Stable Offline Value Function Learning with Bisimulation-based Representations

## Abstract

Abstract In reinforcement learning, offline value function learning is the procedure of using an offline dataset to estimate the expected discounted return from each state when taking actions according to a fixed target policy. The stability of this procedure, i.e., whether it converges to its fixed-point, critically depends on the representations of the state-action pairs. Poorly learned representations can make value function learning unstable, or even divergent. Therefore, it is critical to stabilize value function learning by explicitly shaping the state-action representations. Recently, the class of bisimulation-based algorithms have shown promise in shaping representations for control. However, it is still unclear if this class of methods can stabilize value function learning. In this work, we investigate this question and answer it affirmatively. We introduce a bisimulation-based algorithm called kernel representations for offline policy evaluation ( krope ). krope uses a kernel to shape state-action representations such that state-action pairs that have similar immediate rewards and lead to similar next state-action pairs under the target policy also have similar representations. We show that krope : 1) learns stable representations and 2) leads to lower value error than baselines. Our analysis provides new theoretical insight into the stability properties of bisimulation-based methods and suggests that practitioners can use these methods for stable and accurate evaluation of offline reinforcement learning agents.

## 1 Introduction

Learning the value function of a policy is a critical component of many reinforcement learning (rl) algorithms (Sutton & Barto, ). While value function learning algorithms such as temporal-difference learning (td) have been successful, they can be unreliable. In particular, the deadly triad, i.e., the combination of off-policy updates, function approximation, and bootstrapping, can make td-based methods diverge (Sutton & Barto, ; Tsitsiklis & Van Roy, ; Baird, ). Function approximation is a critical component of value function learning since it determines the representations of state-action pairs, which in turn defines the space of expressible value functions. Depending on how this value function space is represented, value function learning algorithms may diverge (Ghosh & Bellemare, ). That is, the value function learning algorithm may not converge to its fixed-point, or may even diverge away from it. In this work, we investigate how to explicitly learn state-action representations to stabilize value function learning.

In seeking such representations, we turn to $\pi$-bisimulation algorithms. These algorithms define a metric to capture *behavioral similarity* between state-action pairs such that similarity is based on immediate rewards received and the similarity of next state-action pairs visited by $\pi$ (Castro, ). The algorithms then use this metric to learn representations such that state-action pairs that are similar under this metric have similar representations. Ultimately, the goal of $\pi$-bisimulation methods is to learn representations such that state-actions pairs with similar values under $\pi$ also have similar representations (see Figure 1). While these algorithms have shown promise in improving the expected return of rl algorithms, it remains unclear whether they contribute to stability (Castro et al., ; Zhang et al., ; Castro et al., ). In this paper, we aim to understand whether $\pi$-bisimulation-based representations stabilize value function learning.

Figure: Figure 1: The figure illustrates the native state-action representations $\mathcal{X}$ and $\pi$-bisimulation representations $\Phi_{\pi-\textsc{bisim}}$. $\pi$-bisimulation algorithms use a similarity function $k$ that captures similarity between state-action pairs based on immediate rewards and similarity of next state-action pairs under $\pi$ to shape their representations. Ultimately, the goal of $\pi$-bisimulation methods is to learn representations such that state-actions pairs with similar values under $\pi$ also have similar representations. The function $k$ outputs high values within the blue (and orange) state-actions but low values between blue and orange state-actions. Therefore, the blue (and orange) state-actions have similar representations, but different representations between the distinct colors.
Refer to caption: extracted/5973682/images/phi_to_bisim.png

In this work, we focus on offline value function learning. Given a fixed, offline dataset generated by unknown and possibly multiple behavior policies, the goal is to estimate the value function of a fixed, target policy. We introduce kernel representations for offline policy evaluation (krope), a bisimulation-based representation learning algorithm. krope defines a kernel that captures similarity between state-action pairs based on immediate rewards received and similarity of next state-action pairs under the *target* policy. It then shapes the state-action representations such that state-action pairs that are similar according to this kernel have similar representations. We use krope as the representative algorithm for the class of bisimulation-based representation learning algorithms to investigate the following question:

Can bisimulation-based representation learning stabilize offline value function learning?

Through theoretical and empirical analysis, we answer this question affirmatively and make the following contributions:

- 1.
We introduce kernel representations for offline policy evaluation (krope) for stable and accurate offline value function learning (Section 3).
- 2.
We prove that krope’s representations stabilize least-squares policy evaluation (lspe), a popular value function learning algorithm (Sections 3.2).
- 3.
We prove that krope representations are Bellman complete, another indication of stability (Sections 3.3).
- 4.
We empirically validate that krope representations lead to more stable and accurate offline value function learning compared to non-bisimulation baselines (Section 4).
- 5.
We empirically analyze the sensitivity of the krope learning procedure under the deadly triad. These experiments shed light on when representation pre-training may be easier than direct value function learning with lspe (Appendix C.3.1).

## 2 Background

In this section, we present our problem setup and discuss prior work.

### 2.1 Problem Setup and Notation

We consider the infinite-horizon
Markov decision process (mdp) framework (Puterman, ), $\mathcal{M}=\langle\mathcal{S},\mathcal{A},r,P,\gamma,d_{0}\rangle$,
where $\mathcal{S}$ is the state space, $\mathcal{A}$ is the action space,
$r:\mathcal{S}\times\mathcal{A}\to[-1,1]$ is the deterministic reward function,
$P:\mathcal{S}\times\mathcal{A}\to\Delta(\mathcal{S})$ is the transition dynamics function, $\gamma\in[0,1)$ is the discount factor, and $d_{0}\in\Delta(\mathcal{S})$ is the initial state distribution, where $\Delta(X)$ represents the set of all probability distributions over a set $X$. We refer to the joint state-action space as ${\mathcal{X}}:=\mathcal{S}\times\mathcal{A}$. The
agent acting according to policy $\pi:\mathcal{S}\to\Delta(\mathcal{A})$ in the mdp generates a trajectory: $S_{0},A_{0},R_{0},S_{1},A_{1},R_{1},...$, where $S_{0}\sim d_{0}$, $A_{t}\sim\pi(\cdot|S_{t})$,
$R_{t}:=r(S_{t},A_{t})$, and $S_{t+1}\sim P(\cdot|S_{t},A_{t})$ for
$t\geq 0$.

We define the action-value function of a policy $\pi$ for a given state-action pair as $q^{\pi}(s,a):=\mathbb{E}_{\pi}[\sum_{t=0}^{\infty}\gamma^{t}r(S_{t},A_{t})|S_{
0}=s,A_{0}=a]$, i.e., the expected discounted return when starting from state $s$ with initial action $a$ and then following policy $\pi$. The Bellman evaluation operator $\mathcal{T}^{\pi}:\mathbb{R}^{\mathcal{X}}\to\mathbb{R}^{\mathcal{X}}$ is defined as $(\mathcal{T}^{\pi}f)(s,a):=r(s,a)+\gamma\mathbb{E}_{S^{\prime}\sim P(\cdot|s,a
),A^{\prime}\sim\pi}[f(S^{\prime},A^{\prime})],\forall f\in\mathbb{R}^{
\mathcal{X}}$. Accordingly, the action-value function satisfies the Bellman equation, i.e., $q(s,a)=r(s,a)+\gamma\mathbb{E}_{P,{\pi_{e}}}[q(S^{\prime},A^{\prime})]$.

It will be convenient to consider the matrix notation equivalents of the above functions. Since a policy $\pi$ induces a Markov chain on $\mathcal{X}$, we can denote the transition matrix of this Markov chain by $P^{\pi}\in\mathbb{R}^{|\mathcal{X}|\times|\mathcal{X}|}$. Here, each entry $P^{\pi}(i,j)$ is the probability of transitioning from state-actions $i$ to $j$. Similarly, we have the action-value function $q^{\pi}\in\mathbb{R}^{|\mathcal{X}|}$ and reward vector $r\in\mathbb{R}^{|\mathcal{X}|}$, where the entry $q^{\pi}(i)$ and $r(i)$ are the expected discounted return from state-action $i$ under $\pi$ and reward received at state-action $i$ respectively.

In this work, we study the representations of the state-action space. We use $\phi:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}^{d}$ to denote the state-action representations, which maps state-action pairs into a $d$-dimensional Euclidean space. We denote the matrix of all the state-action features as $\Phi\in\mathbb{R}^{|\mathcal{X}|\times d}$, where each row is the state-action feature $\phi(s,a)\in\mathbb{R}^{d}$ for state-action pair $(s,a)$. When dealing with the offline dataset $\mathcal{D}$, $\Phi$’s dimensions are $|\mathcal{D}|\times\ d$, where $|\mathcal{D}|$ is the number state-actions in the dataset $\mathcal{D}$. Note that $\Phi$ can be the native state-action features of the mdp, or the output of some representation learning algorithm, or the penultimate features of the action-value function when using a neural network. Throughout this paper, we will view $\phi$ as an encoder or state-action abstraction (Li et al., ). Note that the state-action abstraction view enables us to view $\phi$ as a state-action aggregator from the space of state-actions $\mathcal{X}$ to the space of state-action groups $\mathcal{X}^{\phi}$.

### 2.2 Offline Policy Evaluation and Value Function Learning

In offline policy evaluation (ope), the goal is to evaluate a fixed target policy, ${\pi_{e}}$, using a fixed dataset of $m$ transition tuples $\mathcal{D}:=\{(s_{i},a_{i},s_{i}^{\prime},r_{i})\}_{i=1}^{m}$. In this work, we evaluate ${\pi_{e}}$ by estimating the *action-value function $q^{{\pi_{e}}}$* using $\mathcal{D}$.
Crucially, $\mathcal{D}$ may have been generated by a set of unknown behavior policies that are different from ${\pi_{e}}$, which means that simply averaging the discounted returns in $\mathcal{D}$ will produce an inconsistent estimate of $q^{{\pi_{e}}}$.
In our theoretical results, we make the standard coverage assumption that $\forall s\in\mathcal{S},\forall a\in\mathcal{A}$ if ${\pi_{e}}(a|s)>0,$ then the state-action pair $(s,a)$ has non-zero probability of appearing in $\mathcal{D}$ (Sutton & Barto, ; Precup et al., ).

We measure the accuracy of the value function estimate with the mean squared value error (msve).
Let $\hat{q}^{{\pi_{e}}}$ be the estimate returned by a value function learning method using $\mathcal{D}$.
The msve of this estimate is defined as $\operatorname{\textsc{msve}}[\hat{q}^{{\pi_{e}}}]\coloneqq\mathbb{E}_{(S,A)
\sim\mathcal{D}}[(\hat{q}^{{\pi_{e}}}(S,A)-q^{{\pi_{e}}}(S,A))^{2}]$. In environments with continuous state-action spaces, where it is impossible to compute $q^{\pi_{e}}$ for all state-actions, we adopt a common evaluation procedure from the ope literature of measuring the mse across only the initial state-action distribution, i.e., $\operatorname{\textsc{mse}}[\hat{q}^{{\pi_{e}}}]\coloneqq\mathbb{E}_{S_{0}\sim
d
_{0},A_{0}\sim{\pi_{e}}}[(\hat{q}^{{\pi_{e}}}(S_{0},A_{0})-q^{{\pi_{e}}}(S_{0}
,A_{0}))^{2}]$. For this procedure, we assume access to $d_{0}$ (Voloshin et al., ; Fu et al., ). While in practice $q^{{\pi_{e}}}$ is unknown, it is standard for the sake of empirical analysis to estimate $q^{{\pi_{e}}}$ by executing unbiased Monte Carlo rollouts of ${\pi_{e}}$ or computing $q^{\pi_{e}}$ exactly using dynamic programming in tabular environments (Voloshin et al., ; Fu et al., ).

##### Least-Squares Policy Evaluation

Least-squares policy evaluation (lspe) is a value function learning algorithm, which models the action-value function as a linear function: $\hat{q}_{\theta}^{{\pi_{e}}}(s,a):=\phi(s,a)^{\top}\theta$, where $\theta\in\mathbb{R}^{d}$ (Nedic & Bertsekas, ). lspe iteratively learns $\theta$ with the following updates per iteration step $t$:

$$ $\theta_{t+1}\leftarrow(\mathbb{E}_{\mathcal{D}}[\Phi^{\top}\Phi])^{-1}\mathbb{ E}_{\mathcal{D},{\pi_{e}}}[\Phi^{\top}(r+\gamma P^{{\pi_{e}}}\Phi\theta_{t})],$ (1) $$

where the expectations are taken with respect to the randomness of the dataset $\mathcal{D}$ and ${\pi_{e}}$.
Note that $\mathbb{E}[\Phi^{\top}\Phi]$ is the feature covariance matrix. Assuming lspe converges, it will converge to the same fixed-point as td(0) (Szepesvari, ), which we denote as $\theta_{\text{{lspe}}}$. In this work, we follow a two-stage approach to applying lspe: we first obtain the encoder $\phi$ either through representation learning or using the native features of the mdp, and then feed the obtained $\phi$ along with $\mathcal{D}$ and ${\pi_{e}}$ as input to lspe, which outputs $\hat{q}_{\theta}^{{\pi_{e}}}$ (Nedic & Bertsekas, ; Chang et al., ). This two-stage approach of learning a linear function on top of fixed representations is called the linear evaluation protocol (Chang et al., ; Farebrother et al., ; ; Grill et al., ; He et al., ). This protocol enables us to cleanly analyze the nature of the learned representations within the context of well-understood value function learning algorithms such as lspe. In Appendix A, we include the pseudo-code for lspe.

### 2.3 Stable, Realizable, and Generalizable Representations

We define stability of lspe and related td-methods following Ghosh & Bellemare ():

###### Definition 1 (Stability) .

lspe is said to be stable if for any initial $\theta_{0}\in\mathbb{R}^{d}$, $\lim_{t\to\infty}\theta_{t}=\theta_{\text{{lspe}}}$ when $\theta_{t}$ is updated according to Equation (1).

When determining the stability of lspe, we have following proposition from prior work:

###### Proposition 1 ( Asadi et al. ( 2024 ); Wang et al. ( 2021a ) ) .

lspe is stable if and only if the spectral radius of $(\mathbb{E}[\Phi^{\top}\Phi])^{-1}(\gamma\mathbb{E}[\Phi^{\top}P^{{\pi_{e}}}
\Phi])$, i.e., its maximum absolute eigenvalue, is less than $1$.

Therefore, the stability of lspe largely depends on the representations $\Phi$ and the distribution shift between the data distribution of $\mathcal{D}$ and ${\pi_{e}}$. In this work, we study the stability of lspe for a fixed distribution of $\mathcal{D}$ and learn $\Phi$. If a given $\Phi$ stabilizes lspe, we say $\Phi$ is a *stable representation*.

In addition to stability, we also care about the realizability and generalizability of $\Phi$. We say $\Phi$ is a *realizable* representation if $q^{{\pi_{e}}}\in\text{Span}(\Phi),$ where $\text{Span}(\Phi)$ is the subspace of all expressible action-value functions with $\Phi$. Note that even if $\Phi$ is a realizable and stable representation, lspe may not recover the $q^{{\pi_{e}}}$ solution (Sutton & Barto, ). While generalization can have multiple interpretations, we say $\Phi$ *generalizes* well if the state-action features that are close in the representation space also have similar $q^{\pi_{e}}$ values (Lyle et al., ; Lan et al., ).

###### Definition 1 (Stability) .

###### Proposition 1 ( Asadi et al. ( 2024 ); Wang et al. ( 2021a ) ) .

### 2.4 Related Works

In this section, we discuss the most relevant prior literature on ope and representation learning.

