# Online Stochastic Gradient Methods Under Sub-Weibull Noise and the Polyak-
Łojasiewicz Condition

Seunghyun Kim, Liam Madden, and Emiliano Dall’Anese S. Kim and L. Madden are
with the Department of Applied Mathematics at the University of Colorado
Boulder; E. Dall’Anese is with the Department of Electrical, Computer and
Energy Engineering and with the Department of Applied Mathematics at the
University of Colorado Boulder. This work was supported by the National
Science Foundation (NSF) CAREER award 1941896 and the NSF award 1923298.

###### Abstract

This paper focuses on the online gradient and proximal-gradient methods with
stochastic gradient errors. In particular, we examine the performance of the
online gradient descent method when the cost satisfies the Polyak-Łojasiewicz
(PL) inequality. We provide bounds in expectation and in high probability
(that hold iteration-wise), with the latter derived by leveraging a sub-
Weibull model for the errors affecting the gradient. The convergence results
show that the instantaneous regret converges linearly up to an error that
depends on the variability of the problem and the statistics of the sub-
Weibull gradient error. Similar convergence results are then provided for the
online proximal-gradient method, under the assumption that the composite cost
satisfies the proximal-PL condition. In the case of static costs, we provide
new bounds for the regret incurred by these methods when the gradient errors
are modeled as sub-Weibull random variables. Illustrative simulations are
provided to corroborate the technical findings.

##  I Introduction

This paper considers online gradient descent and the online proximal-gradient
methods for dynamic optimization and learning [1, 2, 3, 4, 5, 6, 7, 8].
Because of their computational tractability, these are attractive first-order
methods for solving a number of learning and optimization tasks where data
points and functions are processed on-the-fly and without storage. Online
gradient and proximal-gradient descent are powerful methods also in the
context of online stochastic optimization [9, 10], stochastic learning [11,
12], and feedback-based optimization [13, 14].

We examine the performance of online gradient and proximal-gradient descent in
the presence of _inexact_ gradient information, and when the cost to be
minimized satisfies the _Polyak-Łojasiewicz (PL) condition_ [15]. Formally, we
consider a optimization problem of the form

$\displaystyle\min_{{\bf x}\in\mathbb{R}^{n}}F_{t}({\bf x}):=f_{t}({\bf
x})+g_{t}({\bf x})$ (1)

where $t\in\mathbb{N}$ is the time index,
$f_{t}:\mathcal{D}\rightarrow\mathbb{R}$ is a continuously differentiable
function with a Lipschitz-continuous gradient at each time $t$,
$\mathcal{D}\subseteq\mathbb{R}^{n}$ an open and non-empty convex set, and
$g_{t}:\mathcal{D}\to\mathbb{R}\cup\\{+\infty\\}$ is a closed, convex and
proper function for all $t$, possibly not differentiable. Accordingly, we
consider two main cases:

_c1)_ $g_{t}({\bf x})\equiv 0$, ${\bf x}\mapsto f_{t}({\bf x})$ satisfies the
PL inequality for all $t$, and an inexact gradient is available; and,

_c2)_ ${\bf x}\mapsto F_{t}({\bf x})$ satisfies the proximal-PL inequality
[15], and an inexact gradient is available.

We note that strong convexity implies the PL inequality. However, functions
that satisfy the PL inequality are not necessarily convex; instead, they
satisfy the notion of invexity [15]. Examples of cost functions that satisfy
the PL inequality includes least squares (LS) and logistic regression, with
applications that span learning and feedback-based optimization. On the other
hand, prime examples of costs that satisfy the proximal-PL condition are the
LS with a sparsity-promoting regularize and an indicator function for a
polyhedral set (see, e.g., [15] for additional examples).

The analysis is performed in terms of the instantaneous regret
$r_{t}:=F_{t}({\bf x}_{t})-F_{t}^{*}$, where $F_{t}({\bf x}_{t})$ is the cost
achieved at time $t$ by the point ${\bf x}_{t}$ produced by the algorithm and
$F_{t}^{*}$ is the optimal value function (that one would have achieved if the
problem (1) was solved to convergence at time $t$).

Motivating examples for considering stochastic gradient information are drawn
from a variety of applications in learning and data-driven control; for
example: i) settings where bandit and zeroth-order methods are utilized to
estimate the gradient from (one or a few) functional evaluations [16, 17]; ii)
feedback-based optimization of networked systems, where errors in the gradient
are due to measurement errors and asynchronous measurements [18, 13, 14]; and,
iii) online stochastic optimization settings [9, 10].

_Prior works_. Online (projected) gradient descent methods with exact gradient
information have been investigated, and we refer to the representative works
[4, 6, 19] as well as to references therein. A regret analysis was performed
in, e.g., [5, 6, 7] (see also pertinent references therein), and the excess-
risk was analyzed in [2]. Inexact gradient information was considered in,
e.g., [8, 5], where bounds in expectation on the regret incurred by the
inexact online gradient descent were derived, and in [3] where the distance
from the unique trajectory of optimizers was bounded in expectation.
Convergence results in expectation were provided in the context of online
stochastic optimization in, e.g., [9, 10]. Convergence guarantees for online
stochastic gradient methods where drift and noise terms satisfy sub-Gaussian
assumptions were provided in [20]. Online projected gradient methods with sub-
Weibull gradient error and a strongly convex cost are analyzed in [14].

We also acknowledge representative prior works on inexact and stochastic
gradient and proximal-gradient methods for batch optimization in, e.g., [21,
22, 23, 24, 25, 26, 27, 28] (see also references therein). In particular,
almost sure convergence to a first-order stationary point is proved assuming
only strong smoothness and a weak assumption on the noise in [27]; mean
convergence under the PL inequality is shown in, e.g, [29]. High-probability
convergence results assuming strong smoothness and norm sub-Gaussian noise
were provided in e.g., [28], and in [30] for strongly convex functions in the
non-smooth setting. Finally, we also acknowledge prior works that investigate
geometric conditions implying linear convergence of proximal gradient
algorithms [31, 32, 33]. These works are for static optimization.

_Contributions_. We consider the cases _c1)_ and _c2)_ described above, and
offer the following main contributions.

_(i)_ We provide new bounds for the instantaneous regret $r_{t}$ in
_expectation_ and in _high probability_ for the inexact online gradient
descent, when the cost satisfies the PL inequality. The high-probability
convergence results are derived by adopting a sub-Weibull model [34] for the
gradient error. We also provide an _almost sure_ result for the asymptotic
behavior of the regret $r_{t}$.

_(ii)_ Similarly, we provide new bounds for the instantaneous regret $r_{t}$
in _expectation_ and in _high probability_ for the inexact online proximal-
gradient descent method.

_(iii)_ For the case of static costs, our bounds provide contributions over
[15, 21, 22, 23, 24, 25, 26, 27, 28, 27, 29, 28] by considering a sub-Weibull
model for the gradient error. In terms of bounds in expectation, this paper
extends the results of in the context of static optimization to an online
setting where the cost changes over time.

To better highlight the merits of the bounds, is important to mention that the
sub-Weibull distribution allows one to consider a variety of gradient error
models in a unified manner; in fact, the sub-Weibull distribution includes
sub-Gaussian distributions and sub-exponential distributions as sub-cases, as
well as random variables whose probability density function has a finite
support [35]. The bounds we derived can be customized to sub-Gaussian and sub-
exponential distributions and for random variables with finite support by
simply tuning the parameters of the sub-Weibull model. Furthermore, [36]
showed that intermittent updates can also be modeled using sub-Weibull random
variables.

The rest of the paper is organized as follows. Section II introduces relevant
definitions and assumptions, and Section III presents the main results for
online gradient descent. Section IV focuses on the online proximal-gradient
method, and Section V provides numerical results. Section VI concludes the
paper.

##  II Preliminaries

We start by introducing relevant definitions and assumptions that will be
utilized throughout the paper111 _Notation_. Upper-case (lower-case) boldface
letters will be used for matrices (column vectors); $(\cdot)^{\top}$ denotes
transposition. For given column vectors ${\bf x},{\bf y}\in\mathbb{R}^{n}$,
$\langle{\bf x},{\bf y}\rangle$ denotes the inner product and $\|{\bf
x}\|:=\sqrt{\langle{\bf x},{\bf x}\rangle}$. Given a differentiable function
$f:{\cal D}\rightarrow\mathbb{R}$, defined over a domain ${\cal
D}\subseteq\mathbb{R}^{n}$ that is nonempty, $\nabla f({\bf x})$ denotes the
gradient of $f$ at ${\bf x}$ (taken to be a column vector).
$\mathcal{O}(\cdot)$ refers to the big-O notation, whereas $o(\cdot)$ refers
to the little-o notation. For a given random variable $\xi\in\mathbb{R}$,
$\mathbb{E}[\xi]$ denotes the expected value of $\xi$, and
$\mathbb{P}\left[\xi\leq\epsilon\right]$ denotes the probability of $\xi$
taking values smaller than or equal to $\epsilon$; furthermore,
$\|\xi\|_{p}:=\mathbb{E}[|\xi|^{p}]^{1/p}$, for any $p\geq 1$. Finally, $e$
will denote Euler’s number..

###  II-A Modeling and Definitions

We consider functions $\\{f_{t}\\}_{t\in\mathbb{N}}$ and
$\\{g_{t}\\}_{t\in\mathbb{N}}$, defined over an open ball
$\mathcal{D}:=\\{{\bf x}\in\mathbb{R}^{n}:\|{\bf x}\|<r$} for some $r>0$, that
satisfy the following assumptions.

###### Assumption 1

The function ${\bf x}\mapsto f_{t}({\bf x})$ is continuously differentiable
and has a Lipschitz-continuous gradient over $\mathcal{D}$ for all $t$; i.e.,
$\exists~{}L>0$ such that $\|\nabla f_{t}({\bf x})-\nabla f_{t}({\bf y})\|\leq
L\|{\bf x}-{\bf y}\|$ for any ${\bf x},{\bf y}\in{\cal D}$, for all $t$.
$\square$

###### Assumption 2

For every $t\in\mathbb{N}$, the function ${\bf x}\mapsto g_{t}({\bf x})$ is
convex, proper, and lower semi-continuous, possibly non-differentiable over
$\mathcal{D}$. $\square$

Recall that the following inequality follows from the Lipschitz-continuity of
the gradient of $f_{t}$:

$\displaystyle f_{t}({\bf y})\leq f_{t}({\bf x})+\langle\nabla f_{t}({\bf
x}),{\bf y}-{\bf x}\rangle+\frac{L}{2}\|{\bf y}-{\bf x}\|^{2}\,,$ (2)

$\forall\,{\bf x},{\bf y}\in{\cal D}$; this inequality will be utilized
throughout the paper. Let ${\cal X}^{*}_{t}:=\arg\min_{{\bf
x}\in\mathbb{R}^{n}}F_{t}({\bf x})$, be the set of global minimizers of the
problem (1) at time $t$, and let $F_{t}^{*}:=F_{t}({\bf x}_{t}^{*})$, with
${\bf x}_{t}^{*}\in{\cal X}^{*}_{t}$. The following is assumed.

###### Assumption 3

The set ${\cal X}^{*}_{t}$ is non-empty for all $t$ and ${\cal
X}^{*}_{t}\subset\mathcal{D}$; furthermore, $-\infty<F_{t}^{*}$ for all $t$.
$\square$

The temporal variability of the problem (1) could be measured based on how
fast its optimal solutions or optimal value functions change; see, for
example, [8, 5, 7] and references therein. More precisely, one can consider
the change in the optimal value function as:

$\displaystyle\sigma_{t}:=|F_{t}^{*}-F_{t-1}^{*}|\,.$ (3)

It will also be convenient to utilize the additional metrics
$\tilde{\phi}_{t}({\bf x}):=|F_{t}({\bf x})-F_{t-1}({\bf x})|$ and

$\displaystyle\phi_{t}$ $\displaystyle:=\sup_{{\bf
x}\in\mathcal{D}}\tilde{\phi}_{t}({\bf x})\,.$ (4)

For future developments, it is also convenient to define
$\psi_{t}:=\phi_{t}+\sigma_{t}$,
$\tilde{\psi}_{t}:=\tilde{\phi}_{t}+\sigma_{t}$, and
$\bar{\psi}:=\sup_{t\in\mathbb{N}}\psi_{t}$. These metrics will be utilized to
characterize the convergence of the online gradient and proximal-gradient
methods. Whenever $g_{t}\equiv 0$ (this will be the main setting of Section
III), we will use the notation $f_{t}^{*}=\min_{{\bf
x}\in\mathcal{D}}f_{t}({\bf x})$ whenever convenient (in this case, it is
clear that $F_{t}^{*}=f_{t}^{*}$); the definitions of $\sigma_{t}$,
$\tilde{\phi}_{t}$, and $\phi_{t}$ remain unchanged. We also note that the
case where $\sigma_{t}=0$ and $\phi_{t}=0$ for all $t$ corresponds to a static
optimization problem (where the cost function does not change over time).

We next recall the definition of the PL inequality and its generalization to
composite cost functions [15].

######  Definition 1 (Polyak-Łojasiewicz (PL) Inequality)

A continuously differentiable function $f:\mathcal{D}\rightarrow\mathbb{R}$
satisfies the PL inequality over $\mathcal{D}$ if the following holds for some
$\mu>0$:

$\displaystyle 2\mu(f({\bf x})-f^{*})\leq\|\nabla f({\bf
x})\|^{2},\,\forall~{}{\bf x}\in\mathcal{D}$ (5)

where $f^{*}$ is the optimal value function.

It is important to note that the PL inequality implies the quadratic bound
$f({\bf x})-f^{*}\geq\frac{\mu}{2}\|{\bf x}-{\bf x}^{*}\|^{2}$ for any global
minimizer ${\bf x}^{*}$. As shown in [15], strong convexity implies the PL
inequality. However, functions that satisfy the PL inequality are not
necessarily convex, instead, they satisfy the notion of invexity.

