Learning mirror maps in policy mirror descent

## Abstract

Abstract Policy Mirror Descent (PMD) is a popular framework in reinforcement learning, serving as a unifying perspective that encompasses numerous algorithms. These algorithms are derived through the selection of a mirror map and enjoy finite-time convergence guarantees. Despite its popularity, the exploration of PMD’s full potential is limited, with the majority of research focusing on a particular mirror map—namely, the negative entropy—which gives rise to the renowned Natural Policy Gradient (NPG) method. It remains uncertain from existing theoretical studies whether the choice of mirror map significantly influences PMD’s efficacy. In our work, we conduct empirical investigations to show that the conventional mirror map choice (NPG) often yields less-than-optimal outcomes across several standard benchmark environments. Using evolutionary strategies, we identify more efficient mirror maps that enhance the performance of PMD. We first focus on a tabular environment, i.e. Grid-World, where we relate existing theoretical bounds with the performance of PMD for a few standard mirror maps and the learned one. We then show that it is possible to learn a mirror map that outperforms the negative entropy in more complex environments, such as the MinAtar suite. Our results suggest that mirror maps generalize well across various environments, raising questions about how to best match a mirror map to an environment’s structure and characteristics.

## 1 Introduction

Policy gradient (PG) methods (Williams & Peng, ; Sutton et al., ; Konda & Tsitsiklis, ; Baxter & Bartlett, ) are some of the most widely-used mechanisms for policy optimization in reinforcement learning (RL). These algorithms are gradient-based methods that optimize over a class of parameterized policies and have become a popular choice for RL problems, both in theory (Kakade, ; Peters & Schaal, ; Bhatnagar et al., ; Schulman et al., ; Mnih et al., ; Schulman et al., ; Lan, ) and in practice Shalev-Shwartz et al. (); Berner et al. (); Ouyang et al. ().

Among PG methods, some of the most successful algorithms are those that employ some form of regularization in their updates, ensuring that the newly updated policy retains some degree of similarity to its predecessor. This principle has been implemented in different ways. For instance, trust region policy optimization (TRPO) Schulman et al. () imposes a Kullback-Leibler divergence Kullback & Leibler () hard constraint for its updates, while proximal policy optimization (PPO) (Schulman et al., ) uses a clipped objective to penalize large updates. A framework that has recently attracted attention and belongs to this heuristic is that of policy mirror descent (PMD) (Tomar et al., ; Lan, ; Xiao, ; Kuba et al., ; Vaswani et al., ; Alfano et al., ), which applies mirror descent Nemirovski & Yudin () to RL to regularize the policy updates.

PMD consists of a wide class of algorithms, each derived by selecting a *mirror map* that introduces distinct regularization characteristics. In recent years, PMD has been investigated through numerical experiments (Tomar et al., ), but it has mainly been analysed from a theoretical perspective. To the best of our knowledge, research has been mostly focused either on the particular case of the negative entropy mirror map, which generates the natural policy gradient (NPG) algorithm (Kakade, ; Agarwal et al., ), or on finding theoretical guarantees for a generic mirror map.

NPG has been proven to converge to the optimal policy, up to a difference in expected return (or *error floor*), in several settings, e.g. using tabular, linear, general parameterization (Agarwal et al., ) or regularizing rewards (Cen et al., ). It has been shown that NPG benefits from implicit regularization (Hu et al., ), that it can exploit optimism (Zanette et al., ; Liu et al., ), and some of its variants have been evaluated in simulations (Vaswani et al., , ).
When considering other specific mirror maps investigated in the RL literature, the Tsallis entropy has been noted for enhancing performance in offline settings (Tomar et al., ) and for offering improved sample efficiency in online settings (Li & Lan, ), when compared to the negative entropy.

There is a substantial body of theoretical research focused on the general case of mirror maps. This research demonstrates that PMD achieves convergence to the optimal policy under the same conditions as NPG, as evidenced by various studies (Xiao, ; Lan, ; Yuan et al., ; Alfano et al., ). Except for the setting where we have access to the true value of the policy, convergence guarantees in stochastic settings are subject to an error floor due to the inherent randomness or bias within the algorithm. In the majority of PMD analyses involving generic mirror maps, both the convergence rate and the error floor show mild dependence on the specific choice of mirror map. Typically, the effect of the mirror map appears explicitly as a multiplicative factor in the convergence rate and it appears implicitly in the error floor. These analyses often rely on *upper bounds*, meaning they may not accurately reflect the algorithms’ actual performance in applications.

In this work, we contribute to the literature with an empirical investigation of PMD, with the objectives of finding a mirror map that consistently outperforms the negative entropy mirror map and of understanding how the theoretical guarantees of PMD relate to simulations. We first consider a set of tabular environments, i.e. Grid-World (Oh et al., ), where we compare the empirical results to the theoretical guarantees given by Xiao (), which we can compute as we have full control over the environment. In this setting, we provide a learned mirror map which outperforms the negative entropy in all tested environments. Our experiments suggest that the error floor appearing in prototypical PMD convergence guarantees is not a good performance indicator: our learned mirror map achieves the best value while presenting the worst theoretical error floor, implying that the *upper* bounds typically considered in the PMD literature are loose with respect to the choice of the mirror map. Additionally, our experiments indicate that having small policy updates leads to smoother value improvements over time with less instances of performance degradation, as suggested by the monotonic policy improvement property given by Xiao (). We then consider two non-tabular settings, i.e. the Basic Control Suite and the MinAtar Suite, which are more realistic but also more complex, and therefore prevent us from computing the exact theoretical guarantees. We learn a mirror map for each of these environments and, also in this case, show that the learned mirror maps lead to a higher performance of PMD than the negative entropy. Moreover, we show that the learned mirror maps generalize well to other tasks, by testing each of the learned mirror maps on all the other environments.

To establish our findings, we employ the standard formulation of PMD (Xiao, ) for the tabular setting, and we used a generalized version, Approximate Mirror Policy Optimization (AMPO) (Alfano et al., ), for the non-tabular setting. To allow optimization over the space of mirror maps, we introduce parameterization schemes for mirror maps, one for the tabular and one for the non-tabular setting. Specifically, we propose a parameterization for $\omega$-potentials, which have been shown to induce a wide class of mirror maps (Krichene et al., ). We use evolution strategies (ES) to search for the mirror map that maximizes the performance of PMD and AMPO over an environment.

AMPO (Alfano et al., ) is a recently-proposed PMD framework designed to integrate general parameterization schemes, in our case neural networks, and arbitrary mirror maps. It benefits from theoretical guarantees, as Alfano et al. () show that AMPO has quasi-monotonic updates as well as sub-linear and linear convergence rates, depending on the step-size schedule. These desirable properties make AMPO particularly suitable for our numerical investigation.

ES are a type of population-based stochastic optimization algorithm that leverages random noise to generate a diverse pool of candidate solutions, and have been successfully applied to a variety of tasks (Real et al., ; Salimans et al., ; Such et al., ). The main idea consists in iteratively selecting higher-performing individuals, w.r.t. a fitness function, resulting in a gradual convergence towards the optimal solution. ES algorithms are gradient-free and have been shown to be well-suited for optimisation problems where the objective function is noisy or non-differentiable and the search space is large or complex (Beyer, ; Lu et al., , ). We use ES to search over the parameterized classes of mirror maps we introduce, by defining the fitness of a particular mirror map as the value of the last policy outputted by PMD and AMPO, for fixed hyper-parameters.

The rest of the paper is organized as follows. In Section 2, we introduce the setting of RL as well as
the PMD and AMPO algorithms. We describe the methodology behind our numerical experiments in Section 3, which are then discussed in Section 4. Finally, we give our conclusions in Section 5.

## 2 Preliminaries

### 2.1 Reinforcement Learning

Define a discounted Markov Decision Process (MDP) as the tuple $\mathcal{M}=(\mathcal{S},\mathcal{A},P,r,\gamma,\mu)$, where $\mathcal{S}$ and $\mathcal{A}$ are respectively the state and action spaces, $P(s^{\prime}\mid s,a)$ is the transition probability from state $s$ to $s^{\prime}$ when taking action $a$, $r(s,a)\in[0,1]$ is the reward function, $\gamma$ is a discount factor, and $\mu$ is a starting state distribution.
A policy $\pi\in(\Delta(\mathcal{A}))^{\mathcal{S}}$, where $\Delta(\mathcal{A})$ is the probability simplex over $\mathcal{A}$, represents the behavior of an agent on an MDP, whereby at state $s\in\mathcal{S}$ the agents takes actions according to the probability distribution $\pi(\cdot\mid s)$.