##### Representations for Offline rl and ope .

There are several works that have shown shaping representations can be effective for offline rl (Yang & Nachum, ; Islam et al., ; Nachum & Yang, ; Zang et al., ; Arora et al., ; Uehara et al., ; Chen & Jiang, ; Pavse & Hanna, ). Ghosh & Bellemare () presented a theoretical understanding of how various representations can stabilize td learning. However, they did not discuss bisimulation-based representations. Kumar et al. (); Ma et al. (); He et al. () promote the stability of td-based methods by increasing the rank of the representations to prevent representation collapse. However, as we show in Section 4, these types of representations can still lead to inaccurate ope. On the other hand, krope mitigates representation collapse and leads to accurate ope. Chang et al. () introduced bcrl to learn Bellman complete representations for stable ope. While in theory, bc representations are desirable, we found that bcrl is sensitive to hyperparameter tuning. In contrast, we show that krope is more robust to hyperparameter tuning. Pavse & Hanna () showed that bisimulation-based representations mitigate the divergence of fqe; however, they did not provide an explanation for divergence mitigation. Our work provides theoretical insight into the stability properties of bisimulation-based algorithms.

##### Bisimulation-based Representation Learning.

Recently, there has been lot of interest in $\pi$-bisimulation algorithms for better generalization (Ferns et al., ; ; Ferns & Precup, ; Castro, ; Zang et al., ). These algorithms measure similarity between two state-action pairs based on immediate rewards received and the similarity of next state-action pairs visited by $\pi$. These algorithms first define a distance metric that captures this $\pi$-bisimilarity, and then use this metric to learn representations such that $\pi$-bisimular states have similar representations (Castro et al., ; Castro, ; Zhang et al., ; Castro et al., ; Chen & Pan, ; Kemertas & Jepson, ). Zhang et al. (); Castro () introduced a $\pi$-bisimulation learning algorithm but assume that the transition dynamics are either deterministic or Gaussian. Gelada et al. () introduced a method closely related to bisimulation methods but required a reconstruction loss to work in practice. Castro et al. () introduced mico which allows for stochastic transition dynamics and no reconstruction loss, but was difficult to theoretically analyze. To overcome this difficulty, Castro et al. () took a kernel perspective of $\pi$-bisimulation methods, which made their algorithm amenable to theoretical analysis. To the best of our knowledge, no works have studied the stability properties of $\pi$-bisimulation algorithms. In our work, we address this gap in the literature. We first extend Castro et al. ()’s kernel-based formulation from states to state-actions, and then show that this formulation stabilizes offline value function learning. The proofs for krope’s basic theoretical properties (Section 3.1) follow those by Castro et al. (). Our stability-related theoretical results (Sections 3.2 and 3.3) and empirical analysis (Section 4) are novel to this work.

## 3 Kernel Representations for Offline Policy Evaluation

We now present our bisimulation-based representation learning algorithm, kernel representations for ope (krope). We present the desired krope kernel, define the krope operator, present its theoretical properties, prove stability properties of krope representations, and present a practical learning algorithm to learn them. We defer the proofs to Appendix B.

### 3.1 krope Kernel and Operator

Prior $\pi$-bisimulation works define similarity between states in terms of the immediate rewards received and similarity of next states under $\pi$ (see Figure 1) (Castro, ). In this work, we follow Castro et al. () and define a kernel $k^{{\pi_{e}}}:\mathcal{X}\times\mathcal{X}\to\mathbb{R}$ that captures this notion of similarity under ${\pi_{e}}$, but for pairs of state-actions. We refer to $k^{\pi_{e}}$ as the krope kernel.

$$ $\displaystyle k^{{\pi_{e}}}(s_{1},a_{1};s_{2},a_{2})=k_{1}(s_{1},a_{1};s_{2},a _{2})+\gamma k_{2}(k^{{\pi_{e}}})(P^{\pi_{e}}(\cdot|s_{1},a_{1}),P^{\pi_{e}}( \cdot|s_{2},a_{2})].$ (2) $$

where $k_{1}(s_{1},a_{1};s_{2},a_{2}):=1-\frac{|r(s_{1},a_{1})-r(s_{2},a_{2})|}{\left
|r_{\text{max}}-r_{\text{min}}\right|}$ and $k_{2}(k^{\pi_{e}})(P^{\pi_{e}}(\cdot|s_{1},a_{1}),P^{\pi_{e}}(\cdot|s_{2},a_{2
})):=\mathbb{E}_{s_{1}^{\prime},a_{1}^{\prime}\sim P^{\pi_{e}}(\cdot|s_{1},a_{
1}),s_{2}^{\prime},a_{2}^{\prime}\sim P^{\pi_{e}}(\cdot|s_{2},a_{2})}[k^{\pi_{
e}}(s_{1}^{\prime},a_{1}^{\prime};s_{2}^{\prime},a_{2}^{\prime})]$. Here, $k_{1}$ measures short-term similarity based on rewards received and $k_{2}$ measures long-term similarity between probability distributions by measuring similarity between samples of the distributions according to $k^{\pi_{e}}$ (Castro et al., ).

Given this definition of the krope kernel, we now present an operator that converges to $k^{\pi_{e}}$:

###### Definition 2 ( krope operator) .

Given a target policy ${\pi_{e}}$, the krope operator $\mathcal{F}^{\pi_{e}}:\mathbb{R}^{\mathcal{X}\times\mathcal{X}}\to\mathbb{R}^{
\mathcal{X}\times\mathcal{X}}$ is defined as follows: for each kernel $k:\mathcal{X}\times\mathcal{X}\to\mathbb{R}$, $\forall(s_{1},a_{1};s_{2},a_{2})\in\mathcal{X}\times\mathcal{X},$

$$ $\mathcal{F}^{\pi_{e}}(k)(s_{1},a_{1};s_{2},a_{2}):=\underbrace{\vphantom{ \mathbb{E}_{s_{1}^{\prime}}[k(s_{1}^{\prime})]}k_{1}(s_{1},a_{1};s_{2},a_{2})} _{\text{short-term similarity}}+\gamma\underbrace{\mathbb{E}_{s_{1}^{\prime},s _{2}^{\prime}\sim P,a_{1}^{\prime},a_{2}^{\prime}\sim\pi_{e}}[k(s_{1}^{\prime} ,a_{1}^{\prime};s_{2}^{\prime},a_{2}^{\prime})]}_{\text{long-term similarity}}$ (3) $$

where $s_{1}^{\prime}\sim P(s_{1}^{\prime}|s_{1},a_{1}),s_{2}^{\prime}\sim P(s_{2}^{
\prime}|s_{2},a_{2}),a_{1}^{\prime}\sim{\pi_{e}}(\cdot|s_{1}^{\prime}),a_{2}^{
\prime}\sim{\pi_{e}}(\cdot|s_{2}^{\prime})$, and $k_{1}(s_{1},a_{1};s_{2},a_{2}):=1-\frac{|r(s_{1},a_{1})-r(s_{2},a_{2})|}{\left
|r_{\text{max}}-r_{\text{min}}\right|}$ is a positive semidefinite kernel.

Proposition 2, proved in Appendix B.1, tells us that for some initial kernel $k$, repeatedly applying $\mathcal{F}^{\pi_{e}}$ to it will result in convergence $k^{{\pi_{e}}}$. Ultimately, $k^{\pi_{e}}$ outputs a high (or low) similarity measure for two state-action pairs if their action-values under ${\pi_{e}}$ are similar (or dissimilar). This intuition is formalized in Lemma 3, which states that the absolute action-value difference between any two state-action pairs under ${\pi_{e}}$ is upper-bounded by the distance function induced by $k^{\pi_{e}}$ plus an additive constant. Since krope’s contraction, metric space completeness, and fixed-point uniqueness properties are similar to Castro et al. ()’s kernel, we defer the details to Appendix B.1.

###### Definition 2 ( krope operator) .

### 3.2 Stability of krope Representations

In the previous section, we defined the krope kernel. Ultimately, however, we are interested in *representations* that satisfy the relationship in Equation (2). We modify Equation (2) accordingly by giving $k^{\pi_{e}}$ some functional form in terms of state-action representations. We do so with the dot product: $\langle u,v\rangle=u^{\top}v,\forall u,v\in\mathbb{R}^{d}$, i.e., $k^{\pi_{e}}(s_{1},a_{1};s_{2},a_{2})=\phi(s_{1},a_{1})^{\top}\phi(s_{2},a_{2})$. With this setup, we write Equation (2) in matrix notation and define the krope representations as follows:

###### Definition 3 ( krope Representations) .

Consider state-action representations $\Phi\in\mathbb{R}^{|\mathcal{X}|\times d}$ that are embedded in $\mathbb{R}^{d}$ with $k^{\pi_{e}}(s_{1},a_{1};s_{2},a_{2})=\phi(s_{1},a_{1})^{\top}\phi(s_{2},a_{2})$. We say $\Phi$ is a krope representation if it satisfies the following:

$$ $\mathbb{E}_{\mathcal{D}}[\Phi\Phi^{\top}]=\mathbb{E}_{\mathcal{D}}[K_{1}]+ \gamma\mathbb{E}_{\mathcal{D},{\pi_{e}}}[P^{{\pi_{e}}}\Phi(P^{{\pi_{e}}}\Phi)^ {\top}]$ (4) $$

where each entry of $K_{1}\in\mathbb{R}^{|\mathcal{X}|\times|\mathcal{X}|}$ represents the short-term similarity, $k_{1}$, between every pair of state-actions, i.e., $K_{1}(s_{1},a_{1};s_{2},a_{2}):=1-\frac{|r(s_{1},a_{1})-r(s_{2},a_{2})|}{\left
|r_{\text{max}}-r_{\text{min}}\right|}$.

Given this definition, we present our novel result proving the stability of krope representations:

###### Theorem 1 .

Theorem 1, proved in Appendix B.2, tells us that krope representations stabilize ope with lspe. Intuitively, they are stable since when $(\mathbb{E}[\Phi^{\top}\Phi]))^{-1}\mathbb{E}[\gamma\Phi^{\top}P^{{\pi_{e}}}\Phi]$’s spectral radius is less than $1$, each update to $\theta_{t}$ in Equation (1) is non-expansive. That is, each update brings $\theta_{t}$ closer to $\theta_{\textsc{lspe}}$.

###### Definition 3 ( krope Representations) .

###### Theorem 1 .

### 3.3 Connection to Bellman Completeness

In this section, we draw a novel connection between krope representations and Bellman completeness. Chen & Jiang () proved a similar result but focused on bisimulation representations instead of krope representations, which are $\pi$-bisimulation-like representations. We say a function class $\mathcal{F}$ is Bellman complete if it is complete under the Bellman operator: $\mathcal{T}^{{\pi_{e}}}f\subseteq\mathcal{F},\forall f\in\mathcal{F}$. For instance, suppose $\mathcal{F}$ is the class of linear functions spanned by $\Phi$, $\mathcal{F}:=\{f\in\mathbb{R}^{\mathcal{X}}:f:=\Phi w\},w\in\mathbb{R}^{d}$. Then if $\mathcal{T}^{{\pi_{e}}}f,\forall f\in\mathcal{F}$ is also a linear function within the span of $\Phi$, we say $\Phi$ is a Bellman complete representation. Bellman completeness is an alternative condition for stability and is typically assumed to ensure to data-efficient policy evaluation (Wang et al., ; Szepesvári & Munos, ; Chang et al., ).

Let the induced distance function due to $k^{\pi_{e}}$ be

$$ $\forall x,y\in\mathcal{X}:\;d_{\text{{krope}}}(x,y):=k^{{\pi_{e}}}(x,x)+k^{{ \pi_{e}}}(y,y)-2k^{{\pi_{e}}}(x,y).$ $$

We now present our second main result. It states that krope representations are Bellman complete:

###### Theorem 2 .

###### Theorem 2 .

### 3.4 krope Learning Algorithm

In this section, we present an algorithm that learns the krope representations from data. We include the pseudo-code of krope in Appendix A. The krope learning algorithm uses an encoder $\phi_{\omega}:\mathcal{S}\times\mathcal{A}\to\mathbb{R}^{d}$, which is parameterized by weights $\omega$ of a function approximator. It then parameterizes the kernel with the dot product, i.e, $\tilde{k}_{\omega}(s_{1},a_{1};s_{2},a_{2}):=\phi_{\omega}(s_{1},a_{1})^{\top}
\phi_{\omega}(s_{2},a_{2})$ (see Equation (4)). Finally, the algorithm then minimizes the following loss function, which is similar to how the value function is learned in deep rl (Mnih et al., ):

$$ $\mathcal{L}_{\text{{krope}}}(\omega):=\mathbb{E}_{\mathcal{D}}\Biggl{[}\biggl{ (}\underbrace{1-\frac{\left|r(s_{1},a_{1})-r(s_{2},a_{2})\right|}{\left|r_{ \text{max}}-r_{\text{min}}\right|}+\gamma\mathbb{E}_{{\pi_{e}}}[\tilde{k}_{ \bar{\omega}}(s_{1}^{\prime},a_{1}^{\prime};s_{2}^{\prime},a_{2}^{\prime})]}_{ \text{target estimate}}-\underbrace{\vphantom{\bigg{|}}\tilde{k}_{\omega}(s_{1 },a_{1};s_{2},a_{2})}_{\text{current estimate}}\biggr{)}^{2}\Biggr{]},$ (5) $$

where the state-action pairs are sampled from $\mathcal{D}$, and $\bar{\omega}$ are weights of the target network that are periodically copied from $\omega$ (Mnih et al., ). In this work, we use krope as an auxiliary task, which introduces only a learning rate as an additional hyperparameter. We note that this fixed-point optimization procedure is similar to how the action-value function is learned in other rl fixed-point algorithms such as fitted q-evaluation (fqe) (Le et al., ).

## 4 Empirical Results

In this section, we present our empirical study designed to answer the following questions.

- 1.
Does krope lead to stable representations with good realization and generalization?
- 2.
Do krope representations lead to stable msve and low msve?

### 4.1 Empirical Setup

In this section, we describe the main details of our empirical setup. For further details such as datasets, policies, hyperparameters, and evaluation protocol please refer to Appendix C.

##### Baselines

Our primary representation learning baseline is fitted q-evaluation (fqe) (Le et al., ). fqe is the most fundamental deep rl ope algorithm that learns representations of state-actions to predict the long-term performance of a policy. While fqe is typically used as an ope algorithm, it can also be viewed as a value-predictive representation learning algorithm (Lehnert & Littman, ). More specifically, consider its loss function: $\mathbb{E}_{(s,a,s^{\prime})\sim\mathcal{D}}\left[\left(r(s,a)+\gamma\mathbb{E
}_{a^{\prime}\sim{\pi_{e}}}[q_{\bar{\xi}}(s^{\prime},a^{\prime})]-q_{\xi}(s,a)
\right)^{2}\right]$, where $q_{\xi}(s,a):=\phi_{\xi^{\prime}}(s,a)^{\top}w$ and $\xi=\{\xi^{\prime},w\}$. We view $\xi$ as the neural network weights of an action-value neural network and $w$ as the linear weights of the network applied on the output of the penultimate layer $\phi_{\xi^{\prime}}(s,a)$ of the neural network. Then minimizing this loss function shapes the representations $\phi_{\xi^{\prime}}(s,a)$ to predict the expected future discounted return. As noted in Section 2, we follow the linear evaluation protocol where $\phi_{\xi^{\prime}}$ is shaped by different auxiliary tasks and is then used with lspe for ope since it helps us understand the properties of the representations within the context of a well-understood value function learning algorithm (Grill et al., ; Chang et al., ; Farebrother et al., ; Wang et al., ). We provide the pseudocode of this setup in Appendix A. We also note that in Appendix C, we present results of performing ope using fqe instead lspe, and find that krope still reliably produces stable ope estimates.

We consider the following four non-bisimulation auxiliary representation learning algorithms that are typically paired with fqe for stability: 1) krope (ours), 2) bcrl-exp-na, which simultaneously optimizes three objectives: immediate reward prediction, next-state self-prediction loss, and maximization of the exploratory nature of $\mathbb{E}[\Phi^{\top}\Phi]$ by improving its condition number (Chang et al., ), 3) Absolute dr3 regularizer (Kumar et al., ; Ma et al., ), which promotes stability by minimizing the feature co-adaptation between successive features, i.e., $\phi(s,a)^{\top}\phi(s^{\prime},a^{\prime})$, and 4) beer regularizer (He et al., ), which is similar to dr3 but lower bounds $\phi(s,a)^{\top}\phi(s^{\prime},a^{\prime})$. In all cases, the penultimate layer features of fqe’s action-value encoder $\phi_{\xi^{\prime}}$ are fed into lspe for ope. Our experiments focus on analyzing the properties of $\phi_{\xi^{\prime}}$. Note that since bcrl was not designed as an auxiliary task (Chang et al., ), we evaluate it as a non-auxiliary (na) task algorithm. We provide additional details on the baselines in Appendix C.

