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For $u_{y}:\mathscr{Y}\to[V_{f}/\Gamma]$, $\epsilon_{*}\circ u_{y}$ is the
classifying morphism for the frame bundle of $u^{*}_{y}L_{\epsilon}$. We fix a
trivialization for
$\omega_{\underline{\mathscr{C}}}^{\log}|_{\underline{\mathscr{U}}}$ once and
for all. Composed with this trivialization, the $\kappa_{y}$ in the definition
of $\mathrm{LG}^{\mathrm{{b}f}}(\mathscr{Y},X_{f})$ becomes a trivialization
of $u^{*}_{y}L_{\epsilon}$, which is equivalent to a lifting of
$\epsilon_{*}\circ u_{y}$ to $\operatorname{Spec}(\mathbb{C})$. Thus, by the
fibered diagram, $(u_{y},\kappa_{y})$ is equivalent to a morphism
$\mathscr{Y}\to X=[V_{f}/G]$. Since $[V_{f}/G]\to[V_{f}/\Gamma]$ is
representable, a morphisms $\mathscr{Y}\to[V_{f}/\Gamma]$ is representable if
and only if the composed morphism $\mathscr{Y}\to[V_{f}/\Gamma]$ is.

This establishes the equivalence of categories for the left vertical arrow.
The case for the right vertical arrow is similar, and the rest is
straightforward. ∎

Let

$\underline{h}:\underline{\mathscr{U}}\longrightarrow\mathbb{P}^{N-1}\quad\text{and}\quad
h_{\eta}:\mathscr{U}_{\eta}\longrightarrow X_{f}$

be obtained from $\underline{\xi}|_{\underline{\mathscr{U}}}$ and
$\xi_{\eta}|_{\underline{\mathscr{U}}}$ respectively by applying the vertical
arrows in Lemma 6.12. Then using the commutativity and naturality in Lemma
6.12, $\alpha_{\eta}|_{\underline{\mathscr{U}}}$ reads that
$\underline{h}|_{\eta}=\underline{h_{\eta}}$, where
$\underline{h_{\eta}}:\underline{\mathscr{U}}|_{\eta}\to\mathbb{P}^{N-1}$ is
the map induced by the composition of
$\mathscr{U}_{\eta}\overset{h_{\eta}}{\to}X_{f}\overset{F}{\to}\mathbb{P}^{N-1}$.

Applying Lemma 6.12, the valuative criterion for
$(\underline{\xi}|_{\underline{\mathscr{U}}},\xi_{\eta}|_{\underline{\mathscr{U}}},\alpha_{\eta}|_{\underline{\mathscr{U}}})$
translates into the following

######  Lemma 6.13.

Up to isomorphisms there exists at most one
$(\mathscr{U},\Sigma^{\mathscr{U}},h)$, such that

  1. (1)

$\Sigma^{\mathscr{U}}\subset\mathscr{U}\to T$ is (nonproper) family of pointed
balanced twisted curves and $h:\mathscr{U}\to X_{f}$ is a representable
morphism, extending
$(\mathscr{U}_{\eta},\Sigma^{\mathscr{U}_{\eta}}:=\mathscr{U}_{\eta}\cap\Sigma^{\mathscr{C}_{\eta}},h_{\eta})$,

  2. (2)

the coarse moduli of $\mathscr{U}$ is isomorphic to $\underline{\mathscr{U}}$,
extending the identification of the coarse moduli of $\mathscr{U}_{\eta}$ and
$\underline{\mathscr{U}}|_{\eta}$ coming from
$\alpha_{\eta}|_{\underline{\mathscr{U}}}$,

  3. (3)

the map $\underline{\mathscr{U}}\to\mathbb{P}^{N-1}$ induced by the
composition of
$\mathscr{U}\overset{h}{\to}X_{f}\overset{F}{\to}\mathbb{P}^{N-1}$ is equal to
$\underline{h}$.

When $\underline{X}=V/\\!\\!/_{\theta}G$ is projective and $S$ is full,
possibly after finite base change such a triple
$(\mathscr{U},\Sigma^{\mathscr{U}},h)$ always exists.

###### Proof.

First note that once (1) and (2) are satisfied, (3) is automatic since
$\underline{\mathscr{U}}|_{\eta}$ is dense in $\underline{\mathscr{U}}$ and
$\underline{\mathscr{U}}$ is reduced. Let $\underline{X}_{f}$ be the coarse
moduli of $X_{f}$. Let
$\underline{h}^{\prime}_{\eta}:\underline{\mathscr{U}}|_{\eta}\to\underline{X}_{f}$
be the map induced by $h_{\eta}$, and let
$\underline{F}:\underline{X}_{f}\to\mathbb{P}^{N-1}$ be the map induced by
$F$.

Given any $(\mathscr{U},\Sigma^{\mathscr{U}},h)$ satisfying (1) and (2), $h$
induces an

$\underline{h}^{\prime}:\underline{\mathscr{U}}\longrightarrow\underline{X}_{f},$

extending $\underline{h}^{\prime}_{\eta}$. Conversely, given such an
$\underline{h}^{\prime}$ extending $\underline{h}^{\prime}_{\eta}$, possibly
after finite base change there exists a unique
$(\mathscr{U},\Sigma^{\mathscr{U}},h)$ up to unique isomorphism satisfying (1)
and (2) such that $\underline{h}^{\prime}$ is the induced map between coarse
moduli spaces. Indeed, this follows from the proof for [AV02, Proposition
6.0.1]. More precisely, the $f_{1}:C_{1}\to\mathbf{M}$ in [AV02] constructed
in Step 1 is replaced by the given
$(\underline{\mathscr{U}},\Sigma^{\underline{\mathscr{U}}},\underline{h}^{\prime})$.
The remaining steps are constructions that are local on the coarse moduli of
the curves.

To finish the proof, it remains to study the existence and uniqueness of the
$\underline{h}^{\prime}$ extending $\underline{h}^{\prime}_{\eta}$. Since
$\underline{\mathscr{U}}|_{\eta}$ is dense in $\underline{\mathscr{U}}$,
$\underline{\mathscr{U}}$ is reduced, and $\underline{X}_{f}$ is separated,
$\underline{h}^{\prime}$ is unique if it exists. When $\underline{X}$ is
projective and $S$ is full, $\underline{X}_{f}=\underline{X}$ and
$\underline{F}$ is a closed embedding. By continuity the given map
$\underline{\mathscr{U}}\to\mathbb{P}^{N-1}$ factors through
$\underline{X}_{f}$. Hence in this case $\underline{h}^{\prime}$ always
exists. This completes the proof. ∎

##  Appendix A Comparison with old stability condition

In this appendix, we compare the $\Omega$-stability condition with the
stability condition in [CLLL19] for the Fermat quintic. Recall the setup of
prestable MSP fields from Section 2.