Our objective is for the agent to find a policy that maximizes the expected discounted cumulative reward for the starting state distribution $\mu$. That is, we want to find

$$ $\pi^{\star}\in\mathop{\mathrm{argmax}}_{\pi\in(\Delta(\mathcal{A}))^{\mathcal{ S}}}\mathbb{E}_{s\sim\mu}[V^{\pi}(s)].$ (1) $$

Here $V^{\pi}:\mathcal{S}\rightarrow\mathbb{R}$ denotes the value function associated with policy $\pi$ and is defined as

$$ $V^{\pi}(s):=\mathbb{E}\Big{[}\sum\nolimits_{t=0}^{\infty}\gamma^{t}r(s_{t},a_{ t})\mid\pi,s_{0}=s\Big{]},$ $$

where $s_{t}$ and $a_{t}$ are the current state and action at time $t$ and the expectation is taken over the trajectories generated by $a_{t}\sim\pi(\cdot|s_{t})$ and $s_{t+1}\sim P(\cdot|s_{t},a_{t})$.

Similarly to the value function, we define the $Q$-function associated with a policy $\pi$ as

$$ $Q^{\pi}(s,a):=\mathbb{E}\Big{[}\sum\nolimits_{t=0}^{\infty}\gamma^{t}r(s_{t},a _{t})\mid\pi,s_{0}=s,a_{0}=a\Big{]},$ $$

where the expectation is once again taken over the trajectories generated by the policy $\pi$. When the state and action spaces are finite, the $Q$-function can be expressed as $Q^{\pi}=(I-\gamma P^{\pi})^{-1}r$, where $P^{\pi}$ is a square matrix where the position $((s,a),(s^{\prime},a^{\prime}))$ is occupied by $\pi(a^{\prime}|s^{\prime})P(s^{\prime}|s,a)$. We also define the discounted state visitation distribution as

$$ $\displaystyle d_{\mu}^{\pi}(s):=(1-\gamma)\mathbb{E}_{s_{0}\sim\mu}\Big{[}\sum \nolimits_{t=0}^{\infty}\gamma^{t}P(s_{t}=s\mid\pi,s_{0})\Big{]},$ $$

where $P(s_{t}=s\mid\pi,s_{0})$ represents the probability of the agent being in state $s$ at time $t$ when following policy $\pi$ and starting from $s_{0}$. The probability distribution over states $d_{\mu}^{\pi}(s)$ represents the proportion of time spent on state $s$ when following policy $\pi$.

### 2.2 Tabular case: Policy Mirror Descent

We review the PMD framework, starting from mirror maps Bubeck (). Let $\mathcal{X}\subseteq\mathbb{R}^{\mathcal{A}}$ be a convex set. A mirror map $h:\mathcal{X}\rightarrow\mathbb{R}$ is a strictly convex, continuously differentiable and essentially smooth function(^1^11A function $h$ is *essentially smooth* if $\lim_{x\rightarrow\partial\mathcal{X}}\lVert\nabla h(x)\rVert_{2}=+\infty$, where $\partial\mathcal{X}$ denotes the boundary of $\mathcal{X}$.) that satisfies $\nabla h(\mathcal{X})=\mathbb{R}^{\mathcal{A}}$. In particular, we consider mirror maps belonging to the $\omega$-potential mirror map class, which contains most mirror maps used in the literature.

###### Definition 2.1 ( $\omega$ -potential mirror map Krichene et al. ( 2015 ) ) .

For $u\in(-\infty,+\infty]$, $\omega\leq 0$, an *$\omega$-potential* is defined as an increasing $C^{1}$-diffeomorphism $\phi:(-\infty,u)\rightarrow(\omega,+\infty)$ such that

$$ $\lim_{x\rightarrow-\infty}\phi(x)=\omega,\;\lim_{x\rightarrow u}\phi(x)=+ \infty,\;\int_{0}^{1}\phi^{-1}(x)dx\leq\infty.$ $$

For any $\omega$-potential $\phi$, we define the associated mirror map $h_{\phi}$ as

$$ $h_{\phi}(\pi_{s})=\sum\nolimits_{a\in\mathcal{A}}\int_{1}^{\pi(a\mid s)}\phi^{ -1}(x)dx.$ $$

When $\phi(x)=e^{x}$ we recover the negative entropy mirror map, which is the standard choice of mirror map in the RL literature (Agarwal et al., ; Tomar et al., ; Hu et al., ; Vaswani et al., , ), while we recover the $\ell_{2}$-norm when $\phi(x)=x$ (see Section A.1). The *Bregman divergence* (Bregman, ; Censor & Zenios, ) induced by the mirror map $h$ is defined as

$$ $\mathcal{D}_{h}(x,y):=h(x)-h(y)-\langle\nabla h(y),x-y\rangle,$ $$

where $\mathcal{D}_{h}(x,y)\geq 0$ for all $x,y\in\mathcal{Y}$.
Given a starting policy $\pi^{0}$, a learning rate $\eta$ and a mirror map $h$, PMD can be formalized as an iterative algorithm: for all iterations $t\geq 0$,

$$ $\pi^{t+1}\in\mathop{\mathrm{argmax}}\nolimits_{\pi\in(\Delta(\mathcal{A}))^{ \mathcal{S}}}\mathbb{E}_{s\sim d^{t}_{\mu}}[\eta_{t}\langle Q^{t}_{s},\pi_{s} \rangle-\mathcal{D}_{h}(\pi_{s},\pi_{s}^{t})],$ (2) $$

where we used the shorthand: $V^{t}:=V^{\pi^{t}}$, $Q^{t}:=Q^{\pi^{t}}$, $d^{t}_{\mu}:=d^{\pi^{t}}_{\mu}$ and $y_{s}:=y(s,\cdot)\in\mathbb{R}^{\mathcal{A}}$, for any function $y:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}$.
PMD benefits from several theoretical guarantees and, in particular, Xiao () shows that PMD enjoys quasi-monotonic updates and convergence to the optimal policy. We give here a slight modification of the statements of these results. Specifically, we do not upper-bound a term regarding the distance between subsequent policies in the result on quasi-monotonic updates, which we use to draw connections between theory and practice. Additionally, to obtain the convergence rate for PMD with constant step-size in the setting where we do not have access to the true Q-function, we combine the analyses on the sublinear convergence of PMD and linear convergence of inexact PMD given by Xiao (). We provide a proof in Appendix B.

###### Theorem 2.2 ( Xiao ( 2022 ) ) .

Following update (2), we have that, for all $t\geq 0$

$$ $V^{t+1}(\mu)-V^{t}(\mu)\geq-\frac{1}{1-\gamma}\max_{s\in\mathcal{S}}\big{ \lVert}\widehat{Q}^{t}_{s}-Q^{t}_{s}\big{\rVert}_{\infty}\max_{s\in\mathcal{S} }\big{\lVert}\pi^{t+1}_{s}-\pi^{t}_{s}\big{\rVert}_{1},$ (3) $$

where $\lVert\cdot\rVert_{\infty}$ and $\lVert\cdot\rVert_{1}$ represent the $\ell_{\infty}$ and the $\ell_{1}$ norms, respectively, and $\widehat{Q}$ is an estimate of the true $Q$-function. Additionally, at each iteration $T>0$, we have

$$ $\displaystyle V^{\star}(\mu)-\sum_{t<T}\mathbb{E}[V^{t}(\mu)]\leq\frac{1}{T} \left(\frac{\mathbb{E}_{s\sim d_{\mu}^{\star}}[\mathcal{D}_{h}(\pi^{\star}_{s} ,\pi^{0}_{s})]}{\eta_{t}(1-\gamma)}+\frac{1}{(1-\gamma)^{2}}\right)+4\frac{ \underset{t<T,s\in\mathcal{S}}{\max}\big{\lVert}\widehat{Q}^{t}_{s}-Q^{t}_{s} \big{\rVert}_{\infty}}{(1-\gamma)^{2}}.$ (4) $$