######  Definition 2 (Proximal-PL Condition)

Let $f:\mathcal{D}\rightarrow\mathbb{R}$ be a continuously differentiable
function and $g:\mathcal{D}\rightarrow\mathbb{R}$ be a convex function. The
function $F({\bf x}):=f({\bf x})+g({\bf x})$ satisfies the proximal-PL
condition if the following holds:

$\displaystyle 2\mu(F({\bf x})-F^{*})\leq\mathcal{A}_{g}({\bf x},\xi),$ (6)

for all ${\bf x}\in{\cal D}$ and for some $\mu>0$, where $\xi>0$ and

$\displaystyle\mathcal{A}_{g}({\bf x},\xi):=-\frac{2}{\xi}\min\limits_{\bf
y}\Big{\\{}\langle\nabla f({\bf x}),{\bf y}-{\bf
x}\rangle+\frac{1}{2\xi}\|{\bf y}-{\bf x}\|^{2}$ $\displaystyle+g({\bf
y})-g({\bf x})\Big{\\}}.$ (7)

###  II-B Sub-Weibull random variables

In this section, we introduce the definition of sub-Weibull random variable
(rv), which will be utilized to model the errors incurred by the inexact
online gradient methods.

######  Definition 3 (Sub-Weibull rv [34])

A random variable $X\in\mathbb{R}$ is sub-Weibull if $\exists\,\theta>0$ such
that (s.t.) one of the following conditions is satisfied:

  1. (i)

$\exists\,\,K_{1}>0$ s.t. ${\mathbb{P}}[|X|\geq\epsilon]\leq
2e^{-\left(\epsilon/K_{1}\right)^{1/\theta}}$, $\forall\,\epsilon~{}>~{}0$.

  2. (ii)

$\exists\,\,K_{2}>0$ s.t. $\|X\|_{k}\leq K_{2}k^{\theta}$, $\forall\,k\geq 1$.
$\Box$

The parameters $K_{1},K_{2}$ differ by a constant that depends on $\theta$. In
particular, if (ii) holds with parameter $K_{2}$, then (i) holds with
$K_{1}=\left(2e/\theta\right)^{\theta}K_{2}$. In this paper, we use the short-
hand notation $X\sim\mathrm{subW}(\theta,K)$ to indicate that $X$ is a sub-
Weibull rv according to Definition 3(ii).

The coefficient $\theta$ is related to the rate of decay of the tails; in
particular, the tails become heavier as the parameter $\theta$ grows larger.
We note that the sub-Weibull class includes sub-Gaussian and sub-exponential
rvs as sub-cases; in particular, if $\theta=1/2$ and $\theta=1$ we have sub-
Gaussian and sub-exponential rvs, respectively. Furthermore, if a rv has a
distribution with finite support, it belongs to the sub-Gaussian class (by
Hoeffding’s inequality [35, Theorem 2.2.6]) and, thus, to the sub-Weibull
class.

The following lemmas will be utilized throughout the paper to derive the main
results.

###### Lemma II.1

(Closure of sub-Weibull class [36]) Let
$X_{i}\sim\mathrm{subW}(\theta_{i},K_{i})$, $i=1,2$, based on Definition
3(ii).

  1. (a)

_Product by scalar:_ Let $a\in{\mathbb{R}}$, then
$aX_{i}\sim\mathrm{subW}(\theta_{i},|a|K_{i})$.

  2. (b)

_Sum by scalar:_ Let $a\in{\mathbb{R}}$, then
$a+X_{i}\sim\mathrm{subW}(\theta_{i},|a|+K_{i})$.

  3. (c)

_Sum:_ Let $\\{X_{i},i=1,2\\}$ be possibly dependent; then,
$X_{1}+X_{2}\sim\mathrm{subW}(\max\\{\theta_{1},\theta_{2}\\},K_{1}+K_{2})$.

###### Lemma II.2

(Inclusion [34]) Let $X\sim\mathrm{subW}(\theta,K)$ for some $\theta,K>0$. Let
$\theta^{\prime},K^{\prime}$ be s.t. $\theta^{\prime}\geq\theta$,
$K^{\prime}\geq K$. Then, $X\sim\mathrm{subW}(\theta^{\prime},K^{\prime})$.
$\Box$

###### Lemma II.3

(Powers of sub-Weibull rvs [36]) Let $X\sim\mathrm{subW}(\theta,K)$ for some
$\theta,K>0$, and let $a>0$. Then,
$X^{a}\sim\mathrm{subW}(a\theta,K^{a}\max\\{1,a^{a\theta}\\})$. $\Box$

We note that the definition of sub-Weibull rvs and their properties do not
require their mean to be zero. We conclude this section with the following
high probability bound for a sub-Weibull rv.

######  Lemma II.4 (High probability bound)

Let $X\sim\mathrm{subW}(\theta,K)$ according to Definition 3(ii), for some
$\theta,K>0$. Then, for any $\delta\in(0,1)$, the bound:

$|X|\leq\
K\log^{\theta}\left(2\delta^{-1}\right)\left(\frac{2e}{\theta}\right)^{\theta}$
(8)

holds with probability $1-\delta$. $\Box$

##  III Online Stochastic Gradient Descent

We start by considering the case where $g_{t}\equiv 0$ for all $t$;
accordingly, the problem (1) reduces here to:

$\displaystyle\min_{{\bf x}}f_{t}({\bf x})$ (9)

where we recall that $t\in\mathbb{N}$ is the time index. We consider the
following _inexact online gradient descent_ (OGD):

${\bf x}_{t+1}={\bf x}_{t}-\eta\,{\bf v}_{t}$ (10)

where $\eta>0$ is a given step-size, ${\bf v}_{t}:=\nabla f_{t}({\bf
x}_{t})+{\bf e}_{t}$ is the approximate gradient, and ${\bf e}_{t}$ is a
stochastic error. We are interested in studying the performance of (10) when
the function $f_{t}$ satisfies the PL inequality (5), and the error $\|{\bf
e}_{t}\|$ follows a sub-Weibull distribution. A discussion on the sub-Weibull
model as well as the PL inequality in the context of problems in learning and
feedback-based optimization is provided in Section III-B. The main convergence
results are presented next.

###  III-A Convergence in expectation and in high probability

Since $g_{t}\equiv 0$, the instantaneous regret at time $t$ boils down here to
$r_{t}=f_{t}({\bf x}_{t})-f_{t}^{*}$. Throughout this section, we assume that
the gradient error has a sub-Weibull distribution, as formalized next.

######  Assumption 4 (Sub-Weibull norm gradient error)

The error is distributed as
$\|\mathbf{e}_{t}\|\sim\mathrm{subW}(\theta,K_{t})$, for some $\theta>0$ and
$K_{t}>0$.

We note that if each individual entry of the random vector $\mathbf{e}_{t}$
follows a sub-Weibull distribution, then $\|{\bf e}_{t}\|$ is a sub-Weibull
rv. This can be proved by using [36, Lemma 3.4] and part (c) of Proposition
II.1. In the following, we state the main results concerning the convergence
of (10).

######  Theorem III.1 (Convergence of the stochastic OGD)

Let Assumptions 1, 3, and 4 hold, and assume that the map ${\bf x}\mapsto
f_{t}({\bf x})$ satisfies the PL inequality, for some $\mu>0$, for all $t$.
Let $\\{{\bf x}_{i}\\}_{i=0}^{t}$ be a sequence generated by (10) with
$\eta=1/L$. The following bounds hold for (10):

  1. 1.

For all $t\in\mathbb{N}$:

$\displaystyle\mathbb{E}[r_{t}]\leq\zeta^{t}r_{0}+\sum_{\tau=1}^{t}\zeta^{t-\tau}\left(\frac{1}{2L}\mathbb{E}[\|{\bf
e}_{\tau-1}\|^{2}]+\psi_{\tau}\right)$ (11)

where $\zeta:=(1-\frac{\mu}{L})$.

  2. 2.

For any $\delta\in(0,1)$, then the following bound holds with probability
$1-\delta$:

$\displaystyle r_{t}\leq
h(\theta,\delta)\left(\zeta^{t}r_{0}+\sum_{\tau=1}^{t}\zeta^{t-\tau}\left(\frac{4^{\theta}}{2L}K_{\tau-1}^{2}+\psi_{\tau}\right)\right)$
(12)

where
$h(\theta,\delta):=\log^{2\theta}(2\delta^{-1})\left(\frac{e}{\theta}\right)^{2\theta}$.

######  Corollary III.2 (Asymptotic convergence)

Under the same assumptions of Theorem III.1, it holds that

$\displaystyle\limsup_{t\rightarrow\infty}r_{t}\leq\frac{1}{2\mu}\bar{e}+\frac{L}{\mu}\bar{\psi}\,\,\,\,\textrm{a.s.}$
(13)

where $\bar{e}=\sup_{t}\\{\mathbb{E}[\|{\bf e}_{t}\|^{2}]\\}$.

Before providing examples of applications and the proof of the results, some
remarks are in order.

######  Remark 1 (Static optimization [15])

When the optimization problem (9) is time-invariant (i.e., $f_{t}(x)=f(x)$ for
all $t\in\mathbb{N}$), then, (11) is similar to [15, Thm. 4] (where a
different step-size was used). However, relative to [15], we provide the
following bound in high probability

$\displaystyle r_{t}\leq
h(\theta,\delta)\left(\zeta^{t}r_{0}+\sum_{\tau=1}^{t}\zeta^{t-\tau}\frac{4^{\theta}}{2L}K_{\tau-1}^{2}\right)$
(14)

which holds with probability $1-\delta$ for any $\delta\in(0,1)$; this bound
can be derived from (12) by setting $\psi_{\tau}=0$ for all $\tau=1,\ldots,t$.
$\square$

######  Remark 2 (Alternative bound in expectation)

An alternative bound in expectation can be expressed as

$\displaystyle\mathbb{E}[r_{t}]\leq\zeta^{t}r_{0}+\sum_{\tau=1}^{t}\zeta^{t-\tau}\left(\frac{1}{2L}\mathbb{E}[\|{\bf
e}_{\tau-1}\|^{2}]+\mathbb{E}[\tilde{\psi}_{\tau}]\right)$ (15)

where $\tilde{\psi}_{\tau}=\sigma_{\tau}+\tilde{\phi}_{\tau}$, and
$\tilde{\phi}_{\tau}:=|F_{\tau}({\bf x}_{\tau})-F_{\tau-1}({\bf x}_{\tau})|$
(where the expectation $\mathbb{E}[\tilde{\psi}_{\tau}]$ is taken with respect
to the error ${\bf e}_{\tau-1}$, conditioned on a filtration). This leads to a
tighter bound relative to (11). $\square$

######  Remark 3 (Markov’s inequality)

An alternative high probability bound can be obtained by using (11) and
Markov’s inequality. However, the resulting bound would have a dependence
$\delta^{-1}$; on the other hand, our bound has a $\log(\delta^{-1})$
dependence on $\delta$. $\square$

###  III-B Remarks on applications and error model

In this section, we provide some examples of applications that are relevant to
our setting.

######  Example 1 (Online least-squares)

A function $f_{t}({\bf x})=h_{t}({\bf A}_{t}{\bf x})$, with
$h_{t}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ a $\nu$-strongly convex function
and ${\bf A}_{t}\in\mathbb{R}^{d\times n}$ satisfies the PL inequality [15].
This class includes the least-squares (LS) problem by setting $f_{t}({\bf
x})=\frac{1}{2}\|{\bf A}_{t}{\bf x}-{\bf b}_{t}\|^{2}$. Note that, when the
matrix ${\bf A}_{t}$ is not full-column rank, one can utilize the results of
this paper to establish linear convergence of OGD for the under-determined LS
problem.

######  Example 2 (Online Logistic regression)

The logistic regression cost $f_{t}({\bf
x})=\sum_{i=1}^{d}\log(1+\exp(b_{i,t}{\bf a}_{i,t}^{\top}{\bf x}))$, with
$b_{i,t}\in\mathbb{R}$ and ${\bf a}_{i,t}\in\mathbb{R}^{n}$, satisfies the PL
inequality [15].

######  Example 3 (Optimization of LTI systems)

Consider the algebraic representation of a stable linear time-invariant system
${\bf y}_{t}={\bf G}{\bf x}+{\bf H}{\bf w}_{t}$, where ${\bf x}$ is the vector
of controllable inputs and ${\bf w}_{t}$ are unknown exogenous disturbances.
Suppose that $f_{t}({\bf x})=\frac{1}{2}\|{\bf G}{\bf x}+{\bf H}{\bf
w}_{t}-\bar{{\bf y}}_{t}\|^{2}$ with $\bar{{\bf y}}_{t}$ a time-varying
reference. Since ${\bf w}_{t}$ is unknown, one way to compute $\nabla
f_{t}({\bf u}_{t})$ is ${\bf v}_{t}={\bf G}^{\top}(\hat{{\bf y}}_{t}-\bar{{\bf
y}}_{t})$, where $\hat{{\bf y}}_{t}$ is a (noisy) measurement of the output
${\bf y}_{t}$ [18, 14].

######  Example 4 (Training of neural networks)

We refer the reader to recent discussions on the PL inequality in the context
of training of neural networks in, e.g., [37]. The proposed framework may
capture the case where stochastic gradient methods are utilized to train a
neural network in an online fashion.

In terms of gradient information, the error ${\bf e}_{t}$ may arise in the
following (application-specific) scenarios:

_(i)_ A subset of the data points available at time $t$ are utilized to
compute the gradient; for instance, in the Examples 1-2, one may utilize the
data points $\\{{\bf a}_{i,t},b_{i,t}\\}_{i\in{\cal S}_{t}}$, with $|{\cal
S}_{t}|<d$.

_(ii)_ Bandit and zeroth-order methods are utilized to estimate the gradient
[16, 17].