######  Lemma A.1.

Let $\frac{1}{5}<A<\frac{2}{5}$. Let

(A.1)
$\xi=(\mathscr{C},\Sigma^{\mathscr{C}},\mathscr{L},\mathscr{N},\varphi,\rho,\mu,\nu)$

be a prestable MSP field such that

$(\varphi,\mu)\neq 0,\quad(\mu,\nu)\neq 0,\quad(\rho,\nu)\neq 0$

hold everywhere on $\mathscr{C}$. Then the automorphism group of $\xi$ is
finite if and only if the $\mathbb{Q}$-line bundle
$\mathscr{L}\otimes\mathscr{N}^{\otimes
2}\otimes(\omega_{\mathscr{C}}^{\log})^{\otimes A}$ has positive degree on
each irreducible component of $\mathscr{C}$.

###### Proof.

By Theorem 2.4, (2) implies that that $\xi$ has finitely many automorphisms.

Conversely, suppose that $\mathscr{L}\otimes\mathscr{N}^{\otimes
2}\otimes(\omega_{\mathscr{C}}^{\log})^{\otimes A}$ has non-positive degree on
some irreducible component. We would like to show that $\mathrm{Aut}(\xi)$ is
infinite. Obviously it suffices to consider the case when $\mathscr{C}$ is
smooth and irreducible and $\mathrm{Aut}(\mathscr{C},\Sigma^{\mathscr{C}})$ is
infinite. Thus from now on we assume

(A.2) $\deg(\mathscr{L}\otimes\mathscr{N}^{\otimes
2}\otimes(\omega_{\mathscr{C}}^{\log})^{\otimes A})\leq 0.$

We make several observations.

  1. (i)

$\deg(\mathscr{L}\otimes\mathscr{N}^{\otimes
2}\otimes(\omega_{\mathscr{C}}^{\log})^{\otimes(1/5)})\leq 0$. Indeed
$\deg(\omega_{\mathscr{C}}^{\log})\in\mathbb{Z}$ and (A.1) implies
$\deg(\mathscr{L}),\deg(\mathscr{N})\in\frac{1}{5}\mathbb{Z}$.

  2. (ii)

$\deg(\mathscr{L}\otimes\mathscr{N})\in\mathbb{Z}$. Indeed, suppose
$q\in\mathscr{C}$ is an orbifold point. If $\mathscr{L}\otimes\mathscr{N}$ has
nontrivial monodromy at $q$. Then $\mu(q)=0$. Then $\varphi(q)\neq 0$ and
$\nu(q)\neq 0$. But this implies that both $\mathscr{L}$ and $\mathscr{N}$
have nontrivial monodromy at $q$, a contradiction. Hence
$\mathscr{L}\otimes\mathscr{N}$ has trivial monodromy at every orbifold point
of $\mathscr{C}$. Thus its degree must be an integer.

  3. (iii)

$\deg(\mathscr{L}\otimes\mathscr{N})\geq 0$. Indeed, if
$\deg(\mathscr{L}\otimes\mathscr{N})<0$, then $\mu\equiv 0$. Thus $\nu$ is
nonvanishing and $\mathscr{N}\cong\mathcal{O}_{\mathscr{C}}$. Then we have
$\deg(\mathscr{L})<0$, and thus $\varphi_{i}\equiv 0$ for all $i$. This
contradicts to the assumption that $(\varphi,\mu)$ is nonvanishing.

  4. (iv)

$\deg(\mathscr{N})\geq 0$. Indeed, if $\deg(\mathscr{N})<0$, then $\nu\equiv
0$ and $\rho$ is nonvanishing. Since
$\mathrm{Aut}(\mathscr{C},\Sigma^{\mathscr{C}})$ is infinite, this implies
that $\deg(\mathscr{L})=\frac{1}{5}\deg(\omega_{\mathscr{C}}^{\log})\leq 0$.
This contradicts to (iii).

  5. (v)

$\deg(\mathscr{L}\otimes\mathscr{N})=0$. Indeed, otherwise by (ii) and (iii)
we must have $\deg(\mathscr{L}\otimes\mathscr{N})\geq 1$. Then using (i) and
$\deg(\omega^{\log}_{\mathscr{C}})\geq-2$, we must have $\deg(\mathscr{N})<0$,
contradicting to (iv).

Since $\mathrm{Aut}(\mathscr{C},\Sigma^{\mathscr{C}})$ is infinite, we are in
one of the following three cases:

#### Case 1: $\deg(\omega_{\mathscr{C}}^{\log})=0$

In this case, $\omega_{\mathscr{C}}^{\log}$ is trivial and any trivialization
is fixed by $\mathrm{Aut}(\mathscr{C},\Sigma^{\mathscr{C}})$. By (iv) and (v)
above, (A.2) implies $\deg(\mathscr{L})=\deg(\mathscr{N})=0$. Thus each field
is either nonvanishing or identically zero. Hence it is easy to see that $\xi$
has infinitely many automorphisms.

#### Case 2: $\deg(\omega_{\mathscr{C}}^{\log})<0$ and
$\deg(\mathscr{L}^{\otimes(-5)}\otimes\omega_{\mathscr{C}}^{\log})\geq 0$

Combining (i) and (v), we have
$\deg(\mathscr{L}^{\otimes(-5)}\otimes\omega_{\mathscr{C}}^{\log})=0$. Then
$\deg(\mathscr{L})\not\in\mathbb{Z}$, and hence
$\Sigma^{\mathscr{C}}\neq\emptyset$. Then $\Sigma^{\mathscr{C}}$ is a single
orbifold point and $\deg(\mathscr{L})=-\deg(\mathscr{N})=-\frac{1}{5}$, using
(v). Thus $\varphi\equiv 0$ and $\mu$ is an isomorphism
$\mathscr{L}\otimes\mathscr{N}\cong\mathcal{O}_{\mathscr{C}}$. This way, $\xi$
is equivalent to the datum

$(\mathscr{C},\Sigma^{\mathscr{C}},\mathscr{L},\rho,\nu),$

where $\rho\in
H^{0}(\mathscr{C},\mathscr{L}^{\otimes(-5)}\otimes\omega_{\mathscr{C}}^{\log})$
and $\nu\in H^{0}(\mathscr{C},\mathscr{L}^{\otimes(-1)})$. Note that for
degree reason, $\nu$ must have some zeros and $\rho$ is nonvanishing. If
$\nu\equiv 0$, then any automorphism of $(\mathscr{C},\Sigma^{\mathscr{C}})$
lifts to an automorphism of $\xi$. Otherwise, it is easy to see that an
automorphism of $(\mathscr{C},\Sigma^{\mathscr{C}})$ lifts to an automorphism
of $\xi$ if and only if it fixes the rational section $\rho/\nu^{5}$ of
$\omega_{\mathscr{C}}^{\log}$. Choose coordinate $z$ on the coarse moduli of
$\mathscr{C}$, which is isomorphic to $\mathbb{P}^{1}$, such that the image of
$\Sigma^{\mathscr{C}}$ is at infinity. Then $\rho/\nu^{5}$ is a constant
multiple of the pullback of $\operatorname{d}\\!z$. Hence it is left invariant
by any automorphism of $\mathscr{C}$ that induces $z\mapsto
z+a,a\in\mathbb{C}$ on the coarse moduli. Hence $\mathrm{Aut}(\xi)$ is always
infinite.