The statement of Theorem 2.2 is similar to many results in the literature on PMD and NPG (Agarwal et al., ; Xiao, ; Lan, ; Hu et al., ). That is, the convergence guarantee involves two terms, i.e. a convergence rate, which involves the Bregman divergence between the optimal policy and the starting policy, and an error floor, which involves the estimation error $\max_{s\in\mathcal{S}}\lVert\widehat{Q}^{t}_{s}-Q^{t}_{s}\rVert_{\infty}$. Given that by setting $\eta_{0}=\mathbb{E}_{s\sim d_{\mu}^{\star}}[\mathcal{D}_{h}(\pi^{\star}_{s},
\pi^{0}_{s})](1-\gamma)$ we obtain the convergence rate $2(T(1-\gamma)^{2})^{-1}$, and that the error floor has no explicit dependence on the mirror map, Equation (4) suggests that the mirror map has a mild influence on the performance of PMD. The only way the mirror map seems to affect the convergence guarantee is, implicitly, by changing the path of the algorithm and therefore influencing the estimation error. Similar observations can be made for several results in the PMD literature that share the same structure of convergence guarantees. On the other hand, Equation (3) suggests that mirror maps that prevent large updates of the policy cause the PMD algorithm to be less prone to performance degradation, as the lower bound is close to 0 when the policy update distance $\max_{s\in\mathcal{S}}\lVert\pi^{t+1}_{s}-\pi^{t}_{s}\rVert_{1}$ is small. One of the contributions of our work is to challenge and refute the conclusion that the mirror map has little influence on the convergence of PMD, highlighting a gap between theoretical guarantees, which are *based on upper bounds*, and the actual performance observed in PMD-based methodologies. Our empirical studies reveal that the choice of mirror map significantly influences both the speed of convergence and the minimum achievable error floor in PMD. Additionally, we provide evidence that a mirror map that prevents large updates throughout training leads to a better performance, as suggested by (3).

###### Definition 2.1 ( ω 𝜔 \omega italic_ω -potential mirror map Krichene et al. ( 2015 ) ) .

###### Theorem 2.2 ( Xiao ( 2022 ) ) .

### 2.3 Non-tabular case: Approximate Mirror Policy Optimization

AMPO is a theoretically sound framework for deep reinforcement learning based on mirror descent, as it inherits the quasi-monotonic updates and convergence guarantees from the tabular case (Alfano et al., ). Given a parameterized function class $\mathcal{F}^{\Theta}=\{f^{\theta}:\mathcal{S}\times\mathcal{A}\rightarrow
\mathbb{R},\theta\in\Theta\}$, an initial scoring function $f^{\theta^{0}}$, a step-size $\eta$ and an $\omega$-potential mirror map $h_{\phi}$, AMPO can be described, for all iterations $t$, by the two-step update

$$ $\pi^{t}(a\mid s)=\sigma(\phi(\eta f^{t}(s,a)+\lambda_{s}^{t}))\qquad\forall s \in\mathcal{S},a\in\mathcal{A},$ (5) $$

$$ $\displaystyle\theta^{t+1}\in\mathop{\mathrm{argmin}}_{\theta\in\Theta}\mathbb{ E}_{s\sim d^{t}_{\mu},a\sim\pi(\cdot|s)}\big{[}\big{(}f^{\theta}(s,a)-Q^{t}(s, a)-\eta^{-1}\max(\eta f^{t}(s,a)+\lambda_{s}^{t},\phi^{-1}(0))\big{)}^{2}\big{]}$ (6) $$

where $\lambda_{s}^{t}\in\mathbb{R}$ is a normalization factor to ensure $\pi^{t}_{s}\in\Delta(\mathcal{A})$ for all $s\in\mathcal{S}$, $f^{t}:=f^{\theta^{t}}$, and $\sigma(z)=\max(z,0)$ for $z\in\mathbb{R}$. Theorem 1 by Krichene et al. () ensures that there always exists a normalization constant $\lambda_{s}^{t}\in\mathbb{R}$. As shown by Alfano et al. (), AMPO recovers the standard formulation of PMD in (2) in the tabular setting. Assuming for simplicity that $\phi(x)>0$ for all $x\in\mathbb{R}$, the minimization problem in (6) implies that, at each iteration $t$, $f^{t}$ is an approximation of the sum of the $Q$-functions up to that point, that is $f^{t}\simeq\sum_{i=0}^{t-1}Q^{i}$. Therefore, the scoring function $f^{t}$ serves as an estimator of the value of an action.

## 3 Methodology

Denote by $\mathcal{H}$ the class of $\omega$-potentials mirror maps. Our objective is to search for the mirror map that maximizes the value of the last policy outputted by our mirror descent based algorithms, for a fixed time horizon $T$. That is, we want to find

$$ $h^{\star}\in\mathop{\mathrm{argmax}}_{h\in\mathcal{H}}\mathbb{E}\left[V^{T}( \mu)\right],$ (7) $$

where the expectation is taken over the randomness of the policy optimization algorithm and the policy updates are based on the mirror map $h$.
Depending on the setting, we parameterize the mirror map by parameterizing either $\phi^{-1}$ or $\phi$ as monotonically increasing functions. One of the primary objectives of this work it to motivate further research into the choice of mirror maps, as influenced by the choice of the function $\phi$, moving beyond the conventional use of $\phi(x)=e^{x}$. This is achieved by examining how different choices of $\phi$ influence the trajectory of training and demonstrating that, in many cases, there is a mirror map that outperforms the negative entropy by a large margin.

### 3.1 Tabular setting

As tabular setting we adopt a discounted and infinite horizon version of Grid-World (Oh et al., ), which is a large class of tabular MDPs. We parameterize $\phi^{-1}$ as a one layer neural network with 126 hiden units, where all kernels are non-negative and the activation functions are equally split among the following convex and concave monotonic non-linearities: $x^{3}$, $(x)_{+}^{2}$, $(x)_{+}^{1/2}$, $(x)_{+}^{1/3}$, $\log((x)_{+}+10^{-3})$ and $e^{x}$, where $(x)_{+}=\max(x,0)$. To ensure that we are able to recover the negative entropy and the $\ell_{2}$-norm, we add $ax+b\log(x)$ to the final output, where $a,b\geq 0$

To search for the best mirror map within this class, we employ a slight variation of the OpenAI-ES strategy (Salimans et al., ), adapted to the multi-task setting (Jackson et al., ). Denote by $\psi$ the parameters of the mirror map and by $F(\psi)$ the objective function in (7). Given a distribution of tasks $\mathcal{E}$, we estimate the gradient $\nabla_{\psi}F(\psi)$ as

$$ $E_{\epsilon\sim\mathcal{N}(0,I_{d})}\left[E_{e\sim\mathcal{E}}\left[\frac{ \epsilon}{2\sigma}(F_{e}(\psi+\sigma\epsilon)-F_{e}(\psi-\sigma\epsilon)) \right]\right],$ $$

where $\mathcal{N}(0,I_{d})$ is the multivariate normal distribution, $d$ is the number of parameters, and $\sigma>0$ is a hyperparameter regulating the variance of the perturbations. To account for different reward scales across tasks, we perform a rank transformation of the objective functions, whereby, for each sampled task $e$, we return 1 for the higher performing member between $\psi+\sigma\epsilon$ and $\psi-\sigma\epsilon$, and 0 for the other.

### 3.2 Non-tabular setting

As non-tabular environments, we consider the Basic Control Suite (BCS) and the MinAtar suite. Given the higher computational cost of simulations on these environments w.r.t. to the tabular setting, we replace the neural network parameterization for the mirror map with one with fewer parameters, in order to reduce the dimension of the mirror map class we search over.

We define the parameterized class $\Phi=\{\phi_{\psi}:\mathbb{R}\rightarrow[0,1],\psi\in\mathbb{R}^{n}_{+}\}$, with

$$ $\displaystyle\phi_{\psi}(x)=\begin{cases}0&\text{if }x\leq 0,\\ \frac{x}{\psi_{1}n}&\text{if }0<x\leq\psi_{1},\\ \frac{j}{n}+\frac{x-\sum_{i=1}^{j}\psi_{i}}{n\psi_{j+1}}&\text{if }\sum_{i=1}^ {j}\psi_{i}<x\leq\sum_{i=1}^{j+1}\psi_{i},\\ 1&\text{if }x>1,\end{cases}$ $$

Figure: Figure 1: A plot visually demonstrating the parameterization for $\phi$.
Refer to caption: extracted/5651986/plots/phi_param.png

where $1\leq j\leq n-1$. In other words, $\phi$ is defined as a piece-wise linear function with $n$ steps where, for all $j\leq n$, $\phi(\sum_{i=1}^{j}\psi_{i})=j/n$ and subsequent points are interpolated with a straight segment. This is illustrated in Figure 1.
We note that the $\omega$-potentials within $\Phi$ violate some of the constraints in Definition 2.1, as they are non-decreasing instead of increasing and $\lim_{x\rightarrow\infty}\phi(x)=1$ for all $\phi\in\Phi$. In Section A.2, we show that, if $\phi\in\Phi$, we can construct an $\omega$-potential $\phi^{\prime}$ that satisfies the constraints in Definition 2.1 such that $\phi$ and $\phi^{\prime}$ induce the same policies along the path of AMPO.

