeta_l)-f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_l)-f_t(\Bbeta^*)\big|\Bbeta_{l} \right]\right]\\
&\leq \ca\bar{\tau}^2\frac{\ku}{\kl^2}\frac{\ku b_0}{\kl b_1}\frac{d}{n}b_0^2\sqrt{\sum_{l=0}^{t} \left(\frac{l+1+\cb}{t+1+\cb}\right)^{\frac{\ca}{6}}\frac{C^*}{(l+\cb)^3}  }\\
&\leq \ca\bar{\tau}^2\frac{\ku}{\kl^2}\frac{\ku b_0}{\kl b_1}\frac{\sqrt{C^*}}{t+1+\cb}\frac{d}{n}b_0^2.
\end{align*}

Putting together the pieces, we conclude that

\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq C^*\bar{\tau}^2\frac{\ku}{\kl^2}\frac{\ku^2b_0^2}{\kl^2b_1^2}\frac{1}{t+1+\cb} \frac{d}{n} b_0^2.

Case Two: small $\ca$ In this setting, with $\ca<12$, the second term satisfies

\begin{align*}
&~~~\sum_{l=0}^t \left(1-\frac{\ca}{12(l+1+\cb )}\right)\cdots\left(1-\frac{\ca}{12(t+\cb )}\right)\frac{\ca^2}{(l+\cb )^2}\\
&\leq\sum_{l=0}^t \left(\frac{l+\cb+1}{t+\cb+1}\right)^{\frac{\ca}{12}}\frac{\ca^2}{(l+\cb )^2}\leq \frac{\ca^2}{(t+\cb+1)^{\frac{\ca}{12}}}.
\end{align*}

Under the event  $\cup_{l=0}^t\{\ltwo{\Bbeta_{l}-\Bbeta^*}^2\leq \frac{c}{(l+\cb)^{\frac{\ca}{12}}} \ltwo{\Bbeta_{0}-\Bbeta^*}^2\}$, we have

\begin{align*}
\|f_l(\Bbeta_l)-f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_l)-f_t(\Bbeta^*)\big|\Bbeta_{l} \right] \|_{\Psi_2}^2\leq C^*\bar{\tau}\frac{\ku}{n}\frac{1}{(l+\cb)^{\frac{\ca}{12}}} \ltwo{\Bbeta_{0}-\Bbeta^*}^2 .
\end{align*}

Thus, with probability exceeding $1-\exp(-cd)$, it holds

\begin{align*}
&~~~\frac{b_0}{\kl}\sum_{l=0}^t \left(1-\frac{\ca}{12(l+1+\cb )}\right)\cdots\left(1-\frac{\ca}{12(t+\cb )}\right)\frac{\ca}{l+\cb }\times\left[f_l(\Bbeta_l)-f_l(\Bbeta^*)-\EE\left[f_l(\Bbeta_l)-f_t(\Bbeta^*)\big|\Bbeta_{l} \right]\right]\\
&\leq \bar{\tau}\ca\frac{\sqrt{d}}{\sqrt{n}}\frac{\sqrt{\ku}}{\kl}b_0\cdot\ltwo{\Bbeta_{0}-\Bbeta^*}\cdot\sqrt{\sum_{l=0}^{t} \left(\frac{l+1+\cb}{t+1+\cb}\right)^{\frac{\ca}{6}}\frac{C^*}{(l+\cb)^{2+\frac{\ca}{12}}}  }\\
&\leq \bar{\tau}\ca\frac{\sqrt{d}}{\sqrt{n}}\frac{\sqrt{\ku}}{\kl}b_0\cdot\ltwo{\Bbeta_{0}-\Bbeta^*}\cdot \frac{\sqrt{C^*}}{(t+1+\cb)^{\frac{\ca}{12}}}.
\end{align*}

To sum up, we establish the upper bound

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq\left(\frac{1+\cb}{t+1+\cb}\right)^{\frac{\ca}{12}}\ltwo{\Bbeta_{0}-\Bbeta^*}^2+C_3^2\bar{\tau}^2\frac{\ku}{\kl^2}\frac{\ku^2b_0^2}{\kl^2b_1^2}\frac{d}{n}b_0^2\cdot \frac{\ca^2}{(t+\cb+1)^{\frac{\ca}{12}}}\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\bar{\tau}\frac{\sqrt{d}}{\sqrt{n}}\frac{\sqrt{\ku}}{\kl}b_0\cdot\ltwo{\Bbeta_{0}-\Bbeta^*}\cdot \frac{\ca\sqrt{C^*}}{(t+1+\cb)^{\frac{\ca}{12}}}.
\end{align*}