#### Case 3:
$\deg(\omega_{\mathscr{C}}^{\log})<0,\deg(\mathscr{L}^{\otimes(-5)}\otimes\omega_{\mathscr{C}}^{\log})<0$

Thus $\rho\equiv 0$ and $\nu$ is nonvanishing. Thus
$\mathscr{N}\cong\mathcal{O}_{\mathscr{C}}$ and $\xi$ is equivalent to

$(\mathscr{C},\Sigma^{\mathscr{C}},\mathscr{L},\varphi_{1},\ldots,\varphi_{5},\mu).$

This is equivalent to a map to $\mathbb{P}^{5}$ of degree
$\deg(\mathscr{L})\geq 0$. But by (iv) and (v) we have $\deg(\mathscr{L})\leq
0$. Hence the map has degree zero. Hence $\mathrm{Aut}(\xi)$ is infinite. ∎

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# Recency Dropout for Recurrent Recommender Systems

Bo Chang, Can Xu, Matthieu Lê, Jingchen Feng, Ya Le, Sriraj Badam, Ed Chi,
Minmin Chen  Google, Inc.
Mountain View, CA, USA
{bochang, canxu, matthieule, jingchenfeng, elainele, srirajdutt, edchi,
<EMAIL_ADDRESS>
###### Abstract.

Recurrent recommender systems have been successful in capturing the temporal
dynamics in users’ activity trajectories. However, recurrent neural networks
(RNNs) are known to have difficulty learning long-term dependencies. As a
consequence, RNN-based recommender systems tend to overly focus on short-term
user interests. This is referred to as the recency bias, which could
negatively affect the long-term user experience as well as the health of the
ecosystem. In this paper, we introduce the recency dropout technique, a simple
yet effective data augmentation technique to alleviate the recency bias in
recurrent recommender systems. We demonstrate the effectiveness of recency
dropout in various experimental settings including a simulation study, offline
experiments, as well as live experiments on a large-scale industrial
recommendation platform.

††copyright: none††conference: ; ††booktitle: ;

##  1\. Introduction

With the ever-expanding corpus of contents across the web applications,
recommender systems are increasingly relied upon to help users find the needle
in the haystack—to discover the small fraction of items that match users’
areas of interest. The user-item interactions on a platform are naturally
sequentially dependent; what the users are interested in next is highly
dependent on what they have consumed in the past. In order to surface the
right item to the right user at the right time, recommender systems need to
have a holistic understanding of the users’ areas of interest and preferences,
often based on their past activities on the platform. Incorporating sequential
information has been shown to improve the performance of recommender systems
(Koenigstein et al., 2011; Campos et al., 2014; Koren, 2009; He and McAuley,
2016; Tang and Wang, 2018; Kang and McAuley, 2018; Chen et al., 2018b; Yuan et
al., 2019; Garg et al., 2019). In particular, recurrent neural networks (RNNs)
are widely used to capture the temporal dynamics in users’ activity
trajectories (Hidasi et al., 2015; Beutel et al., 2018; Wu et al., 2017; Tang
et al., 2019; Chen et al., 2019).

Despite the successes of recurrent recommender systems, RNNs are known to have
difficulty capturing long-term dependencies in the input sequence. It often
fails to utilize input information far into the past due to the vanishing and
exploding gradient when trained using backpropagation through time (BPTT)
(Pascanu et al., 2013; Bengio et al., 1994). A long line of research has been
dedicated to alleviating this issue using techniques such as special
initialization schemes and model architecture changes (Hochreiter and
Schmidhuber, 1997; Cho et al., 2014; Le et al., 2015; Chen et al., 2018a;
Arjovsky et al., 2016; Wisdom et al., 2016; Vorontsov et al., 2017; Chang et
al., 2019).

The lack of long-term dependencies prompts an RNN-based recommender system to
rely heavily on items a user recently interacted with for recommendation and
further recommend items that are similar to them. In other words, the system
overly focuses on users’ short-term interests and ignores the long-term ones;
this is referred to as the _recency bias_ in this paper. The bias can cause
the recommender systems to pigeonhole users to their most recent interests.

In this paper, we introduce a data augmentation technique tailored to
recurrent recommender systems to better surface users’ long-term interests.
Data augmentation techniques are widely used in various machine learning
fields (Shorten and Khoshgoftaar, 2019; Fadaee et al., 2017; Kobayashi, 2018).
Many of them perform some form of random transformations on the input data.
For example, for image data, the transformations could include random scaling,
rotation, flipping, etc. The extra noise introduced to the input data acts as
regularization and makes the model generalize better.

We propose a simple yet effective data augmentation technique named _recency
dropout_. It removes the most recent user activities from the input sequence
to the recurrent recommender system. This in turn forces the model to make use
of long-term user interests in the data and mitigates issues caused by the
recency bias.

The main contributions of this work include:

  * •

We quantitatively study the recency bias in recurrent recommender systems,
focusing on a REINFORCE recommender system.

  * •

We propose recency dropout, a simple yet effective data augmentation
technique, to alleviate the recency bias in recurrent recommender systems.

  * •

Applying the Jacobian analysis, which is used to study the long-term
dependencies for RNNs, we illustrate that with recency dropout, the training
of RNN becomes better conditioned and the vanishing gradient issue is
alleviated.

  * •

We demonstrate the benefits of the recency dropout technique in large-scale
live experiments on a commercial recommendation platform serving billions of
users and millions of items.

  * •

Using the number of daily active users on the platform as a holistic
measurement of improved user experience on the platform, we observe that
capturing long-term user interests leads to improved long-term user
experience.

##  2\. Related Work

###  2.1. Data Augmentation

Data augmentation techniques are widely used in various machine learning
fields to improve the performance of the model. These techniques often
artificially enlarge the training dataset by performing random transformations
on the input data and act as a form of regularization to alleviate overfitting
and improve generalization of the model.

Take image classification tasks in computer vision as an example. Before being
fed into a neural network, the input images are often transformed by, for
instance, horizontally flipping, random cropping, random tilting, and altering
the brightness to create multiple instantiations (Shorten and Khoshgoftaar,
2019). Data augmentations are also commonly used in other fields, such as word
substitution in natural language processing (Fadaee et al., 2017; Kobayashi,
2018), and randomly masking blocks of frequency channels and blocks of time
steps in speech recognition (Park et al., 2019).

###  2.2. Long-Term Dependencies

Modeling long-term dependencies using RNNs is challenging; the main difficulty
arises as the gradient backpropagated through time (BPTT) suffers from
exponential growth or decay, a dilemma commonly referred to as the exploding
or vanishing gradient (Bengio et al., 1994; Pascanu et al., 2013).

It has been a long-standing research topic to capture long-term dependencies
using RNNs, and various approaches have been proposed. The gating mechanism is
designed to facilitate gradient propagation and is applied in long short-term
memory networks (LSTM) (Hochreiter and Schmidhuber, 1997) and gated recurrent
units (GRU) (Cho et al., 2014). These models however can still suffer from the
same problem of not being able to provably account for long-range dependent
patterns in sequences (Belletti et al., 2018).