##### Domains

We conduct our evaluation on a variety of domains: 1) Garnet mdps, which are a class of tabular stochastic mdps that are randomly generated given a fixed number of states and actions (Archibald et al., ); 2) $4$ dm Control environments: CartPoleSwingUp, CheetahRun, FingerEasy, WalkerStand (Tassa et al., ); and 3) $9$ d4rl datasets (Fu et al., ; ). The first domain enables us to analyze the algorithms’ performance across a wide range of stochastic tabular mdps. The second and third set of domains test the algorithms in continuous higher-dimensional state-action environments. Due to space constraints, we defer the d4rl results to Appendix C.3.2.

### 4.2 Analyzing Fundamental Properties of the Learned Representations

In this set of experiments on the Garnet mdps domain, we answer our first question of whether krope representations lead to stable representations with good realization and generalization properties. We present the results in Figure 2. Our Garnet mdps were generated with $8$ states and $5$ actions, with a total of $|\mathcal{X}|=40$ state-actions, and each native $(s,a)$ representation is a $1$-hot vector. In these experiments, the native representation is fed into a linear encoder with a bias component and no activation function. All algorithms are trained for $500$ epochs and we report the results by evaluating the final learned representations for different latent dimensions $d$.

Figure: (a) Spectral Radius
Refer to caption: extracted/5973682/results/randommdp/RandomMDP_spectral.jpg

##### Stability.

Based on Theorem 1, a representation is stable if it induces a spectral radius of $(\mathbb{E}[\Phi^{\top}\Phi])^{-1}(\gamma\mathbb{E}[\Phi^{\top}P^{{\pi_{e}}}
\Phi])$ that is less than $1$. In Figure 2(a), we present the fraction of runs that result in such representations. We find that up till $d=30$, $100\%$ of krope and beer runs have spectral radius less than $1$. We also find that bcrl-exp-na produces stable representations up till $d=40$. At $d=50$, all algorithms produce unstable representations. These results suggest that krope, beer, and bcrl-exp-na are reliable in producing stable representations when projecting state-actions into low dimensions. When $d\geq|\mathcal{X}|$, the covariance matrix $\mathbb{E}[\Phi^{\top}\Phi]$ is more likely to be a singular matrix, which implies higher chance of instability.

##### Realizability.

A basic criterion for learning $q^{\pi_{e}}$ is realizability. That is, we want ${\epsilon}:=\|\Phi\hat{w}-q^{{\pi_{e}}}\|^{2}_{2}$, where $\hat{w}:=\operatorname*{arg\,min}_{w}\|\Phi w-q^{{\pi_{e}}}\|_{2}^{2}$, to be low. In our experiments, we compute ${\epsilon}$ and plot it as a function of $d$ in Figure 2(b). A critical message from our results is that stability and realizability do not always go hand-in-hand. While bcrl-exp-na has favorable spectral radius properties (Figure 2(a)), it has poor realizability, which will negatively affect its ope accuracy. krope, on the other hand, has favorable stability and realizability properties up till $d=30$. When $d\geq 40$, the realizability error is 0 0 for all algorithms since the subspace spanned by $\Phi$ is large enough to contain the true action-value function (Ghosh & Bellemare, ). While the realizability error is 0 0 for $d\geq 40$, the representations can be unstable (Figure 2(a)).

##### Generalization.

Finally, we say that the representations have generalized well when state-actions that have similar $q^{\pi_{e}}$ values are close to each other in the representation space (Lyle et al., ). We assess generalization by measuring the orthogonality: $1-\frac{|\langle\phi(s_{1},a_{1}),\phi(s_{2},a_{2})\rangle|}{\|\phi(s_{1},a_{1
})\|\|\phi(s_{2},a_{2})\|}$ (Wang et al., ) between every state-action pair, $(s_{1},a_{1};s_{2},a_{2})$, and the absolute action-value difference: $|q^{\pi_{e}}(s_{1},a_{1})-q^{\pi_{e}}(s_{2},a_{2})|$. We then compute the Pearson correlation between these values for every pair and plot the correlation for each $d$ in Figure 2(c). A correlation coefficient close to $1$ indicates that the representations generalize well. We find that krope representations satisfy this property almost perfectly since it specifically tries to learn representations such that state-action pairs with similar values under ${\pi_{e}}$ are similar. We observe that the other algorithms typically have zero or even negative correlation. A negative correlation indicates that state-actions with different action-values may be similar in latent space, which can result in higher realizability error (Figure 2(b)). A near-zero correlation but low realizability error such as in the case of fqe implies that accurate offline value prediction is still possible but that it generalizes poorly, which may slow down convergence to its ope solution (Lyle et al., ).

### 4.3 Offline Policy Evaluation

In this set of experiments, we conduct experiments on four DM control environments to answer our second empirical question: whether krope representations lead to stable and low msve?
We also evaluate bcrl-na, which is bcrl without the exploration maximization regularizer. To stabilize training for all algorithms, we use wide neural networks with layernorm (Gallici et al., ; Ota et al., ). Note that while wide networks and layernorm stabilize training, they may not lead to stable lspe under the linear evaluation protocol. During representation learning, we periodically evaluate the learned representations for ope using lspe. The corresponding (normalized) squared value errors are presented in Figure 3.

Figure: (a) CartPoleSwingUp
Refer to caption: extracted/5973682/results/dmcontrol/ope/CartPoleSwingUp_ope.jpg

In general, we find that krope representations lead to low and stable msve. On the other hand, we find that the other auxiliary tasks inconsistently produce stable ope estimates across all environments. For example, the performance of the abs-dr3 and beer regularizer suggests that explicitly trying to increase the rank of the features of the penultimate layer may hurt stability, and even if the ope error is stable, it can hurt accuracy. We also make a similar observation for bcrl. However, in this case, we attribute poor performance to difficulty in optimizing the bcrl objective. In fact, in Figure 4(c), we will see that bcrl is sensitive to hyperparameter tuning. We also observe results consistent with a known result that bcrl-exp-na performs better than bcrl-na indicating the known result that exploration maximization of the covariance matrix helps produce stable representations (Chang et al., ). Finally, while fqe achieves lower ope error than krope on WalkerStand, it is very unstable on CartPoleSwingUp and Finger Easy, which motivates the need to shape the representations for stable and accurate ope. We note that in the WalkerStand instance, the higher error of krope is unsurprising since Lemma 3 suggests that krope may lose realizability (see Appendix B.1). We refer the reader to Appendix C.1 for details of each algorithm. We also conduct the same experiment on $9$ d4rl datasets and reach the similar conclusions (see Appendix C.3.2).

Figure: (a) Feature Co-adaptation
Refer to caption: extracted/5973682/results/dmcontrol/featcoadapt/CartPoleSwingUp_coadapt.jpg

#### 4.3.1 Analyzing Stability-Related Metrics

While krope performs well on the downstream task of ope, it is important to analyze upstream stability-related metrics during the course of learning. These results give insight into the properties of the learned representations. We present the results in Figure 4 for CartPoleSwingUp and defer the remaining results to the Appendix C.

Feature co-adaptation. The feature-co-adaptation metric, i.e., $\sum_{(s,a,s^{\prime})\in\mathcal{D},a^{\prime}\sim{\pi_{e}}}\phi(s,a)^{\top}
\phi(s^{\prime},a^{\prime})$ was shown to correlate with instability as high values can indicate representation collapse (Kumar et al., ). From Figure 4(a), we find that krope mitigates representation collapse of fqe (lowers feature co-adaptation from $10^{6}$ to $10^{5}$). Its relatively moderate value indicates that it is able to balance generalization and maintain distinctness between state-action representations. While low co-adaptation is preferred, extremely low values such as those achieved by abs-dr3 do not necessarily lead to accurate ope (see Figure 3). These results tell us which algorithms are more prone to representation collapse.

Condition number. Another metric that we analyze is the condition number of the covariance matrix $M:=\mathbb{E}[\Phi^{\top}\Phi]$. A low condition number indicates numerical stability of the learning algorithm, i.e., solutions recovered by lspe (see Equation (1)) or general td algorithms (Asadi et al., ) are less likely to change with small perturbations to $M$ (Chang et al., ). From Figure 4(b), we find that krope achieves lower condition number than fqe (from $10^{4}$ to $10^{1}$). As excepted, we observe that bcrl-exp-na achieves an even lower condition since it explicitly optimizes for a well-conditioned covariance matrix ($10^{4}$ to $10^{3}$). While there is no ideal value, these results indicate the reliability of the representations in enabling td-based algorithms to recover their value function solution. We observe qualitatively similar results for other environments (see Appendix C).

Hyperparameter Sensitivity. In ope, hyperparameter tuning can be challenging since it may be infeasible to get access to ground truth performance of ${\pi_{e}}$ (Fu et al., ). Therefore, we prefer algorithms that are robust to hyperparameter tuning, i.e, they reliably produce accurate ope estimates for a wide range of hyperparameters. In Figure 4(c), we present the performance profile for each algorithm across *all* hyperparameter combinations and *all* trials (Agarwal et al., ). We tune the hyperparameters discussed in Appendix C.1. We find that $100\%$ krope runs across all instances produce msve $\leq 1$, which is not the case with other algorithms.

## 5 Limitations and Future Work

In this section, we discuss limitations and future work. A shortcoming of our work is that krope’s learning algorithm is susceptible to instability since it is a semi-gradient method (Sutton & Barto, ). Moreover, its fixed-point optimization means it does not solve any objective function (Feng et al., ). In our work, we employed commonly-used techniques such as layernorm and wide neural networks to mitigate instability (Ota et al., ; Gallici et al., ). While these techniques potentially side-step the issue, the consequences of a semi-gradient method may still exist. In Appendix C.3.1, we present an empirical analysis to gauge when krope’s learning algorithm may be unstable. We find that while *individual* off-policy transitions can determine the instability of fixed-point and semi-gradient algorithms such as fqe, *pairs* of off-policy transitions can determine krope’s instability. Since we are unlikely to have control over the distribution over pairs of transitions in practice, we need to resort to fundamental changes to the algorithm. One potential change is based on that by Feng et al. (). Their key insight is to leverage the Legendre-Fenchel transformation from optimization theory and replace the fixed-point loss function of semi-gradient methods with an equivalent expression that avoids semi-gradient learning (Rockafellar & Wets, ). However, a drawback with this approach is that the new learning objective is a minimax procedure, which can be challenging to optimize in practice. In future work, we will explore the viability of this approach to design a provably convergent version of krope.

## 6 Conclusion

In this work, we tackled the problem of stabilizing offline value function learning in reinforcement learning. We introduced a bisimulation-based representation learning algorithm, kernel representations for ope (krope), that shapes the state-action representations to stabilize this procedure. Theoretically, we showed that krope representations are stable from two perspectives: 1) non-expansiveness, i.e., they lead to value function learning updates that enable convergence to a fixed-point and 2) Bellman completeness, i.e., they satisfy a condition for data-efficient policy evaluation. Empirically, we showed that krope leads to more stable and accurate offline value function learning than baselines. Our work showed that bisimulation-based representation learning effectively stabilizes long-term performance evaluations of offline reinforcement learning agents.

## Acknowledgements

The authors thank Abhinav Narayan Harish, Adam Labiosa, Andrew Wang, Anshuman Senapati, David Abel, Kyle Domico, Ishan Durugkar, and Prabhat Nagarajan for feedback on earlier versions of this draft, and thank Fanghui Liu for inspiring discussions on kernel methods. Y. Chen is partially supported by NSF CCF-1704828 and NSF CCF-2233152.

## Appendix A Background

In this section, we present the theoretical background.

### A.1 Bisimulation Metrics

In this section, we present background information on bisimulations and its associated metrics. Our proposed representation learning algorithm is a bisimulation-based algorithm. Bisimulation abstractions are those under which two states with identical reward functions and that lead to identical groups of next states under any action are classified as similar (Ferns et al., ; ; Ferns & Precup, ). Bisimulations are the strictest forms of abstractions. In practice, the exact bisimulation criterion is difficult to satisfy computationally and statistically. A more relaxed version of this notion is the *$\pi$-bisimulation metrics*. These metrics capture the similarity between two states such that two states are considered similar if they have identical *expected* reward functions and *expected* transitions to identical groups of next states under a *fixed policy $\pi$* (Castro, ).

We first give the definition of bisimulation.

###### Definition 4 .

(Li et al., ) An abstraction $\phi:\mathcal{S}\to\mathcal{S}^{\phi}$ over the state space $\mathcal{S}$ is a bisimulation if for any action $a$ and any abstract state $s^{\phi}\in\mathcal{S}^{\phi}$, $\phi$ is such that for any two states $s_{1},s_{2}\in\mathcal{X}$, $\phi(s_{1})=\phi(s_{2})$ implies that $r(s_{1},a)=r(s_{2},a)$ and $\sum_{s^{\prime}\in s^{\phi}}P(s^{\prime}|s_{1},a)=\sum_{s^{\prime}\in s^{\phi
}}P(s^{\prime}|s_{2},a)$.

Below we define $\pi$-bisimulations for state-actions instead of states:

###### Definition 5 .

(Castro, ) An abstraction $\phi:\mathcal{X}\to\mathcal{X}^{\phi}$ over the state-action space $\mathcal{X}$ is a $\pi$-bisimulation for a fixed policy $\pi$ if for any two state-actions $x,y\in\mathcal{X}$ and abstract state-action $x^{\phi}\in\mathcal{X}^{\phi}$, $\phi$ is such that $\phi(x)=\phi(y)$ implies that $r(x)=r(y)$ and $\sum_{x^{\prime}\in x^{\phi}}P^{\pi}(x^{\prime}|x)=\sum_{x^{\prime}\in x^{\phi
}}P^{\pi}(x^{\prime}|y)$.

The above definitions are based on exact groupings between state-actions. This strictness motivates the use of bisimulation and $\pi$-bisimulation metrics, which we define below.

###### Theorem 3 .

(Ferns et al., ) Let $\mathcal{M}(\mathcal{S})$ be the space of bounded pseudometrics on the state-space $\mathcal{S}$. Then define $\mathcal{B}:\mathcal{M}(\mathcal{S})\to\mathcal{M}(\mathcal{S})$ such that for each $d\in\mathcal{M}(\mathcal{S})$:

$$ $\mathcal{B}(d)(s_{1},s_{2})=\max_{a\in\mathcal{A}}(|r(s_{1},a)-r(s_{2},a)|+ \gamma\mathcal{W}(d)(P(\cdot|s_{1},a),P(\cdot|s_{2},a)),$ $$

where $\mathcal{W}$ is Wasserstein distance between the two distributions under metric $d$. Then $\mathcal{B}$ has a unique fixed point, $d^{*}$, and $d^{*}$ is a bisimulation metric.

Similarly, we have the $\pi$-bisimulation metric:

###### Theorem 4 .

(Castro, ) Let $\mathcal{M}(\mathcal{X})$ be the space of bounded pseudometrics on the state-action space $\mathcal{X}$ and $\pi$ be a fixed policy. Then define $\mathcal{B}:\mathcal{M}(\mathcal{X})\to\mathcal{M}(\mathcal{X})$ such that for each $d\in\mathcal{M}(\mathcal{S})$:

$$ $\mathcal{B}(d)(x,y)=|r(x)-r(y)|+\gamma\mathcal{W}(d)(P^{\pi}(\cdot|x),P^{\pi}( \cdot|y)),$ $$

where $\mathcal{W}$ is Wasserstein distance between the two distributions under metric $d$. Then $\mathcal{B}$ has a unique fixed point, $d^{*}$, and $d^{*}$ is a $\pi$-bisimulation metric.