Therefore, by choosing sufficiently small $\ca<(\kl/\ku)(b_1/b_0)(\kl/\sqrt{\ku})\sqrt{n/d}\ltwo{\Bbeta_{0}-\Bbeta^*}/\bar{\tau}b_0$ and $\ca<\cb\leq1$, we obtain

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq\frac{C^*}{(t+1+\cb)^{\frac{\ca}{12}}}\ltwo{\Bbeta_{0}-\Bbeta^*}^2.
\end{align*}

§.§.§ Proof of Theorem <ref>

Prior to demonstrating Theorem <ref>, it is imperative to establish the ensuing lemma.

Assume the same conditions and stepsize schemes as in Theorem <ref>. In the first phase, we have $\EE\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq (1-c_1(\kl/\ku)/(\max\{\tau^2,(1-\tau)^2\} ))^{t+1}\ltwo{\Bbeta_{0}-\Bbeta^*}^2$; in the second, $\EE\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq (1-c_1(\kl/\ku)/(b_1^2/b_0^2))^{t+1}\ltwo{\Bbeta_{t_1}-\Bbeta^*}^2$; and in phrase three, $\EE\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq C\frac{\ku}{\kl^2}\frac{1}{t+1-t_2+\cb} \frac{d}{n}b_0^2$.

Note that the expectation can be expressed as

\begin{align*}
\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\right]&=\EE\left[\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\big| \Bbeta_{t}\right]\right]\\
&=\EE\left[\EE\left[\ltwo{\Bbeta_{t}-\Bbeta^*}^2-2\eta_t\inp{\Bbeta_{t}-\Bbeta^*}{\g_t}+\eta_t^2\ltwo{\g_t}^2\big| \Bbeta_{t}\right]\right]
\end{align*}

We first consider the inside conditional expectation. According to the sub-gradient definition and considering that $\eta_t$ is independent of $(\X_{t},Y_{t})$, we have

\begin{equation}
\begin{split}
&~~~\EE\left[\ltwo{\Bbeta_{t}-\Bbeta^*}^2-2\eta_t\inp{\Bbeta_{t}-\Bbeta^*}{\g_t}+\eta_t^2\ltwo{\g_t}^2\big| \Bbeta_{t}\right]\\
&\leq\ltwo{\Bbeta_{t}-\Bbeta^*}^2-2\eta_t\EE\left[f_t(\Bbeta_t)-f_t(\Bbeta^*)\big|\Bbeta_{t}\right]+\eta_t^2 \EE\left[\ltwo{\g_t}^2\big| \Bbeta_{t}\right]
\end{split}
\label{eq:2-2}
\end{equation}

The sub-gradient value at $0$ won't affect the expectation calculations, so for convenience, we let $\partial \rho_{Q,\tau}(x)|_{x=0}=0$. Then the sub-gradient at $\Bbeta_{t}$ could be written as

\begin{align*}
n_t\g_t=\sum_{i=1}^{n_t}\X_i^{(t)}\cdot \left(\tau1_{Y_i^{(t)}<\inp{\X_i^{(t)}}{\Bbeta_{t}}} + (1-\tau)1_{Y_i^{(t)}>\inp{\X_i^{(t)}}{\Bbeta_{t}}}\right).
\end{align*}

Its length is bounded with

\begin{align*}
&n_t^2\ltwo{\g_t}^2=\sum_{i=1}^{n_t}\ltwo{\X_i^{(t)}}^2\cdot \left(\tau^21_{Y_i^{(t)}<\inp{\X_i^{(t)}}{\Bbeta_{t}}} + (1-\tau)^21_{Y_i^{(t)}>\inp{\X_i^{(t)}}{\Bbeta_{t}}}\right)\\
&+\sum_{i\neq j} \inp{\X_i^{(t)}}{\X_j^{(t)}}\cdot\left(\tau1_{Y_i^{(t)}<\inp{\X_i^{(t)}}{\Bbeta_{t}}} + (1-\tau)1_{Y_i^{(t)}>\inp{\X_i^{(t)}}{\Bbeta_{t}}}\right)\cdot \left(\tau1_{Y_j^{(t)}<\inp{\X_j^{(t)}}{\Bbeta_{t}}} + (1-\tau)1_{Y_j^{(t)}>\inp{\X_j^{(t)}}{\Bbeta_{t}}}\right).
\end{align*}