_(iii)_ In an online stochastic optimization setting [9], i.e. when
$f_{t}({\bf x})=\mathbb{E}\left[\ell_{t}({\bf x},{\bf z})\right]$ for a given
loss $\ell_{t}:\mathbb{R}^{n}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ and a
random variable ${\bf z}$, the approximate gradient ${\bf v}_{t}$ may be
computed using a single sample or a mini-batch.

_(iv)_ In measurement-based algorithms as in Example 3, measurement errors and
asynchronous measurements render the computation of the gradient inexact.

###  III-C Proofs

In this section, we present the proof of Theorem III.1.

We start by using (2) with ${\bf y}={\bf x}_{t+1}$ and ${\bf x}={\bf x}_{t}$,
where ${\bf x}_{t}$ and ${\bf x}_{t+1}$ are generated by (10); this allows us
to obtain:

$\displaystyle f_{t}({\bf x}_{t+1})$ $\displaystyle\leq f_{t}({\bf
x}_{t})-\frac{1}{L}\langle\nabla f_{t}({\bf x}_{t}),{\bf
v}_{t}\rangle+\frac{1}{2L}\|{\bf v}_{t}\|^{2}$ (16a) $\displaystyle=f_{t}({\bf
x}_{t})-\frac{1}{L}\langle\nabla f_{t}({\bf x}_{t}),\nabla f_{t}({\bf
x}_{t})+\mathbf{e}_{t}\rangle$ $\displaystyle\quad+\frac{1}{2L}\|\nabla
f_{t}({\bf x}_{t})+\mathbf{e}_{t}\|^{2}$ (16b) $\displaystyle=f_{t}({\bf
x}_{t})-\frac{1}{L}\|\nabla f_{t}({\bf x}_{t})\|^{2}-\frac{1}{L}\langle\nabla
f_{t}({\bf x}_{t}),\mathbf{e}_{t}\rangle$
$\displaystyle\quad+\frac{1}{2L}\|\nabla f_{t}({\bf
x}_{t})+\mathbf{e}_{t}\|^{2}$ (16c) $\displaystyle=f_{t}({\bf
x}_{t})-\frac{1}{L}\|\nabla f_{t}({\bf x}_{t})\|^{2}-\frac{1}{L}\langle\nabla
f_{t}({\bf x}_{t}),\mathbf{e}_{t}\rangle$
$\displaystyle\quad+\frac{1}{2L}\langle\nabla f_{t}({\bf
x}_{t})+\mathbf{e}_{t},\nabla f_{t}({\bf x}_{t})+\mathbf{e}_{t}\rangle$ (16d)
$\displaystyle=f_{t}({\bf x}_{t})-\frac{1}{L}\|\nabla f_{t}({\bf
x}_{t})\|^{2}-\frac{1}{L}\langle\nabla f_{t}({\bf
x}_{t}),\mathbf{e}_{t}\rangle$ $\displaystyle\quad+\frac{1}{2L}(\|\nabla
f_{t}({\bf x}_{t})\|^{2}+2\langle\nabla f_{t}({\bf
x}_{t}),\mathbf{e}_{t}\rangle+\|\mathbf{e}_{t}\|^{2})$ (16e)
$\displaystyle=f_{t}({\bf x}_{t})-\frac{1}{2L}\|\nabla f_{t}({\bf
x}_{t})\|^{2}+\frac{1}{2L}\|\mathbf{e}_{t}\|^{2}\,.$ (16f)

Next, adding $-f_{t}^{*}$ on both sides and using the PL inequality (5), one
gets:

$\displaystyle f_{t}({\bf x}_{t+1})-f_{t}^{*}\leq-\frac{1}{2L}\|\nabla
f_{t}({\bf x}_{t})\|^{2}+\frac{1}{2L}\|\mathbf{e}_{t}\|^{2}+f_{t}({\bf
x}_{t})-f_{t}^{*}$ $\displaystyle\leq-\frac{\mu}{L}(f_{t}({\bf
x}_{t})-f_{t}^{*})+f_{t}({\bf
x}_{t})-f_{t}^{*}+\frac{1}{2L}\|\mathbf{e}_{t}\|^{2}\,.$ (17)

Next, adding $-f_{t+1}^{*}$ and $f_{t+1}({\bf x}_{t+1})$ on both sides, using
the definition of regret $r_{t}$, and applying the definitions of $\zeta$ and
$\psi_{t}$ we obtain the stochastic inequality

$r_{t}\leq\zeta\,r_{t-1}+\frac{1}{2L}\|{\bf e}_{t-1}\|^{2}+\psi_{t},$

which holds almost surely. Unraveling, we get

$\displaystyle
r_{t}\leq\kappa_{t}+\frac{1}{2L}\sum_{i=1}^{t}\zeta^{t-i}\|\mathbf{e}_{i-1}\|^{2}$
(18)

where $\kappa_{t}:=\zeta^{t}r_{0}+\sum_{i=1}^{t}\zeta^{t-i}\psi_{i}$ for
brevity. Taking the expectation on both sides, we get (11).

For the high-probability bound (12), recall that
$\|\mathbf{e}_{i}\|\sim\mathrm{subW}(\theta,K_{i})$; by Lemma II.3, setting
$a=2$ we get that $\|\mathbf{e}_{i}\|^{2}$ is a sub-Weibull rv and, in
particular,
$\|\mathbf{e}_{i}\|^{2}\sim\mathrm{subW}(2\theta,4^{\theta}K_{i}^{2})$. Next,
using the closure properties (a), (b), and (c) in Lemma II.1 and the fact that
$\zeta>0$, we have that the right-hand-side of (18) is a sub-Weibull rv; in
particular,

$\displaystyle\kappa_{t}+\frac{1}{2L}\sum_{i=1}^{t}\zeta^{t-i}\|\mathbf{e}_{i-1}\|^{2}\sim\mathrm{subW}\left(2\theta,K^{\prime}\right)\,.$
(19)

where

$K^{\prime}=\kappa_{t}+\frac{4^{\theta}}{2L}\sum_{i=1}^{t}\zeta^{t-i}K_{i-1}^{2}$

Using Lemma II.4, the high-probability bound (12) follows.

The proof of Corollary III.2 follows similar steps as in [36, Corollary 4.8],
and is omitted.

##  IV Stochastic Proximal-Gradient Method

We now turn the attention to the time-varying problem (1), with the cost
satisfying the Assumptions 1-3. Throughout this section, we further assume
that the cost function $F_{t}({\bf x})$ satisfies the proximal-PL inequality
(6), for a given $\mu>0$. As discussed in [15], an important example of cost
satisfying the proximal-PL inequality is the $\ell_{1}$-regularized least
squares problem; additional examples of costs include (see the discussion in
[15, Appendix F]):

  1. 1.

$F_{t}({\bf x})=f_{t}({\bf A}{\bf x})+g_{t}({\bf x})$, with $f_{t}$ strongly
convex, $g_{t}$ the indicator function for a polyhedral set, and ${\bf A}$ a
given matrix.

  2. 2.

The case where $f_{t}$ is convex, and $F_{t}$ satisfies the quadratic growth
condition.

  3. 3.

The case where $F_{t}$ satisfies the Kurdyka-Łojasiewicz inequality or the
proximal exponential bound.

Consider then the stochastic online proximal-gradient method (OPGM), which
involves the following step:

$\displaystyle{\bf x}_{t+1}=\mathrm{prox}_{\frac{1}{L}g_{t}}\left\\{{\bf
x}_{t}-\frac{1}{L}{\bf v}_{t}\right\\}\,,\,\,\,\,t\in\mathbb{N}$ (20)

where ${\bf v}_{t}$ is again an estimate of $\nabla f_{t}({\bf x}_{t})$,
$\mathrm{prox}_{\frac{1}{L}g_{t}}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$
denotes the proximal operator, and the step-size is taken to be $1/L$.

We are now interested in analyzing the behavior of (20) in terms of regret
$r_{t}=F_{t}({\bf x}_{t})-F_{t}^{*}$, where we recall that $F_{t}^{*}$ is the
optimal value function, when the function $F_{t}$ satisfies proximal-PL
inequality and the error $\|{\bf e}_{t}\|$ follows a sub-Weibull distribution.
The main convergence result for (20) is stated next.

######  Theorem IV.1 (Convergence of the stochastic OPGM)

Let Assumptions 1–4 hold. Assume further that the function ${\bf x}\mapsto
F_{t}({\bf x})$ satisfies the proximal-PL inequality for some $\mu>0$, for all
$t$. Let $\\{{\bf x}_{i}\\}_{i=0}^{t}$ be a sequence generated by (20). Then:

  1. 1.

For all $t\in\mathbb{N}$:

$\displaystyle\mathbb{E}[r_{t}]\leq\zeta^{t}r_{0}+\sum_{\tau=1}^{t}\zeta^{t-\tau}\left(2D\mathbb{E}[\|{\bf
e}_{\tau-1}\|]+\psi_{\tau}\right)$ (21)

where $\zeta=(1-\frac{\mu}{L})$ and $D$ is the diameter of $\mathcal{D}$.

  2. 2.

If $\delta\in(0,1)$, then with probability $1-\delta$:

$\displaystyle r_{t}\leq
h_{p}(\theta,\delta)\left(\zeta^{t}r_{0}+\sum_{\tau=1}^{t}\zeta^{t-\tau}\left(2DK_{\tau-1}+\psi_{\tau}\right)\right)$
(22)

where
$h_{p}(\theta,\delta):=\log^{\theta}(2\delta^{-1})\left(\frac{2e}{\theta}\right)^{\theta}$.

A result for the asymptotic convergence of the OPGM similar to Corollary III.2
can be derived too, but it is omitted to avoid repetitive arguments. Similar
considerations as in Remark 1 can also be drawn.

We note that when $g_{t}$ is the indicator function for a bounded polyhedron,
the constant $D$ can be replaced by the diameter of the polyhedron. We also
note that tighter bounds could be derived by introducing a filtered
probability space (this will be clear in the proof, where the inner product
between the iterates and the gradient error appears); however, this is left as
a future extension.

To outline the proof of the theorem, we first note that step (20) is
equivalent to

$\displaystyle{\bf x}_{t+1}=\arg\min_{{\bf x}}\langle{\bf v}_{t},{\bf x}-{\bf
x}_{t}\rangle$ $\displaystyle+\frac{L}{2}\|{\bf x}-{\bf
x}_{t}\|^{2}+g_{t}({\bf x})-g_{t}({\bf x}_{t}).$ (23)

We also recall the definition of $\mathcal{A}_{g_{t}}({\bf x}_{t},1/L)$ in
(2), and define $\tilde{\mathcal{A}}_{g_{t}}({\bf x}_{t},1/L)$ as:

$\displaystyle\tilde{\mathcal{A}}_{g_{t}}({\bf x}_{t},1/L)$
$\displaystyle:=-2L\min\limits_{{\bf y}}\\{\langle{\bf v}_{t},{\bf y}-{\bf
x}_{t}\rangle+\frac{L}{2}\|{\bf y}-{\bf x}_{t}\|^{2}$ $\displaystyle\hskip
79.6678pt+g_{t}({\bf y})-g_{t}({\bf x}_{t})\\}\,.$ (24)

Lastly, for any ${\bf x}\in\mathcal{D}$, we recall that
$\mathbf{e}_{t}\in\mathbb{R}^{n}$ is the gradient error, i.e., ${\bf
v}_{t}=\nabla f_{t}({\bf x})+\mathbf{e}_{t}$, and $\|{\bf
e}_{t}\|\sim\mathrm{subW}(\theta,K_{t})$.

_Proof of Theorem IV.1._ We start by recalling that $F_{t+1}({\bf
x}_{t+1})=f_{t+1}({\bf x}_{t+1})+g_{t+1}({\bf x}_{t+1})$; adding and
subtracting $F_{t}({\bf x}_{t+1})$ on the right-hand-side, and using the
definition (4), we get

$\displaystyle F_{t+1}({\bf x}_{t+1})\leq\phi_{t+1}+f_{t}({\bf
x}_{t+1})+g_{t}({\bf x}_{t+1})+g_{t}({\bf x}_{t})-g_{t}({\bf x}_{t})$
$\displaystyle\leq\phi_{t+1}+f_{t}({\bf x}_{t})+\langle\nabla f_{t}({\bf
x}_{t}),{\bf x}_{t+1}-{\bf x}_{t}\rangle$
$\displaystyle\quad+\frac{L}{2}\|{\bf x}_{t+1}-{\bf x}_{t}\|^{2}+g_{t}({\bf
x}_{t+1})+g_{t}({\bf x}_{t})-g_{t}({\bf x}_{t})$ (25)

where we used (2) in the last step. Next, we add and subtract ${\bf e}_{t}$ in
the inner product and use the definition ${\bf v}_{t}=\nabla f_{t}({\bf
x})+\mathbf{e}_{t}$ to obtain:

$\displaystyle F_{t+1}({\bf x}_{t+1})$ $\displaystyle\leq F_{t}({\bf
x}_{t})+\langle{\bf v}_{t},{\bf x}_{t+1}-{\bf x}_{t}\rangle+\frac{L}{2}\|{\bf
x}_{t+1}-{\bf x}_{t}\|^{2}$ $\displaystyle+g_{t}({\bf x}_{t+1})-g_{t}({\bf
x}_{t})-\langle\mathbf{e}_{t},{\bf x}_{t+1}-{\bf x}_{t}\rangle+\phi_{t+1}$
(26) $\displaystyle\leq F_{t}({\bf
x}_{t})-\frac{1}{2L}\tilde{\mathcal{A}}_{g_{t}}({\bf
x}_{t},1/L)+\phi_{t+1}-\langle\mathbf{e}_{t},{\bf x}_{t+1}-{\bf x}_{t}\rangle$
(27)

where we used the definition (IV). Adding and subtracting
$\mathcal{A}_{g_{t}}({\bf x}_{t},1/L)$ on the right-hand-side, we get