Identity and orthogonal initialization is another proposed solution to the
exploding or vanishing gradient problem (Le et al., 2015; Chen et al., 2018a).
Arjovsky et al. (2016) advocate going beyond initialization and forcing the
weight matrices to be orthogonal throughout the entire learning process
(Wisdom et al., 2016; Vorontsov et al., 2017). However, orthogonal weight
matrices alone do not prevent exploding and vanishing gradients, due to the
nonlinear nature of deep neural networks (Pennington et al., 2017; Gilboa et
al., 2019).

Different from RNNs, the attention and self-attention mechanism can
effectively capture long-term dependencies by focusing on the relevant part of
the sequence and computing a weighted average of the input (Bahdanau et al.,
2015; Vaswani et al., 2017). They have been successfully applied to sequential
recommender systems (Ying et al., 2018; Kang and McAuley, 2018; Tang et al.,
2019). They, however, do present significant challenges to be utilized in
industrial recommender systems with long user histories due to the increased
serving cost in the order of $O(M^{2})$ compared to $O(M)$ for RNNs, where $M$
is the length of the input sequence.

###  2.3. Calibration and Diversity of Recommender Systems

Recommender systems are known to be subject to a strong feedback loop effect;
as the system makes recommendations based heavily on users’ recent activities,
the user feedback in turn further reinforces the system’s recency bias. This,
as a consequence, can narrow down users’ areas of interest and prevent them
from exploring other contents available on the platform that are of potential
interest. Pigeonholing users to a narrow set of interests can negatively
affect the users, creating echo chambers or filter bubbles, as well as the
ecosystem, missing the opportunities to surface contents from certain
providers (Nguyen et al., 2014; Pardos and Jiang, 2020; Celis et al., 2019).

To mitigate this effect, the notion of calibrated recommendation is proposed,
which encourages the distribution of genres or popularity of the recommended
items to be calibrated with that of a user’s past activities (Steck, 2018;
Kaya and Bridge, 2019; Zhao et al., 2021). On a high level, it ensures the
various areas of interest of a user are reflected proportionally according to
his/her historical activities and avoids over-amplifying the main or popular
areas of interest of the user.

Another direction is to promote diversity of the recommendation (Lathia et
al., 2010; Ziegler et al., 2005; Vargas et al., 2011), using methods including
determinantal point processes (Chen et al., 2018c; Gartrell et al., 2016;
Wilhelm et al., 2018) and submodular optimization (Ashkan et al., 2014, 2015;
Qin and Zhu, 2013). See Kunaver and Požrl (2017) and reference therein for a
survey on this research direction.

##  3\. Background

In this section, we give an overview of the REINFORCE recommender system (Chen
et al., 2019), and examine the recency bias of the system.

###  3.1. REINFORCE Recommender System

Figure 1. A diagram illustrating the REINFORCE recommender system and the
recency dropout technique. The item features of an input sequence
$(a_{1},\ldots,a_{t-1})$ are passed to an RNN, the last hidden state of which
is used as the latent user state representation $s_{t}$. Together with the
label context, a softmax is applied to obtain the policy
$\pi_{\theta}(a_{t}|s_{t})$. When recency dropout is applied, the most recent
$N$ items are removed from the input sequence, as shown in the grey box in the
figure ($N=3$ in this illustrative example). As a result, we use a truncated
user activity history $(a_{1},\ldots,a_{t-N-1})$ to construct the policy
$\pi_{\theta}(a_{t}|s_{t-N})$, as denoted by the dashed arrow. This encourages
the model to make better use of long-term user interests and alleviates the
recency bias.

Let ${\mathcal{I}}$ be the item corpus on the platform. We consider the
following sequential recommendation problem: at serving time, given a sequence
of a user’s historical activities on the platform
${\mathcal{H}}_{1:(t-1)}=(a_{1},a_{2},\ldots,a_{t-1})$, where each item
$a_{i}\in{\mathcal{I}}$ is the one that the user interacted with at time $i$,
the goal is to recommend a set of items to the user. In particular, we return
a distribution $\pi\left(a_{t}\middle|{\mathcal{H}}_{1:(t-1)}\right)$
supported on ${\mathcal{I}}$; the set of recommended items are generated from
this distribution.

We consider the REINFORCE recommender system (Chen et al., 2019), where the
recommendation problem is translated into a Markov decision process (MDP):
$\left({\mathcal{S}},{\mathcal{A}},{\bm{P}},R,\rho_{0},\gamma\right)$. Here
${\mathcal{S}}$ is a continuous latent state space describing the user state
and context, ${\mathcal{A}}$ is a discrete action space containing items
available for recommendation,
${\bm{P}}:{\mathcal{S}}\times{\mathcal{A}}\times{\mathcal{S}}\rightarrow[0,1]$
is the state transition probability,
$R:{\mathcal{S}}\times{\mathcal{A}}\rightarrow\mathbb{R}$ is the reward
function, where $r(s,a)$ is the immediate reward obtained by performing action
$a$ at state $s$, $\rho_{0}$ is the initial state distribution, and
$\gamma\geq 0$ is the discount factor for future rewards.

A recommender agent is built by parameterizing and learning a softmax policy

(1)
$\pi_{\theta}(a|s)=\frac{\exp(s^{\top}v_{a}/T)}{\sum_{a^{\prime}\in{\mathcal{A}}}\exp(s^{\top}v_{a^{\prime}}/T)},$

which defines a distribution over the action space ${\mathcal{A}}$
conditioning on the user state $s\in{\mathcal{S}}$. In our case, the action
space is the item corpus ${\mathcal{A}}={\mathcal{I}}$. Here, $v_{a}$ denotes
the learned representation for item $a$, and $T>0$ is the temperature that
adjusts the entropy of the learned policy. The policy parameters $\theta$ are
learned using REINFORCE (Williams, 1992) so as to maximize the expected
cumulative reward over the interaction trajectories,

(2) $\max_{\theta}\mathbb{E}_{\tau\sim\pi_{\theta}}\left[R(\tau)\right],\mbox{
where }R(\tau)=\sum_{t=0}^{|\tau|}r(s_{t},a_{t}).$

Note that the expectation is taken over the trajectories of
$\tau=(s_{0},a_{0},s_{1},\ldots)$ obtained by acting according to the latest
learned policy: $s_{0}\sim\rho_{0}$, $a_{t}\sim\pi_{\theta}(\cdot|s_{t})$, and
$s_{t+1}\sim{\bm{P}}(\cdot|s_{t},a_{t})$.

The recommender agent uses a recurrent neural network (RNN) to model the state
transition, i.e., the transition of latent state $s_{i}$ to $s_{i+1}$ after
taking the action of $a_{i}$. Therefore, by rolling out the RNN steps, the
learned latent state representation at time $t$ can be written as

(3)
$s_{t}=f_{\theta}^{\mathrm{RNN}}\left({\mathcal{H}}_{1:(t-1)}\right)=f_{\theta}^{\mathrm{RNN}}\left(a_{1},\ldots,a_{t-1}\right).$

Figure 1 shows a diagram of the model architecture of the REINFORCE
recommender system.