Taking the $\Bbeta_t$ conditional expectation on each side of the above equation leads to

\begin{align*}
&n_t^2\EE\left[\ltwo{\g_t}^2\big| \Bbeta_{t}\right]\leq \max\{\tau^2,(1-\tau)^2\}\ku n_td+\frac{n_t(n_t-1)}{2}\EE\left[\inp{\X_i^{(t)}}{\X_j^{(t)}}\right.\\
&~~~~~\left.\times\left(H_\xi(\inp{\X_i^{(t)}}{\Bbeta_{t}-\Bbeta^*})-H_\xi(0)\right) \cdot\left(H_\xi(\inp{\X_j^{(t)}}{\Bbeta_{t}-\Bbeta^*})-H_\xi(0)\right)\big|\Bbeta_{t}\right].
\end{align*}

Notice that $\X_i^{(t)}$ follows Assumption <ref> and then the transformed vector $\Z_i^{(t)}:=\bSigma^{-\frac{1}{2}}\X_i^{(t)}$ is isotropic. Thus we have

\begin{align*}
&~~~E:=\EE\left[\inp{\X_i^{(t)}}{\X_j^{(t)}}\cdot\left(H_\xi(\inp{\X_i^{(t)}}{\Bbeta_{t}-\Bbeta^*})-H_\xi(0)\right) \cdot\left(H_\xi(\inp{\X_j^{(t)}}{\Bbeta_{t}-\Bbeta^*})-H_\xi(0)\right)\big|\Bbeta_{t}\right]\\
&=\EE\left[\inp{\bSigma^{\frac{1}{2}}\Z_i^{(t)}}{\bSigma^{\frac{1}{2}}\Z_j^{(t)}}\times\left(H_\xi(\inp{\Z_i^{(t)}}{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)})-H_\xi(0)\right) \left(H_\xi(\inp{\Z_j^{(t)}}{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)})-H_\xi(0)\right)\big|\Bbeta_{t}\right].
\end{align*}

There exists a set of unit length orthogonal vectors $\{\e_1:=\frac{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)}{\ltwo{ \bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)}},\e_2,\dots,\e_d\}$ such that any two of them satisfy $\e_i^{\top}\e_j=0$ with $i\neq j$. Then the vector $\Z_i$ can be decomposed into $d$ independent random vectors

\begin{align*}
\Z_i^{(t)}=\frac{\inp{\Z_i^{(t)}}{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)} }{\ltwo{\bSigma^{\frac{1}{2}}(\Bbeta_t-\Bbeta^*)}^2}\bSigma^{\frac{1}{2}}(\Bbeta_t-\Bbeta^*)+\inp{\Z_{i}^{(t)}}{\e_2}\e_2+\dots +\inp{\Z_{i}^{(t)}}{\e_d}\e_d.
\end{align*}

And $\Z_j^{(t)}$ could be written in a parallel way. It is worth noting that the term $H_\xi(\inp{\Z_i^{(t)}}{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)})-H_\xi(0)$ is independent of the last $d-1$ components of the decomposition and $\EE\inp{\Z_i^{(t)}}{\e_l}=\boldsymbol{0}$ holds with $l=2,\dots,d$. Thus we have

\begin{align*}
&~~~~\left.\times\left(H_\xi(\inp{\Z_i^{(t)}}{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)})-H_\xi(0)\right) \cdot\left(H_\xi(\inp{\Z_j^{(t)}}{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)})-H_\xi(0)\right)\big|\Bbeta_{t}\right].
\end{align*}

There are two methods to upper bound the above equation, which are presented as follows.