$\displaystyle F_{t+1}({\bf x}_{t+1})$ $\displaystyle\leq F_{t}({\bf
x}_{t})+\frac{1}{2L}|\mathcal{A}_{g_{t}}({\bf
x}_{t},1/L)-\tilde{\mathcal{A}}_{g_{t}}({\bf x}_{t},1/L)|$
$\displaystyle-\frac{1}{2L}\mathcal{A}_{g_{t}}({\bf
x}_{t},1/L)+\phi_{t+1}-\langle\mathbf{e}_{t},{\bf x}_{t+1}-{\bf
x}_{t}\rangle\,.$ (28)

Let $\varepsilon_{t}:=|\mathcal{A}_{g_{t}}({\bf
x}_{t},1/L)-\tilde{\mathcal{A}}_{g_{t}}({\bf x}_{t},1/L)|$ for brevity; using
the definition of $\mathcal{A}_{g_{t}}({\bf x}_{t},1/L)$ in (2) and
subtracting $F_{t+1}^{*}$ on both sides, we get:

$\displaystyle F_{t+1}({\bf x}_{t+1})-F_{t+1}^{*}\leq F_{t}({\bf
x}_{t})-F_{t+1}^{*}-\frac{\mu}{L}(F_{t}({\bf x}_{t})-F_{t}^{*})$
$\displaystyle\quad+\frac{1}{2L}\varepsilon_{t}-\langle\mathbf{e}_{t},{\bf
x}_{t+1}-{\bf x}_{t}\rangle+\phi_{t+1}$ (29a) $\displaystyle\leq F_{t}({\bf
x}_{t})-F_{t}^{*}-\frac{\mu}{L}(F_{t}({\bf
x}_{t})-F_{t}^{*})+\frac{1}{2L}\varepsilon_{t}-\langle\mathbf{e}_{t},{\bf
x}_{t+1}-{\bf x}_{t}\rangle$ $\displaystyle\quad+\phi_{t+1}+\sigma_{t+1}$
(29b) $\displaystyle=(1-\frac{\mu}{L})(F_{t}({\bf
x}_{t})-F_{t}^{*})+\psi_{t+1}+\frac{1}{2L}\varepsilon_{t}-\langle\mathbf{e}_{t},{\bf
x}_{t+1}-{\bf x}_{t}\rangle$ (29c) $\displaystyle=(1-\frac{\mu}{L})(F_{t}({\bf
x}_{t})-F_{t}^{*})+\psi_{t+1}+\frac{1}{2L}\varepsilon_{t}+\|\mathbf{e}_{t}\|D$
(29d)

where we used the definition of $\sigma_{t+1}$ and $D$ is the diameter of
$\mathcal{D}$.

We now bound $\frac{1}{2L}\varepsilon_{t}$; from the definitions (2) and (IV),
we have that

$\displaystyle\frac{1}{2L}\varepsilon_{t}=$ $\displaystyle\Big{|}\min_{{\bf
y}}\left\\{\langle\nabla f_{t}({\bf x}_{t}),{\bf y}-{\bf
x}_{t}\rangle+\frac{L}{2}\|{\bf y}-{\bf x}_{t}\|^{2}+g_{t}({\bf y})\right\\}$
$\displaystyle\hskip 5.69046pt-\min_{{\bf z}}\left\\{\langle{\bf v}_{t},{\bf
z}-{\bf x}_{t}\rangle+\frac{L}{2}\|{\bf z}-{\bf x}_{t}\|^{2}+g_{t}({\bf
z})\right\\}\Big{|}.$ (30)

From (23), one can notice that the minimizer of $\langle{\bf v}_{t},{\bf
z}-{\bf x}_{t}\rangle+\frac{L}{2}\|{\bf z}-{\bf x}_{t}\|^{2}+g_{t}({\bf z})$
is ${\bf x}_{t+1}$ (the constant term $g_{t}({\bf x}_{t})$ does not modify the
minimizer); thus, substituting ${\bf z}$ with ${\bf x}_{t+1}$ we get

$\displaystyle\frac{1}{2L}\varepsilon_{t}=$ $\displaystyle\Big{|}\min_{{\bf
y}}\left\\{\langle\nabla f_{t}({\bf x}_{t}),{\bf y}-{\bf
x}_{t}\rangle+\frac{L}{2}\|{\bf y}-{\bf x}_{t}\|^{2}+g_{t}({\bf y})\right\\}$
$\displaystyle-\langle{\bf v}_{t},{\bf x}_{t+1}-{\bf
x}_{t}\rangle-\frac{L}{2}\|{\bf x}_{t+1}-{\bf x}_{t}\|^{2}-g_{t}({\bf
x}_{t+1})\Big{|}.$ (31)

Next, one has that $\min_{{\bf y}}\\{\langle\nabla f_{t}({\bf x}_{t}),{\bf
y}-{\bf x}_{t}\rangle+\frac{L}{2}\|{\bf y}-{\bf x}_{t}\|^{2}+g_{t}({\bf
y})\\}\leq\langle\nabla f_{t}({\bf x}_{t}),{\bf y}-{\bf
x}_{t}\rangle+\frac{L}{2}\|{\bf y}-{\bf x}_{t}\|^{2}+g_{t}({\bf y})$ for any
${\bf y}\in\mathcal{D}$, and, thus

$\displaystyle\frac{1}{2L}\varepsilon_{t}\leq$
$\displaystyle\Big{|}\langle\nabla f_{t}({\bf x}_{t}),{\bf y}-{\bf
x}_{t}\rangle+\frac{L}{2}\|{\bf y}-{\bf x}_{t}\|^{2}+g_{t}({\bf y})$
$\displaystyle-\langle{\bf v}_{t},{\bf x}_{t+1}-{\bf
x}_{t}\rangle-\frac{L}{2}\|{\bf x}_{t+1}-{\bf x}_{t}\|^{2}-g_{t}({\bf
x}_{t+1})\Big{|}$ (32)

for any ${\bf y}\in\mathcal{D}$. Pick ${\bf y}={\bf x}_{t+1}$; then, we have
that:

$\displaystyle\frac{1}{2L}\varepsilon_{t}\leq$
$\displaystyle\Big{|}\langle\nabla f_{t}({\bf x}_{t}),{\bf x}_{t+1}-{\bf
x}_{t}\rangle-\langle{\bf v}_{t},{\bf x}_{t+1}-{\bf x}_{t}\rangle\Big{|}$
(33a) $\displaystyle=$ $\displaystyle\Big{|}\langle{\bf e}_{t},{\bf
x}_{t}-{\bf x}_{t+1}\rangle\Big{|}\leq\|\mathbf{e}_{t}\|D.$ (33b)

Therefore, letting $r_{t}=F_{t}({\bf x}_{t})-F_{t}^{*}$ for brevity, we get
the stochastic recursion

$\displaystyle r_{t+1}\leq$
$\displaystyle(1-\frac{\mu}{L})r_{t}+\psi_{t+1}+2D\|\mathbf{e}_{t}\|$ (34)

which holds almost surely. By applying recursively (34) from $\tau=0$ to
$\tau=t$, we get

$\displaystyle
r_{t}\leq\zeta^{t}r_{0}+\sum_{i=1}^{t}\zeta^{t-i}\left(2D\|\mathbf{e}_{i-1}\|+\psi_{i}\right)\,.$
(35)

Taking the expectation on both sides of (35), the bound (21) follows.

To show (22), recall first that $\|{\bf
e}_{t}\|\sim\mathrm{subW}(\theta,K_{t})$, and let
$\kappa_{t}:=\zeta^{t}r_{0}+\sum_{i=1}^{t}\zeta^{t-i}\psi_{i}$ for brevity so
that (35) can be rewritten as
$r_{t}\leq\kappa_{t}+2D\sum_{i=1}^{t}\zeta^{t-i}\|\mathbf{e}_{i-1}\|$. Using
the closure properties (a), (b), and (c) in Lemma II.1 and the fact that
$\zeta>0$ and $\kappa_{t}\geq 0$, we have that the right-hand-side of this
inequality is a sub-Weibull rv; in fact,

$\displaystyle\kappa_{t}+2D\sum_{i=1}^{t}\zeta^{t-i}\|\mathbf{e}_{i-1}\|\sim\mathrm{subW}\left(\theta,K^{{}^{\prime\prime}}\right)\,.$
(36)

where $K^{{}^{\prime\prime}}=\kappa_{t}+2D\sum_{i=1}^{t}\zeta^{t-i}K_{i-1}$.
Using Lemma II.4, the high-probability bound (12) follows.

##  V Illustrative Numerical Results

We provide two illustrative numerical experiments. The first one is based on a
time-varying LS regression problem; then, we consider a problem related to
real-time demand response in power grids.

Least-squares problem. We consider a time-varying LS regression problem, with
the following cost at time $t$:

$\displaystyle f_{t}({\bf x})=\frac{1}{2}\|{\bf A}{\bf x}-{\bf b}_{t}\|^{2}$
(37)

where ${\bf A}\in\mathbb{R}^{d\times n}$ and ${\bf b}_{t}\in\mathbb{R}^{d}$;
this cost satisfies the PL inequality, as shown in [15].

We consider the case $n=10$ and $d=20$. The matrix ${\bf A}$ is generated by
defining its singular value decomposition; for its left and right-singular
vectors, we sampled two orthogonal matrices, ${\bf U}\in\mathbb{R}^{d\times
d}$ and ${\bf V}\in\mathbb{R}^{n\times n}$, and we let its singular values be
equally spaced from $\mu=0.1$ to $L=1$. We generated ${\bf b}_{t}$ as ${\bf
b}_{t}={\bf A}{\bf x}_{t}^{*}+{\bf r}_{t}$, where the optimal parameter ${\bf
x}_{t}^{*}$ evolves via a random walk, i.e., ${\bf x}_{t}^{*}={\bf
x}_{t-1}^{*}+{\bf s}$ with ${\bf s}\sim\mathcal{N}(\mathbf{0},0.1{\bf I})$,
and ${\bf r}_{t}$ is a Gaussian vector $\mathcal{N}(\mathbf{0},10^{-3}{\bf
I})$ (we set ${\bf x}_{0}^{*}$ to the vector of all ones).

Figure 1: Inexact OGD: Evolution of average regret obtained experimentally,
the empirical $3$-standard deviation confidence interval, and the theoretical
bound.

We corrupt the gradient with a random vector ${\bf e}_{t}$, which is modelled
as a Gaussian vector $\mathcal{N}(\mathbf{0},10^{-3}{\bf I})$; we note that,
if ${\bf e}_{t}$ is a Gaussian vector, then $\|{\bf e}_{t}\|^{2}$ is a sub-
Weibull random variable. The regret is computed using a Monte Carlo approach,
with 100 tests. Accordingly, Figure 1 illustrates the evolution of the
expected regret obtained by averaging the trajectories of the instantaneous
regret $r_{t}$ over the various runs, the empirical $3-\sigma$ confidence
interval, and the theoretical bound (15). The figure validates the convergence
results for the inexact OGD and, since ${\bf b}_{t}$ continuously changes, the
average $r_{t}$ exhibits a plateau.

Real-time demand response problem. We consider an example in the context of a
power distribution grid serving residential houses or commercial facilities.
We consider $n$ controllable distributed energy resources (DERs) providing
services to the main grid; precisely, consider the setting where the vector
${\bf x}$ collects the active power outputs of the DERs, and assume the
algebraic relationship $p_{0,t}={\bf a}_{x}^{\top}{\bf x}+{\bf
a}_{w}^{\top}{\bf w}_{t}$ for the net active power at the point of common
coupling, where ${\bf a}_{x}\in\mathbb{R}^{n}$ and ${\bf
a}_{w}\in\mathbb{R}^{w}$ are sensitivity coefficients, and ${\bf
w}_{t}\in\mathbb{R}^{w}$ is a vector collecting active powers of
uncontrollable devices; in particular, ${\bf a}_{x}$ and ${\bf a}_{w}$ can be
set to the vector of all ones when line losses are negligible, or they are
derived based on a linearized model for the power flow equations in case of
resistive lines [18]. Consider the following time-varying optimization problem
for real-time management of DERs:

$\min_{{\bf x}}\,\frac{1}{2}\left({\bf a}_{x}^{\top}{\bf x}+{\bf
a}_{w}^{\top}{\bf w}_{t}-p_{0,t}^{\mathrm{ref}}\right)^{2}+\mathbb{I}_{\\{{\bf
B}{\bf x}\leq{\bf c}\\}}$ (38)

where $p_{0,t}^{\mathrm{ref}}$ is a time-varying reference point for the net
active power at the point of common coupling $p_{0,t}$, and
$\mathbb{I}_{\\{{\bf B}{\bf x}\leq{\bf c}\\}}$ is the set indicator function
for the set $\\{{\bf x}\in\mathbb{R}^{n}:{\bf B}{\bf x}\leq{\bf c}\\}$
modeling box constraints for the active powers. For example, $p_{0,t}$ may be
an automatic control generation (ACG) signal, a flexible ramping signal, or a
demand response setpoint. We note that the cost (38) satisfies the proximal-PL
inequality [15]. The main challenge behind applying a proximal-gradient
descent to (38) is that the vector ${\bf w}_{t}$ is unknown; we therefore
consider the approach of, e.g., [18], where measurements of $p_{0,t}$ are
utilized to estimate the gradient in lieu of the model ${\bf a}_{x}^{\top}{\bf
x}+{\bf a}_{w}^{\top}{\bf w}_{t}$. Precisely, we compute the approximate
gradient as

${\bf v}_{t}={\bf a}_{x}(\hat{p}_{0,t}-p_{0,t}^{\mathrm{ref}})$ (39)

where $\hat{p}_{0,t}$ is a measurement of $p_{0,t}$ collected at time $t$.
Since measurements of $p_{0,t}$ may be affected by errors or by outliers,
${\bf v}_{t}$ does not in general coincide with the true gradient ${\bf
a}_{x}({\bf a}_{x}^{\top}{\bf x}_{t}+{\bf a}_{w}^{\top}{\bf
w}_{t}-p_{0,t}^{\mathrm{ref}})$.