Using the above metrics, prior works have introduced several representation learning algorithms to learn representations such that the distance between representations in latent space model the above distance metrics (Castro et al., ; ; Zhang et al., ; Kemertas & Aumentado-Armstrong, ; Pavse & Hanna, ).

###### Definition 4 .

###### Definition 5 .

###### Theorem 3 .

###### Theorem 4 .

### A.2 Reproducing Kernel Hilbert Spaces

Let $\mathcal{X}$ be a finite set and define a function $k:\mathcal{X}\times\mathcal{X}\to\mathbb{R}$ to be a positive semidefinite kernel if it is symmetric and positive semidefinite. We then have for any $\{x_{1},x_{2},...,x_{n}\}\in\mathcal{X}$ and $\{c_{1},c_{2},...,c_{n}\}\in\mathbb{R}$:

$$ $\sum_{i=1}^{n}\sum_{j=1}^{n}c_{i},c_{j}k(x_{i},x_{j})\geq 0$ $$

Note that if the above inequality is strictly greater than zero whenever $\{c_{1},\ldots,c_{n}\}$ has at least one nonzero, we say the kernel is positive definite. Given a kernel $k$ on $\mathcal{X}$ with the reproducing property, we can construct a space of functions $\mathcal{H}_{k}$ referred to as a reproducing kernel Hilbert space (rkhs) with the following steps:

- 1.
Construct a vector space of real-valued functions on $\mathcal{X}$ of the form $\{k(x,\cdot):x\in\mathcal{X}\}$.
- 2.
Equip this space with an inner product given by $\langle k(x,\cdot),k(y,\cdot)\rangle_{\mathcal{H}_{k}}=k(x,y)$.
- 3.
Take the completion of the vector space with respect to the above inner product.

Our resulting vector space $\mathcal{H}_{k}$ is then an rkhs.

It is often convenient to write $\psi(x):=k(x,\cdot)\in\mathcal{H}_{k}$, which is called the feature map and is an embedding of $x$ in $\mathcal{H}_{k}$. One can also embed probability distributions into $\mathcal{H}_{k}$. That is, $\Phi:\mathcal{P}(\mathcal{X})\to\mathcal{H}_{k}$, which maps probability distributions over $\mathcal{X}$ to $\mathcal{H}_{k}$. We define $\Phi(\mu)=\mathbb{E}_{X\sim\mu}[\psi(X)]$, which is the mean embedding in $\mathcal{H}_{k}$ under $\mu$.

Given these embeddings in the Hilbert space, we can quantify the distances between elements in $\mathcal{X}$ and $\mathcal{P}(\mathcal{X})$ in terms of the embeddings.

###### Definition 6 .

Given a positive semidefinite kernel $k$, define $\rho_{k}$ as its induced distance:

$$ $\rho_{k}:=\|\psi(x)-\psi(y)\|_{\mathcal{H}_{k}}.$ $$

By expanding the inner product, the squared distance can be written in terms of $k$:

$$ $\rho^{2}_{k}(x,y)=k(x,x)+k(y,y)-2k(x,y).$ $$

Similarly, we have distances on $\mathcal{P}(\mathcal{X})$ using $\Phi$:

###### Definition 7 .

(Gretton et al., ) Let $k$ be a kernel on $\mathcal{X}$ and $\Phi:\mathcal{P}(\mathcal{X})\to\mathcal{H}_{k}$ be as defined above. Then the Maximum Mean Discrepancy (MMD) is a pseudo metric on $\mathcal{P}(\mathcal{X})$ defined by:

$$ $\text{MMD}(k)(\mu,\nu)=\|\Phi(\mu)-\Phi(\nu)\|_{\mathcal{H}_{k}}.$ $$

The core usage of the rkhs is to precisely characterize the nature of the krope kernel. In practice, we deal with neural network representations, which are embedded in Euclidean space. Therefore, our goal is to learn representations in Euclidean space that approximate the properties of representations in the rkhs. For more details on the rkhs, we refer readers to Castro et al. () and Gretton et al. ().

###### Definition 6 .

###### Definition 7 .

### A.3 Algorithm Pseudocode

In this section, we present the pseudocode for lspe and for our fqe + auxiliary task with lspe for ope setup.

Figure: Algorithm 1 lspe

Figure: Algorithm 2 fqe + representation learning auxiliary task with lspe for ope

## Appendix B Theoretical Results

In this section, we present the proofs of our main and supporting theoretical results. The first set of proofs in Section B.1 show that krope is a valid operator. While new to our work, the proofs follow those by Castro et al. (). The next set of proofs in Section B.2 prove the stability of krope representations and are novel to our work. For presentation purposes, it will often be convenient to refer to a state-action pair as $x\in\mathcal{X}$ instead of $(s,a)$.

### B.1 krope Operator Validity

We now present the proofs demonstrating the validity of the krope operator. All the proofs in this sub-section model those by Castro et al. (). The primary difference is that our operator is for state-actions instead of states.

###### Lemma 1 .

Let $\mathcal{K}(\mathcal{X})$ be the space of positive semidefinite kernels on $\mathcal{X}$. The krope operator $\mathcal{F}^{\pi_{e}}$ is a contraction with modulus $\gamma$ in $\|\cdot\|_{\infty}$.

###### Proof.

Let $k_{1},k_{2}\in\mathcal{K}(\mathcal{X})$. We then have:

$$ $\displaystyle\|\mathcal{F}^{\pi_{e}}(k_{1})-\mathcal{F}^{\pi_{e}}(k_{2})\|_{\infty}$ $\displaystyle=\max_{(x,y)\in\mathcal{X}\times\mathcal{X}}|\mathcal{F}^{\pi_{e} }(k_{1})(x,y)-\mathcal{F}^{\pi_{e}}(k_{2})(x,y)|$ $\displaystyle=\gamma\max_{(x,y)\in\mathcal{X}\times\mathcal{X}}|\mathbb{E}_{X^ {\prime}\sim P^{\pi_{e}}(\cdot|x),Y^{\prime}\sim P^{\pi_{e}}(\cdot|y)}[k_{1}(X ^{\prime},Y^{\prime})]-\mathbb{E}_{X^{\prime}\sim P^{\pi_{e}}(\cdot|x),Y^{ \prime}\sim P^{\pi_{e}}(\cdot|y)}[k_{2}(X^{\prime},Y^{\prime})]|$ $\displaystyle=\gamma\max_{(x,y)\in\mathcal{X}\times\mathcal{X}}|\mathbb{E}_{X^ {\prime}\sim P^{\pi_{e}}(\cdot|x),Y^{\prime}\sim P^{\pi_{e}}(\cdot|y)}[k_{1}(X ^{\prime},Y^{\prime})-k_{2}(X^{\prime},Y^{\prime})]|$ $\displaystyle\leq\gamma\|k_{1}-k_{2}\|_{\infty}.$ $$

This completes the proof of the lemma.
∎

###### Lemma 2 .

Let $\mathcal{K}(\mathcal{X})$ be the space of positive semidefinite kernels on $\mathcal{X}$. Then the metric space $(\mathcal{K}(\mathcal{X}),\|\cdot\|_{\infty})$ is complete.

###### Proof.

To show that $\mathcal{K}(\mathcal{X})$ is complete it suffices to show that every Cauchy sequence $\{k_{n}\}_{n\geq 0}$ has a limiting point in $\mathcal{K}(\mathcal{X})$. Since $\mathcal{X}$ is a finite, the space of function $\mathbb{R}^{\mathcal{X}\times\mathcal{X}}$ is a finite-dimensional vector space, which is complete under $\|\cdot\|_{\infty}$. Thus, the limiting point $k\in\mathbb{R}^{\mathcal{X}\times\mathcal{X}}$ of the Cauchy sequence $\{k_{n}\}_{n\geq 0}$ lies in $\mathbb{R}^{\mathcal{X}\times\mathcal{X}}$. Moreover, since we are considering only positive semidefinite kernel elements in the Cauchy sequence and they uniformly converge to $k\in\mathbb{R}^{\mathcal{X}\times\mathcal{X}}$, $k$ must also be positive semidefinite. Thus, $\mathcal{K}(\mathcal{X})$ is complete under $\|\cdot\|_{\infty}$.
∎

###### Proposition 2 .

The krope operator $\mathcal{F}^{\pi_{e}}$ has a unique fixed point in $\mathcal{K}(\mathcal{X})$. That is, there is a unique kernel $k^{\pi_{e}}\in\mathcal{K}(\mathcal{X})$ satisfying

$$ $k^{{\pi_{e}}}(s_{1},a_{1};s_{2},a_{2})=1-\frac{|r(s_{1},a_{1})-r(s_{2},a_{2})| }{\left|r_{\text{max}}-r_{\text{min}}\right|}+\gamma\mathbb{E}_{s_{1}^{\prime} ,s_{2}^{\prime}\sim P,a_{1}^{\prime},a_{2}^{\prime}\sim\pi_{e}}[k^{{\pi_{e}}}( s_{1}^{\prime},a_{1}^{\prime};s_{2}^{\prime},a_{2}^{\prime})].$ $$

###### Proof.

Due to Lemmas 1 and 2, $\mathcal{F}^{\pi_{e}}$ is a contraction in a complete metric space. Therefore, by Banach’s fixed point theorem, the unique fixed point $k^{{\pi_{e}}}$ exists.
∎

Before presenting the remaining results, we define the distance metric $d_{\text{{krope}}}:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$ induced by the krope kernel $k^{{\pi_{e}}}$ as follows:

$$ $\forall x,y\in\mathcal{X}:\;d_{\text{{krope}}}(x,y):=k^{{\pi_{e}}}(x,x)+k^{{ \pi_{e}}}(y,y)-2k^{{\pi_{e}}}(x,y).$ $$

###### Proposition 3 .

The krope similarity metric $d_{\textsc{krope}}$ satisfies:

$$ $\forall x,y\in\mathcal{X},d_{\textsc{krope}}(x,y)=|r(x)-r(y)|+\gamma\text{MMD} ^{2}(k^{\pi_{e}})(P^{\pi_{e}}(\cdot|x),P^{\pi_{e}}(\cdot|y)).$ $$

###### Proof.

To see this fact, we can write out the squared Hilbert space distance:

$$ $\displaystyle d_{\textsc{krope}}(x,y)$ $\displaystyle=\|\psi^{\pi_{e}}(x)-\psi^{\pi_{e}}(y)\|^{2}_{\mathcal{H}^{\pi_{e }}_{k}}$ $\displaystyle=k^{\pi_{e}}(x,x)+k^{\pi_{e}}(y,y)-2k^{\pi_{e}}(x,y)$ $\displaystyle\begin{aligned} =|r(x)-r(y)|+\gamma\langle\Phi(P^{\pi_{e}}(\cdot| x)),\Phi(P^{\pi_{e}}(\cdot|x))\rangle_{\mathcal{H}^{\pi_{e}}_{k}}+\gamma \langle\Phi(P^{\pi_{e}}(\cdot|y)),\Phi(P^{\pi_{e}}(\cdot|y))\rangle_{\mathcal{ H}^{\pi_{e}}_{k}}\\ -2\gamma\langle\Phi(P^{\pi_{e}}(\cdot|x)),\Phi(P^{\pi_{e}}(\cdot|y))\rangle_{ \mathcal{H}^{\pi_{e}}_{k}}\end{aligned}$ $\displaystyle=|r(x)-r(y)|+\gamma\text{MMD}^{2}(k^{\pi_{e}})(P^{\pi_{e}}(\cdot| x),P^{\pi_{e}}(\cdot|y)),$ $$

where the third line uses

$$ $k^{\pi_{e}}(x,x)=\gamma\mathbb{E}_{X_{1}^{\prime},X_{2}^{\prime}\sim P^{\pi_{e }}(\cdot|x)}[k^{{\pi_{e}}}(X_{1}^{\prime},X_{2}^{\prime})]=\gamma\langle\Phi(P ^{\pi_{e}}(\cdot|x),P^{\pi_{e}}(\cdot|x)\rangle_{\mathcal{H}^{\pi_{e}}_{k}}.$ $$

This completes the proof.
∎

###### Lemma 3 .

We have
$|q^{{\pi_{e}}}(x)-q^{{\pi_{e}}}(y)|\leq d_{\text{{krope}}}(x,y)+C,$
where $C=\frac{1}{2}\sum_{n\geq 0}\gamma^{n}(\Delta_{n}^{{\pi_{e}}}(x)+\Delta_{n}^{{
\pi_{e}}}(y))$ and $\Delta_{n}^{{\pi_{e}}}(x)=\mathbb{E}_{X^{{}^{\prime}}\sim(P^{{\pi_{e}}}(\cdot|
x))^{n}}\big{[}\mathbb{E}_{X_{1}^{{}^{\prime\prime}},X_{2}^{{}^{\prime\prime}}
\sim P^{{\pi_{e}}}(\cdot|X^{\prime})}\big{[}|r(X_{1}^{{}^{\prime\prime}})-r(X_
{2}^{{}^{\prime\prime}})|\big{]}\big{]}$.

###### Proof.

We will prove this with induction. We first define the relevant terms involved. We consider the sequences of functions $\{k_{m}\}_{m\geq 0}$ and $\{q_{m}\}_{m\geq 0}$, where $k_{0},q_{0}=0$. Since $\mathcal{F}^{\pi_{e}}$ and $\mathcal{T}^{\pi_{e}}$ are contraction mappings, we know that $\lim_{m\to\infty}k_{m}=k^{\pi_{e}}$ and $\lim_{m\to\infty}q_{m}=q^{\pi_{e}}$ as $\mathcal{F}^{\pi_{e}}$ and $\mathcal{T}^{\pi_{e}}$ are applied respectively at each iteration $m$. At the $m$th application of the operators, we have the corresponding kernel function $k_{m}$ along with its induced distance function $d_{m}(x,y)=k_{m}(x,x)+k_{m}(y,y)-2k_{m}(x,y)$. We will now prove the following for all $m$:

$$ $|q_{m}(x)-q_{m}(y)|\leq d_{m}(x,y)+\frac{1}{2}\sum^{m}_{n\geq 0}\gamma^{n}( \Delta_{n}^{{\pi_{e}}}(x)+\Delta_{n}^{{\pi_{e}}}(y))$ (6) $$

where $\Delta_{n}^{{\pi_{e}}}(x)=\mathbb{E}_{X^{{}^{\prime}}\sim(P^{{\pi_{e}}}(\cdot|
x))^{n}}[\mathbb{E}_{X_{1}^{{}^{\prime\prime}},X_{2}^{{}^{\prime\prime}}\sim P
^{{\pi_{e}}}(\cdot|X^{\prime})}[|r(X_{1}^{{}^{\prime\prime}})-r(X_{2}^{{}^{
\prime\prime}})|]]$.