To effectively learn the hyperparameters, we employ the Separable Covariance Matrix Adaptation Evolution Strategy (sep-CMA) (Ros & Hansen, ), a variant of the popular algorithm CMA-ES (Hansen & Ostermeier, ).
CMA-ES is, essentially, a second-order method adapted for gradient free optimization. At every generation, it samples $n$ new points from a normal distribution, parameterized by a mean vector $m_{k}$ and a covariance matrix $C_{k}$. That is, the samples for generation $k$, $x^{k}_{1},\dots,x^{k}_{n}$, are distributed i.i.d. according to $x^{k}_{i}\sim\mathcal{N}(m_{k},C_{k})$. For each generation $k$, denote the $m\leq n$ best performing samples as $x^{*k}_{1},\dots,x^{*k}_{m}$. Then update the mean vector as $m_{k+1}=\sum_{i=1}^{m}w_{i}x^{*k}_{i}$, where $\sum_{i=1}^{m}w_{i}=1$, so that the next generation is distributed around the weighted mean of the best performing samples. $C_{k+1}$ is also updated to reflect the covariance structure of $x^{*k}_{1},\dots,x^{*k}_{m}$, in a complex way beyond the scope of this text. Due to the need to update $C_{k}$ based on covariance information, CMA-ES exhibits a quadratic scaling behavior with respect to the dimensionality of the search space, potentially hindering its efficiency in high-dimensional settings. To improve computational efficiency, we adopt sep-CMA, which introduces a diagonal constraint on the covariance matrix and reduces the computational complexity of the algorithm.

## 4 Experiments

In this section, we discuss the results of our numerical experiments. We start by presenting the tabular setting, where we track errors in order to understand what properties are desirable in a mirror map, and proceed by showing our results in the non-tabular setting.

### 4.1 Tabular setting: Grid-World

#### Model architecture and training

We define the tabular policy as a one-layer neural network with a softmax head, which takes as input a one-hot encoding of the environment state and outputs a distribution over the action space. We train the policy using the PMD update in (2), where we solve the minimization problem through stochastic gradient descent and estimate the $Q$-function through generalized advantage estimation (GAE) Schulman et al. (). We perform a simple grid-search over the hyperparameters to maximize the performance for the negative entropy and the $\ell_{2}$-norm mirror maps, in order to have a fair comparison. We report the chosen hyperparameters in Appendix C. The training procedure is implemented in Jax, using evosax (Lange, ) for the evolution. We run on four A40 GPUs, and the optimization process takes roughly 12 hours.

#### Environment

We adopt the version of Grid-World implemented by Jackson et al. (), and adapt it to the discounted and infinite horizon setting. We learn a mirror map by training PMD on a continuous distribution of Grid-World environments, and test PMD with the learned mirror map on five held-out configurations from previous publications Oh et al. (); Chevalier-Boisvert et al. () and on 256 randomly sampled configurations. For all PMD iterations $t$, we track two quantities that appear in Theorem 2.2, that is the estimation error  $\max_{s\in\mathcal{S}}\lVert\widehat{Q}^{t}_{s}-Q^{t}_{s}\rVert_{\infty}$, and the distance between policy updates $\max_{s\in\mathcal{S}}\lVert\pi^{t+1}_{s}-\pi^{t}_{s}\rVert_{1}$. To obtain the true $Q$-function, we compute the transition matrix $P^{\pi^{t}}$ and use the formula specified in Section 2, that is $Q^{t}=(I-\gamma P^{\pi^{t}})^{-1}r$. PMD is run for $128$ iterations and $2^{18}\simeq 250k$ total environment steps for all Grid-World configurations.

Figure: Figure 2: Comparison between the learned map and the negative entropy and $\ell_{2}$-norm mirror maps across a range of held-out configurations of Grid-World. We display the average over 256 runs and report the standard error as a shaded region. The column “Random tasks”, reports the averaged metrics for 256 randomly sampled configurations of Grid-World.
Refer to caption: extracted/5651986/image.png

We show the results of our simulations in Figure 2. As shown by the first row, the learned mirror map, the $\ell_{2}$-norm, and the negative entropy consistently rank first, second and third, respectively, in all tested configurations, in terms of final value. These results advocate the effectiveness of our methodology, as we are able to find a mirror map that outperforms the benchmark mirror maps in all tested environments. Moreover, we have that the $\ell_{2}$-norm consistently outperforms the negative entropy, further proving that the negative entropy is not always the best choice of mirror map. Another shared property among all training curves, is that the learned mirror map presents a slower convergence in the initial iterations w.r.t. the $\ell_{2}$-norm and to the negative entropy, as testified by a lower value, but convergence to a higher value in the long run. Lastly, we note that in all environments most of the value improvement happens in the first 50k environment steps.

To gain a better understanding of why the learned mirror maps outperforms the negative entropy and the $\ell_{2}$-norm, and to draw connections with the theoretical results outlined in Section 2.2, we report the estimation error and the distance between policy updates for all mirror maps. The first conclusion that we draw is that a smaller estimation error does not seem to be related to a higher performance. On the contrary, the second row of Figure 2 shows how the three mirror maps consistently have the same ranking in both value and estimation error, which is exactly the opposite of what Theorem 2.2 would suggest. On the other hand, the first and third rows of Figure 2 show how the lower bound on the performance improvement in (3) brings some valid insight on the behaviour of PMD during its first iterations, which are the ones that bring the largest improvement. In all configurations, we have that in the initial iterations of PMD the learned mirror map induces the smallest policy update distances as well as the performance curve with fewer dips in value, while both the negative entropy and the $\ell_{2}$-norm induce larger policy update distances and performance curves with several dips in value. This observation confirms the behaviour described by (3), whereby a small distance between policy updates prevents large performance degradation in policy updates.

### 4.2 Non-tabular setting: Basic Control and MinAtar suites

Figure: Figure 3: Comparison between the learned mirror map, the $\ell_{2}$-norm and the negative entropy across a range of standard environments. The top plots present the performance of AMPO for all mirror maps, reporting the average over 100 realizations and a shaded region denoting the standard error around the average. The middle plots report the $\omega$-potentials that induce the mirror maps. The bottom plots report the policy distribution according to (5), for each mirror map and score scales. The score scales are obtained by multiplying the vector $[1,\dots,|\mathcal{A}|]$ by a variable $c\in[0,4]$.
Refer to caption: extracted/5651986/image2.png

#### Model architecture and training

We define the scoring function as a deep neural network, which we train using the AMPO update in (6) and (5), where (6) is solved through Adam and the $Q$-function is estimated through GAE. We optimize the hyper-parameters of AMPO for the negative entropy mirror map for each suite, using the hyper-parameter tuning framework Optuna (Akiba et al., ). This is done to ensure we are looking at a fair benchmark of performance using the negative entropy mirror map. We report the chosen hyper-parameters in Appendix C. We then initialize the parameterized mirror map to be an approximation of the negative entropy(^2^22We achieve this by assigning $\psi_{i}\propto\log(i/(i-1))$, for $i=2,\dots,n$, and setting $\psi_{1}\propto 3\log(10)$.) and run Sep-CMA-ES on each environment separately. The whole training procedure is implemented in JAX, using gymnax environments (Lange, ) and evosax (Lange, ) for the evolution. We run on 8 GTX 1080Ti GPUs, and the optimization process takes roughly 48 hours for a single environment.

#### Environments

We test AMPO on the Basic Controle Suite (BCS) and the MinAtar Suite. For BCS, we run the evolution for 600 generation, each with 500k timesteps. For MinAtar, we run the evolution for 500 generation with 1M timesteps, then run 100 more generations with 10M timesteps.

Our empirical results are illustrated in Figure 3, where we show the performance of AMPO for the learned mirror map, for the negative entropy and for the $\ell_{2}$-norm. For all environments and mirror maps, we use the hyper-parameters returned by Optuna for AMPO with the negative entropy. Our learned mirror map leads to a better overall performance in all environments, apart from Acrobot, where it ties with the $\ell_{2}$-norm. We observe the largest improvement in performance on Asterix and SpaceInvaders, where the average return for the learned mirror map is more than double the one for the negative entropy. Figure 3 also suggests that different mirror maps may result in different error floors, as shown by the performance curves in Acrobot, Asterix and SpaceInvaders, where the negative entropy converges to a lower point than the learned mirror map.