The algorithm is adapted to the offline (batch) training setup, commonly seen
in industrial recommender systems, by applying a technique known as the off-
policy correction. In particular, the training data ${\mathcal{D}}$ consists
of users’ historical activities in the format of
$\left({\mathcal{H}}_{1:(t-1)},a_{t},R_{t}\right)$ tuples, where $a_{t}$ is
the item consumed by the user at time $t$, and $R_{t}$ is the discounted
cumulative reward associated with $a_{t}$. The loss function can be written as

(4)
${\mathcal{L}}=\frac{1}{|{\mathcal{D}}|}\sum_{\left({\mathcal{H}}_{1:(t-1)},a_{t},R_{t}\right)\in{\mathcal{D}}}L\left({\mathcal{H}}_{1:(t-1)},a_{t},R_{t}\right),$

where for each tuple $\left({\mathcal{H}}_{1:(t-1)},a_{t},R_{t}\right)$, the
loss is

(5)
$L\left({\mathcal{H}}_{1:(t-1)},a_{t},R_{t}\right)=\frac{\pi_{\theta}(a_{t}|s_{t})}{\beta(a_{t}|s_{t})}R_{t}\log\pi_{\theta}(a_{t}|s_{t}).$

The policy $\pi_{\theta}$ is defined in Equation 1 and the state
representation $s_{t}$ is computed according to Equation 3. The behavior
policy $\beta(a|s)$ is an estimate of the historical policy, and the ratio of
$\pi_{\theta}(a|s)/\beta(a|s)$ is known as the importance weight, which is
used to address the data bias caused by the mismatch between the updated
policy $\pi_{\theta}$ and the historical policy $\beta$ that generated the
training data. We refer the interested readers to Chen et al. (2019) for more
details about the REINFORCE recommender system.

###  3.2. Recency Bias

The latent state representation is encoded by an RNN as in Equation 3, which
is known to have difficulty capturing long-term dependencies in the input
sequence. As a result, _the recommender system tends to forget about users’
activities long in the past and promote contents that are more similar to what
a user has recently consumed_. We refer to this as the _recency bias_ of the
recommender system.

To quantitatively study the recency bias of the REINFORCE recommender system,
we examine the degree of “similarity” between the policy
$\pi_{\theta}(\cdot|s_{t})$ and $a_{t-k}$, the item that the user interacted
with $k$ steps ago. We here measure similarity by the providers and topic
clusters111The topic cluster for each item is produced by: 1) taking the item
co-occurrence matrix, where the $(i,j)$-th entry counts the number of times
item $i$ and $j$ were interacted by the same user consecutively; 2) performing
matrix factorization to generate an embedding for each item; 3) using
$k$-means to cluster the learned embeddings into $10,000$ clusters; 4)
assigning the nearest cluster to each item.  of the items, assuming items from
the same provider or topic cluster are more similar. More concretely, we
investigate the probability mass assigned by the policy
$\pi_{\theta}(\cdot|s_{t})$ on items that are uploaded by the same provider of
$a_{t-k}$; in other words, we study the following quantity

(6) $d_{\mathrm{provider}}(k)\\\
=\sum_{a\in{\mathcal{I}}}\pi_{\theta}(a|s_{t})\mathbb{I}\\{a\text{ and
}a_{t-k}\text{ are from the same provider}\\}$

as a function of the time difference $k\geq 1$, where $\mathbb{I}$ is the
indicator function. We can similarly define $d_{\mathrm{cluster}}(k)$ based on
the topic cluster. If the functions $d_{\mathrm{provider}}(k)$ and
$d_{\mathrm{cluster}}(k)$ quickly decay to zero as $k$ increases, it indicates
that the recommender system is strongly recency biased; on the other hand, if
the functions are relatively flat, it means that the system is capable of
capturing users’ short-term as well as long-term interests.

In Figures 2(a) and 2(b), the magenta curves correspond to
$d_{\mathrm{provider}}(k)$ and $d_{\mathrm{cluster}}(k)$, respectively. It
shows that $d_{\mathrm{provider}}(1)\approx 0.06$ and
$d_{\mathrm{provider}}(k)<0.02$ for $k>100$; that is, there is a 6%
probability that an item that is similar to what a user most recently
interacted with, whereas the probability drops to 2% for a past item more than
100 time-steps ago. The pattern of $d_{\mathrm{cluster}}(k)$ is similar. This
indicates that there is indeed recency bias in the REINFORCE recommender
system. We will revisit this figure in Section 6.1 to discuss the effect of
applying recency dropout.

(a) $d_{\mathrm{provider}}(k)$ as a function of $k$.

(b) $d_{\mathrm{cluster}}(k)$ as a function of $k$.

Figure 2. Recency bias in the REINFORCE recommender system. It shows the
probability mass that the policy $\pi_{\theta}(\cdot|s_{t})$ assigns on items
from the same provider or cluster of $a_{t-k}$. On the x-axis is the time
difference $k$ on a log scale; on the y-axis are $d_{\mathrm{provider}}(k)$
and $d_{\mathrm{cluster}}(k)$ defined in Equation 6.

##  4\. Recency Dropout

In this section, we introduce a simple data augmentation or regularization
technique named _recency dropout_. It encourages the model to better capture
long-term dependencies in users’ areas of interest to alleviate the recency
bias.

Motivated by other data augmentation techniques discussed in Section 2.1, the
proposed method performs random alterations to the input data. _During
training, we remove the most recent $N$ items from the user’s activity
history_, where $N$ is a random variable drawn from a distribution supported
on non-negative integers $\mathbb{Z}_{\geq 0}$.

Recall that the input data are tuples
$\left({\mathcal{H}}_{1:(t-1)},a_{t},R_{t}\right)$, representing a user’s
activity history up to time $t$, next action at time $t$, and the associated
discounted cumulative reward. With the proposed recency dropout technique,
instead of minimizing $L\left({\mathcal{H}}_{1:(t-1)},a_{t},R_{t}\right)$
defined in Equation 5, we minimize
$L\left({\mathcal{H}}_{1:(t-N-1)},a_{t},R_{t}\right)$. In other words, part of
the user’s activity history $(a_{t-N},\dots,a_{t-1})$ is ablated from the
training data. This is similar to randomly masking out blocks of the input
data proposed by Park et al. (2019). Figure 1 illustrates the idea of recency
dropout on the REINFORCE recommender system.

Conceptually, we are asking the model to make a recommendation of $a_{t}$
without using the most recent information, which forces the recommender system
to better leverage “older” activities ${\mathcal{H}}_{1:(t-N-1)}$ instead of
relying solely on the most recent ones.

The number of activities to drop $N$ could be sampled from a variety of
discrete distributions. For example, $N$ could follow a discrete uniform
distribution on $[N_{\min},N_{\max}]$. Other discrete distributions are also
valid choices, including the Poisson distribution, negative binomial
distribution, hypergeometric distribution, etc. We also considered a
degenerate distribution as a special case; i.e., $N$ equals a fixed value
$N_{\mathrm{fixed}}$ almost surely.