The base case $m=0$ follows immediately since the LHS is zero while the RHS can be non-zero. We now assume the induction hypothesis in Equation (6) is true. We then consider iteration $m+1$:

$$ $\displaystyle|q_{m+1}(x)-q_{m+1}(y)|$ $\displaystyle=\left|r(x)+\gamma\mathbb{E}_{X^{\prime}\sim P^{\pi_{e}}(\cdot|x) }[q_{m}(X^{\prime})]-r(y)-\gamma\mathbb{E}_{Y^{\prime}\sim P^{\pi_{e}}(\cdot|y )}[q_{m}(Y^{\prime})]\right|$ $\displaystyle\leq|r(x)-r(y)|+\gamma\mathbb{E}_{X^{\prime}\sim P^{\pi_{e}}( \cdot|x),Y^{\prime}\sim P^{\pi_{e}}(\cdot|y)}[|q_{m}(X^{\prime})-q_{m}(Y^{ \prime})|]$ $\displaystyle\leq|r(x)-r(y)|+\gamma\mathbb{E}_{X^{\prime}\sim P^{\pi_{e}}( \cdot|x),Y^{\prime}\sim P^{\pi_{e}}(\cdot|y)}\left[d_{m}(X^{\prime},Y^{\prime} )+\frac{1}{2}\sum^{m}_{n=0}\gamma^{n}(\Delta_{n}^{{\pi_{e}}}(X^{\prime})+ \Delta_{n}^{{\pi_{e}}}(Y^{\prime}))\right]$ $\displaystyle=|r(x)-r(y)|+\gamma\mathbb{E}_{X^{\prime}\sim P^{\pi_{e}}(\cdot|x ),Y^{\prime}\sim P^{\pi_{e}}(\cdot|y)}\left[d_{m}(X^{\prime},Y^{\prime})+\frac {1}{2}\sum^{m+1}_{n=1}\gamma^{n}(\Delta_{n}^{{\pi_{e}}}(x)+\Delta_{n}^{{\pi_{e }}}(y))\right]$ $$

where we have used the fact that $\mathbb{E}_{X^{\prime}\sim P^{\pi_{e}}(\cdot|x)}[\Delta_{n}^{\pi_{e}}(X^{
\prime})]=\Delta_{n+1}^{\pi_{e}}(x)$. We can then proceed from above as follows:

$$ $\displaystyle=|r(x)-r(y)|+\gamma\mathbb{E}_{X^{\prime}\sim P^{\pi_{e}}(\cdot|x ),Y^{\prime}\sim P^{\pi_{e}}(\cdot|y)}\left[d_{m}(X^{\prime},Y^{\prime})+\frac {1}{2}\sum^{m+1}_{n=1}\gamma^{n}(\Delta_{n}^{{\pi_{e}}}(x)+\Delta_{n}^{{\pi_{e }}}(y))\right]$ $\displaystyle\begin{aligned} \leq&|r(x)-r(y)|+\gamma\mathbb{E}_{X^{\prime}\sim P ^{\pi_{e}}(\cdot|x),Y^{\prime}\sim P^{\pi_{e}}(\cdot|y)}[d_{m}(X^{\prime},Y^{ \prime})]\\ &+\frac{1}{2}\mathbb{E}_{\begin{subarray}{c}X_{1}^{\prime},X_{2}^{\prime}\sim P ^{\pi_{e}}(\cdot|x)\\ Y_{1}^{\prime},Y_{2}^{\prime}\sim P^{\pi_{e}}(\cdot|y)\end{subarray}}[|r(X_{1} ^{\prime})-r(X_{2}^{\prime})|+|r(Y_{1}^{\prime})-r(Y_{2}^{\prime})|]\\ &+\frac{1}{2}\sum^{m+1}_{n=1}\gamma^{n}(\Delta_{n}^{{\pi_{e}}}(x)+\Delta_{n}^{ {\pi_{e}}}(y))\end{aligned}$ $\displaystyle=|r(x)-r(y)|+\gamma\mathbb{E}_{X^{\prime}\sim P^{\pi_{e}}(\cdot|x ),Y^{\prime}\sim P^{\pi_{e}}(\cdot|y)}[d_{m}(X^{\prime},Y^{\prime})]+\frac{1}{ 2}\sum^{m+1}_{n=0}\gamma^{n}(\Delta_{n}^{{\pi_{e}}}(x)+\Delta_{n}^{{\pi_{e}}}( y))$ $\displaystyle=d_{m+1}(x,y)+\frac{1}{2}\sum^{m+1}_{n=0}\gamma^{n}(\Delta_{n}^{{ \pi_{e}}}(x)+\Delta_{n}^{{\pi_{e}}}(y))$ $$

We thus have $|q_{m+1}(x)-q_{m+1}(y)|\leq d_{m+1}(x,y)+\frac{1}{2}\sum^{m+1}_{n=0}\gamma^{n}
(\Delta_{n}^{{\pi_{e}}}(x)+\Delta_{n}^{{\pi_{e}}}(y))$, which completes the proof.
∎

Lemma 3 tells us that the krope state-actions that are close in latent space also have similar action-values upto a constant $C:=\frac{1}{2}\sum^{m+1}_{n=0}\gamma^{n}(\Delta_{n}^{{\pi_{e}}}(x)+\Delta_{n}^
{{\pi_{e}}}(y))$. Intuitively, $\Delta_{n}^{{\pi_{e}}}(x)$ is the expected absolute reward difference between two trajectories at the $n$th step after ${\pi_{e}}$ is rolled out from $x$. If the transition dynamics and ${\pi_{e}}$ are deterministic, we have $C=0$ (Castro, ; Zhang et al., ). Note that while the deterministic transition dynamics assumption is eliminated, the bound suggests that krope may hurt accuracy of $\hat{q}^{{\pi_{e}}}$ since when $d_{\text{{krope}}}(x,y)=0$, we get $|q^{\pi_{e}}(x)-q^{\pi_{e}}(y)|\leq C$. This indicates that two state-actions that may have different action-values are considered the same under krope. This implies that while $x$ and $y$ should have different representations, they actually may have the same representation.

###### Lemma 1 .

###### Proof.

###### Lemma 2 .

###### Proof.

###### Proposition 2 .

###### Proof.

###### Proposition 3 .

###### Proof.

###### Lemma 3 .

###### Proof.

### B.2 krope Stability

In this section we present our main results. We present supporting theoretical results in Section B.2.1 and main theoretical results in Section B.3. To the best of our knowledge, even the supporting proofs in Section B.2.1 are new.

#### B.2.1 Supporting Theoretical Results

We present the following definitions that we refer to in our proofs.

###### Definition 8 (Bellman completeness (Chen & Jiang, 2019 ) ) .

The function class $\mathcal{F}$ is said to be Bellman complete if
$\forall f\in\mathcal{F}$, it holds that $\mathcal{T}^{{\pi_{e}}}f\in\mathcal{F}$. That is $\sup_{f\in\mathcal{F}}\inf_{g\in\mathcal{F}}\|g-\mathcal{T}^{{\pi_{e}}}f\|_{
\infty}=0$, where $\mathcal{F}\subset\mathcal{X}\to[\frac{r_{\text{min}}}{1-\gamma},\frac{r_{
\text{max}}}{1-\gamma}]$, and $\mathcal{T}^{{\pi_{e}}}$ is the Bellman operator.

###### Definition 9 (Piece-wise constant functions (Chen & Jiang, 2019 ) ) .

Given a state-action abstraction $\phi$, let $\mathcal{F}^{\phi}\subset\mathcal{X}\to[\frac{r_{\text{min}}}{1-\gamma},\frac{
r_{\text{max}}}{1-\gamma}]$. Then $f\in\mathcal{F}^{\phi}$ is said to be a piece-wise constant function if $\forall x,y\in\mathcal{X}$ where $\phi(x)=\phi(y)$, we have $f(x)=f(y)$.

###### Proposition 4 .

If a state-action abstraction function $\phi:\mathcal{X}\to\mathcal{X}^{\phi}$ is a ${\pi_{e}}$-bisimulation abstraction, then $\mathcal{F}^{\phi}$ is Bellman complete, that is, $\sup_{f\in\mathcal{F}^{\phi}}\inf_{f^{\prime}\in\mathcal{F}^{\phi}}\|f^{\prime
}-\mathcal{T}^{{\pi_{e}}}f\|_{\infty}=0$.

###### Proof.

We first define ${\pi_{e}}$-bisimulation Castro (). Note that Castro () considered only state abstractions, while we consider state-action abstractions. $\phi$ is considered a ${\pi_{e}}$-bisimulation abstraction if it induces a mapping between $\mathcal{X}$ and $\mathcal{X}^{\phi}$ such that for any $x,y\in\mathcal{X}$ such that $x,y\in\phi(x)$, we have:

- 1.
$r(x)=r(y)$
- 2.
$\forall x^{\phi}\in\mathcal{X}^{\phi},\sum_{x^{\prime}\in x^{\phi}}P^{\pi_{e}}
(x^{\prime}|x)=\sum_{x^{\prime}\in x^{\phi}}P^{\pi_{e}}(x^{\prime}|y)$

Given our ${\pi_{e}}$-bisimulation abstraction function $\phi$, we can group state-actions actions according to its definition above. Once we have this grouping, according to Definition 9, $\phi$ induces a piece-wise constant (pwc) function class $\mathcal{F}^{\phi}$. Note that by definition of $\phi$ we have:

$$ $\displaystyle{\epsilon}_{r}:=\max_{x_{1},x_{2}:\phi(x_{1})=\phi(x_{2})}|r(x_{1 })-r(x_{2})|=0$ $\displaystyle{\epsilon}_{p}:=\max_{x_{1},x_{2}:\phi(x_{1})=\phi(x_{2})}\left| \sum_{x^{\prime}\in x^{\phi}}P^{\pi_{e}}(x^{\prime}|x_{1})-\sum_{x^{\prime}\in x ^{\phi}}P^{\pi_{e}}(x^{\prime}|x_{2})\right|=0,\forall x^{\phi}\in\mathcal{X}^ {\phi}.$ $$

Once we have $\phi$, we consider the following to show Bellman completeness. Our proof closely follows the proof of Proposition 20 from Chen & Jiang (). First recall the definition of Bellman completeness from Definition 8: $\forall f\in\mathcal{F},\forall\mathcal{T}^{{\pi_{e}}}f\in\mathcal{G}$, $\sup_{f\in\mathcal{F}}\inf_{g\in\mathcal{G}}\|g-\mathcal{T}^{{\pi_{e}}}f\|_{
\infty}=0$. Given that the smallest value $\forall f\in\mathcal{F},\forall\mathcal{T}^{{\pi_{e}}}f\in\mathcal{G}$, $\sup_{f\in\mathcal{F}}\inf_{g\in\mathcal{G}}\|g-\mathcal{T}^{{\pi_{e}}}f\|_{\infty}$ can take on is zero, we will prove our claim by showing that $\forall f\in\mathcal{F},\forall\mathcal{T}^{{\pi_{e}}}f\in\mathcal{G}$, $\sup_{f\in\mathcal{F}}\inf_{g\in\mathcal{G}}\|g-\mathcal{T}^{{\pi_{e}}}f\|_{\infty}$ is upper-bounded by zero when $\phi$ is a ${\pi_{e}}$-bisimulation.

We will prove the upper bound by showing that there exists a function $f^{\prime}\in\mathcal{F}^{\phi}$ such that $\|f^{\prime}-\mathcal{T}^{{\pi_{e}}}f\|_{\infty}\leq 0$, which implies that $\inf_{f^{\prime}\in\mathcal{F}^{\phi}}\|f^{\prime}-\mathcal{T}^{{\pi_{e}}}f\|_
{\infty}\leq 0$.

We now construct such a $f^{\prime}\in\mathcal{F}^{\phi}$. We first define the following terms for a given abstract state-action $x^{\phi}\in\mathcal{X}^{\phi}$: $x_{+}:=\operatorname*{arg\,max}_{x\in\phi^{-1}(x^{\phi})}(\mathcal{T}^{{\pi_{e
}}}f)(x)$ and $x_{-}:=\operatorname*{arg\,min}_{x\in\phi^{-1}(x^{\phi})}(\mathcal{T}^{{\pi_{e
}}}f)(x)$. We can then define $f^{\prime}$ as follows:

$$ $f^{\prime}(x):=\frac{1}{2}((\mathcal{T}^{{\pi_{e}}}f)(x_{+})+(\mathcal{T}^{{ \pi_{e}}}f)(x_{-})),\forall x\in x^{\phi}.$ $$

And since this holds true for $\forall x\in x^{\phi}$, $f_{1}^{\prime}$ is piece-wise constant function. We can then upper bound $\|f^{\prime}-\mathcal{T}^{{\pi_{e}}}f\|_{\infty}$ as follows:

$$ $\displaystyle f_{1}^{\prime}(x)-(\mathcal{T}^{{\pi_{e}}}f)(x)$ $\displaystyle\leq\frac{1}{2}((\mathcal{T}^{{\pi_{e}}}f)(x_{+})+(\mathcal{T}^{{ \pi_{e}}}f)(x_{-}))-(\mathcal{T}^{{\pi_{e}}}f)(x_{-})$ $\displaystyle=\frac{1}{2}((\mathcal{T}^{{\pi_{e}}}f)(x_{+})-(\mathcal{T}^{{\pi _{e}}}f)(x_{-}))$ $\displaystyle=\frac{1}{2}(r(x_{+})+\gamma\mathbb{E}_{x^{\prime}_{+}\sim P^{{ \pi_{e}}}(x_{+})}[f^{{\pi_{e}}}(x^{\prime}_{+})]-r(x_{-})-\gamma\mathbb{E}_{x^ {\prime}_{-}\sim P^{{\pi_{e}}}(x_{-})}[f^{{\pi_{e}}}(x^{\prime}_{-})])$ $\displaystyle\leq\frac{\gamma}{2}\left|\mathbb{E}_{x^{\prime}_{+}\sim P^{{\pi_ {e}}}(x_{+})}[f^{{\pi_{e}}}(x^{\prime}_{+})]-\mathbb{E}_{x^{\prime}_{-}\sim P^ {{\pi_{e}}}(x_{-})}[f^{{\pi_{e}}}(x^{\prime}_{-})]\right|$ (1) $\displaystyle=\frac{\gamma}{2}\left|\sum_{x^{\prime}\in\mathcal{X}}[f^{{\pi_{e }}}(x^{\prime})(P^{\pi_{e}}(x^{\prime}|x_{+})-P^{\pi_{e}}(x^{\prime}|x_{-}))]\right|$ $\displaystyle=\frac{\gamma}{2}\left|\sum_{x^{\phi}\in\mathcal{X}^{\phi}}\left( \sum_{x^{\prime}\in x^{\phi}}f^{{\pi_{e}}}(x^{\prime})P^{\pi_{e}}(x^{\prime}|x _{+})-\sum_{x^{\prime}\in x^{\phi}}f^{{\pi_{e}}}(x^{\prime})P^{\pi_{e}}(x^{ \prime}|x_{-})\right)\right|$ $\displaystyle=\frac{\gamma}{2}\left|\sum_{x^{\phi}\in\mathcal{X}^{\phi}}f^{\pi _{e}}(x^{\phi})\left(\sum_{x^{\prime}\in x^{\phi}}P^{\pi_{e}}(x^{\prime}|x_{+} )-\sum_{x^{\prime}\in x^{\phi}}P^{\pi_{e}}(x^{\prime}|x_{-})\right)\right|$ (2) $\displaystyle=\frac{\gamma}{2}\left|\sum_{x^{\phi}\in\mathcal{X}^{\phi}}f^{\pi _{e}}(x^{\phi})\left(\Pr(x^{\phi}|x_{+})-\Pr(x^{\phi}|x_{-})\right)\right|$ (3) $\displaystyle\leq\frac{\gamma}{2}\left\|\Pr(x^{\phi}|x_{+})-\Pr(x^{\phi}|x_{-} )\right\|_{1}\cdot\|f^{\pi_{e}}(x^{\phi})\|_{\infty}$ (4) $\displaystyle\leq 0$ ${\epsilon}_{p}=0$ $$

where $\Pr$ denotes probability, (1) is due to $\max_{x_{1},x_{2}:\phi(x_{1})=\phi(x_{2})}|r(x_{1})-r(x_{2})|=0$, (2) is due to $f^{\pi_{e}}(x^{\phi})=f^{\pi_{e}}(x),\forall x\in x^{\phi}$ since pwc, (3) is due to $\Pr(x^{\phi}|x)=\sum_{x^{\prime}\in x^{\phi}}P^{\pi_{e}}(x^{\prime}|x)$, and (4) is due to Hölder’s, $\|f(g)g(x)\|_{1}\leq\|f(x)\|_{1}\|g(x)\|_{\infty}$.