The second and third row of Figure 3 illustrate the properties of the learned mirror map, in comparison to the negative entropy and the $\ell_{2}$-norm. In particular, the second row shows the corresponding $\omega$-potential, while the third row shows the policy distribution induced by the mirror map according to (5), depending on the scores assigned by the scoring function to each action. A shared property among all the learned mirror maps is that they all lead to assigning 0 0 probability to the worst actions for relatively small score scales, whilst the negative entropy always assigns positive weights to all actions. In more complex environments, where the evaluation of a certain action may be strongly affected by noise or where the optimal state may be combination locked Misra et al. (), this behaviour may lead to a critical lack of exploration. However, it appears that this is not the case in these environments, and we hypothesise that by setting the probability of the worst actions to 0 0 the learned mirror maps avoid wasting samples and hence can converge to the optimal policy more rapidly.

**Table 1: The table contains, for each entry, the value of the final policy outputted by AMPO trained on the environment corresponding to the column with the mirror map learned on the environment corresponding to the row. The last row represents the performance of AMPO with the negative entropy for the corresponding column environments. The value is averaged over 100 runs. Green cells correspond to a value higher than that associated to the negative entropy.**
|  | Acrobot | CartPole | Asterix | Freeway | SpaceInvaders |
| --- | --- | --- | --- | --- | --- |
| Acrobot | -88.49 | 476.41 | 24.03 | 56.00 | 144.01 |
| CartPole | -83.76 | 499.93 | 27.26 | 52.26 | 100.07 |
| Asterix | -103.55 | 490.86 | 30.22 | 58.56 | 122.00 |
| Freeway | -82.51 | 457.47 | 3.20 | 58.21 | 143.93 |
| SpaceInvaders | -78.29 | 489.56 | 4.36 | 22.27 | 170.24 |
| Negative entropy | -105.63 | 359.14 | 17.80 | 53.69 | 81.77 |

Our last result consists in testing each learned mirror map across the other environments we consider. In Table 1, we report the value of the final policy outputted by AMPO for all learned maps, plus the negative entropy, and for all environments, averaged over 100 runs. The table shows that the learned mirror maps generalize well to different environments and to different sets of hyper-parameters, which are shared within BCS and MinAtar but not between them. In particular, we have that the mirror maps learned in Acrobot and Asterix outperform the negative entropy in all environments, those learned in CartPole and Freeway outperform the negative entropy in 4 out of 5 environments, and that learned in SpaceInvaders outperforms the negative entropy in 3 out of 5 environments. These results show that our methodology can be useful in practice, as it benefits from good generalization across tasks.

## 5 Conclusion

Our study presents an empirical examination of PMD, where we successfully test the possibility of learning a mirror map that outperforms the negative entropy in both the tabular and non-tabular settings. In particular, we have shown that the learned mirror maps perform well on a set of configurations in Grid-World and that they can generalize to different tasks in BCS and in the MinAtar suite. Additionally, we have compared the theoretical findings established in the literature and the actual performance of PMD methods in the tabular setting, highlighting how the estimation error is not a good indicator of performance and validating the intuition that small policy updates lead to less instances of performance degradation. Our findings indicate that the choice of mirror map significantly impacts PMD’s effectiveness, an aspect not adequately reflected by existing convergence guarantees.

Our research introduces several new directions for inquiry. From a theoretical perspective, obtaining convergence guarantees that reflect the impact of the mirror map is an area for future exploration. On the practical side, investigating how temporal awareness (Jackson et al., ) or specific environmental challenges, such as exploration, robustness to noise, or credit assignment (Osband et al., ), can inform the choice of a mirror map to improve performance represents another research focus.

## Appendix A Further discussion on ω 𝜔 \omega italic_ω -potentials

### A.1 Negative entropy and ℓ 2 subscript ℓ 2 \ell_{2} roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT -norm

If $\phi(x)=e^{x-1}$, then the associated mirror map $h_{\phi}$ is the negative entropy. We have that

$$ $\displaystyle\int_{0}^{1}\phi^{-1}(x)dx$ $\displaystyle=\int_{0}^{1}\log(x)dx=[x\log(x)-x]^{1}_{0}=-1\leq+\infty.$ $$

The mirror map $h_{\phi}$ becomes the negative entropy, up to a constant, as

$$ $\displaystyle h_{\phi}(\pi_{s})$ $\displaystyle=\sum_{a\in\mathcal{A}}\int_{1}^{\pi(a\mid s)}\log(x)dx=|\mathcal {A}|-1+\sum_{a\in\mathcal{A}}\pi(a\mid s)\log(\pi(a\mid s)).$ $$

If $\phi(x)=x$, then the associated mirror map $h_{\phi}$ is the $\ell_{2}$-norm. We have that

$$ $\displaystyle\int_{0}^{1}\phi^{-1}(x)dx$ $\displaystyle=\int_{0}^{1}xdx=\left[\frac{x^{2}}{2}\right]^{1}_{0}=\frac{1}{2} \leq+\infty.$ $$

The mirror map $h_{\phi}$ becomes the $\ell_{2}$-norm, up to a constant, as

$$ $\displaystyle h_{\phi}(\pi_{s})$ $\displaystyle=\sum_{a\in\mathcal{A}}\int_{1}^{\pi(a\mid s)}xdx=\frac{1}{2}\sum _{a\in\mathcal{A}}\pi(a\mid s)^{2}-1.$ $$

### A.2 Parametric ϕ italic-ϕ \phi italic_ϕ

We show here that the parametric class of $\omega$-potentials we introduce in Section 3 results in a well defined algorithm when used for AMPO, even if it breaks some of the constraints in Definition 2.1. In particular, $\omega$-potentials within $\Phi$ are not $C^{1}$-diffeomorphisms and are only non-decreasing. We can afford not meeting the first constraint as the proof for Theorem 1 by Krichene et al. (), which establishes the existence of the normalization constant in Equation (5), only requires Lipschitz-continuity. As to the second constraint, we build an augmented class $\Phi^{\prime}$ that contains increasing $\omega$-potentials and show that it results in the same updates for AMPO as $\Phi$.

Let $\Delta_{n}$ be the $n$-dimensional probability simplex. We define the parameterized class $\Phi^{\prime}=\{\phi^{\prime}_{\psi}:\mathbb{R}\rightarrow[0,1],\psi\in\Delta_
{n}\}$, with

$$ $\displaystyle\phi^{\prime}_{\psi}(x)=\begin{cases}e^{x}-1&\text{if }x\leq 0,\\ \frac{x}{\psi_{1}n}&\text{if }0<x\leq\psi_{1},\\ \frac{j}{n}+\frac{x-\sum_{i=1}^{j}\psi_{i}}{n\psi_{j+1}}&\text{if }\sum_{i=1}^ {j}\psi_{i}<x\leq\sum_{i=1}^{j+1}\psi_{i},\\ x&\text{if }x>1,\end{cases}$ $$

where $1\leq j\leq n-1$. The functions within $\Phi^{\prime}$ are increasing and continuous. We start by showing that $\phi^{\prime}_{\psi}\in\Phi^{\prime}$ and $\phi_{\psi}\in\Phi$ are equivalent for Equation (5), for the same parameters $\psi$. At each iteration $t$, we have that

$$ $\displaystyle\pi^{t+1}(a\mid s)$ $\displaystyle=\sigma(\phi^{\prime}(\eta f^{t+1}(s,a)+(\lambda^{\prime})_{s}^{t +1}))$ $\displaystyle=\sigma(\phi(\eta f^{t+1}(s,a)+(\lambda^{\prime})_{s}^{t+1}))$ $\displaystyle=\sigma(\phi(\eta f^{t+1}(s,a)+\lambda_{s}^{t+1})).$ $$

This due to the following two facts. For $x\leq 0$, $\sigma(\phi^{\prime}(x))$ and $\sigma(\phi(x))$ are the same function. Also, $\sigma(\phi^{\prime}(\eta f^{t+1}(s,a)+(\lambda^{\prime})_{s}^{t+1}))$ has to be less or equal to 1, as a result of the projection, meaning that $\eta f^{t+1}(s,a)+(\lambda^{\prime})_{s}^{t+1}\leq 1$, where $\sigma(\phi^{\prime}(\cdot))$ and $\sigma(\phi(\cdot))$ are equivalent.