In the same vein with other data augmentation techniques, recency dropout is
only applied during training. At serving time, the full user activity history
${\mathcal{H}}_{1:(t-1)}$ is used as the input to the model.

Note that the proposed recency dropout method should not be confused with the
dropout regularization (Srivastava et al., 2014) that randomly removes hidden
units of a neural network during training, and methods that apply the dropout
regularization to RNNs (Gal and Ghahramani, 2016; Semeniuta et al., 2016).

##  5\. Simulation Study

Simulation studies have often been used as a tool to evaluate information
retrieval and recommender systems (Yao et al., 2021; Szlávik et al., 2011;
Carterette et al., 2011; Diaz and Arguello, 2009). To better illustrate the
recency bias in recurrent recommender systems, we design and present a
simulation study in this section.

The purpose of the simulation is to construct a setting where users have long-
term areas of interest that span uniformly among topic clusters, yet a
recurrent recommender system tends to focus on recent or short-term interests.
Furthermore, we demonstrate how recency dropout effectively mitigates this
issue.

###  5.1. Simulation Setup

We notice that in the user activity history we collected on an industrial
recommendation platform, consecutive items tend to be more similar: there is a
30% probability that $a_{t-1}$ and $a_{t}$ are in the same topic cluster and a
26% probability that them are uploaded by the same provider, where $a_{t-1}$
and $a_{t}$ are the items a user interacted with consecutively at time $t-1$
and $t$. Therefore, we design a generative model that simulates sequences of
user activities that reflect this property; to that end, we use a Markov chain
with a structured transition matrix as the generative model. We further
illustrate that when the learning aims at recovering a user’s next interaction
based on the activity history, such as the learning objective described in
equation 5, the recency bias naturally arises.

Let us consider a set of items ${\mathcal{I}}$, which can be partitioned into
$K$ clusters ${\mathcal{C}}_{k}$ for $k=1,\ldots,K$, such that
${\mathcal{I}}=\bigcup_{k=1}^{K}{\mathcal{C}}_{k}$ and
${\mathcal{C}}_{k}\cap{\mathcal{C}}_{k^{\prime}}=\emptyset$ if $k\neq
k^{\prime}$. In other words, each item is associated with a unique cluster.
Conceptually, the clusters could represent a notion of users’ areas of
interest, such as topics, genres, content providers, etc. We set
${\mathcal{I}}=\\{0,\ldots,99\\}$ and there are $K=10$ clusters, each has the
same number of items. Without loss of generality, we assume
${\mathcal{C}}_{1}=\\{0,\ldots,9\\},{\mathcal{C}}_{2}=\\{10,\ldots,19\\},\ldots,{\mathcal{C}}_{10}=\\{90,\ldots,99\\}$.
We further assume a user’s activities follow a discrete-time Markov chain. The
transition matrix is denoted by
${\bm{P}}=(p_{ij})\in[0,1]^{|{\mathcal{I}}|\times|{\mathcal{I}}|}$, where
$p_{ij}$ describes the probability that the user will interact with item $j$
next given he/she just interacted with item $i$. The transition matrix is
designed such that consecutive interactions tend to be on items from the same
cluster. In particular, the $(i,j)$-th entry of the transition matrix
${\bm{P}}$ is

(7) $p_{ij}=\left\\{\begin{array}[]{ll}0.7/10=0.07&\text{ if $i$ and $j$ are
in the same cluster;}\\\ 0.3/90=0.0033&\text{ otherwise. }\end{array}\right.$

It holds that $\sum_{j\in{\mathcal{I}}}p_{ij}=1$ for all $i\in{\mathcal{I}}$.
Note that, for simplicity, we assume a user can repeatedly interact with the
same item, which is a valid assumption for some (e.g., music streaming or
e-commerce) but not all platforms.

At any time step, the probability that a user stays in the same cluster is
$0.7$ and that a user transits to a new cluster is $0.3$. The number of steps
it takes for a user to transit to a new cluster follows a geometric
distribution with $p=0.3$, the expectation of which is $1/0.3\approx 3.3$.
Because of the symmetry of the transition matrix, its stationary distribution
is a uniform distribution among the item set ${\mathcal{I}}$, which can be
regarded as the “long-term” user interests. On the other hand, the most recent
item in the activity history can be thought of as the “short-term” user
interests. For the next item prediction task, it is apparent that the “short-
term” user interests are more predictive of the user’s next interaction. The
proposed recency dropout technique instead encourages the recommender system
to recover the “long-term” user interests as well. As we will show in the
experiment section, capturing long-term user interests leads to improved long-
term user experience.

###  5.2. Effects of Recency Dropout

We generate sequences under the simulation setup described above; each has 100
items. A simple sequence model is trained on the generated sequences to
predict the last item in the sequence given the 99 interactions before that:
the model first performs an embedding lookup for items in the input sequence,
then passes the embeddings to a GRU followed by two hidden ReLU layers;
finally, softmax is applied to obtain the predictive distribution over
${\mathcal{I}}$. We train the model for 500 steps with a batch size of 128. A
separate batch of 1,000 sequences is also generated for evaluation.

We first qualitatively study the effect of recency dropout. Figure 3(a) shows
an illustrative example of the prediction of the baseline model on 10
evaluation sequences: the predictive distribution is highly concentrated on
items in the same cluster as the last item. As a reference, the last items of
the sequences in the batch are $(46,12,4,92,17,90,52,74,64,9)$. For example,
the first row has the predictive distribution concentrated on items in
$[40,50)$, which are in the same cluster as item 46. In comparison, we also
train a model with recency dropout with $N\sim{\mathcal{U}}[0,5)$. Figure 3(b)
shows the predictive distribution. It is clear that the distribution becomes
flatter and closer to the stationary distribution (uniform) of the Markov
chain, thus better represents users’ long-term interests. Meanwhile, it still
reflects the short-term interests, but not as strongly as the baseline model.

(a) Predictive distribution for the baseline model.

(b) Predictive distribution with recency dropout.

Figure 3. Predictive distributions on simulated data. It shows that the
predictive distribution of the baseline model is highly concentrated on items
in the same cluster as the last item in the activity history. With recency
dropout, the distribution becomes flatter and closer the stationary
distribution of the Markov chain, thus better represents users’ long-term
interests.

We also provide a quantitative analysis of the effects of recency dropout. Two
variants are considered: (1) random dropout: $N$ follows a discrete uniform
distribution ${\mathcal{U}}[0,N_{\max}]$; (2) fixed dropout: $N$ takes a fixed
value $N_{\mathrm{fixed}}$. In order to make the two variants comparable, we
report the expected number of dropout for both, that is,
$\mathbb{E}(N)=N_{\max}/2=N_{\mathrm{fixed}}$. The evaluation metrics in
consideration are as follows:

  1. (1)

the mean average precision at 1 (mAP@1);

  2. (2)

the mean average precision at 10 (mAP@10);

  3. (3)

the entropy of the predictive distribution;

  4. (4)

the Kullback–Leibler (KL) divergence between the stationary distribution
(uniform) and the predictive distribution.