* Bound $H_\xi(\inp{\Z_i^{(t)}}{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)})-H_\xi(0)$ with constant $1$. Then we have

\begin{align*}
E&\leq \frac{\ku}{\ltwo{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)}^2}\EE\left[|\inp{\Z_i^{(t)}}{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)}||\inp{\Z_j^{(t)}}{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)}|\big|\Bbeta_{t}\right]\\
&\leq \ku.
\end{align*}

In this way, we have

\begin{align*}
\EE\left[\ltwo{\g_t}^2\big|\Bbeta_{t}\right]\leq \max\{\tau^2,(1-\tau)^2\}\ku\frac{d}{n_t}+\ku.
\end{align*}

* Bound $H_\xi(\inp{\Z_i^{(t)}}{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)})-H_\xi(0)$ using conditions in Assumption <ref>, which implies $H_\xi(\inp{\Z_i^{(t)}}{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)})-H_\xi(0)\leq \frac{1}{b_1}|\inp{\Z_{i}^{(t)}}{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)}|$.
Then we have

\begin{align*}
E&\leq \frac{\ku}{b_1^2\ltwo{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)}^2}\EE\left[|\inp{\Z_i^{(t)}}{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)}|^2|\inp{\Z_j^{(t)}}{\bSigma^{\frac{1}{2}}(\Bbeta_{t}-\Bbeta^*)}|^2\big|\Bbeta_{t}\right]\\
&\leq \frac{1}{b_1^2}\ku^2\ltwo{\Bbeta_{t}-\Bbeta^*}^2.
\end{align*}

This method leads to

\begin{align*}
\EE\left[\ltwo{\g_t}^2\big|\Bbeta_t\right]\leq \max\{\tau^2,(1-\tau)^2\}\ku\frac{d}{n_t}+\frac{\ku^2}{b_1^2}\ltwo{\Bbeta_{t}-\Bbeta^*}^2.
\end{align*}

Thus, altogether, we have

\begin{align}
\EE\left[\ltwo{\g_t}^2\big|\Bbeta_t\right]\leq \max\{\tau^2,(1-\tau)^2\}\ku\frac{d}{n_t}+\min\left\{\ku,\frac{\ku^2}{b_1^2}\ltwo{\Bbeta_{t}-\Bbeta^*}^2\right\}.
\label{eq:2-3}
\end{align}

Then we are ready to analyze the convergence dynamics in each phase.

First Phase Analysis. The convergence is proved via induction. $\ltwo{\Bbeta_{0}-\Bbeta^*}\leq D_0$ is already guaranteed. We suppose $\EE\ltwo{\Bbeta_{t}-\Bbeta^*}\leq D_t$ holds and we are going to prove the iterate at $t+1$.
In the first phase, equation (<ref>) could be bounded with

\begin{align*}
\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\big| \Bbeta_{t} \right] &\leq \ltwo{\Bbeta_{t}-\Bbeta^*}^2-\eta_t\sqrt{\kl}\sqrt{\frac{2}{\pi}}\ltwo{\Bbeta_t-\Bbeta^*}+2\eta_t\gamma+2\eta_t^2\ku\\
\end{align*}

where $\EE\left[\ltwo{\g_t}^2\big|\Bbeta_t\right]\leq2\ku$ follows from equation (<ref>) in the first phase region and $f_t(\Bbeta_{t})-f_t(\Bbeta^*)\geq \rho_{Q,\tau}(\inp{\X_t}{\Bbeta_t-\Bbeta^*})-\rho_{Q,\tau}(\xi_t)-\rho_{Q,\tau}(-\xi_t)$ is used. Take expectation over $\Bbeta_{t}$ and then we have

\begin{align*}
\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\right]&\leq \EE\left[\ltwo{\Bbeta_{t}-\Bbeta^*}^2-\frac{1}{3}\eta_t\sqrt{\kl}\ltwo{\Bbeta_t-\Bbeta^*}+2\eta_t^2\ku\right]\leq \left(1-\frac{\kl}{\ku}\right)D_t^2.
\end{align*}

Second Phase Analysis. In this region, equation (<ref>) yields $\EE\left[\ltwo{\g_t}^2\big|\Bbeta_t\right]\leq 2\frac{\ku^2}{b_1^2}\ltwo{\Bbeta_{t}-\Bbeta^*}^2$ and Lemma <ref> provides $\EE[f_t(\Bbeta_{t})-f_t(\Bbeta^*)|\Bbeta_{t}]\geq\frac{\kl}{b_0}\ltwo{\Bbeta_{t}-\Bbeta^*}^2$. Hence, equation (<ref>) is upper bounded with

\begin{align*}
\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\big| \Bbeta_{t} \right] &\leq \ltwo{\Bbeta_{t}-\Bbeta^*}^2-\eta_t\frac{\kl}{12b_0}\ltwo{\Bbeta_t-\Bbeta^*}^2+\eta_t^2\frac{\ku}{b_1^2}\ltwo{\Bbeta_{t}-\Bbeta^*}^2\\
\end{align*}