Since $\phi$ and $\phi^{\prime}$ induce the same policy, share that same normalization constant, and have the same value at 0, they also induce the same expression for Equation (6).

## Appendix B Proof of Theorem 2.2

In this section we outline how we use the proofs given by Xiao () in order to obtain Theorem 2.2.

### B.1 Preliminary Lemmas

We start by presenting two preliminary lemmas, which are ubiquitous in the literature on PMD. The first characterizes the difference in value between two policies, while the second characterizes the PMD update.

###### Lemma B.1 (Performance difference lemma, Lemma 1 in Xiao ( 2022 ) ) .

For any policy $\pi,\pi^{\prime}\in\Delta(\mathcal{A})^{\mathcal{S}}$ and $\mu\in\Delta(\mathcal{S})$,

$$ $\displaystyle V^{\pi}(\mu)-V^{\pi^{\prime}}(\mu)$ $\displaystyle=\frac{1}{1-\gamma}\mathbb{E}_{s\sim d^{\pi}_{\mu}}\left[\langle Q ^{\pi^{\prime}}_{s},\pi_{s}-\pi^{\prime}_{s}\rangle\right].$ $$

###### Lemma B.2 (Three-point decent lemma, Lemma 6 in Xiao ( 2022 ) ) .

Suppose that $\mathcal{C}\subset\mathbb{R}^{m}$ is a closed convex set, $f:\mathcal{C}\rightarrow\mathbb{R}$ is a proper, closed (^3^33A convex function $f$ is proper if $\mathrm{dom\,}f$ is nonempty and for all $x\in\mathrm{dom\,}f$, $f(x)>-\infty$. A convex function is closed, if it is lower semi-continuous.) convex function, $\mathcal{D}_{h}(\cdot,\cdot)$ is the Bregman divergence generated by a mirror map $h$.
Denote $\mathrm{rint\,dom\,}h$ as the relative interior of $\mathrm{dom\,}h$.
For any $x\in\mathrm{rint\,dom\,}h$, let

$$ $\displaystyle x^{+}\in\arg\min_{u\,\in\,\mathrm{dom\,}h\,\cap\,\mathcal{C}}\{f (u)+\mathcal{D}_{h}(u,x)\}.$ $$

Then $x^{+}\in\mathrm{rint\,dom\,}h\cap\mathcal{C}$ and for any $u\in\mathrm{dom\,}h\cap\mathcal{C}$,

$$ $\displaystyle f(x^{+})+\mathcal{D}_{h}(x^{+},x)\leq f(u)+\mathcal{D}_{h}(u,x)- \mathcal{D}_{h}(u,x^{+}).$ $$

###### Lemma B.1 (Performance difference lemma, Lemma 1 in Xiao ( 2022 ) ) .

###### Lemma B.2 (Three-point decent lemma, Lemma 6 in Xiao ( 2022 ) ) .

### B.2 Quasi-monotonic updates

We first prove that PMD enjoys quasi-monotonic updates (Equation (3)), that is PMD updates have an upper bound on how much they can deteriorate performance.

###### Proposition B.3 (Lemma 11 in Xiao ( 2022 ) ) .

At each time $t\geq 0$, we have

$$ $\langle\eta_{t}\widehat{Q}_{s}^{t},\pi^{t+1}_{s}-\pi^{t}_{s}\rangle\geq 0.$ $$

Additionally, we have that

$$ $V^{t+1}(\mu)-V^{t}(\mu)\geq-\frac{1}{1-\gamma}\max_{s\in\mathcal{S}}\big{ \lVert}\widehat{Q}^{t}_{s}-Q^{t}_{s}\big{\rVert}_{\infty}\max_{s\in\mathcal{S} }\big{\lVert}\pi^{t+1}_{s}-\pi^{t}_{s}\big{\rVert}_{1}.$ (8) $$

###### Proof.

Using Lemma B.2 with $x^{+}=\pi_{s}^{t+1}$, $\mathcal{C}=\Delta(\mathcal{A})$, $f(u)=\langle\widehat{Q}_{s}^{t},u\rangle$, $x=\pi^{t}$ and $u=\pi^{t+1}$, we obtain

$$ $\displaystyle\langle\eta_{t}\widehat{Q}_{s}^{t},\pi^{t}_{s}-\pi^{t+1}_{s} \rangle\leq\mathcal{D}_{h}(\pi^{t}_{s},\pi^{t}_{s})-\mathcal{D}_{h}(\pi^{t+1}_ {s},\pi^{t}_{s})-\mathcal{D}_{h}(\pi^{t}_{s},\pi^{t+1}_{s}).$ (9) $$

By rearranging terms and noticing $\mathcal{D}_{h}(\pi^{t}_{s},\pi^{t}_{s})=0$, we have

$$ $\displaystyle\langle\eta_{t}\widehat{Q}_{s}^{t},\pi^{t+1}_{s}-\pi^{t}_{s} \rangle\geq\mathcal{D}_{h}(\pi^{t+1}_{s},\pi^{t}_{s})+\mathcal{D}_{h}(\pi^{t}_ {s},\pi^{t+1}_{s})\geq 0.$ (10) $$

Equation (8) can be obtained using the performance difference lemma and (10):

$$ $\displaystyle V^{t+1}(\mu)-V^{t}(\mu)$ $\displaystyle=\frac{1}{1-\gamma}\mathbb{E}_{s\sim d^{t+1}_{\mu}}\left[\langle Q ^{t}_{s},\pi^{t+1}_{s}-\pi^{t}_{s}\rangle\right]$ $\displaystyle=\frac{1}{1-\gamma}\mathbb{E}_{s\sim d^{t+1}_{\mu}}\left[\langle \widehat{Q}^{t}_{s},\pi^{t+1}_{s}-\pi^{t}_{s}\rangle\right]$ $\displaystyle\quad+\frac{1}{1-\gamma}\mathbb{E}_{s\sim d^{t+1}_{\mu}}\left[ \langle\widehat{Q}^{t}_{s}-Q^{t}_{s},\pi^{t+1}_{s}-\pi^{t}_{s}\rangle\right]$ $\displaystyle\geq-\frac{1}{1-\gamma}\mathbb{E}_{s\sim d^{t+1}_{\mu}}\left[\big {\lVert}\widehat{Q}^{t}_{s}-Q^{t}_{s}\big{\rVert}_{\infty}\big{\lVert}\pi^{t+1 }_{s}-\pi^{t}_{s}\big{\rVert}_{1}\right].$ $$

∎

###### Proposition B.3 (Lemma 11 in Xiao ( 2022 ) ) .

###### Proof.

### B.3 Convergence guarantee

We can now prove the convergence guarantee reported in Equation (4). The proof of our result is obtained combining the analysis from the sublinear convergence guarantee with the analysis for the inexact linear convergence guarantee given by (Xiao, , Theorems 8 and 13). For two different time $t,t^{\prime}\geq 0$, denote the expected Bregman divergence between the policy $\pi^{t}$ and policy $\pi^{t^{\prime}}$, where the expectation is taken over the discounted state visitation distribution of the optimal policy $d_{\mu}^{\star}$, by

$$ $\mathcal{D}^{t}_{t^{\prime}}:=\mathbb{E}_{s\sim d_{\mu}^{\star}}\left[\mathcal {D}_{h}(\pi^{t}_{s},\pi^{t^{\prime}}_{s})\right].$ $$

Similarly, denote the expected Bregman divergence between the optimal policy $\pi^{\star}$ and $\pi^{t}$ by

$$ $\mathcal{D}^{\star}_{t}:=\mathbb{E}_{s\sim d_{\mu}^{\star}}\left[\mathcal{D}_{ h}(\pi^{\star}_{s},\pi^{t}_{s})\right].$ $$

###### Theorem B.4 (Theorems 8 and 13 in Xiao ( 2022 ) ) .

Consider the PMD update in (2), at each iteration $T\geq 0$, we have

$$ $\displaystyle V^{\star}(\mu)-\sum_{t<T}\mathbb{E}[V^{t}(\mu)]\leq\frac{1}{T} \left(\frac{\mathbb{E}[\mathcal{D}^{\star}_{0}]}{\eta_{t}(1-\gamma)}+\frac{1}{ (1-\gamma)^{2}}\right)+\frac{4}{(1-\gamma)^{2}}\mathbb{E}\left[\max_{s\in \mathcal{S}}\big{\lVert}\widehat{Q}^{t}_{s}-Q^{t}_{s}\big{\rVert}_{\infty} \right].$ $$