Similarly, we can show the other way around: $(\mathcal{T}^{{\pi_{e}}}f)(x)-f_{1}^{\prime}(x)\leq 0$ by giving the symmetric argument starting with $(\mathcal{T}^{{\pi_{e}}}f)(x)-f_{1}^{\prime}(x)\leq(\mathcal{T}^{{\pi_{e}}}f)(
x_{+})-\frac{1}{2}((\mathcal{T}^{{\pi_{e}}}f)(x_{+})+(\mathcal{T}^{{\pi_{e}}}f
)(x_{-}))$. Therefore, when $\phi$ is a ${\pi_{e}}$-bisimulation, we have $\sup_{f\in\mathcal{F}^{\phi}}\inf_{f^{\prime}\mathcal{F}^{\phi}}\|f^{\prime}-
\mathcal{T}^{{\pi_{e}}}f\|_{\infty}=0$.
∎

###### Lemma 4 .

Define the matrix $K_{1}\in\mathbb{R}^{|\mathcal{X}|\times|\mathcal{X}|}$ such that each entry is the short-term similarity, $k_{1}$, between every pair of state-actions, i.e., $K_{1}(s_{1},a_{1};s_{2},a_{2}):=1-\frac{1}{\left|r_{\text{max}}-r_{\text{min}}
\right|}|r(s_{1},a_{1})-r(s_{2},a_{2})|$. Then $K_{1}$ is a positive semidefinite matrix.

###### Proof.

Proposition 2.21 from Paulsen & Raghupathi () states that any kernel $k$ is positive semidefinite if it takes the form: $k(a,b)=\min\{a,b\}$ where $a,b\in[0,\infty)$.

First, recall that $r(s,a)\in[-1,1]$, we then have each entry in the $K_{1}$ matrix of the following kernel form $K_{1}(x,y)=1-\frac{1}{2}|x-y|$. We can then re-write $k_{1}$ as follows:

$$ $\displaystyle k_{1}(x,y)$ $\displaystyle=1-\frac{1}{2}|x-y|$ $\displaystyle=1+\frac{1}{2}\min\{-x,-y\}+\frac{1}{2}\min\{x,y\}$ $\displaystyle=\underbrace{\frac{1}{2}\min\{1-x,1-y\}}_{k_{a}}+\underbrace{ \frac{1}{2}\min\{1+x,1+y\}}_{k_{b}}.$ $$

That is,

$$ $\displaystyle k_{1}(x,y)$ $\displaystyle=k_{a}(x,y)+k_{b}(x,y).$ $$

Since $x\in[-1,1]$, each term in the $\min$ function is non-negative. Thus, $k_{a}$ and $k_{b}$ are both positive semidefinite kernels, which means $k_{1}$ is also a positive semidefinite kernel. We then have that $K_{1}$ is a positive semidefinite matrix.
∎

###### Lemma 5 .

Given a finite set $\mathcal{X}$ and a kernel $k$ defined on $\mathcal{X}$, let $K=(k(x,y))_{x,y\in\mathcal{X}}\in\mathbb{R}^{|\mathcal{X}|\times|\mathcal{X}|}$ be the corresponding kernel matrix. If $K$ is full-rank and $\text{MMD}(k)(p,q)=0$ for two probability distributions $p$ and $q$ on $\mathcal{X}$, then $p=q$.

###### Proof.

From Gretton et al. (), we have the definition of MMD between two probability distributions $p,q$ given kernel $k$:

$$ $\text{MMD}(k)(p,q):=\|\mathbb{E}_{x\sim p}[k(x,\cdot)]-\mathbb{E}_{x\sim q}[k( x,\cdot)]\|_{\mathcal{H}_{k}}.$ $$

Now when $\text{MMD}(k)(p,q)=0$, we have:

$$ $\displaystyle 0$ $\displaystyle=\|\mathbb{E}_{x\sim p}[k(x,\cdot)]-\mathbb{E}_{x\sim q}[k(x, \cdot)]\|_{\mathcal{H}_{k}},$ $$

which implies

$$ $\displaystyle 0$ $\displaystyle=\|\mathbb{E}_{x\sim p}[k(x,\cdot)]-\mathbb{E}_{x\sim q}[k(x, \cdot)]\|_{2}$ $$

since all norms are equivalent in a finite-dimensional Hilbert space.
With $p$ and $q$ viewed as vectors in $\mathbb{R}^{|\mathcal{X}|}$, the above equality means

$$ $\displaystyle 0$ $\displaystyle=\|Kp-Kq\|_{2}.$ $$

Hence, $K(p-q)=0$. Since $K$ is full rank by assumption, we conclude that $p=q$.
∎

###### Lemma 6 .

Suppose we have a reproducing kernel $k$ defined on the finite space $\mathcal{X}$, which produces a reproducing kernel Hilbert space (rkhs) ${\mathcal{H}_{k}}$, with the induced distance function $d$ such that $d(x,y)=k(x,x)+k(y,y)-2k(x,y),\forall x,y\in\mathcal{X}$. When $d(x,y)=0$, then $k(x,\cdot)=k(y,\cdot)$.

###### Proof.

When $d(x,y)=0$, we have $2k(x,y)=k(x,x)+k(y,y)$. Therefore, we the following equalities:

$$ $\displaystyle k(x,x)+k(y,y)$ $\displaystyle=2k(x,y)$ $\displaystyle k(x,x)-k(x,y)$ $\displaystyle=k(x,y)-k(y,y)$ $\displaystyle\langle k(x,\cdot),k(x,\cdot)\rangle_{\mathcal{H}_{k}}-\langle k( x,\cdot),k(y,\cdot)\rangle_{\mathcal{H}_{k}}$ $\displaystyle=\langle k(x,\cdot),k(y,\cdot)\rangle_{\mathcal{H}_{k}}-\langle k (y,\cdot),k(y,\cdot)\rangle_{\mathcal{H}_{k}}$ (1) $\displaystyle\langle k(x,\cdot),k(x,\cdot)-k(y,\cdot)\rangle_{\mathcal{H}_{k}}$ $\displaystyle=\langle k(x,\cdot)-k(y,\cdot),k(y,\cdot)\rangle_{\mathcal{H}_{k}}$ (2) $\displaystyle\langle k(x,\cdot),k(x,\cdot)-k(y,\cdot)\rangle_{\mathcal{H}_{k}}$ $\displaystyle=\langle k(y,\cdot),k(x,\cdot)-k(y,\cdot)\rangle_{\mathcal{H}_{k}}$ (3) $\displaystyle\implies k(x,\cdot)=k(y,\cdot)$ $$

where (1), (2), and (3) are is due to rkhs definition, linearity of inner product, and symmetry of inner product respectively.
∎

###### Proposition 5 .

Let $x_{1},\ldots,x_{n}\in(0,\infty)$ be $n$ distinct and strictly
positive numbers. Let $K\in^{n\times n}$ be the matrix with
entries $K_{ij}=\min\{x_{i},x_{j}\}$. Then $K$ is a positive definite
matrix.

###### Proof.

By Proposition 2.21 in Paulsen & Raghupathi (), the matrix $K$ is positive semidefinite,
so we only need to show that $K$ is full rank. WLOG assume that $0<x_{1}<x_{2}<\cdots<x_{n}$.
We prove by induction on $n$. The base case with $n=1$ clearly holds.
Suppose the claim holds for $n-1$ numbers. Now consider $n$ numbers.
Let $\alpha=(\alpha_{1},\ldots,\alpha_{n})^{\top}\in^{n}$. It
suffices to show that $K\alpha=0$ implies $\alpha=0$. We write $K$
in block matrix form as

$$ $K=x_{1}J_{n}+\begin{bmatrix}0&0&\cdots&0\\ 0&x_{2}-x_{1}&\cdots&x_{2}-x_{1}\\ \vdots&\cdots&\ddots&\cdots\\ 0&x_{2}-x_{1}&\cdots&x_{n}-x_{1}\end{bmatrix}=x_{1}\begin{bmatrix}1&1&\cdots&1 \\ \mathbf{1}&\mathbf{1}&\cdots&\mathbf{1}\end{bmatrix}+\begin{bmatrix}0&0&\cdots &0\\ \mathbf{0}&\mathbf{u}_{2}&\cdots&\mathbf{u}_{n}\end{bmatrix},$ $$

where $J_{n}$ is the $n$-by-$n$ all one matrix, $\mathbf{1}\in^{n-1}$
the all one vector, $\mathbf{0}\in^{n-1}$ the all zero vector, and $\mathbf{u}_{i}\in^{n-1},i=2,\ldots,n$.
It follow that

$$ $0=K\alpha=\begin{bmatrix}x_{1}\sum_{i=1}^{n}\alpha_{i}\\ \left(x_{1}\sum_{i=1}^{n}\alpha_{i}\right)\mathbf{1}+\sum_{i=2}^{n}\alpha_{i} \mathbf{u_{i}}\end{bmatrix},$ $$

that is,

$$ $\displaystyle x_{1}\sum_{i=1}^{n}\alpha_{i}$ $\displaystyle=0,$ (7) $\displaystyle\left(x_{1}\sum_{i=1}^{n}\alpha_{i}\right)\mathbf{1}+\sum_{i=2}^{ n}\alpha_{i}\mathbf{u_{i}}$ $\displaystyle=\mathbf{0}.$ (8) $$

Plugging equation (7) into equation (8),
we get $\sum_{i=2}^{n}\alpha_{i}\mathbf{u_{i}}=\mathbf{0}$. By the
induction hypothesis, the $(n-1)$-by-$(n-1)$ matrix

$$ $\begin{bmatrix}\mathbf{u}_{2}&\cdots&\mathbf{u}_{n}\end{bmatrix}=\big{[}\min \left\{x_{i}-x_{1},x_{j}-x_{1}\right\}\big{]}_{i,j=2,\ldots,n}$ $$

has full rank since the $(n-1)$ numbers $x_{2}-x_{1},\ldots,x_{n}-x_{1}$
are distinct and strictly positive. Therefore, we must have $\alpha_{2}=\cdots=\alpha_{n}=0$.
Plugging back into equation (7) and using $x_{1}>0$,
we obtain $\alpha_{1}=0$.
∎

###### Definition 8 (Bellman completeness (Chen & Jiang, 2019 ) ) .

###### Definition 9 (Piece-wise constant functions (Chen & Jiang, 2019 ) ) .

###### Proposition 4 .

###### Proof.

###### Lemma 4 .

###### Proof.

###### Lemma 5 .

###### Proof.

###### Lemma 6 .

###### Proof.

###### Proposition 5 .

###### Proof.

### B.3 Main krope Theoretical Results

We now present the main theoretical contributions of our work.

See 1

###### Proof.

Recall from Definition 3, we have:

$$ $\mathbb{E}[\Phi\Phi^{\top}]=K_{1}+\gamma\mathbb{E}[P^{{\pi_{e}}}\Phi(P^{{\pi_{ e}}}\Phi)^{\top}],$ $$

where $K_{1}\in\mathbb{R}^{|\mathcal{X}|\times|\mathcal{X}|}$ such that each entry is the short-term similarity, $k_{1}$, between every pair of state-actions i.e. $K_{1}(s_{1},a_{1};s_{2},a_{2}):=1-\frac{|r(s_{1},a_{1})-r(s_{2},a_{2})|}{\left
|r_{\text{max}}-r_{\text{min}}\right|}$.

From this definition, we can proceed by left and right multiplying $\Phi^{\top}$ and $\Phi$ respectively to get:

$$ $\mathbb{E}[\Phi^{\top}\Phi\Phi^{\top}\Phi]=\mathbb{E}[\Phi^{\top}K_{1}\Phi]+ \gamma\mathbb{E}[\Phi^{\top}P^{{\pi_{e}}}\Phi(P^{{\pi_{e}}}\Phi)^{\top}\Phi].$ $$

Notice that $B:=\mathbb{E}[\Phi^{\top}\Phi]$ is the feature covariance matrix and $C:=\mathbb{E}[\Phi^{\top}P^{{\pi_{e}}}\Phi]$ is the cross-covariance matrix. By making the appropriate substitutions, we get:

$$ $BB^{\top}=\mathbb{E}[\Phi^{\top}K_{1}\Phi]+\gamma CC^{\top}.$ $$

We can then left and right multiply by $B^{-1}$ and $B^{-\top}$ to get the following where $L:=\gamma B^{-1}C$:

$$ $I=B^{-1}\mathbb{E}[\Phi^{\top}K_{1}\Phi]B^{-\top}+\frac{1}{\gamma}LL^{\top}.$ $$

Rearranging terms gives

$$ $I-\frac{1}{\gamma}LL^{\top}=B^{-1}\mathbb{E}[\Phi^{\top}K_{1}\Phi]B^{-\top}.$ $$

From Lemma 4, we know that $K_{1}$ is positive semidefinite, which means that $B^{-1}\mathbb{E}[\Phi^{\top}K_{1}\Phi]B^{-\top}$ is also positive semidefinite. Therefore, the eigenvalues of LHS above must also be greater than or equal to zero. Letting $\lambda$ be the eigenvalue of $L$, we know that that the following must hold:

$$ $\displaystyle 1-\frac{\lambda^{2}}{\gamma}\geq 0\implies|\lambda|\leq\sqrt{ \gamma}.$ $$

Since $\gamma<1$, the spectral radius of $L=(\mathbb{E}[\Phi^{\top}\Phi])^{-1}(\gamma\mathbb{E}[\Phi^{\top}P^{{\pi_{e}}}
\Phi])$ is always less than $1$. Thus, krope representations are stable. Finally, since krope representations are stable and due to Proposition 1, krope representations stabilize lspe.
∎

See 2

###### Proof.

Our proof strategy is to show that the abstraction function $\phi$ due to krope is a ${\pi_{e}}$-bisimulation, which implies it is Bellman complete due to Proposition 4.

According to Propsition 3, $d_{\text{{krope}}}(x,y)=|r(x)-r(y)|+\gamma\text{MMD}(k^{\pi_{e}})(P^{\pi_{e}}(
\cdot|x),P^{\pi_{e}}(\cdot|y))$. When $d_{\text{{krope}}}(x,y)=0$ for any two state-actions, it implies that $r(x)=r(y)$ and $\text{MMD}(k^{\pi_{e}})(P^{\pi_{e}}(\cdot|x),P^{\pi_{e}}(\cdot|y))=0$.

For $\phi$ to be a ${\pi_{e}}$-bisimulation, we need $\forall x^{\phi}\in\mathcal{X}^{\phi},\sum_{x^{\prime}\in x^{\phi}}P^{\pi_{e}}
(x^{\prime}|x)=\sum_{x^{\prime}\in x^{\phi}}P^{\pi_{e}}(x^{\prime}|y)$ to be true for any $x,y\in\mathcal{X}$ such that $\phi(x)=\phi(y)$. While $\text{MMD}(k^{\pi_{e}})(P^{\pi_{e}}(\cdot|x),P^{\pi_{e}}(\cdot|y))=0$, it is possible that $P^{\pi_{e}}(\cdot|x)\neq P^{\pi_{e}}(\cdot|y)$. However, as we will show, under the assumption that the abstract rewards $r^{\phi}$ are distinct $\forall x^{\phi}\in\mathcal{X}^{\phi}$, we do have $\forall x^{\phi}\in\mathcal{X}^{\phi},\sum_{x^{\prime}\in x^{\phi}}P^{\pi_{e}}
(x^{\prime}|x)=\sum_{x^{\prime}\in x^{\phi}}P^{\pi_{e}}(x^{\prime}|y)$. Before we proceed, we make the following technical assumption on the reward function: $r(x)\in(-1,1),\forall x\in\mathcal{X}$. The exclusion of the rewards $-1$ and $1$ allows us to use Proposition 5 to show that the krope kernel is positive definite instead of positive semi-definite.