Metrics (1) and (2) measure the accuracy of the recommendation; metric (3)
indicates the diversity of the predictive distribution; metric (4) measures
the “calibration” between the predictive distribution and the users’ long-term
interests (Steck, 2018).

(a) mAP@1.

(b) mAP@10.

(c) Entropy.

(d) KL divergence.

Figure 4. Evaluation of two variants of recency dropout on the simulated
dataset. The blue curves correspond to the random dropout variant, and red
curves correspond to the fixed dropout variant. On the x-axis is the expected
number of dropout $\mathbb{E}(N)=N_{\max}/2=N_{\mathrm{fixed}}$, and
$\mathbb{E}(N)=0$ corresponds to the baseline model without using recency
dropout.

We evaluate models trained with various $\mathbb{E}(N)$ values; each
configuration is trained 10 times with different random seeds. The evaluation
batch has 1,000 samples, and the mean and standard error of each metric is
reported as a function of $\mathbb{E}(N)$. The model with $\mathbb{E}(N)=0$ is
the baseline model without using recency dropout. Figure 4 shows the results
of the quantitative analysis: blue curves correspond to the random dropout
variant, and red curves correspond to the fixed dropout variant. The two
variants have similar trends in all metrics. Compared with the baseline model,
the recency dropout models have slightly worse mAP when the expected number of
dropout is small; when it is large, the decrease in mAP becomes more
prominent. Meanwhile, diversity and calibration are improved as the number of
dropout increases. This indicates that the model has better coverage of users’
long-term interests and recommends a more diverse set of items. In summary,
recency dropout improves diversity and calibration, without hurting much of
mAP when the number of dropout is relatively small. One can see as the number
of fixed dropout $N_{\mathrm{fixed}}$ reaches $4$ or $5$, the KL divergence
shown in Figure 4(d) reaches 0, suggesting the model recovers the stationary
distribution and fully captures users’ long-term interests. As explained in
the simulation setup, the expected steps it takes for a user to interact with
a new cluster is $3.3$, which explains the observed results on the KL
divergence.

Another observation is that, the fixed dropout variant (blue curves in Figures
4(a) and 4(b)) has lower mAP than random dropout (red curves). We argue that
this is because the task of fixed dropout is harder than random dropout. The
red curve in Figure 5 denotes the probability that two items that are $k$
steps apart are in different clusters222This probability can be computed
recursively. Let $p_{k}$ be the probability that two items that are $k$ steps
apart are in the SAME cluster, and $q_{k}:=1-p_{k}$. According to Equation 7,
we have $p_{1}=0.7$ and $q_{1}=0.3$. Furthermore, the following recurrence
relation holds for $k>1$: $p_{k}=p_{k-1}p_{1}+q_{k-1}q_{1}/(K-1)$, where
$K=10$ is the total number of clusters. Figure 5 shows $q_{k}$ as a function
of $k$., as a function of $k$. This can be thought of as the “difficulty” of
the task when the most recent $k$ items are removed from the input sequence.
It is evident that this function is concave. Using Jensen’s inequality, on
average, fixed dropout is more difficult than random dropout.

Figure 5. Comparison of random dropout and fixed dropout. The red curve shows
the probability that two items that are $k$ steps apart are in different
clusters, which can be thought of as a proxy of the difficulty of the task.
When comparing fixed dropout with $N_{\mathrm{fixed}}=5$ against random
dropout with $N\sim{\mathcal{U}}[0,10]$. the former has higher expected
difficulty by Jensen’s inequality, as indicated by the blue stars. For fixed
dropout, the expected difficulty is the point on the curve at
$k=N_{\mathrm{fixed}}=5$, whereas for random dropout, it is the center of mass
of the curve because $N$ is uniformly distributed.

###  5.3. Jacobian Analysis

Improving the trainability of RNN has been a long-standing research topic.
Fundamentally, the difficulty is due to the vanishing and exploding gradient
issue when training the RNN using backpropagation through time (BPTT) (Pascanu
et al., 2013). Let $h_{t}$ be the hidden state of an RNN at time $t$, and
$1\leq t\leq T$ where $T$ is the total number of steps. Training an RNN
requires the computation of the gradient of the loss ${\mathcal{L}}$ with
respect to the hidden state $h_{t}$:

(8) $\frac{\partial{\mathcal{L}}}{\partial
h_{t}}=\frac{\partial{\mathcal{L}}}{\partial h_{T}}\prod_{t\leq
i<T}\frac{\partial h_{i+1}}{\partial h_{i}}.$

Iteratively multiplying $\partial h_{i+1}/\partial h_{i}$ causes the gradient
to be exponentially vanishing if the largest eigenvalue is less than 1;
otherwise, the gradient will exponentially explode. This is known as the
vanishing and exploding gradient issue, which causes RNNs to have difficulty
capturing long-term dependencies in the input sequence. Ideally, we want the
eigenvalues all to be close to 1 to avoid the vanishing or exploding gradient
issue (Chang et al., 2019).

In this section, we study the connection between recency dropout and the
eigenvalues of the Jacobian matrix between RNN hidden states. Intuitively,
recency dropout forces the model to leverage information from a long time ago,
which is unattainable if the RNN has severe vanishing or exploding gradient
issues.

Figure 6. The effect of randomized dropout on mitigating the vanishing
gradient issue of RNN. The x-axis represents the time difference $k$ and the
y-axis denotes the average eigenvalue of the Jacobian matrix between hidden
states that are $k$ steps apart, i.e., $\partial h_{T}/\partial h_{T-k}$,
where $T$ is the total number of steps; in this simulation study, $T=100$. The
grey dashed line represents the value of 1; the randomized dropout model has
eigenvalues that are closer to the grey line, which indicates the vanishing
gradient issue is less severe than the baseline model.

Figure 6 shows the average eigenvalue of the Jacobian matrix $\partial
h_{T}/\partial h_{T-k}$ computed on a batch of 1,000 evaluation sequences, as
a function of $k$. The magenta curve denotes the baseline model and the blue
curve corresponds to a recency dropout model with $N\sim{\mathcal{U}}[0,5)$.
Note that the y-axis is on a log scale; visually the two curves are both close
to linear, indicating that the gradients vanish exponentially. Compared with
the baseline model, the dropout model has eigenvalues closer to 1, suggesting
that recency dropout alleviates the vanishing gradient issue and makes the RNN
better capture long-term dependencies in the input sequences.

##  6\. Offline and Live Experiments

To measure the effectiveness of recency dropout, we evaluate it offline on a
large-scale dataset containing millions of users and tens of millions of
items, with billions of feedback between them. We also verify our approach in
live experiments on a commercial recommendation platform serving billions of
users.

###  6.1. Offline Experiments

We extract hundreds of millions of user trajectories from a commercial
recommendation platform. Each contains a user’s historical activity on the
platform and the reward, as described in Section 3.1. We keep at most 500
historical activities for users with at least one positive interaction. Among
the collected trajectories, 1% are held out for evaluation. The action space
or item corpus contains the most popular 10 million items on the platform. Our
goal is to build a recommender system that chooses the next item from the 10
million corpus in order to maximize the cumulative long-term reward for users.