Figure 2: Demand response application: non-controllable power ${\bf
a}_{w}^{\top}{\bf w}_{t}$ and reference point $p_{0,t}^{\mathrm{ref}}$ for the
active power $p_{0,t}$.  Figure 3: Demand response application: Evolution of
average regret obtained experimentally; the zoomed area also provides the
empirical $3-\sigma$ confidence interval.

As an example, we consider the case where $N=500$ DERs are controlled; the
limits for the active power of each device are $[-50,50]$ kW for energy
storage resources and $[0,50]$ kW for solar inverters. We consider the case
where $p_{0,t}^{\mathrm{ref}}$ follows the trajectory shown in Figure 2; real
data with a granularity of one second is taken from [38] to generate the non-
controllable powers ${\bf w}_{t}$, with the net power ${\bf a}_{w}^{\top}{\bf
w}_{t}$ plotted in Figure 2 as well. The sensitivity vector ${\bf a}_{w}$ is
computed as in [38]. A Gaussian random variable with zero mean and variance
$10$ kW is utilized to generate the measurement error affecting
$\hat{p}_{0,t}$.

Figure 3 illustrates the evolution of the regret $r_{t}$, averaged over $50$
experiments, in logarithmic scale. One can notice a linear decrease of the
average regret during the first iteration of the algorithms; the regret then
exhibit variations that are due to the considerable time-variability of the
cost function (due to the large swings in the non-controllable powers ${\bf
w}_{t}$). The plot also provides a zoomed version (in linear scale), where the
$3$-standard deviation confidence interval is also reported.

##  VI Conclusions

In this paper, we showed that cost function achieved by the online
(proximal-)gradient method exhibits a linear convergence to the optimal value
functions within an error, for functions satisfying the (proximal-)PL
inequality, and when inexact gradient information is available. We derived
bounds in expectation and in high probability, where for the latter we
utilized a sub-Weibull model for the gradient errors. The convergence results
are applicable to a number of learning and feedback-optimization tasks, where
the cost functions may not be SC, but satisfies the PL inequality. Our results
also provide new insights on the convergence of the (proximal-)gradient method
for time-varying functions and with exact gradient information, and for the
case of static optimization with inexact gradient information. The gradient
error model is general, and it allows one to consider various sources of
inaccuracy and gradient estimation techniques.

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# On the intrinsic wave-particle duality of coordinate maps

Tony Lyons  Department of Computing & Mathematics, South East Technological
University, Waterford, Ireland<EMAIL_ADDRESS>
###### Abstract.

The relative motion of material, point-like observers is analysed in terms of
coordinate maps between the respective rest-frame of each observer. Under the
assumption these maps are $C^{2}$-regular, conservation laws are deduced,
which in turn are found to yield wave equations familiar as scalar and spinor
fields. Wave-particle dualism is then understood as a property of these
coordinate maps as opposed to an inherent property of the point-like observers
themselves. Additionally it is shown that the relative acceleration of these
observers can only be realised in this context by the introduction of
additional potential fields. It is shown that if each observer is to be
associated with an inertial frame of reference, then the potential field is
necessarily a massless gauge field.

###### Contents

  1. 1 Introduction
  2. 2 Relative motion and coordinate transformations
    1. 2.1 Local Lorentz transformations
    2. 2.2 Regularity & conservation laws
    3. 2.3 Coordinate maps as scalar fields
    4. 2.4 Commutators
    5. 2.5 Coordinate maps as spinor fields
  3. 3 Relative acceleration
    1. 3.1 Acceleration via potential fields
    2. 3.2 Relativity of acceleration
  4. 4 Conclusion

##  1\. Introduction

The wave-particle duality associated with massive particles, first postulated
by de Broglie in his seminal work [4, 5] has provided the theoretical basis
for the later development of wave mechanics (cf. [18, 19]) and its attendant
successes in describing a vast array of microscopic phenomena. Nevertheless,
the interpretation of the wave-particle duality has remained somewhat
ambiguous since its introduction, in the sense that a material point particle
must be associated with an extended wave-like character. While the theoretical
interpretation of the wave-particle duality has remained a vexing issue, its
experimental confirmation was realised almost immediately for electrons by
Davisson & Germer (cf. [3]). Wave-particle duality is now routinely confirmed
for more substantial material bodies, see for instance [1] which
experimentally confirms wave-particle duality for $\mathrm{C_{60}}$ molecules,
while [9, 15, 17, 21, 24] provide further confirmation of wave-particle
dualism for matter. The aim of the current work is to investigate the wave-
particle duality from the perspective of relational mechanics, whereby all
motion is treated in strictly relative terms. It is demonstrated that a
consistent application of the relativity principle (see [13] for instance)
suggests an apparent contradiction with the exclusively point-like
interpretation of massive particles.

The interpretation of wave-particle dualism as an inherent feature of space-
time coordinate transformations is motivated by the assumption that all
observers are associated with a rest-frame whose coordinates provide a valid
frame work for the description of physical laws. The maps between these
reference frames are considered as maps with $C^{2}$-regularity with respect
to the space-time coordinates of their domain. This regularity is found to
ensure the existence of conservation laws for the energy-momentum of each
material particle when observed from any frame of reference. Moreover, a
consistent implementation of the nonlinear relativistic energy-momentum
constraint combined with this $C^{2}$-regularity is also found to necessitate
the introduction of appropriate potentials to describe the case of non-uniform
relative motion between observers and their respective rest-frames.

A reformulation of the conservation laws associated with the
$C^{2}$-regularity of the coordinate maps and the energy-momentum constraint
is found to reproduce the Klein-Gordon wave equation. This wave-equation is
found to be an intrinsic property of the space-time mappings as opposed to a
feature of the material bodies themselves. It is noted that this is entirely
consistent with Wigner’s observation that the fundamental particles of nature
must belong to some representation of the Lorentz group (cf. [23]). The
intrinsic wave character of the coordinate mapping is found to reproduce the
Dirac equation (cf. [7, 8]) as a reformulation of the inverse function theorem
associated with the coordinate transformation. Finally, the case of observers
in a state of relative acceleration is found to necessitate the introduction
of potential fields, familiar as massless gauge fields. Thus it appears wave-
particle dualism is essentially a physical manifestation of the relativity
principle for interacting observers.

##  2\. Relative motion and coordinate transformations

Consider the case of two observers $\mathscr{O}$ and $\mathscr{O}^{\prime}$,
and associated with each observer are their respective rest frames $\Omega$
and $\Omega^{\prime}$. The location of $\mathscr{O}$ with reference to
$\Omega$ is always of the form $(t,0,0,0)$ for $t\in\mathds{R}$ while the
location of $\mathscr{O}^{\prime}$ with reference to this frame is of the form
$(t,x,y,z)\in\mathds{R}^{4}$, where $x$, $y$ and $z$ may be functions of $t$,
depending on the state of relative motion between the two observers.
Conversely, in the frame $\Omega^{\prime}$ the space-time coordinates of the
observer $\mathscr{O}^{\prime}$ are always of the form $(\tau,0,0,0)$ with
$\tau\in\mathds{R}$, while the coordinates of $\mathscr{O}$ are now of the
form $(\tau,\xi,\upsilon,\zeta)$, where again $\xi$, $\upsilon$ and $\zeta$
may be functions of $\tau$ depending on the state of relative motion between
the two observers.

It is clear that to every space-time location in $\Omega$ there are
corresponding space-time coordinates in the frame $\Omega^{\prime}$, and we
say $(t,x,y,z)\sim(\tau,\xi,\upsilon,\zeta)$ if these coordinates represent
the same event as observed in the frames $\Omega$ and $\Omega^{\prime}$
respectively. Since every coordinate in $\Omega$ has a counterpart in
$\Omega^{\prime}$ it follows that there exist maps from one frame to the other
of the form

(1) $\displaystyle{\Phi^{\prime}}$
$\displaystyle:\Omega\rightarrow\Omega^{\prime}$
$\displaystyle{\Phi^{\prime}}$
$\displaystyle:\left(t,x,y,z\right)\mapsto\left(\Theta,\Xi,\Upsilon,\mathcal{Z}\right)\big{|}_{(t,x,y,z)}=\left(\tau,\xi,\upsilon,\zeta\right)$

with the inverse map of the form

(2) $\displaystyle{\Phi}$ $\displaystyle:\Omega^{\prime}\rightarrow\Omega$
$\displaystyle{\Phi}$
$\displaystyle:\left(\tau,\xi,\upsilon,\zeta\right)\mapsto\left(\mathrm{T},\mathrm{X},\mathrm{Y},\mathrm{Z}\right)\big{|}_{(\tau,\xi,\upsilon,\zeta)}=\left(t,x,y,z\right).$

In the following it shall be assumed that ${\Phi^{\prime}}\in
C^{2}\left(\Omega\right)$ and ${\Phi}\in C^{2}\left(\Omega^{\prime}\right)$.
Moreover, given

$({\Phi}\circ{\Phi^{\prime}})(t,x,y,z)=(t,x,y,z)\quad({\Phi^{\prime}}\circ{\Phi})(\tau,\xi,\upsilon,\zeta)=(\tau,\xi,\upsilon,\zeta)$

then the aforementioned regularity requirements ensures the inverse function
theorem applies (see [16] for instance):

(3) $\begin{bmatrix}\Theta_{t}&\Xi_{t}&\Upsilon_{t}&\mathcal{Z}_{t}\\\
\Theta_{x}&\Xi_{x}&\Upsilon_{x}&\mathcal{Z}_{x}\\\
\Theta_{y}&\Xi_{y}&\Upsilon_{y}&\mathcal{Z}_{y}\\\
\Theta_{z}&\Xi_{z}&\Upsilon_{z}&\mathcal{Z}_{z}\end{bmatrix}\begin{bmatrix}\mathrm{T}_{\tau}&\mathrm{X}_{\tau}&\mathrm{Y}_{\tau}&\mathrm{Z}_{\tau}\\\
\mathrm{T}_{\xi}&\mathrm{X}_{\xi}&\mathrm{Y}_{\xi}&\mathrm{Z}_{\xi}\\\
\mathrm{T}_{\upsilon}&\mathrm{X}_{\upsilon}&\mathrm{Y}_{\upsilon}&\mathrm{Z}_{\upsilon}\\\
\mathrm{T}_{\zeta}&\mathrm{X}_{\zeta}&\mathrm{Y}_{\zeta}&\mathrm{Z}_{\zeta}\end{bmatrix}=\begin{bmatrix}1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{bmatrix},$

where subscripts denote partial differentiation with respect to the
corresponding variables. In the following it shall be assumed that the
$(x,y,z)$-axes of $\Omega$ align with the corresponding
$(\xi,\upsilon,\zeta)$-axes of $\Omega^{\prime}$, with no rotation of one
frame relative to another taking place throughout the course of their relative
motion. To simplify the following analysis, it is assumed the motion of
$\mathscr{O}^{\prime}$ relative to $\Omega$ takes place along the $x$-axis of
$\Omega$, and as such it shall only be necessary to consider the $(t,x)$
coordinates of $\mathscr{O}^{\prime}$ in this frame, in most cases. Likewise,
it is assumed the motion of $\mathscr{O}$ relative to the frame
$\Omega^{\prime}$ shall take place along the $\xi$-axis of this reference
frame and so it is again only necessary to refer to the
$(\tau,\xi)$-coordinates of $\mathscr{O}$ in $\Omega^{\prime}$ for the most
part.

###  2.1. Local Lorentz transformations

The principle of general relativity implies the coordinate systems $\Omega$
and $\Omega^{\prime}$ should be treated equivalently from a physical point of
view (see [13] for further discussion of this principle as it relates to the
description of acceleration in relative terms). In this regard the relative
motion between the pair may be described in the same fashion in the reference
frame $\Omega$ or $\Omega^{\prime}$. At some instant the observer
$\mathscr{O}^{\prime}$ may occupy the space-time location $(t,x,y,z)\in\Omega$
and is moving with some velocity $\mathbf{v}$ relative to $\mathscr{O}$ (and
thus $\Omega$), while at the same instant the observer $\mathscr{O}$ occupies
some space-time location $(\tau,\xi,\upsilon,\zeta)\in\Omega^{\prime}$ and is
moving with velocity $-\mathbf{v}$. The relative motion of $\mathscr{O}$ and
$\mathscr{O}^{\prime}$ is described as uniform relative motion if $\mathbf{v}$
is constant throughout $\Omega$ and $\Omega^{\prime}$, otherwise, the relative
velocity $\mathbf{v}$ has instantaneous values at various stages of the
relative motion. However it is always the case that if a point
$(\tau,\xi,\upsilon,\zeta)\in\Omega^{\prime}$ moves with velocity $\mathbf{v}$
relative to $\Omega$ then the corresponding point $(t,x,y,z)\in\Omega$ is
found to move with velocity $-\mathbf{v}$ relative to the frame
$\Omega^{\prime}$, instantaneously.