Once we group state-actions $x,y\in\mathcal{X}$ together such that $d_{\text{{krope}}}(x,y)=0$, we have the corresponding abstraction function $\phi:\mathcal{X}\to\mathcal{X}^{\phi}$. Accordingly, $\phi$ induces a Markov reward process, $\mathcal{M}^{\phi}:=\langle\mathcal{X}^{\phi},r^{\phi},P^{\phi},\gamma\rangle$ where $r^{\phi}$ is the abstract reward function $r^{\phi}:\mathcal{X}^{\phi}\to(-1,1)$ and $P^{\phi}$ is the transition dynamics on the abstract mrp i.e. $P^{\phi}(\cdot|x^{\phi})$. We can also consider the abstract krope kernel, $k^{\phi}(x^{\phi},y^{\phi})$, which measures the krope relation on $\mathcal{X}^{\phi}$. Note that all these quantities are a function of ${\pi_{e}}$. We drop the notation for clarity. By this construction, we have:

$$ $\displaystyle r^{\phi}(x^{\phi})=r(x),\forall x\in x^{\phi}$ Since all rewards are equal within $x^{\phi}$ $\displaystyle k^{\phi}(x^{\phi},\cdot)=k(x,\cdot),\forall x\in x^{\phi}$ Lemma 6 $$

Now, under the assumption that all abstract rewards $r^{\phi}(x^{\phi})$ are distinct $\forall x^{\phi}\in\mathcal{X}^{\phi}$, we have that the kernel matrix $K^{\phi}\in\mathbb{R}^{\mathcal{X}^{\phi}\times\mathcal{X}^{\phi}}$ where each entry $k^{\phi}(x^{\phi},y^{\phi})$ is positive definite. To see this fact, consider that:

$$ $k^{\phi}(x^{\phi},y^{\phi})=k_{1}^{\phi}(x^{\phi},y^{\phi})+\gamma\mathbb{E}_{ X^{\phi}\sim P^{\phi}(\cdot|x^{\phi}),Y^{\phi}\sim P^{\phi}(\cdot|y^{\phi})}[k ^{\phi}(X^{\phi},Y^{\phi})],$ (9) $$

where $k_{1}^{\phi}(x^{\phi},y^{\phi}):=1-\frac{1}{r^{\phi}_{\text{max}}-r^{\phi}_{
\text{min}}}|r^{\phi}(x^{\phi})-r^{\phi}(y^{\phi})|$. From Lemma 4, we know that $k_{1}^{\phi}$ is positive semidefinite. However, under the assumption that all abstract rewards $r^{\phi}$ are distinct, Proposition 5 tells us that $k_{1}^{\phi}$ is positive definite. Given that $k^{\phi}(x^{\phi},y^{\phi})$ (Equation (9)) is just a summation of positive definite kernels, $k^{\phi}$ is positive definite, which means $K^{\phi}$ is positive definite.

We now consider when the MMD is zero. Again, by construction, we have the following when $\text{MMD}(k^{\pi_{e}})(P^{\pi_{e}}(\cdot|x),P^{\pi_{e}}(\cdot|y))=0$. For clarity, we use $k$ instead of $k^{\pi_{e}}$.

$$ $\displaystyle 0$ $\displaystyle=\|\mathbb{E}_{X^{\prime}\sim P^{\pi_{e}}(\cdot|x)}[k(X^{\prime}, \cdot)]-\mathbb{E}_{X^{\prime}\sim P^{\pi_{e}}(\cdot|y)}[k(X^{\prime},\cdot)] \|_{\mathcal{H}_{k}}$ $\displaystyle=\|\mathbb{E}_{X^{\prime}\sim P^{\pi_{e}}(\cdot|x)}[k(X^{\prime}, \cdot)]-\mathbb{E}_{X^{\prime}\sim P^{\pi_{e}}(\cdot|y)}[k(X^{\prime},\cdot)] \|_{2}$ $\displaystyle=\|\sum_{x^{\prime}\in\mathcal{X}}P^{\pi_{e}}(x^{\prime}|x)k(x^{ \prime},\cdot)-\sum_{x^{\prime}\in\mathcal{X}}P^{\pi_{e}}(x^{\prime}|y)k(x^{ \prime},\cdot)\|_{2}$ $\displaystyle=\left\|\sum_{x^{\phi}\in\mathcal{X}^{\phi}}\sum_{x^{\prime}\in x ^{\phi}}P^{\pi_{e}}(x^{\prime}|x)k(x^{\prime},\cdot)-\sum_{x^{\phi}\in\mathcal {X}^{\phi}}\sum_{x^{\prime}\in x^{\phi}}P^{\pi_{e}}(x^{\prime}|y)k(x^{\prime}, \cdot)\right\|_{2}$ $\displaystyle=\left\|\sum_{x^{\phi}\in\mathcal{X}^{\phi}}k^{\phi}(x^{\phi}, \cdot)\sum_{x^{\prime}\in x^{\phi}}P^{\pi_{e}}(x^{\prime}|x)-\sum_{x^{\phi}\in \mathcal{X}^{\phi}}k^{\phi}(x^{\phi},\cdot)\sum_{x^{\prime}\in x^{\phi}}P^{\pi _{e}}(x^{\prime}|y)\right\|_{2}$ (1) $\displaystyle=\left\|\sum_{x^{\phi}\in\mathcal{X}^{\phi}}k^{\phi}(x^{\phi}, \cdot)\Pr(x^{\phi}|x)-\sum_{x^{\phi}\in\mathcal{X}^{\phi}}k^{\phi}(x^{\phi}, \cdot)\Pr(x^{\phi}|y)\right\|_{2}$ $\Pr$ denotes probability $$

where $(1)$ is due to $k^{\phi}(x^{\phi},\cdot)=k(x,\cdot),\forall x\in x^{\phi}$
From above, we can see that the kernel and probability distributions are over $\mathcal{X}^{\phi}$. In matrix notation, we can write the above as follows where $p^{\phi}:=\Pr(\cdot|x)$ and $q^{\phi}:=\Pr(\cdot|y)$ are viewed as probability distribution vectors in $\mathbb{R}^{|\mathcal{X}^{\phi}|}$.

$$ $\displaystyle 0=\left\|K^{\phi}p^{\phi}-K^{\phi}q^{\phi}\right\|_{2}$ $\displaystyle\implies p^{\phi}=q^{\phi}$ since $K^{\phi}$ is positive definite, from Lemma 5 . $$

We thus have $\forall x^{\phi}\in\mathcal{X}^{\phi},\sum_{x^{\prime}\in x^{\phi}}P^{\pi_{e}}
(x^{\prime}|x)=\sum_{x^{\prime}\in x^{\phi}}P^{\pi_{e}}(x^{\prime}|y)$ to be true for any $x,y\in\mathcal{X}$ such that $\phi(x)=\phi(y)$. Given this condition holds true and $r(x)=r(y),\forall x,y\in\mathcal{X}$ such that $\phi(x)=\phi(y)$, $\phi$ is a ${\pi_{e}}$-bisimulation. From Proposition 4 we then have that $\phi$ is Bellman complete.

∎

###### Proof.

###### Proof.

## Appendix C Empirical Details

In this section, we provide specific details on the empirical setup and additional results.

### C.1 Empirical Setup

##### General Training Details.

In all the continuous state-action experiments, we use a neural network with $1$ layer and $1024$ neurons using relu activation function and layernorm to represent the encoder $\phi:\mathcal{X}\to\mathbb{R}^{d}$ (Gallici et al., ). We use mini-batch gradient descent to train the network with mini-batch sizes of $2048$ and for $500$ epochs, where a single epoch is a pass over the full dataset. We use the Adam optimizer with learning rate $\{1e^{-5},2e^{-5},5e^{-5}\}$ and weight decay $1e^{-2}$. The target network is updated with a hard update after every epoch. The output dimension $d$ is $\{|\mathcal{X}|/4,|\mathcal{X}|/2,3|\mathcal{X}|/4\}$, where $|\mathcal{X}|$ is the dimension of the original state-action space of the environment. All our results involve analyzing this learned $\phi$. Since fqe outputs a scalar, we add a linear layer on top of the $d$-dimensional vector to output a scalar. The entire network is then trained end-to-end. The discount factor is $\gamma=0.99$. The auxiliary task weight with fqe for all representation learning algorithms is $\alpha=0.1$. When using lspe for ope, we invert the covariance matrix by computing the pseudoinverse.

In the tabular environments, we use a similar setup as above. The only changes are that we use a linear network with a bias component but no activation function and fix the learning rate to be $1e^{-3}$. For experiment in Appendix C.3.1, $\alpha=0.8$. We refer the reader pseudo-code in Appendix A.

##### Evaluation Protocol: ope Error

. As noted earlier, we measure ope error by measuring msve. To ensure comparable and interpretable values, we normalize the msve by dividing with $\operatorname{\textsc{msve}}[q^{\textsc{rand}}]\coloneqq\mathbb{E}_{(S,A)\sim
\mathcal{D}}[(q^{\textsc{rand}}(S,A)-q^{{\pi_{e}}}(S,A))^{2}]$, where $q^{\textsc{rand}}$ is the action-value function of a random-policy. Similarly, in the continuous state-action environments, we normalize by $\operatorname{\textsc{msve}}[q^{\textsc{rand}}]\coloneqq\mathbb{E}_{S_{0}\sim d
_{0},A_{0}\sim{\pi_{e}}}[(q^{\textsc{rand}}(S_{0},A_{0})-q^{{\pi_{e}}}(S_{0},A
_{0}))^{2}]$. Values less than one mean that the algorithm estimates the true performance of ${\pi_{e}}$ better than a random policy.

Evaluation Protocol: Realizability Error. In tabular experiments, we normalize the realizability error. After solving the least-squares problem ${\epsilon}:=\|\Phi\hat{w}-q^{{\pi_{e}}}\|^{2}_{2}$, where $\hat{w}:=\operatorname*{arg\,min}_{w}\|\Phi w-q^{{\pi_{e}}}\|_{2}^{2}$. We divide ${\epsilon}$ by $\frac{1}{|\mathcal{X}|}\sum_{i}|q^{{\pi_{e}}}(x_{i})|$ and plot this value.

Pearson Correlation. The formula for the Pearson correlation used in the main experiments is:

$$ $r=\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i=1}^{n}(x_{ i}-\bar{x})^{2}}\sqrt{\sum_{i=1}^{n}(y_{i}-\bar{y})^{2}}}$ $$

where $\bar{x}$ and $\bar{y}$ are the means of all the $x_{i}$’s and $y_{i}$’s respectively.

##### Custom Datasets.

We generated the datasets by first training policies in the environment using sac (Haarnoja et al., ) and recording the trained policies during the course of training. For each environment, we select $3$ policies, where each contributes equally to generate a given dataset. We set ${\pi_{e}}$ to be one of these policies. The expected discounted return of the policies and datasets for each domain is given in Table 1 ($\gamma=0.99$). In all environments, ${\pi_{e}}=\pi^{1}_{b}$ (see Table 1). The values for the evaluation and behavior policies were computed by running each for $300$ unbiased Monte Carlo rollouts, which was more than a sufficient amount for the estimate to converge. This process results in total of $4$ datasets, each of which consisted of $100$K transitions.

**Table 1: Policy values of the target policy and behavior policy on DM-control (Tassa et al., ).**
| Environments | ${\pi_{e}}$ | $\pi^{1}_{b}$ | $\pi^{2}_{b}$ |
| --- | --- | --- | --- |
| CartPoleSwingUp | $50$ | $20$ | $5$ |
| FingerEasy | $100$ | $71$ | $32$ |
| HalfCheetah | $51$ | $27$ | $2$ |
| WalkerStand | $90$ | $55$ | $40$ |

##### d4rl Datasets.

Due to known discrepancy issues between newer environments of gym(^1^11[https://github.com/Farama-Foundation/D4RL/tree/master](https://github.com/Farama-Foundation/D4RL/tree/master)), we generat our datasets instead of using the publicly available ones. To generate the datasets, we use the publicly available policies (^2^22[https://github.com/google-research/deep_ope](https://github.com/google-research/deep_ope)). For each domain, the expert (and target policy) was the $10$th (last policy) from training. The medium (and behavior policy) was the $5$th policy. We added a noise of $0.1$ to the policies. The values for the evaluation and behavior policies were computed by running each for $300$ unbiased Monte Carlo rollouts , which was more than a sufficient amount for the estimate to converge. We set $\gamma=0.99$. We evaluate on the Cheetah, Walker, and Hopper domains. This generation process for three environments, led to $9$ datasets, each of which consisted of $100$K transitions.

### C.2 Baselines

We provide details of the baselines in this section.

##### bcrl .

Unlike the other algorithms, bcrl is not used as an auxiliary loss with fqe (Chang et al., ). We use the same learning rates as mentioned above for $\{1e^{-5},2e^{-5},5e^{-5}\}$ when training $\phi$. As suggested by prior work, self-predictive algorithms such bcrl work well when network that outputs the predicted next state-action is trained at a faster rate (Tang et al., ; Grill et al., ). Accordingly, we set its learning rate to be $1e^{-4}$. For bcrl-exp, which involves the log determinant regularizer, we set this coefficient to $1e^{-2}$. bcrl’s hyperparameters are: the learning rates for $\phi$, $M$, and $\rho$; the output dimension of $\phi$; and the log determinant coefficient (see Equation (10)).

##### dr3 .

The dr3 regularizer minimizes the total feature co-adaptation by adding the term $\sum_{(s,a,s^{\prime})\in\mathcal{D},a^{\prime}\sim{\pi_{e}}}\phi(s,a)^{\top}
\phi(s^{\prime},a^{\prime})$ as an auxiliary task to the main fqe loss (Kumar et al., ). Ma et al. () introduced an improvement to this auxiliary loss by suggesting that the absolute value of the feature co-adaptation be minimized, i.e., $\sum_{(s,a,s^{\prime})\in\mathcal{D},a^{\prime}\sim{\pi_{e}}}|\phi(s,a)^{\top}
\phi(s^{\prime},a^{\prime})|$. e use $\alpha=0.1$ as its auxiliary task weight. Absolute dr3’s only hyperparameters are the auxiliary task weight $\alpha$ and the $\phi$ output dimension.

##### beer .

He et al. () introduced an alternative regularizer to dr3 rank regularizer since they suggested that the minimization of the unbounded feature co-adaptation can undermine performance. They introduced their bounded rank regularizer beer (see Equation (12) in He et al. ()). beer introduces only the auxiliary task weight $\alpha$ as the additional hyperparameter.

##### krope .

krope’s only hyperparameters are the output dimension of $\phi$ and the learning rate of the krope learning algorithm.

### C.3 Additional Results

In this section, we include additional empirical results.

#### C.3.1 Understanding the Stability of the krope Learning Procedure

In this section analyze how the deadly triad affects krope learning procedure. Since krope’s learning algorithm is a semi-gradient method, it is susceptible to instability (Tsitsiklis & Van Roy, ). As such, these experiments aim to understand when we might expect krope to diverge and when it might mitigate the divergence of fqe.

We conduct our experiments on the Markov reward process in Figure 5(a), which was introduced by Feng et al. (). The mrp consists of $4$ non-terminal states, $1$ terminal state (the box), and only $1$ action. The value function estimate is linear in the weight $w=[w_{1},w_{2},w_{3}]$, so the native features of the states are $[1,0,0]$, $[0,1,0]$ , $[0,0,2]$, and $[0,0,1]$ (clockwise starting from left-most). We set $\gamma=1$, so the optimal weights are $w^{*}=[0.8,1,0]$. In this setup, we say a transition is a *bad transition* if the bootstrapping target is a moving target for the current state (Asadi et al., ). For example, the transition from $w_{3}$ to $2w_{3}$ is a bad transition since updates made to $w_{3}$ may move $2w_{3}$ further away. When this transition is sampled at a frequency that is different from the on-policy distribution, algorithms such as td, lspe, and fqe tend to diverge (Asadi et al., ).

Figure: (a) Divergence Counterexample
Refer to caption: extracted/5973682/images/lihong_divergence.png

To better understand the stability of the learning procedures, we design the following experiment. We start with two datasets $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$. $\mathcal{D}_{1}$ consists of $2000$ on-policy transitions and $5000$ off-policy bad transitions from $w_{3}$ to $2w_{3}$, while $\mathcal{D}_{2}$ consists of only $2000$ on-policy transitions. We then consider four different variations of $\mathcal{D}_{2}$: $\mathcal{D}_{2}^{w_{1}}$, $\mathcal{D}_{2}^{w_{2}}$, $\mathcal{D}_{2}^{w_{3}}$, and $\mathcal{D}_{2}^{2w_{3}}$, where each variation denotes a dataset where we add $5000$ off-policy transitions from the specified state to $\mathcal{D}_{2}$. With this setup, our goal is understand which dataset pairing with $\mathcal{D}_{1}$: $(\mathcal{D}_{1},\mathcal{D}_{2}^{w_{1}})$, $(\mathcal{D}_{1},\mathcal{D}_{2}^{w_{2}})$, $(\mathcal{D}_{1},\mathcal{D}_{2}^{w_{3}})$, and $(\mathcal{D}_{1},\mathcal{D}_{2}^{2w_{3}})$ increases the susceptibility of krope to divergence. Recall that at each training step, krope samples *pairs* of transitions. Thus, in this setup, krope will sample one transition from $\mathcal{D}_{1}$ and the other transition from the other dataset.