###### Proof.

Using Lemma B.2 with $x^{+}=\pi_{s}^{t+1}$, $\mathcal{C}=\Delta(\mathcal{A})$, $f(u)=\langle\widehat{Q}_{s}^{t},u\rangle$, $x=\pi^{t}$ and $u=\pi^{t+1}$, we have that

$$ $\displaystyle\langle\eta_{t}\widehat{Q}_{s}^{t},\pi^{\star}_{s}-\pi^{t+1}_{s} \rangle\leq\mathcal{D}_{h}(\pi^{\star},\pi^{t})-\mathcal{D}_{h}(\pi^{\star}, \pi^{t+1})-\mathcal{D}_{h}(\pi^{t+1},\pi^{t}),$ $$

which can be decomposed as

$$ $\displaystyle\langle\eta_{t}\widehat{Q}_{s}^{t},\pi^{t}_{s}-\pi^{t+1}_{s} \rangle+\langle\eta_{t}\widehat{Q}_{s}^{t},\pi^{\star}_{s}-\pi^{t}_{s}\rangle \leq\mathcal{D}_{h}(\pi^{\star},\pi^{t})-\mathcal{D}_{h}(\pi^{\star},\pi^{t+1} )-\mathcal{D}_{h}(\pi^{t+1},\pi^{t}).$ $$

Taking expectation with respect to the distribution $d_{\mu}^{\star}$ over states and with respect to the randomness of PMD and dividing both sides by $\eta_{t}$, we have

$$ $\mathbb{E}\left[\mathbb{E}_{s\sim d_{\mu}^{\star}}\left[\langle\widehat{Q}_{s} ^{t},\pi^{t}_{s}-\pi^{t+1}_{s}\rangle\right]\right]+\mathbb{E}\left[\mathbb{E} _{s\sim d_{\mu}^{\star}}\left[\langle\widehat{Q}_{s}^{t},\pi^{\star}_{s}-\pi^{ t}_{s}\rangle\right]\right]\leq\frac{1}{\eta_{t}}\mathbb{E}[\mathcal{D}^{\star }_{t}-\mathcal{D}^{\star}_{t+1}-\mathcal{D}^{t+1}_{t}].$ (11) $$

We lower bound the two terms on the left hand side of (11) separately. For the first term, we have that

$$ $\displaystyle\mathbb{E}\bigg{[}\mathbb{E}_{s\sim d_{\mu}^{\star}}$ $\displaystyle\left[\langle\widehat{Q}^{t}_{s},\pi^{t}_{s}-\pi^{t+1}_{s}\rangle \right]\bigg{]}\overset{(a)}{\geq}\frac{1}{1-\gamma}\mathbb{E}\left[\mathbb{E} _{s\sim d_{d^{\star}_{\mu}}^{t+1}}\left[\langle\widehat{Q}_{s}^{t},\pi^{t}_{s} -\pi^{t+1}_{s}\rangle\right]\right]$ $\displaystyle=\frac{1}{1-\gamma}\mathbb{E}\left[\mathbb{E}_{s\sim d_{d^{\star} _{\mu}}^{t+1}}\left[\langle Q_{s}^{t},\pi^{t}_{s}-\pi^{t+1}_{s}\rangle\right] \right]+\frac{1}{1-\gamma}\mathbb{E}\left[\mathbb{E}_{s\sim d_{d^{\star}_{\mu} }^{t+1}}\left[\langle\widehat{Q}_{s}^{t}-Q_{s}^{t},\pi^{t}_{s}-\pi^{t+1}_{s} \rangle\right]\right]$ $\displaystyle\overset{(b)}{=}\mathbb{E}\left[V^{t}({d^{\star}_{\mu}})-V^{t+1}( {d^{\star}_{\mu}})\right]+\frac{1}{1-\gamma}\mathbb{E}\left[\mathbb{E}_{s\sim d _{d^{\star}_{\mu}}^{t+1}}\left[\langle\widehat{Q}_{s}^{t}-Q_{s}^{t},\pi^{t}_{s }-\pi^{t+1}_{s}\rangle\right]\right]$ $\displaystyle\geq\mathbb{E}\left[V^{t}({d^{\star}_{\mu}})-V^{t+1}({d^{\star}_{ \mu}})\right]-\frac{1}{1-\gamma}\mathbb{E}\left[\mathbb{E}_{s\sim d_{d^{\star} _{\mu}}^{t+1}}\left[\big{\lVert}\widehat{Q}^{t}_{s}-Q^{t}_{s}\big{\rVert}_{ \infty}\big{\lVert}\pi^{t+1}_{s}-\pi^{t}_{s}\big{\rVert}_{1}\right]\right]$ $\displaystyle\geq\mathbb{E}\left[V^{t}({d^{\star}_{\mu}})-V^{t+1}({d^{\star}_{ \mu}})\right]-\frac{2}{1-\gamma}\mathbb{E}\left[\max_{s\in\mathcal{S}}\big{ \lVert}\widehat{Q}^{t}_{s}-Q^{t}_{s}\big{\rVert}_{\infty}\right]$ $$

where $(a)$ follows from Lemmas B.3 and the fact that $d_{d^{\star}_{\mu}}^{t+1}(s)\geq(1-\gamma)d_{\mu}^{\star}(s)\;\forall s\in
\mathcal{S}$, and $(b)$ follows from B.1. For the second term, we have that

$$ $\displaystyle\mathbb{E}\left[\mathbb{E}_{s\sim d_{\mu}^{\star}}\left[\langle \widehat{Q}^{t}_{s},\pi^{\star}_{s}-\pi^{t}_{s}\rangle\right]\right]$ $\displaystyle=$ $\displaystyle\mathbb{E}\left[\mathbb{E}_{s\sim d_{\mu}^{\star}}\left[\langle Q ^{t}_{s},\pi^{\star}_{s}-\pi^{t}_{s}\rangle\right]\right]+\mathbb{E}\left[ \mathbb{E}_{s\sim d_{\mu}^{\star}}\left[\langle\widehat{Q}^{t}_{s}-Q^{t}_{s}, \pi^{\star}_{s}-\pi^{t}_{s}\rangle\right]\right]$ $\displaystyle\overset{(b)}{=}$ $\displaystyle\mathbb{E}[V^{\star}(\mu)-V^{t}(\mu)](1-\gamma)+\mathbb{E}\left[ \mathbb{E}_{s\sim d_{\mu}^{\star}}\left[\langle\widehat{Q}^{t}_{s}-Q^{t}_{s}, \pi^{\star}_{s}-\pi^{t}_{s}\rangle\right]\right],$ $\displaystyle\geq$ $\displaystyle\mathbb{E}[V^{\star}(\mu)-V^{t}(\mu)](1-\gamma)-\frac{2}{1-\gamma }\mathbb{E}\left[\max_{s\in\mathcal{S}}\big{\lVert}\widehat{Q}^{t}_{s}-Q^{t}_{ s}\big{\rVert}_{\infty}\right],$ $$

where $(b)$ follows from Lemma B.1.

Plugging the two bounds in (11), dividing both sides by $(1-\gamma)$ and rearranging, we obtain

$$ $\displaystyle\frac{\mathbb{E}[\mathcal{D}^{t+1}_{t}]}{\eta_{t}(1-\gamma)}+ \mathbb{E}[V^{\star}(\mu)-V^{t}(\mu)]$ $\displaystyle\leq\frac{\mathbb{E}[\mathcal{D}^{\star}_{t}-\mathcal{D}^{\star}_ {t+1}]}{\eta_{t}(1-\gamma)}+\frac{\mathbb{E}\left[V^{t}({d^{\star}_{\mu}})-V^{ t+1}({d^{\star}_{\mu}})\right]}{1-\gamma}$ $\displaystyle\quad+\frac{4}{(1-\gamma)^{2}}\mathbb{E}\left[\max_{s\in\mathcal{ S}}\big{\lVert}\widehat{Q}^{t}_{s}-Q^{t}_{s}\big{\rVert}_{\infty}\right].$ $$

Summing up from 0 0 to $T-1$ and dropping some positive terms on the left hand side and some negative terms on the right hand side, we have

$$ $\displaystyle V^{\star}(\mu)-\sum_{t<T}\mathbb{E}[V^{t}(\mu)]\leq\frac{\mathbb {E}[\mathcal{D}^{\star}_{0}]}{\eta_{t}(1-\gamma)}-\frac{\mathbb{E}\left[V^{0}( {d^{\star}_{\mu}})-V^{T}({d^{\star}_{\mu}})\right]}{1-\gamma}+\frac{4}{(1- \gamma)^{2}}\mathbb{E}\left[\max_{s\in\mathcal{S}}\big{\lVert}\widehat{Q}^{t}_ {s}-Q^{t}_{s}\big{\rVert}_{\infty}\right].$ $$

Notice that $\mathbb{E}\left[V^{0}({d^{\star}_{\mu}})-V^{T}({d^{\star}_{\mu}})\right]\leq
\frac{1}{1-\gamma}$ as $r(s,a)\in[0,1]$. By dividing $T$ on both side, we yield the statement.
∎

###### Theorem B.4 (Theorems 8 and 13 in Xiao ( 2022 ) ) .

###### Proof.

## Appendix C Training details

We give the hyper-parameters we use for training in Table 2 and 3.