Since we consider $\mathscr{O}$ and $\mathscr{O}^{\prime}$ to be in relative
motion along the respective $x$ and $\xi$ axes of $\Omega$ and
$\Omega^{\prime}$, the relative velocity of $\mathscr{O}^{\prime}$ with
reference to $\Omega$ is simply of the form $\mathbf{v}=({v},0,0)$, and it is
convenient for now to consider space-time locations of $\mathscr{O}^{\prime}$
with reference to $\Omega$ using only their $(t,x)$-coordinates. If $v$ is the
instantaneous velocity of $\mathscr{O}^{\prime}$ with reference to $\Omega$,
we consider a pair of space-time locations $(t,x)\in\Omega$ and
$(t+dt,x+dx)\in\Omega$, where the coordinate displacements $(dt,dx)$ have
arbitrarily small magnitude. The coordinate map (1) now gives

(4) $d\tau=\Theta_{t}dt+\Theta_{x}dx\qquad d\xi=\Xi_{t}dt+\Xi_{x}dx.$

As such the instantaneous velocity along the trajectory connecting these
space-time locations in each frame are related according to

(5)
$\frac{d\xi}{d\tau}=\frac{\Xi_{t}+\Xi_{x}\frac{dx}{dt}}{\Theta_{t}+\Theta_{x}\frac{dx}{dt}}.$

If the coordinate displacements $(dt,dx)$ are related according to $dx={v}dt$,
then this corresponds to the instantaneous trajectory of
$\mathscr{O}^{\prime}$ with reference to $\Omega$ and as such it must be the
case that

(6)
$0=\frac{d\xi}{d\tau}=\frac{\Xi_{t}+\Xi_{x}{v}}{\Theta_{t}+\Theta_{x}{v}}\implies\Xi_{t}=-{v}\Xi_{x}.$

Alternatively, if the velocity along the trajectory vanishes with reference to
$\Omega$, that is $dx=0$, then the velocity along the trajectory with
reference to $\Omega^{\prime}$ must be $-{v}$ and so it follows that

(7)
$-{v}=\frac{d\xi}{d\tau}=\frac{\Xi_{t}}{\Theta_{t}}\implies\Xi_{t}=-{v}\Theta_{t}.$

Thus equations (6)–(7) ensure that

(8) $\Theta_{t}=\Xi_{x}.$

The Lorentz transformation between reference frames in a state of uniform
relative motion follows from the constancy of the speed of light
$c\approx$2.998\text{\times}{10}^{8}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$
in all frames of reference. Specifically, if $(dt,dx)$ are coordinate
displacements in $\Omega$ related by $dx=cdt$ then the associated coordinate
displacements in the frame $\Omega^{\prime}$ are also related by
$d\xi=cd\tau$. As such, equations (5)–(8) now give

(9)
$c=\frac{d\xi}{d\tau}=\frac{\Xi_{t}+\Xi_{x}c}{\Theta_{t}+\Theta_{x}c}\implies\Theta_{x}=-\frac{{v}}{c^{2}}\Theta_{t}.$

In line with the principle of relativity, the components
$\\{\mathrm{T},\mathrm{X}\\}$ of the inverse map ${\Phi}$ (cf. equation (2))
are related by

(10)
$\mathrm{T}_{\tau}=\mathrm{X}_{\xi}\quad\mathrm{X}_{\tau}={v}\mathrm{T}_{\tau}\quad\mathrm{T}_{\xi}=\frac{{v}}{c^{2}}\mathrm{T}_{\tau}.$

As such, the inverse function theorem now requires

(11) $\begin{bmatrix}\mathrm{T}_{\tau}&{v}\mathrm{T}_{\tau}\\\
\frac{{v}}{c^{2}}\mathrm{T}_{\tau}&\mathrm{T}_{\tau}\end{bmatrix}\begin{bmatrix}\Theta_{t}&-{v}\Theta_{t}\\\
-\frac{{v}}{c^{2}}\Theta_{t}&\Theta_{t}\end{bmatrix}=\begin{bmatrix}1&0\\\
0&1\end{bmatrix}$

from which we deduce

(12) $\mathrm{T}_{\tau}\Theta_{t}=\frac{1}{1-\frac{{v}^{2}}{c^{2}}}.$

Since the map $\mathrm{T}$ is expected to be obtained from the map $\Theta$
under a reverse of velocity, namely ${v}\to-{v}$, from the constraint (12) we
expect
$\mathrm{T}_{\tau}=\Theta_{t}=\pm\frac{1}{\sqrt{1-\frac{{v}^{2}}{c^{2}}}}$,
which also ensures $\mathrm{T}_{t}$ and $\Theta_{\tau}$ are independent of the
direction of $v$. Choosing for the moment the positive solution, the Jacobian
of the coordinate map is given by

(13)
$J=\Theta_{t}^{2}-c^{2}\Theta_{x}^{2}=\Xi_{x}^{2}-\frac{1}{c^{2}}\Xi^{2}_{t}=1,$

and so the coordinate displacements $(dt,dx)$ and $(d\tau,d\xi)$ are related
according to

(14) $d\tau=\frac{dt-\frac{{v}}{c^{2}}dx}{\sqrt{1-\frac{{v}^{2}}{c^{2}}}}\quad
d\xi=\frac{dx-{v}dt}{\sqrt{1-\frac{{v}^{2}}{c^{2}}}},$

which is simply a localised form of the familiar Lorentz transformation.

If $(mc^{2},0,0,0)$ is the constant energy-momentum vector of
$\mathscr{O}^{\prime}$ with reference to its rest-frame $\Omega^{\prime}$,
then its instantaneous energy-momentum with reference to the frame $\Omega$ is
given by

(15)
$\left(E,p,0,0\right)=\left(\frac{mc^{2}}{\sqrt{1-\frac{{v}^{2}}{c^{2}}}},\frac{m{v}}{\sqrt{1-\frac{{v}^{2}}{c^{2}}}},0,0\right),$

from which one easily deduces the relativistic energy-momentum constraint

(16) $E^{2}-p^{2}c^{2}=m^{2}c^{4}.$

Moreover, the local Lorentz transformation may now be written in terms of the
instantaneous energy-momentum vector of $\mathscr{O}^{\prime}$ with reference
to $\Omega$ according to

(17) $d\tau=\frac{1}{mc^{2}}\left(Edt-pdx\right)\quad
d\xi=\frac{1}{mc^{2}}\left(Edx-pc^{2}dt\right),$

which will prove more convenient in what follows. Moreover it is clear that
the energy-momentum of $\mathscr{O}^{\prime}$ with reference to $\Omega$ can
be characterised by

(18) $E=mc^{2}\Theta_{t}\quad p=-mc^{2}\Theta_{x},$

while the energy momentum condition (16) is simply a reformulation of the
condition $J=1$ (cf. equation (13)).

###  2.2. Regularity & conservation laws

Given the regularity conditions ${\Phi^{\prime}}\in C^{2}(\Omega)$ and
${\Phi}\in C^{2}(\Omega^{\prime})$, Schwartz’s theorem (see [6] for instance)
implies ${\Phi^{\prime}}_{tx}={\Phi^{\prime}}_{xt}$ throughout $\Omega$. A
state of relative motion between $\mathscr{O}$ and $\mathscr{O}^{\prime}$ as
characterised by the coordinate transformation ${\Phi^{\prime}}$ requires

(19) $\begin{bmatrix}\Theta_{t}&\Xi_{t}\\\
\Theta_{x}&\Xi_{x}\end{bmatrix}=\frac{1}{mc^{2}}\begin{bmatrix}E&-pc^{2}\\\
-p&E\end{bmatrix}$

whose inverse

(20) $\begin{bmatrix}\mathrm{T}_{\tau}&\mathrm{X}_{\tau}\\\
\mathrm{T}_{\xi}&\mathrm{X}_{\xi}\end{bmatrix}=\frac{1}{mc^{2}}\begin{bmatrix}E&pc^{2}\\\
p&E\end{bmatrix}$

is easily deduced from the inverse function theorem (cf. equation (3)). The
$C^{2}$-regularity of $\Theta$ now ensures that

(21) $\Theta_{tx}=\Theta_{xt}\implies E_{x}+p_{t}=0,$

which simply relates the acceleration to the gradient of the energy in the
vicinity of $\mathscr{O}^{\prime}$ with reference to the frame $\Omega$.
Similarly the $C^{2}-$regularity of $\Xi$ also implies

(22) $\Xi_{xt}=\Xi_{tx}\implies\frac{1}{c^{2}}E_{t}+p_{x}=0,$

which ensures the conservation of energy-momentum of $\mathscr{O}^{\prime}$
along its trajectory through the frame $\Omega$.

Moreover, it follows that equation (19) may also be written according to

(23) $\begin{bmatrix}\Theta_{t}&c^{2}\Theta_{x}\\\
\Theta_{x}&\Theta_{t}\end{bmatrix}=\begin{bmatrix}\Xi_{x}&-\Xi_{t}\\\
-\frac{1}{c^{2}}\Xi_{t}&\Xi_{x}\end{bmatrix}=\frac{1}{mc^{2}}\begin{bmatrix}E&-pc^{2}\\\
-p&E\end{bmatrix},$

which along with the conditions (21)–(22) now yield the following wave
equations:

(24a) $\displaystyle\Theta_{tt}-c^{2}\Theta_{xx}$ $\displaystyle=0$ (24b)
$\displaystyle\Xi_{tt}-c^{2}\Xi_{xx}$ $\displaystyle=0.$

The wave equations (24a)–(24b) must be considered in conjunction with the
nonlinear constraints (13) following from the relativistic energy-momentum
condition.

###  2.3. Coordinate maps as scalar fields

The constraint that must be satisfied by the function $\Theta$ is (13) which
is equivalent to the mass-energy constraint:

$\Theta_{t}^{2}-c^{2}\Theta_{x}^{2}=1\iff E^{2}-p^{2}c^{2}=m^{2}c^{4}.$

In addition the map must satisfy the wave equation (24a) which is simply a
reformulation of the conservation of energy-momentum

$\Theta_{tt}-c^{2}\Theta_{xx}=0\iff\frac{1}{c^{2}}E_{t}+p_{x}=0$

which is itself a consequence of the $C^{2}$–regularity of the maps $\Theta$
and $\Xi$.

We introduce the map $\psi(t,x)\in C^{2}(\Omega)$ defined according to

(25) $\Theta(t,x)=\rho(\psi(t,x))\in C^{2}(\Omega).$

where $\rho(\cdot)$ and $\psi(\cdot,\cdot)$ are as yet otherwise arbitrary.
The condition (13) may now be reformulated according to

(26) $\rho^{\prime}(\psi)^{2}\left[\psi_{t}^{2}-c^{2}\psi_{x}^{2}\right]=1,$

in which case $\rho^{\prime}(\psi)$ and $\psi_{t}^{2}-c^{2}\psi_{x}^{2}$ are
both non-zero in $\Omega$. Meanwhile the conservation equation (24a) may be
written according to

(27)
$\rho^{\prime\prime}(\psi)\left[\psi_{t}^{2}-c^{2}\psi_{x}^{2}\right]+\rho^{\prime}(\psi)\left[\psi_{tt}-c^{2}\psi_{xx}\right]=0,$

Physically, the conditions

$E=mc^{2}\rho^{\prime}(\psi)\psi_{t}\quad
p=-mc^{2}\rho^{\prime}(\psi)\psi_{x}$

with $E$ and $p$ both non-zero (since $\mathscr{O}$ and $\mathscr{O}^{\prime}$
are in a state of relative motion) also suggest $\psi_{t}$ and $\psi_{x}$ are
both non-zero in $\Omega$.

The constraint (26) may be used to reformulate (27) according to

(28)
$\frac{\rho^{\prime\prime}(\psi)}{\rho^{\prime}(\psi)^{3}}+\psi_{tt}-c^{2}\psi_{xx}=0,$

while upon multiplying equation (28) by $\psi_{t}$ one obtains

(29)
$\frac{1}{2}\partial_{t}\left[-\frac{1}{\rho^{\prime}(\psi)^{2}}+\psi_{t}^{2}\right]-c^{2}\psi_{xx}\psi_{t}=0.$

Equation (26) now allows one to substitute for
$\frac{1}{\rho^{\prime}(\psi)^{2}}$, which then ensures

(30)
$\psi_{xt}\psi_{x}-\psi_{xx}\psi_{t}=0\implies\left(\frac{\psi_{x}}{\psi_{t}}\right)_{x}=0.$

In a similar manner, multiplying equation (28) by $\psi_{x}$ also ensures

(31) $\left(\frac{\psi_{x}}{\psi_{t}}\right)_{t}=0,$

in which case $\frac{\psi_{x}}{\psi_{t}}$ is a constant. It follows that
$\psi(t,x)$ is in general of the form

(32) $\psi(t,x)=\psi(\omega t-kx)\equiv\psi(s),$

where $\omega$ and $k$ are constants.

Rewriting equation (26) in terms of $\psi(s)$, it follows at once that

(33) $\omega_{0}^{2}\rho^{\prime}(\psi)^{2}\psi^{\prime}(s)^{2}=1,$

where $\omega_{0}^{2}=\omega^{2}-k^{2}c^{2}\neq 0$. Moreover, the inverse
function theorem (cf. [16]) may be applied to equation (33) thus ensuring

(34) $\pm\omega_{0}(\rho\circ\psi)(s)=\operatorname{id}(s)=\omega t-kx$

in which case $\Theta(t,x)$ is a linear map, specifically

(35) $\Theta(t,x)=\pm\frac{\omega t-kx}{\omega_{0}}.$

Taking the negative sign in equation (35) means events in the frame $\Omega$
are labelled in reverse chronological order compared to their labelling in
$\Omega^{\prime}$. In the current context, the principle of relativity
suggests there is no absolute sense as to which coordinate frame uses the
truly “forward” progressing time coordinate.

Note that $\psi(t,x)$ as defined by equation (25) is quite arbitrary. In
particular if given two maps $\Theta_{1}(t,x)=\rho(\phi(t,x))$ and
$\Theta_{2}(t,x)=\rho(\psi(t,x))$ with $\phi=\kappa\psi$ for some non-zero
constant $\kappa$, then equation (26) must be satisfied in each case, and as
such

(36)
$\rho^{\prime}(\kappa\psi)^{2}\kappa^{2}\left[\psi_{t}^{2}-c^{2}\psi_{x}^{2}\right]=\rho^{\prime}(\psi)^{2}\left[\psi_{t}^{2}-c^{2}\psi_{x}^{2}\right]=1.$

Thus it appears

(37) $\kappa^{2}\rho^{\prime}(\kappa\psi)^{2}=\rho^{\prime}(\psi)^{2}$

from which it must follow

(38) $\rho^{\prime}(\psi)\simeq\frac{1}{\psi},$

up to some multiplicative constant. It follows that the map $\Theta(t,x)$ may
be generally defined according to

(39) $\Theta(t,x)=\frac{\mu}{mc^{2}}\ln\psi(\omega t-kx),$

where $\mu$ is a constant whose unit must clearly be that of action, that is
$[\mu]=$\text{\,}\mathrm{J}\cdot\mathrm{s}$$. The energy-momentum as given by
$E=mc^{2}\Theta_{t}$ and $p=-mc^{2}\Theta_{x}$ are equivalently written
according to the eigenvalue problem

(40) $\mu\psi_{t}=E\psi\qquad-\mu\psi_{x}=p\psi,$

and equation (27) may be reformulated according to

(41) $\frac{1}{c^{2}}\psi_{tt}-\psi_{xx}-\frac{m^{2}c^{2}}{\mu^{2}}\psi=0.$

While it is always desirable to deduce all conditions as far as possible from
first principles, it appears that in the current framework one must actively
choose the value of $\mu$, specifically whether $\mu$ should be chosen as a
real or imaginary constant.