Take expectation over $\Bbeta_t$ and then we obtain

\begin{align*}
\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2 \right]\leq\left(1-\frac{\kl}{\ku}\frac{b_1^2}{b_0^2}\right)\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2 \right].
\end{align*}

Third Phase Analysis. In this region, equation (<ref>) guarantees $\EE\left[\ltwo{\g_t}^2\big|\Bbeta_t\right]\leq 2(\ku^2/\kl^2)(b_0^2/b_1^2)\max\{\tau^2,(1-\tau)^2\}\ku\frac{d}{n_t}$ and Lemma <ref> proves $\EE[f_t(\Bbeta_{t})-f_t(\Bbeta^*)|\Bbeta_{t}]\geq\frac{\kl}{b_0}\ltwo{\Bbeta_{t}-\Bbeta^*}^2$. Hence, equation (<ref>) is upper bounded with

\begin{align*}
&~~~\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\big| \Bbeta_{t} \right] \\
&\leq \ltwo{\Bbeta_{t}-\Bbeta^*}^2-\eta_t\frac{\kl}{12b_0}\ltwo{\Bbeta_t-\Bbeta^*}^2+\eta_t^2\frac{\ku^2 b_0^2}{\kl^2 b_1^2}\max\{\tau^2,(1-\tau)^2\}\ku\frac{d}{n}\\
&\leq\left(1-\frac{\ca}{12(t-t_2+\cb)}\right)\ltwo{\Bbeta_{t}-\Bbeta^*}^2+\frac{\ca}{(t-t_2+\cb)^2}\frac{\ku^2 b_0^2}{\kl^2 b_1^2}\max\{\tau^2,(1-\tau)^2\}\frac{\ku}{\kl^2}\frac{d}{n}b_0^2.
\end{align*}

Take expectation over $\Bbeta_t$ and then we have

\begin{align*}
&~~~\EE\left[\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2 \right]\leq\left(1-\frac{\ca}{12(t-t_2+\cb)}\right)\EE\left[\ltwo{\Bbeta_{t}-\Bbeta^*}^2\right]+\frac{\ca}{(t-t_2+\cb)^2}\max\{\tau^2,(1-\tau)^2\}\frac{\ku^2 b_0^2}{\kl^2 b_1^2}\frac{\ku}{\kl^2}\frac{d}{n}b_0^2\\
&\leq C^*\left(1-\frac{\ca}{24(t-t_2+\cb)}\right)\frac{\ku^2 b_0^2}{\kl^2 b_1^2}\frac{\ku}{\kl^2}\frac{1}{t-t_2+\cb}\frac{d}{n}b_0^2\leq C^*\frac{\ku^2 b_0^2}{\kl^2 b_1^2}\frac{\ku}{\kl^2}\frac{1}{t+1-t_2+\cb}\frac{d}{n}b_0^2.
\end{align*}

Then we are ready to prove Theorem <ref>. According to the expected excess risk bound in Lemma <ref>, we have $\Bbeta^*=\arg\max_{\Bbeta}\EE\sum_{t=0}^{T} f_{t}(\Bbeta_{t})-f_t(\Bbeta)$.  In the first phase, it has

\begin{align*}
\EE\left[f_t(\Bbeta_{t})-f_t(\Bbeta^*)\right]
\end{align*}

where the first line is obtained in the same way as Theorem <ref> and the last line follows from Lemma <ref>. In the second phase and the third phase, according to Lemma <ref>, we have

\begin{align*}
\EE\left[f_t(\Bbeta_{t})-f_t(\Bbeta^*)\right]&=	\EE\left[\EE\left[f_t(\Bbeta_{t})-f_t(\Bbeta^*)\big|\Bbeta_{t}\right]\right]\leq \frac{\ku}{b_1}\EE\ltwo{\Bbeta_t-\Bbeta^*}^2.
\end{align*}