**Table 2: Hyper-parameter settings of PMD and AMPO for different sets of environments**
| Parameter | Grid-World (PMD) | BCS (AMPO) | MinAtar (AMPO) |
| --- | --- | --- | --- |
| Number of environment steps | $2^{18}\simeq 2.5\text{e}5$ | 5e5 | 1e7 |
| Number of environments | 64 | 4 | 256 |
| Unroll length | 32 | 128 | 128 |
| Number of minibatches | 1 | 4 | 8 |
| Number of update epochs | 32 | 16 | 8 |
| Adam learning rate | - | 4e-3 | 7e-4 |
| Sgd learning rate | 40 | - | - |
| Gamma | 0.99 | 0.99 | 0.99 |
| Max grad norm | - | 1.4 | 1 |
| AMPO learning rate | 1e-1 | 0.9 | 0.9 |

**Table 3: Hyper-parameter settings of OpenAI-ES, which we used in the tabular setting, and of Sep-CMA-ES, which we used in the non-tabular case.**
|  | OpenAI-ES | Sep-CMA-ES |
| --- | --- | --- |
| Population Size | 512 | 128 |
| Number of generations | 512 | 600 |
| Sigma init | 0.5 | 2 |
| Sigma Decay | 0.995 | - |
| Learning rate | 0.01 | - |

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 5
presentation: 2
questions: See weaknesses.
soundness: 3
strengths: 1. The author presents a novel approach by replacing the conventional mirror map with a learned mirror map, offering valuable insights for reinforcement learning research.
2. The experiments cover both a tabular setting and a real-world application, enhancing the potential for this work to be applied to real-world problems.
summary: This paper studies the impact of different mirror graphs within the Policy Mirror Descent (PMD) framework in reinforcement learning, traditionally dominated by negentropy graphs. Through empirical testing, the authors identify alternative mirror mappings that significantly improve performance and generalization in different environments, challenging the default choice of mirror mappings in PMD.
weaknesses: 1. There are some typos in this paper, for example, in equation (5), it should be $\pi_{\theta_t}$ instead of $\pi_{t}heta$?
2. The author didn't provide related works of this paper, making it hard to understand and follow. And it is better to provide more training details in Appendix C.
3. It seems that the author only provides comparisons of their method with the negative entropy and d $\ell_2$-norm mirror map. Can the author provide comparisons with other methods?
4. The computational cost of the proposed method appears to be relatively high. It would be beneficial for the author to include a discussion comparing the computational cost of this method to that of existing approaches. This comparison would provide a valuable disscussion for understanding the trade-offs in efficiency and practicality.

---

confidence: 3
contribution: 3
initial_rating: 5
presentation: 4
questions: 1. How does the mirror map learned on e.g. Hopper perform on very different environments (other than Mujoco's)? Is performance/training dynamics significantly different from using the negative entropy mirror map?
2. I find it surprising that using negative entropy or l2 norm does not solve Asterix-MinAtar under 10M steps. I personally recall experimenting with Asterix and rather easily solving it. Could you provide a more detailed description of how hyperparameters were selected for each method, or include ablation studies showing the impact of key hyperparameters?
soundness: 2
strengths: - Empirical validation: the authors provide empirical evidence that challenges the reliance on the negative entropy mirror map in PMD.
- Comparison with theory: the paper highlights a gap between theoretical bounds and actual empirical performance.
- Algorithm: the use of ES makes sense here and is well justified.
- Clarity: the background and methodology sections are well written.
summary: This paper explores the role of mirror maps in the Policy Mirror Descent (PMD) framework in reinforcement learning (RL), which includes various algorithms for policy optimization. Traditionally, the negative entropy mirror map, giving rise to the Natural Policy Gradient (NPG), is the most studied, but the authors argue that this choice is often sub-optimal. Through empirical experiments, they show that alternative, learned mirror maps can enhance PMD's performance across different RL environments, starting with tabular settings like Grid-World and scaling to more complex scenarios like MinAtar and MuJoCo. Their findings suggest that mirror map choice has a more substantial impact on PMD performance than theoretical convergence guarantees indicate. The manuscript highlights the potential for learned mirror maps to generalize across tasks and calls for further theoretical research to understand the influence of mirror maps on PMD's effectiveness.
weaknesses: - Reliance on ES which may not be very scalable (to higher dimensional mirror map parameterization).
- Experiments only include small environments where finding better mirror maps may not lead to significantly superior learning curves. Testing on a broader array of environments, especially those with different reward structures or high-dimensional state spaces would enhance the robustness and applicability of the findings.
- Experiments do not seem statistically significant and may hinder hyper-parameter optimization for some methods and not e.g. for the negative entropy mirror map baseline.

---

confidence: 4
contribution: 3
initial_rating: 8
presentation: 3
questions: Q1: Although the experiments demonstrate that the learned map is more effective, I believe this does not necessarily conflict with the theoretical assertion that these methods share the same bound. Could the authors provide further explanation on this point?

Q2: In lines 165-166, what is the relationship between Bregman divergence and the mirror map $h$? Could this be further clarified?

Q3(minor): There is a typo in Equation 5 (\theta).
soundness: 3
strengths: 1. The research effectively highlights the limitations of existing convergence guarantees in capturing the impact of mirror maps. It provides robust empirical evidence that challenges the conventional choice of mirror maps, offering a fresh perspective on the potential of PMD. 

2. The experimental design is well-conceived and rigorous, aligning closely with the preceding analysis.

3. The paper is well-written, with clear definitions and precise descriptions. Despite covering a substantial amount of theory, it remains accessible and easy to read.
summary: This paper investigates the Policy Mirror Descent in RL, which is known for its finite-time convergence guarantees and encompasses various algorithms by selecting a mirror map. However, the study explores the potential of alternative mirror maps (instead of negative entropy mirror map, which leads to the Natural Policy Gradient method), demonstrating that the conventional choice often results in suboptimal performance. In empirical studies, the authors use evolutionary strategies to identify more effective mirror maps, showing improved performance in tabular and non-tabular environments. The research highlights that the choice of mirror map significantly affects PMD's effectiveness, challenging existing theoretical assumptions and suggesting new directions for future research.
weaknesses: From my perspective, this paper makes a satisfactory and engaging contribution, though there is always room for improvement.

1. The paper offers compelling empirical evidence but does not provide a comprehensive theoretical framework to elucidate why certain mirror maps outperform others. Although developing such a framework is challenging, it represents a promising avenue for future research. 

2. the study's focus on specific benchmark environments may not fully encompass the diversity and complexity of real-world scenarios in reinforcement learning, potentially limiting the generalizability of the findings.

Meta-review: In this paper, the authors argue that the well-studied negative entropy mirror map, which results in natural policy gradient, is often sub-optimal. They empirically show that learning a proper mirror map using extensive search methods, such as evolutionary strategies,   
can result in improved performance in tabular and non-tabular environments. This work highlights the importance of the choice of mirror map in the performance of policy mirror descent algorithms and calls for further research to understand the influence of mirror maps on the effectiveness of these algorithms.

(+) The paper is well-written and easy to follow. The authors used clear definitions and precise descriptions, which helps their work to be accessible to a large audience.
(+) The experimental design is well-conceived and rigorous, and it is closely aligned closely with the analysis.
(+) It properly challenges the conventional choice of mirror maps and offers a fresh perspective on the potential of policy mirror descent algorithms.

Despite some issues (mainly minor) raised by the reviewers, which I strongly recommend that the authors address them, the paper could be useful for the community.

Decision: Accept (Poster)