We first demonstrate that recency dropout alleviates the recency bias in the
REINFORCE recommender system by revisiting Figure 2 in Section 3.2. Recall
that on the y-axis are $d_{\mathrm{provider}}(k)$ and
$d_{\mathrm{cluster}}(k)$, measures of how similar the policy
$\pi_{\theta}(\cdot|s_{t})$ is from item $a_{t-k}$, which the user interacted
with $k$ steps ago, as defined in Equation 6; on the x-axis is $k$, the time
difference. The magenta curves correspond to the baseline model without using
recency dropout, and the blue curves are the recency dropout model with
$N\sim{\mathcal{U}}[0,10)$. Overall, the dropout model has flatter curves than
the baseline model, indicating that the policy captures short-term and long-
term interests more evenly. The difference between the two curves is more
prominent when $k<200$, showing that with recency dropout, the policy no
longer focuses too much on recent user interactions. For $k\geq 200$, the
dropout model has a (slightly) higher value, which implies that more
probability mass is shifted towards longer-term user interests.

For a quantitative evaluation, we use the same set of offline evaluation
metrics as in Section 5.2, that is, the mean average precision mAP@1 and
mAP@10, the entropy, and the KL divergence. Note that the KL divergence is
computed between the distribution of the topic clusters in users’ past
interactions and the topic cluster distribution of the policy. This is to
measure how the recommended topics are calibrated with users’ existing
interests (Steck, 2018).

Similar to the experiment setup in Section 5.2, we also consider the two
variants of recency dropout: random dropout where
$N\sim{\mathcal{U}}[0,N_{\max}]$ and fixed dropout where
$N=N_{\mathrm{fixed}}$. The expected number of dropout is reported
$\mathbb{E}(N)=N_{\max}/2=N_{\mathrm{fixed}}\in\\{0,5,100,200\\}$. Figure 7
summarizes the evaluation metrics as functions of the expected number of
dropout $\mathbb{E}(N)$. The patterns are similar to that of Figure 4. With
recency dropout, the mean average precision becomes lower. This is expected
since the task deviates from the next interaction prediction. Furthermore, the
policy also becomes more diverse, measured by the entropy of the distribution.
Finally, the calibration improves and the policy focuses more on a user’s
long-term interests. The difference in metrics between random dropout and
fixed dropout is also consistent with that in the simulation study, which
suggests that the explanation provided in Figure 5 is plausible.

(a) mAP@1.

(b) mAP@10.

(c) Entropy.

(d) KL divergence.

Figure 7. Offline evaluation of two variants of recency dropout. The blue
curves correspond to the random dropout variant, and the red curves correspond
to the fixed dropout variant. On the x-axis is the expected number of dropout
$\mathbb{E}(N)=N_{\max}/2=N_{\mathrm{fixed}}$, and $\mathbb{E}(N)=0$
corresponds to the baseline model without using recency dropout.

###  6.2. Live Experiments

(a) Overall enjoyment.

(b) Daily active users.

(c) Diversity.

Figure 8. Live experiment results on the homepage. On the x-axis is the date;
on the y-axis is the relative difference in percentage between the experiment
and control.

(a) Overall enjoyment.

(b) Daily active users with more than 2 hours of activities.

(c) Diversity.

Figure 9. Live experiment results on the short-form contents service.

We conduct a series of A/B experiments in a live system serving billions of
users to measure the benefits of the proposed recency dropout technique. The
REINFORCE recommender system is built to retrieve hundreds of candidates from
a corpus of $10$ million items upon each user request. The retrieved
candidates, along with those returned by other sources, are scored and ranked
by a separate ranking system before showing the top results to the user.
Experiments are run for four weeks, during which both the control and
experiment models are trained continuously with new interactions and feedback
being used as training data. We focus our discussion on metrics capturing: (1)
users’ overall enjoyment; (2) the number of daily active users on the
platform; (3) diversity of the user-item interactions, which represents the
number of unique topic clusters the users have interacted with.

We first conduct experiments for recommendations on the homepage (browse page)
of the app or web page, where users are presented with a whole page of items
to choose from. Figure 8 summarizes the live experiment results using recency
dropout with $N\sim{\mathcal{U}}[0,10)$, comparing to a control using the
baseline REINFORCE algorithm during the full four-week experiment period. On
the x-axis is the date, and on the y-axis is the relative difference of a
metric in percentage between the experiment and control. We report the mean
and 95% confidence intervals of the metrics based on the experiment results
during the last 7 days. Relative to the control, the experiment model improves
the overall enjoyment by $+0.12\%$ with a 95% confidence interval of
$(+0.08\%,+0.16\%)$. The number of daily active users also increases by
$+0.05\%\ (+0.00\%,+0.09\%)$, which is known to be a difficult metric to
increase. It indicates an improvement in the long-term user experience. What
is more interesting is that we observe an upward trend in these two metrics
during the experiment phase, as shown in Figure 8, suggesting a user learning
effect, i.e., user states change in response to the recommendation policy.
Finally, consistent with the offline experiment results, the diversity metric
also improves, leading to a $+0.07\%$ increase with a 95% confidence interval
of $(+0.03\%,+0.11\%)$.

We also conduct a second round of experiments on a new service that provides
short-form items to users. Since the contents are much shorter, users are able
to interact with many more items within the time frame they spend on the
platform. To that end, we experiment with much more aggressive dropout; the
experiment model applies recency dropout with $N\sim{\mathcal{U}}[0,100)$. It
is again compared against the baseline REINFORCE recommender system as the
control. Figure 9 shows the live experiment results on the service of short-
form contents. The patterns are similar to that of the experiments on the
landing page. The overall enjoyment on this service improves by $+0.56\%\
(+0.26\%,0.86\%)$; the number of daily active users with more than 2 hours of
activities on this service increases by $+2.95\%\ (+2.03\%,+3.89\%)$; and
diversity improves by $+3.03\%(+2.75\%,+3.30\%)$. It demonstrates that the
benefits of recency dropout are reproducible and transferable across different
services.

##  7\. Conclusion

In this paper, we investigate the recency bias in recurrent recommender
systems, in particular, a REINFORCE recommender system. The recency bias
causes the system to recommend items anchoring towards users’ short-term
interests, ignoring the long-term ones. In order to alleviate the recency bias
and to better reflect users’ long-term interests, we introduce the recency
dropout technique. A simulation study is designed to illustrate the existence
of recency bias and to compare the performance of two variants of recency
dropout models on simulated data. Recency dropout is shown to improve the
diversity and calibration of the policy. The Jacobian analysis also sheds
light on how recency dropout improves RNN gradient propagation by alleviating
the vanishing gradient problem. Similar experiment results are obtained
through large-scale offline experiments as well. Finally, we conduct live
experiments on an industrial recommender platform serving billions of users
and tens of millions of items to verify the benefits and reproducibility of
the proposed technique.

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