The possible choice $\mu=\mu_{1}+i\mu_{2}$ with $\mu_{n}\in\mathds{R}$ for
$n\in\\{1,2\\}$, provides no greater generality since $\Theta(t,x)$ is
expected to be real valued. If $\mu$ is a general complex constant, then it
must be the case that $\ln\psi$ is a complex valued function, with

$\Re{\ln\psi(t,x)}=a_{1}(t,x),\quad\Im{\ln\psi(t,x)}=a_{2}(t,x).$

The condition $\Theta(t,x)$ is real valued now ensures

(42)
$mc^{2}\Theta=\mu_{1}a_{1}-\mu_{2}a_{2}+i\left(\mu_{1}a_{2}+\mu_{2}a_{1}\right),$

is also real valued and as such it is clear that

$a_{1}(t,x)=-\frac{\mu_{1}}{\mu_{2}}a_{2}(t,x).$

It follows that $\Theta(t,x)$ may be written exclusively in terms of
$a_{1}(t,x)$ or $a_{2}(t,x)$ according to

$\Theta(t,x)=\frac{\mu_{1}^{2}+\mu_{2}^{2}}{\mu_{1}mc^{2}}a_{1}(t,x)=-\frac{\mu_{1}^{2}+\mu_{2}^{2}}{\mu_{2}mc^{2}}a_{2}(t,x).$

Thus the possible values of $\mu$ to be considered are either real or
imaginary values, without loss of generality.

It is well established that $\mu=\frac{\hbar}{i}$ is consistent with the
Hamilton-Jacobi equations of classical mechanics (cf. [14]). Moreover it is
noteworthy that once $\mu$ is imaginary the wave-equation (41) has a well-
posed Cauchy problem (cf. [22]), while its solution space is naturally endowed
with an inner-product structure and an associated orthogonal basis, as well as
a Hilbert space structure under appropriate conditions (cf. [2]). In
particular, the $\tau$-difference between events in $\Omega$ of the form
$(t,x_{1})$ and $(t,x_{2})$ is given by

$\int_{x_{1}}^{x_{2}}\Theta_{x}(t,x)dx=\int_{x_{1}}^{x_{2}}\frac{1}{\psi}\psi_{x}dx=\int_{x_{1}}^{x_{2}}e^{-\frac{1}{\mu}(Et-
px)}\partial_{x}e^{\frac{1}{\mu}(Et-px)}dx.$

On the other hand, a general solution of (41) takes the form

$\psi(t,x)=\sum_{n=1}^{N}a_{n}e^{\frac{1}{\mu}(E_{n}t-p_{n}x)},$

while the observer $\mathscr{O}^{\prime}$ associated $\psi(t,x)$ has a unique
energy-momentum once

$\int_{x_{1}}^{x_{2}}\Theta_{x}(t,x)dx=\int_{x_{1}}^{x_{2}}e^{-\frac{1}{\mu}(Et-
px)}\partial_{x}\psi(t,x)dx,$

is time-independent. This is possible only if $e^{-\frac{1}{\mu}(Et-px)}$ form
an orthogonal set of basis functions, thus implying $\mu$ is imaginary.
Moreover, there is significant experimental evidence to support wave-particle
duality for massive particles in nature, something which appears to be
realised only when $\mu$ is imaginary. Thus a clear choice for this constant
is $\mu=\frac{\hbar}{i}$ where
$\hbar=$1.054\,57\text{\times}{10}^{-34}\text{\,}\mathrm{J}\cdot\mathrm{s}$$
is the reduced Planck constant. The eigenvalue problem (40) now becomes

(43) $-i\hbar\psi_{t}=E\psi\qquad i\hbar\psi_{x}=p\psi.$

while equation (41) now yields

(44) $\frac{1}{c^{2}}\psi_{tt}-\psi_{xx}+\frac{m^{2}c^{2}}{\hbar^{2}}\psi=0.$

A general coordinate transformation as given by (3) may always be reduced to
the form (4) by means of appropriate rotations (cf. [23]). Hence equations
(43)–(44) are generalised in a straightforward manner to relative motion in an
arbitrary direction. In particular, it may be shown that the coordinate map
$\Theta(t,x,y,z)$ may always be written in the form

(45) $\Theta(t,x,y,z)=-\frac{i\hbar}{mc^{2}}\ln\psi(t,x,y,z),$

with

(46) $-i\hbar\psi_{t}=E\psi\quad i\hbar\nabla\psi=\mathbf{p}\psi,$

where $\mathbf{p}=(p_{1},p_{2},p_{3})$ are the momentum components of
$\mathscr{O}^{\prime}$ with reference to the frame $\Omega$. The corresponding
wave equation is then simply

(47)
$\frac{1}{c^{2}}\psi_{tt}-\nabla^{2}\psi+\frac{m^{2}c^{2}}{\hbar^{2}}\psi=0,$

the familiar Klein-Gordon wave equation.

###  2.4. Commutators

While our analysis so far has related to maps
$\psi(t,x)=e^{\frac{i}{\hbar}(Et-px)}$ associated with coordinate
transformations characterised by unique energy momentum $(E,p)$, we note that
the solution space of (44) is linear, and as such any linear combination of
such wave-functions is also an admissible solution. In particular, it means
one may construct generalised coordinate maps $\psi$ wherein the energy-
momentum is associated with an _ensemble of possible values_ , something
characteristic of quantum systems.

Moreover, as demonstrated in [14], the classical commutativity of position and
momentum as give by $px-xp=0$ is realised if and only if
$\left(\hat{p}\hat{x}-\hat{x}\hat{p}\right)\psi=i\hbar\psi$ where
$\hat{x}\psi\equiv x\psi$ and $\hat{p}\psi\equiv i\hbar\partial_{x}\psi$.
Equation (43) means we may obtain the classical (i.e. measured) momentum of
$\mathscr{O}^{\prime}$ with respect to $\Omega$ according to
$p=i\hbar\frac{\psi_{x}}{\psi}$. Classical commutativity of position and
momentum now requires

(48) $px-xp=0\implies i\hbar\frac{\psi_{x}}{\psi}x-i\hbar
x\frac{\psi_{x}}{\psi}=0.$

As such we have

(49) $i\hbar\psi_{x}x-i\hbar x\psi_{x}=0\implies
i\hbar\partial_{x}(x\psi)-i\hbar\psi-i\hbar x\psi_{x}=0,$

and so it follows that

(50)
$\left[i\hbar\partial_{x},x\right]\psi=i\hbar\psi\implies[\hat{p},\hat{x}]\psi=i\hbar\psi,$

where $[\cdot,\cdot]$ denotes the usual operator commutator. Thus we conclude
that the position and momentum of $\mathscr{O}^{\prime}$ are not
simultaneously definable with reference to $\Omega$ using $\psi(t,x)$. This is
not to say $\mathscr{O}^{\prime}$ is not a localised particle, in fact
$\mathscr{O}^{\prime}$ always has a well-defined location
$(\tau,0)\in\Omega^{\prime}$ for all $\tau\in\mathds{R}$. Instead it means the
mapping ${\Phi^{\prime}}:\Omega\to\Omega^{\prime}$ or its inverse
${\Phi}:\Omega^{\prime}\to\Omega$ cannot fully localise
$(\tau,0)\in\Omega^{\prime}$ with reference to $\Omega$. However, this does
not contradict any of our earlier observations.

The derivation of the local Lorentz transformation in §2.1 relied on
specifying the velocity of $\mathscr{O}^{\prime}$ with reference to $\Omega$,
which means the ratio $v=\frac{pc^{2}}{E}$ has a well-defined value, and not
necessarily that $\mathscr{O}^{\prime}$ is observed to move between locations
$(t,x)$ and $(t+dt,x+vdt)$ in $\Omega$. Moreover, the Lorentz transformation
(14) does not relate coordinates in $\Omega$ with their counterpart in
$\Omega^{\prime}$, rather it relates coordinate displacements $(dt,dx)$ in
$\Omega$ with corresponding coordinate displacements $(d\tau,d\xi)$ in
$\Omega^{\prime}$, once the frames have a well-defined relative velocity.

Importantly, we realise that in the current frame-work the inability to assign
the observer $\mathscr{O}^{\prime}$ simultaneous position and momentum
eigenvalues $x\in\mathds{R}$ and $p\in\mathds{R}$ with reference to $\Omega$,
is not an inherent feature of this observer. Rather it is an intrinsic feature
of the map from the rest-frame of $\mathscr{O}$ to the rest-frame of
$\mathscr{O}^{\prime}$, and so is more appropriately described as a feature of
maps between space-time coordinate systems.

###  2.5. Coordinate maps as spinor fields

While the inverse function theorem as given by equation (3) is expected to be
true, its explicit form follows from the presupposition that space-time should
be considered as a four-dimensional manifold (which of course we still hope to
accommodate in general). However, as we have seen thus far, the treatment of
the relative motion of $\mathscr{O}$ and $\mathscr{O}^{\prime}$ may be
considered quite generally in two-dimensional space-time, with any conclusions
being extendible to four-dimensional space-time in a relatively simple manner.

In this regard, we continue to investigate the relative motion of
$\mathscr{O}$ and $\mathscr{O}^{\prime}$ via a two component map
$(t,x)\mapsto(\Theta(t,x),\Xi(t,x))$. The coordinate transformation from
$\Omega$ to $\Omega^{\prime}$ and its inverse must satisfy the inverse
function theorem:

(51) $\begin{bmatrix}p_{0}&p_{1}\\\
p_{1}&p_{0}\end{bmatrix}\begin{bmatrix}p_{0}&-p_{1}\\\
-p_{1}&p_{0}\end{bmatrix}=\begin{bmatrix}m^{2}c^{2}&0\\\
0&m^{2}c^{2}\end{bmatrix},$

where $(p_{0},p_{1},0,0)=\left(\frac{E}{c},p,0,0\right)$ is the four-momentum
of $\mathscr{O}^{\prime}$ with respect to $\Omega$, with
$p_{0}=mc\Theta_{t}=mc\Xi_{x}$ and $p_{1}=-mc^{2}\Theta_{x}=-m\Xi_{t}$. Note
that equation (51) may be reformulated as

(52)
$\left[p_{0}\mathds{1}_{2}-p_{1}\sigma_{1}\right]\left[p_{0}\mathds{1}_{2}+p_{1}\sigma_{1}\right]=m^{2}c^{2}\mathds{1}_{2},$

where $\mathds{1}_{2}$ is the $2\times 2$-identity matrix and
$\sigma_{1}=\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}$ is a Pauli spin matrix. It
is clear from equation (43) that equation (52) may be written in terms of
$\psi(t,x)$ according to

(53)
$\left[\partial_{0}\mathds{1}_{2}+\partial_{1}\sigma_{1}\right]\left[\partial_{0}\mathds{1}_{2}-\partial_{1}\sigma_{1}\right]\psi=-\frac{m^{2}c^{2}}{\hbar^{2}}\psi,$

where the right-hand side should be understood to contain a factor of
$\mathds{1}_{2}$. As such the inverse function theorem may be re-formulated
according to

(54)
$\partial_{0}^{2}\psi=\left[\left(\partial_{1}\sigma_{1}\right)^{2}-\frac{m^{2}c^{2}}{\hbar^{2}}\right]\psi=\left[\partial_{1}^{2}-\frac{m^{2}c^{2}}{\hbar^{2}}\right]\psi,$

since $\sigma_{1}^{2}=\mathds{1}_{2}$. Thus it appears one may identify the
differential operator $\partial_{0}$ with a matrix-differential operator
according to

(55)
$\partial_{0}\psi=\left[\alpha\partial_{1}+i\beta\frac{mc}{\hbar}\right]\psi,$

where $\alpha$ and $\beta$ are $2\times 2$-matrices such that

(56)
$\alpha^{2}=\beta^{2}=\mathds{1}_{2}\qquad\alpha\beta+\beta\alpha=\mathbb{0}_{2},$

where $\mathbb{0}_{2}$ is the $2\times 2$-null matrix.

It is clear that a possible scheme to describe relative motion in four-
dimensional space-time is to postulate an extension of (55) according to

(57)
$\partial_{0}\psi=\left[\alpha^{k}\partial_{k}+i\beta\frac{mc}{\hbar}\right]\psi,\quad
k\in\\{1,2,3\\},$

where the Einstein summation convention is applied to repeated upper and lower
indices and

(58)
$(\alpha^{k})^{2}=\beta^{2}=\mathds{1}\qquad\alpha^{k}\beta+\beta\alpha^{k}=\alpha^{k}\alpha^{l}+\alpha^{l}\alpha^{k}=\mathbb{0}.$

Of course, such a procedure was first developed in Dirac’s seminal work on the
relativistic theory of the electron (cf. [7, 8]). It is common practice to
write equation (57) in a more covariant manner according to

(59)
$\left(i\gamma^{\mu}\partial_{\mu}+\frac{mc}{\hbar}\right)\psi=0,\quad\gamma^{0}=\beta,\quad\gamma^{k}=-\beta\alpha^{k}.$

The mass-energy constraint now requires four matrices
$\\{\gamma^{\mu}\\}_{\mu=0}^{4}$ satisfying the anti-commutator relations

(60)
$\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2\eta^{\mu\nu}\mathds{1},\quad\mu,\nu\in\\{0,1,2,3\\},$

where $\eta^{\mu\nu}$ are components of the metric tensor
$\eta=\operatorname{diag}(1,-1,-1,-1)$. It is well established (see [20] p.71
for instance) that the Clifford algebra generated by (60) has exactly sixteen
linearly independent product combinations of the four $\gamma$-matrices, in
which case every representation of (60) is realised by a quadruple of $n\times
n$ matrices with $n\geq 4$.