As a result, according to Lemma <ref>, the regret could finally be bounded with

\begin{align*}
&\textsf{Regret}_{T}=\EE\sum_{t=0}^{T} f_{t}(\Bbeta_{t})-f_t(\Bbeta^*)\\
&~~~~~~~~~~~~~+\frac{\ku}{b_1}\sum_{t=t_1}^{t_2} \left(1-c\frac{b_1^2}{b_0^2}\frac{\kl^2}{\ku^2}\right)^{t-t_1}\ltwo{\Bbeta_{t_1}-\Bbeta^*}^2+C\frac{\ku^3}{\kl^3}\frac{b_0^3}{b_1^3}\frac{d}{n} b_0\sum_{t=t_1}^{T}\frac{1}{t-t_1+\cb }\\
&\leq C\frac{\ku}{\kl}\sqrt{\ku} \ltwo{\Bbeta_{0}-\Bbeta^*}+C\frac{ku^3}{\kl^3}\frac{b_0^2}{b_1^2}\gamma^2/b_1+C\frac{\ku^3}{\kl^3}\frac{b_0^3}{b_1^3}\frac{d}{n}b_0\log\left(\frac{T+1-t_1+\cb }{\cb }\right)\\
&\leq C\frac{\ku^3}{\kl^3}\frac{b_0^2}{b_1^2}\max\{\sqrt{\ku} \ltwo{\Bbeta_{0}-\Bbeta^*}, \gamma^2/b_1\}+C\frac{\ku^3}{\kl^3}\frac{b_0^3}{b_1^3} \frac{d}{n}b_0\log\left(\frac{T+1-t_1+\cb }{\cb }\right).
\end{align*}

The proof is complete.

§.§ Proof of Sequential Learning with Infinite Storage

Same as the previous settings, here the update at $t$ can be characterized as the follows,

\begin{align}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&=\ltwo{\Bbeta_{t}-\Bbeta^*}^2-2\eta_t\inp{\Bbeta_{t}-\Bbeta^*}{\g_t}+\eta_t^2\ltwo{\g_t}^2.
\label{eq:update infinite storage}
\end{align}

We first demonstrate the regularity properties in this setting, which are useful in the convergence proof. According to Lemma 10 in [Shen et al., 2023] and Lemma <ref>, under the event of

\begin{align*}
\bcalE_t&:=\left\{\sup_{\Bbeta_1,\Bbeta_2\in\RR^d} \left|f_t(\Bbeta_1)-f_t(\Bbeta_2)-\EE\left[ f_t(\Bbeta_1)-f_t(\Bbeta_2)\right]\right|\cdot\ltwo{\Bbeta_1-\Bbeta_2}^{-1}\right.\\
&~~~~~~~~~~~~~~~~~~~~~\left.\leq C\max\{\tau,1-\tau\}\sqrt{\ku d/\sum_{l=0}^tn_l}\right\},
\end{align*}

then for all $\Bbeta\in\RR^d$ and its corresponding sub-gradient $\g\in\partial f_t(\Bbeta)$, it has

\begin{align}
f_t(\Bbeta)-f_t(\Bbeta^*)\geq\frac{1}{4}\sqrt{\kl}\ltwo{\Bbeta-\Bbeta^*}- \gamma,\quad 	\ltwo{\g}\leq \sqrt{\ku},
\label{eq:4-1}
\end{align}

and at the same time

\begin{equation}
\begin{split}
f_t(\Bbeta)-f_t(\Bbeta^*)&\geq\frac{1}{12b_0}\kl\ltwo{\Bbeta-\Bbeta^*}^2-C_1\sqrt{\ku d/\sum_{l=0}^{t}n_l}\ltwo{\Bbeta-\Bbeta^*},\\	\ltwo{\g}&\leq 2\frac{1}{b_1}\ku\ltwo{\Bbeta-\Bbeta^*}+ C_2\sqrt{\ku d/\sum_{l=0}^t n_l},
\end{split}
\label{eq:4-2}
\end{equation}

where $C>0$ is some constant independent of dimension or sample size. Specifically, Proposition <ref> shows that $\PP(\bcalE_t)\geq 1-\exp(-C_3d)-\exp\left(-\sqrt{\sum_{l=0}^t n_l/\log\left(\sum_{l=0}^t n_l \right)}\right)$.