##  3\. Relative acceleration

Differentiating the nonlinear constraint on $\Theta$ from equation (13) with
respect to $t$ and with respect to $x$ yields

(61)
$\Theta_{t}\Theta_{tt}-c^{2}\Theta_{x}\Theta_{xt}=0\qquad\Theta_{t}\Theta_{tx}-c^{2}\Theta_{x}\Theta_{xx}=0.$

Multiplying the first of these by $\Theta_{t}$ and the second by
$c^{2}\Theta_{x}$ and adding the resulting equations, it follows that

(62) $\left(\Theta_{t}^{2}-c^{2}\Theta_{x}^{2}\right)\Theta_{tt}=0,$

having also used (24a). In a similar manner one may also deduce the condition

$\left(\Xi_{x}^{2}-\frac{1}{c^{2}}\Xi_{t}^{2}\right)\Xi_{xx}=0.$

The condition (62) now means one of the following conditions must be true:

(63)
$\Theta^{2}_{t}-c^{2}\Theta_{x}^{2}=\frac{1}{mc^{2}}\left(E^{2}-p^{2}c^{2}\right)=0\qquad\Theta_{tt}=\Theta_{xx}=0,$

in which case the condition (13) fails, or the coordinate map
${\Phi^{\prime}}$ must have components such that $\Theta_{t}$ and $\Theta_{x}$
are constant, which is only the case when $\mathscr{O}$ and
$\mathscr{O}^{\prime}$ are in a state of uniform relative motion.

It follows that more general states of relative motion between the observers
which preserve (16) and the $C^{2}$-regularity of the coordinate map
${\Phi^{\prime}}$ and its inverse cannot be accommodated by the coordinate
maps directly. Instead the relative acceleration of $\mathscr{O}$ and
$\mathscr{O}^{\prime}$ must be accommodated by an external potential
$(\phi(t,x),A(t,x))\in C^{2}(\Omega)\times C^{2}(\Omega)$. It is such
potentials which then account for the variation of the energy-momentum of the
observers as they move through the rest-frame of their counterpart.

###  3.1. Acceleration via potential fields

The observer $\mathscr{O}^{\prime}$ moving with reference to $\Omega$ with
unique energy-momentum $(E,p)$, is described by

(64) $\Theta(t,x)=-i\frac{\hbar}{mc^{2}}\ln\psi(t,x),$

where $\psi(t,x)$ is a solution of equation (44). To introduce a relative
acceleration between $\mathscr{O}$ and $\mathscr{O}^{\prime}$ we introduce a
local phase shift in the coordinate map, of the form
$\Theta\rightarrow\Theta+\frac{1}{mc^{2}}\Lambda$ where $\Lambda(t,x)\in
C^{2}(\Omega)$. The _new_ coordinate map ${\Phi^{\prime}}$ now has time-
component

(65)
$\Theta(t,x)+\frac{1}{mc^{2}}\Lambda(t,x)=-i\frac{\hbar}{mc^{2}}\ln\psi(t,x).$

This corresponds to a local phase-rotation of the function
$\psi(t,x)=e^{i(\omega t-kx)}$ according to

(66) $\psi(t,x)\to e^{i\left(\omega
t-kx+\Lambda(t,x)\right)}=e^{\frac{i}{\hbar}\Lambda(t,x)}\psi(t,x).$

The condition $mc^{2}\Theta_{t}=E$ and $-mc^{2}\Theta_{x}=p$ now becomes

(67)
$\mathcal{E}(t,x)=-i\hbar\frac{\psi_{t}}{\psi}+\Lambda_{t}\quad\mathcal{P}(t,x)=i\hbar\frac{\psi_{x}}{\psi}-\Lambda_{x},$

which would indicate the space-time-dependence of the energy-momentum of
$\mathscr{O}^{\prime}$ is directly incorporated within the coordinate
transformation, which contradicts the conclusions deduced from equations (63).

Instead, one introduces a potential field $(\phi(t,x),A(t,x))\in
C^{2}(\Omega)\times C^{2}(\Omega)$ such that

(68)
$\begin{cases}\mathcal{E}\psi=\left(-i\hbar\partial_{t}+q\phi(t,x)\right)\psi=\left(mc^{2}\Theta_{t}+q\phi(t,x)\right)\psi\\\
\mathcal{P}\psi=\left(i\hbar\partial_{x}+qA(t,x)\right)\psi=\left(-mc^{2}\Theta_{x}+qA(t,x)\right)\psi,\end{cases}$

thereby ensuring the $(t,x)$ dependance of $(\mathcal{E},\mathcal{P})$ via
$(\phi,A)$. The constant parameter $q$ represents the coupling strength
between the observer $\mathscr{O}^{\prime}$ and the potential $(\phi,A)$. As
such, if $q=0$, the observer $\mathscr{O}^{\prime}$ does not interact with the
potential and so is in a state of uniform relative motion with reference to
$\Omega$.

If the observer $\mathscr{O}^{\prime}$ does interact with the potential
$(\phi,A)$, then it is still the case that
${\Phi^{\prime}}:\Omega\to\Omega^{\prime}$ is a map between _inertial
reference frames_ , even though the observers are accelerating with respect to
each other. The principle of general relativity is crucial to this observation
since it is this principle which ensures each observer is at rest in its own
right thus occupying an inertial reference frame (see [13] for in-depth
discussion of locally inertial coordinate frames). Since $\Theta(t,x)$ is also
required to satisfy the wave-equation (24a), and the energy-momentum
$(\mathcal{E},\mathcal{P})$ is conserved in the sense of equation (22) (cf.
[11]) it must be the case the potential satisfies

(69) $\frac{1}{c^{2}}\phi_{t}+A_{x}=0,$

the Lorentz gauge condition.

The constraint $\mathcal{E}^{2}-\mathcal{P}^{2}c^{2}=m^{2}c^{4}$ is imposed
since it ensures the rest-energy of $\mathscr{O}^{\prime}$, namely $mc^{2}$,
is independent of its sate of motion relative to $\mathscr{O}$, or
equivalently with reference to $\Omega$. Combined with the eigenvalue problem
(68), this generalised energy-momentum constraint now yields the wave-equation

(70)
$\frac{1}{c^{2}}\left(-i\hbar\partial_{t}+q\phi\right)^{2}\psi-\left(i\hbar\partial_{x}+qA\right)^{2}\psi-m^{2}c^{2}\psi=0.$

It is clear that this wave equation is naturally extended to relative motion
between $\mathscr{O}$ and $\mathscr{O}^{\prime}$ in four-dimensional space-
time according to

(71)
$\frac{1}{c^{2}}\left(-i\hbar\partial_{t}+q\phi\right)^{2}\psi-\left(i\hbar\nabla+q\mathbf{A}\right)^{2}\psi-m^{2}c^{2}\psi=0$

with $\mathbf{A}=(A_{1},A_{2},A_{3})$, which is the Klein-Gordon equation with
minimal coupling to a potential field.

###  3.2. Relativity of acceleration

We assign to the observers $\mathscr{O}$ and $\mathscr{O}^{\prime}$ the rest-
energies and charges $\\{Mc^{2},Q\\}$ and $\\{mc^{2},q\\}$ respectively, which
interact via a potential field $(\phi,A)$. Relativity of acceleration is
assured if there is a reference frame $\Omega$ wherein $\mathscr{O}$ has
_fixed_ energy-momentum $(Mc^{2},0)$ and similarly a reference frame
$\Omega^{\prime}$ where $\mathscr{O}^{\prime}$ has fixed energy-momentum
$(mc^{2},0)$. The conserved energy-momentum of $\mathscr{O}^{\prime}$ with
reference to the frame $\Omega$ is given by

(72) $(\mathcal{E}(t,x),\mathcal{P}(t,x))=(E+q\phi(t,x),p+qA(t,x))$

The energy-momentum of $\mathscr{O}$ with reference to $\Omega$ is

(73)
$(\mathcal{E}(t,0),\mathcal{P}(t,0))=(Mc^{2}+Q\phi(t,0),0+QA(t,0))=(Mc^{2},0)\quad\forall
t\in\mathds{R},$

and so it is required that

(74) $\phi(t,0)=A(t,0)=0\quad\forall t\in\mathds{R}.$

One similarly requires $(\phi^{\prime}(\tau,0),A^{\prime}(\tau,0))=(0,0)$ for
all $\tau\in\mathds{R}$, where $(\phi^{\prime},A^{\prime})$ is the potential
field as observed in the reference frame $\Omega^{\prime}$.

Given a local phase-shift
$\Theta(t,x)\to\Theta(t,x)+\frac{1}{mc^{2}}\Lambda(t,x)$, the coordinate maps
$\Theta(t,x)$ and $\Theta(t,x)+\frac{1}{mc^{2}}\Lambda(t,x)$ satisfy (71), the
governing equations describing the motion of $\mathscr{O}^{\prime}$ with
reference to $\Omega$, if the potential field $(\phi,A)$ transforms according
to

(75) $q\phi\to q\phi-\Lambda_{t}\qquad qA\to qA+\Lambda_{x},$

in which case the eigenvalue problem (68) is invariant under local phase
shifts of the coordinate map as given by (65). That is to say, if
$\psi=e^{\frac{i}{\hbar}mc^{2}\Theta}$, then

(76)
$\displaystyle\left[\frac{1}{c^{2}}\left(-i\hbar\partial_{t}+q\phi-\Lambda_{t}\right)^{2}-\left(i\hbar\partial_{x}+q\phi+\Lambda_{x}\right)^{2}-m^{2}c^{2}\right]e^{\frac{i}{\hbar}\Lambda}\psi$
$\displaystyle\quad=e^{\frac{i}{\hbar}\Lambda}\left[\left(-i\hbar\partial_{t}+q\phi\right)^{2}-\left(i\hbar\partial_{x}+A\phi\right)^{2}-m^{2}c^{2}\right]\psi$
$\displaystyle\qquad=0,$

where the last equality follows from the fact $\psi$ is a solution of (71).
Thus, one may always introduce a local phase shift
$\Theta\to\Theta+\frac{1}{mc^{2}}\Lambda$, such that

(77)
$(q\phi(t,x),qA(t,x))\to(q\phi(t,x)-\Lambda_{t}(t,x),qA(t,x)+\Lambda_{x}(t,x))=(0,0),$

at a specific $(t,x)$. The potential field as observed in the reference frame
$\Omega^{\prime}$ is now given by

(78) $\begin{bmatrix}\phi^{\prime}(\tau,\xi)\\\
A^{\prime}(\tau,\xi)\end{bmatrix}=\frac{1}{mc^{2}}\begin{bmatrix}E&-pc^{2}\\\
-p&E\end{bmatrix}\begin{bmatrix}q\phi(t,x)-\Lambda_{t}(t,x)\\\
qA(t,x)+\Lambda_{x}(t,x)\end{bmatrix}=\begin{bmatrix}0\\\ 0\end{bmatrix}.$

where $(\tau,\xi)\sim(t,x)$. Of course, the issue remains that one cannot
appropriately choose $(t,x)\in\Omega$ such that
${\Phi^{\prime}}(t,x)=(\tau,0)\in\Omega^{\prime}$. Nevertheless, it is clear
in principle, there is always a local phase shift $\Lambda(t,x)$ which will
ensure (74) for $\mathscr{O}^{\prime}$ in $\Omega^{\prime}$ also.

Recall that the conservation equation (22) applied to
$(\mathcal{E},\mathcal{P})$ ensured the Lorentz gauge condition (69). It is
noted that the conservation law (21) is also expected to be satisfied by
$(\mathcal{E},\mathcal{P})$ and so one also deduces

(79)
$(mc^{2}\Theta_{t}+q\phi)_{x}+(-mc^{2}\Theta_{x}+qA)_{t}=0\implies\phi_{x}+A_{t}=0,$

which is clearly gauge invariant under (75) once $\Lambda\in C^{2}(\Omega)$.
Combining equations (75) and (69), one immediately deduces

(80)
$\frac{1}{c^{2}}\phi_{tt}-\phi_{xx}=0\qquad\frac{1}{c^{2}}A_{tt}-A_{xx}=0,$

in which case the potential $(\phi,A)$ is a dynamical field whose excitations
propagate with velocity $c$ relative to all reference frames.

##  4\. Conclusion

The coordinate maps between rest frames of systems of point like observers are
an effective method to analyse the relative motion and interactions of these
observers. It is found that the structure of the wave-equations and nonlinear
constraints applied to these coordinate transformations means the maps may be
identified with scalar and spinor wave-functions familiar from relativistic
quantum mechanics. The governing equations for these wave-functions may in
turn be reduced to the familiar Schrödinger equation of non-relativistic
quantum mechanics by a number of well-established procedures (see [10] for
instance).

Moreover, to ensure consistency, it is found that the relative acceleration of
the observers cannot be accommodated directly by the coordinate maps, instead
it is necessary to introduce potential fields which are found to be familiar
massless gauge fields _by requirement_. In the current framework, this gauge
redundancy is found to have an essential physical role to play as it ensures
each observer may be associated with an inertial coordinate frame throughout
the course of the relative acceleration.

## Declaration

The author declares there are no conflicts of interest to disclose.

## Funding

No funding was received during the preparation of this manuscript.

## Data Availability Statement

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

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