§.§.§ First Phase

We shall also prove the convergence dynamics via induction. The desired inequality at $\Bbeta_0$ is obvious. Then assume we already have the dynamics at $t$, namely, $\ltwo{\Bbeta_{t}-\Bbeta^*}\leq \left(1-\frac{1}{100}\frac{\kl}{\ku}\right)^t\cdot D_0$ and we are going to prove the concentration at $t+1$. For convenience, denote $D_t:= \left(1-\frac{1}{100}\frac{\kl}{\ku}\right)^t\cdot D_0$ and $D_{t+1}:=\left(1-\frac{1}{100}\frac{\kl}{\ku}\right)^{t+1}\cdot D_0$. Definition of sub-gradient and event $\bcalE_t$ together infer that the intermediate term of (<ref>) could be lower bounded with

\begin{align*}
\inp{\Bbeta_{t}-\Bbeta^*}{\g_t}&\geq f_t(\Bbeta_t)-f_t(\Bbeta^*)\\
&\geq \EE\left[f_t(\Bbeta_t)-f_t(\Bbeta^*)\big| \Bbeta_{t} \right]-C\max\{\tau,1-\tau\}\sqrt{\ku d/\sum_{l=0}^t n_l}\ltwo{\Bbeta_t-\Bbeta^*}\\
&\geq \frac{1}{6}\sqrt{\kl}\ltwo{\Bbeta_t-\Bbeta^*},
\end{align*}

where the last line uses expectation calculations, which can be found in the batch learning setting and $n_0\geq C\max\{\tau^2,(1-\tau)^2\} (\ku/\kl) d$. Moreover, Lemma <ref> proves that under $\bcalE_t$, $\ltwo{\g_t}\leq\sqrt{\ku}$. Then (<ref>) could be upper bounded with

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq \ltwo{\Bbeta_{t}-\Bbeta^*}^2-\frac{1}{3}\eta_t\sqrt{\kl}\ltwo{\Bbeta_{t}-\Bbeta^*}+\eta_t^2\ku.
\end{align*}

Then by the induction $\ltwo{\Bbeta_{t}-\Bbeta^*}\leq D_t$ and stepsize definition $\eta_t\in\frac{\sqrt{\kl}}{\ku} D_t\cdot[1/8,5/24]$, and by regarding the right hand side as quadratic function of $\ltwo{\Bbeta_t-\Bbeta^*}$, we have

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq D_t^2-\frac{1}{3}\eta_t\sqrt{\kl} D_t + \eta_t^2\ku\leq \left(1-\frac{1}{50}\frac{\kl}{\ku}\right)^{t+1}\cdot D_0^2,
\end{align*}

which yields $\ltwo{\Bbeta_{t+1}-\Bbeta^*}\leq \left(1-\frac{1}{100}\frac{\kl}{\ku}\right)^{t+1}\cdot D_0$.

§.§.§ Second Phase

In the second phase, the expectation of the loss function could be lower bounded with

\begin{align*}
\EE\left[f_t(\Bbeta_{t})-f_t(\Bbeta^*)\big| \Bbeta_t\right]\geq\frac{1}{12b_0}\kl\ltwo{\Bbeta_{t}-\Bbeta^*}^2.
\end{align*}

Furthermore, the definition of sub-gradient and event $\bcalE_t$ imply that

\begin{align*}
f_t(\Bbeta_{t})-f_t(\Bbeta^*)&\geq \frac{1}{12b_0}\kl\ltwo{\Bbeta_t-\Bbeta^*}^2-C\max\{\tau,1-\tau\}\sqrt{\ku d/\sum_{l=0}^{t}n_l}\cdot\ltwo{\Bbeta_{t}-\Bbeta^*}\\
&\geq \frac{1}{24b_0}\kl\ltwo{\Bbeta_t-\Bbeta^*}^2- C_1b_0\frac{\ku}{\kl}\cdot\frac{d}{\sum_{l=0}^t n_l},
\end{align*}

where Cauchy-Schwarz inequality is used in the last line $\frac{1}{24b_0}\kl\ltwo{\Bbeta_t-\Bbeta^*}^2+ Cb_0\bar{\tau}^2\frac{\ku}{\kl}\frac{d}{\sum_{l=0}^t n_l}\geq\tilde{C}_1\bar{\tau}\sqrt{\ku d/\sum_{l=0}^{t}n_l}\cdot\ltwo{\Bbeta_{t}-\Bbeta^*}$ and $\bar{\tau}:=\max\{\tau,1-\tau\}$. Lemma <ref> shows that

\begin{align*}
\ltwo{\g_t}^2\leq 4\frac{1}{b_1^2}\ku^2\ltwo{\Bbeta_t-\Bbeta^*}^2+C_2\bar{\tau}^2\ku \cdot\frac{d}{\sum_{l=0}^t n_l}.
\end{align*}

Then equation (<ref>) could be upper bounded with

\begin{align*}