In Figures 5(b) and 5(c), we show the training loss of krope only and fqe+krope with krope as an auxiliary task. In both cases, the sampled state-actions are fed into a linear network encoder with a bias component and no activation function which outputs a $d=3$ representation. In case of fqe, this representation is then fed into a linear function to output the scalar value. From Figure 5(b), we find that even though $\mathcal{D}_{1}$ consists of mostly bad transitions, if krope also samples from $\mathcal{D}_{2}^{w_{1}}$ or $\mathcal{D}_{2}^{w_{2}}$, its divergence is mitigated. However, as expected, if krope uses $\mathcal{D}_{2}^{w_{3}}$ or $\mathcal{D}_{2}^{2w_{3}}$, krope diverges since the pairing of samples from $(\mathcal{D}_{1},\mathcal{D}_{2}^{w_{3}})$, and $(\mathcal{D}_{1},\mathcal{D}_{2}^{2w_{3}})$ leads to krope chasing a moving bootstrapped target, resulting in divergence. Building upon this insight, we can then better understand when krope can mitigate fqe’s divergence. From Figure 5(c), we see the expected result that when fqe uses $\mathcal{D}_{1}$ only, it diverges since $\mathcal{D}_{1}$ consists of many bad transitions. However, if fqe uses krope as an auxiliary loss, krope may either reduce the divergence $(\mathcal{D}_{2}^{w_{1}}\text{ and }\mathcal{D}_{2}^{w_{2}})$ or worsen it $(\mathcal{D}_{2}^{w_{3}}\text{ and }\mathcal{D}_{2}^{2w_{3}})$ depending on which dataset is paired with $\mathcal{D}_{1}$. While in practice, it is unclear to determine which situation is more likely, this result sheds light on the fact that the krope learning procedure can potentially mitigate the divergence of fqe’s learning procedure depending on which *pairs* of transitions are sampled.

#### C.3.2 Offline Policy Evaluation on d4rl Datasets

In this section, we present the offline policy evaluation results on the d4rl datasets. The setup is the same as the one used in Section 4. We present the results in Figure 6.

Figure: (a) Cheetah random
Refer to caption: extracted/5973682/results/dmcontrol/ope/d4rl/Cheetah_random_ope.jpg

Qualitatively, we reach the similar conclusions: krope is effective in producing stable and accurate ope estimates. However, in $3/9$ instances, krope does diverge. This divergence is likely related to the discussion in
Section 5 and Appendix C.3.1. Recall that krope is a semi-gradient method, which does not optimize any objective function and is susceptible to divergence (Feng et al., ; Sutton & Barto, ). So while krope *representations* stabilize value function learning, krope’s *learning algorithm* may diverge and not converge to krope representations. However, regardless of this result, krope does improve the stability and accuracy of fqe in all cases.

#### C.3.3 Stability-related Analysis on Custom Datasets

In this section, We include the remaining stability-related metric analysis that was deferred from the main paper.

##### Feature Co-adaptation, Condition Number, and Hyperparameter Sensitivity.

In this subsection, we include all the remaining results related to the stability metrics for all environments.

Figure: (a) CheetahRun
Refer to caption: extracted/5973682/results/dmcontrol/featcoadapt/CheetahRun_coadapt.jpg

Figure: (a) CheetahRun
Refer to caption: extracted/5973682/results/dmcontrol/condnum/CheetahRun_cond.jpg

Figure: (a) CheetahRun
Refer to caption: extracted/5973682/results/dmcontrol/hparam/CheetahRun_hparam.jpg

##### Bellman completeness.

Another metric that is associated with stability is Bellman completeness (bc) (Chang et al., ; Wang et al., ). We find that krope is approximately Bellman complete even though it does not explicitly optimize for it; this finding aligns with our Theorem 2. While bc is difficult to approximate, we can minimize the proxy metric introduced given in Equation (10) (Chang et al., ):

$$ $\mathcal{L}(M,\rho):=\mathbb{E}_{\mathcal{D}}\left\|\begin{bmatrix}M\\ \rho^{\top}\end{bmatrix}\phi(s,a)-\begin{bmatrix}\gamma\mathbb{E}_{s^{\prime} \sim P(\cdot|s,a),a^{\prime}\sim{\pi_{e}}(\cdot|s^{\prime})}[\phi(s^{\prime},a ^{\prime})]\\ r(s,a)\end{bmatrix}\right\|^{2}_{2}$ (10) $$

where $(\rho,M)\in\mathbb{R}^{d\times d}$, $\phi$ is fixed, and $\mathcal{L}(M,\rho)=0$ indicates Bellman completeness. Given the final learned representation, we compute and report the bc error in Table 2. We find that krope is approximately Bellman complete even though it does not explicitly optimize for it; this finding aligns with our Theorem 2. We note that bcrl-exp is less Bellman complete since it also includes the exploratory objective in its loss function, which if maximized can reduce the Bellman completeness. While bcrl is more bc than bcrl-exp, we found that it is less bc in general. We attribute this finding due to the difficulty in explicitly optimizing the bcrl objective which involves multiple neural networks ($M,\rho,\phi)$ and multiple loss functions on different scales (reward, self-prediction, log determinant regularization losses). krope can achieve approximate Bellman completeness without these optimization-related difficulties.

**Table 2: Bellman completeness measure for all algorithms across all domains. Results are averaged across $20$ trials and the deviation shown is the $95\%$ confidence interval. Values are rounded to the nearest single decimal.**
|  | Algorithm |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
| Domain | bcrl + exp | bcrl | beer | dr3 | fqe | krope (ours) |
| CartPoleSwingUp | $0.4\pm 0.1$ | $0.2\pm 0.1$ | $0.1\pm 0.0$ | $0.0\pm 0.0$ | $0.1\pm 0.0$ | $0.0\pm 0.0$ |
| CheetahRun | $3.3\pm 0.6$ | $2.4\pm 0.5$ | $0.7\pm 0.0$ | $0.0\pm 0.0$ | $0.7\pm 0.0$ | $0.2\pm 0.0$ |
| FingerEasy | $1.3\pm 0.6$ | $0.7\pm 0.2$ | $0.9\pm 0.0$ | $137.0\pm 4.4$ | $0.9\pm 0.0$ | $0.2\pm 0.0$ |
| WalkerStand | $10.4\pm 2.0$ | $0.3\pm 0.1$ | $0.5\pm 0.1$ | $66.1\pm 0.6$ | $0.6\pm 0.0$ | $0.1\pm 0.0$ |

#### C.3.4 Using fqe Directly for ope

In our main empirical section (Section 4), we used fqe as a representation learning algorithm on our custom datasets. We adopted the linear evaluation protocol, i.e., an approach of analyzing the penultimate features of the action-value function network and applied lspe on top of these features for ope. This protocol enabled us to better understand the nature of the learned features.

For the sake of completeness, we present results of fqe as an ope algorithm where the action-value network is directly used to estimate the performance of ${\pi_{e}}$. We present the results in Figures 10 and  11. As done in Section 4, we evaluate the performance of fqe and krope based on how they shape the penultimate features of the action-value network. However, when conducting ope, we evaluate two variants: 1) using lspe (-l) and 2) using the same end-to-end fqe action-value network (-e2e).

From Figure 10, we find that there are hyperparameter configurations that can outperform the krope variants. However, based on Figure 11, we find both krope variants are significantly more robust to hyperparameter tuning. This latter result suggests that krope does improve stability with respect to the hyperparameter sensitivity metric as well.

Regardless of fqe’s hyperparameter sensitivity, it is still interesting to observe that when fqe is used as an ope algorithm, it produces reasonably accurate ope estimates. It even outperforms the fqe+krope combination. However, as shown in Section 4, the penultimate features of this same network actually have poor property values such as high feature co-adaptation, high condition numbers, and highly erratic ope estimates with lspe. Given that these features have weak properties, it is unclear why fqe still can lead to accurate ope. The primary difference between using fqe for ope vs. fqe features and lspe for ope is in how the last linear layer is trained. The former is trained by gradient descent while the latter is trained with the iterative lspe algorithm on the fixed features. An interesting future direction will be to explore the learning dynamics of these two approaches.

On a related note, we point out that the training dynamics of fqe are still not well-understood. For example, Fujimoto et al. () show that the fqe loss function poorly correlates with value error. That is, the fqe loss can be high but value error (and ope error) can be low.

Figure: (a) CartPoleSwingUp
Refer to caption: extracted/5973682/results/dmcontrol/ope/fqe/CartPoleSwingUp_ope_fqe.jpg

Figure: (a) CartPoleSwingUp
Refer to caption: extracted/5973682/results/dmcontrol/ope/fqe_hparam/CartPoleSwingUp_fqe_hparam.jpg

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 6
presentation: 2
questions: 1. How can we obtain $\pi_e$ which is needed for calculation in KROPE?

2. In Garnet MDPs domain and 4 DM Control environments, how can we conduct offline value function learning while we don't have any offline datasets?

3. We are interested in understanding the specific implementation of the representation of the (s, a) pair. Specifically, we are wondering whether states and actions should be concatenated before being inputted into the network for representation, or if they should be inputted into separate networks for representation and then concatenated later. Furthermore, we wonder if the implementation will vary depending on whether the states are image-based or physics-based.
soundness: 3
strengths: 1. This paper rigorously proves theoretically that the KROPE's representations stabilize least-squares policy evaluation and KROPE's representations are Bellman complete.

2. The authors have shown that the KROPE method can learn offline value functions with both stability and accuracy through a series of experiments.
summary: The paper introduces Kernel Representations for Offline Policy Evaluation (KROPE), a novel algorithm aimed at stabilizing offline value function learning in reinforcement learning. By utilizing π-bisimulation to shape state-action representations, KROPE ensures that similar state-action pairs are represented similarly, enhancing convergence and reliability. The authors establish theoretical foundations demonstrating KROPE's stability through non-expansiveness and Bellman completeness. Empirical results show that KROPE outperforms non-bisimulation baselines in terms of stability and accuracy.
weaknesses: The experimental part of this paper does not explain how $\pi_e$ is obtained, which will result in the inability to complete MSVE calculations.

---

confidence: 5
contribution: 3
initial_rating: 8
presentation: 4
questions: A few questions, suggestions & typos:

1. L121: is the number "of" state-actions in the dataset...
2. L156: clarifying what "native features of the MDP" mean may improve readibility
3. Proposition 1: Including $\mathcal{D}$ in the expectations (like in Eq. 1) would help the reader make the connection to how stability of LSPE depends on the distribution shift between $\mathcal{D}$ and $\pi_e$.
4. L373-374: I'm not sure about the higher chance of instability in high dimensions due to the covariance matrix. This seems to require some assumptions on the distribution of $\phi$, which I am not sure if these representations necessarily satisfy. Hence, the sentence sounds speculative to me.
5. L367-374: To what extent is the degradation in stability with increasing dimensionality related to (4) being better satisfied? The difference between LHS and RHS of (4) in practice seems like an important proxy that could be studied in the Appendix.
6. Fig. 3: Legend placement is awkward. 
7. L486:  As "expected"
8. L487: Reads like BCRL-EXP-NA achieves lower condition number than KROPE, which doesn't seem right to me (Fig. 4b).
soundness: 4
strengths: 1. I found the paper to be very-well written and a breeze to read.
2. The paper is clear about what the scientific questions being investigated are (L74), what the contributions are and where the key novelties lie. For example, I appreciated the forthcoming note in L215 that the properties discussed in Sec. 3.1 are largely due to prior work, but Sections 3.2, 3.3, and 4 provide novel insights.
3. Given the clear scope of study (stability benefits of the bisimulation-like KROPE kernel in OPE) -- although somewhat narrow -- the theoretical results seem sound and provide clear answers (Theorems 1 and 2).
4. Prior work is discussed clearly and credited appropriately to my knowledge.
5. Experimental setup seems sound, providing additional insights and confirming theoretical results.
6. The empirical results show benefit of the proposed KROPE kernel for OPE, which may be of practical interest for a broader audience.
summary: The paper studies the stability of bisimulation-based representations in the context of offline RL, specifically as they serve to stabilize the LSPE algorithm and guaranatee Bellman completeness. To this end, they introduce a state-action similarity kernel, called KROPE, based on Castro et al. (2022) and Castro et al. (2023) in the spirit of bisimulation for state similarity. KROPE representations are shown to theoretically guarantee off-policy evaluation (OPE) stability for LSPE and yield Bellman-complete representations. Empirically, KROPE is compared to various alternatives in terms of mean-squared value error and various proxy measures aimed at analyzing stability in more depth. The empirical results verify that KROPE representations indeed provide stability benefits for OPE.
weaknesses: I found the discussion and the empirical study around generalization to be somewhat limited. While stability and realizability are important and seem to be at the center of this paper, bisimulation-based representations have attracted a lot of interest from the community due to their generalization capabilities (e.g., via invariance to distractors in the environment). Pearson correlation between value difference and orthogonality (Fig. 2c) is a useful proxy, but I would have expected to see more direct and thorough experiments in settings where bisimulation-like algorithms shine, e.g., value error over "The distracting control suite" by Stone et al. (2021).

---

confidence: 3
contribution: 2
initial_rating: 6
presentation: 3
questions: * It would be great to have more description for a few of the introduced concepts: why is the notion of generalizability introduced in section 2.3 a notion of generalizability? Why do we think Bellman completeness is a desirable property for stability? Is the condition on the injectivity of the abstract reward function in theorem 2 reasonable in practice?
* I think the first paragraph of the introduction could do a better job of orienting readers who aren't familiar with the idea of value representation learning. What does value function learning look like without special representation learning methods, and how do these methods aim to improve it? I.e. something like $q^{\pi}(s, a)$ --> $q^{\pi}(\phi(s, a))$, and making the role of function approximation here more explicit.
* Why is the higher error of KROPE for WalkerStand unsurprising? Lemma 3 discusses that KROPE may be less accurate when the transition dynamics or policy are stochastic, but I believe all the environments (including Walker) are deterministic and the policy should be stochastic across environments.
* In figure 4b, why does KROPE have a lower condition number than BCRL which specifically optimizes for this? The text at the top of page 10 seems to suggest that BCRL achieves a lower condition number even though the values contradict this.
* The description of FQE in section 4.1 is a little confusing. A cartoon of the $\phi$ network emphasizing the last layers and the representation vs. value function outputs might be helpful.
* I think terms like "one-step similarity" and "future similarity" would be more clear than "short-term" and "long-term" in equation (3) and definition 3.
* Minor issues: mismatched bracket at the end of equation (2); strange spacing around the sampled-from symbols beneath equation (3); FQE acronym was used without being defined yet in section 2.4.
soundness: 4
strengths: The paper is well-written, well-organized, and clear about the scope of the contributions and how they relate to prior work. The notation is clearly defined and easy to follow. The experiments compare the proposed representation learning algorithm (KROPE) with several appropriate baseline methods and convincingly show that KROPE supports stable value learning in standard RL benchmark tasks.
summary: This paper explores a method for learning representations of state-action pairs that can stabilize the learning of state-action value functions (e.g. for use in reinforcement learning) from offline datasets. The motivating idea is that if state-action pairs with similar values under a policy $pi$ have similar representations, this will make it easier to learn the true value function. First, the authors adapt the kernel-based $pi$-bisimulation representation proposed by Castro et al. (2023) from state to state-action value functions. Then, they present two novel stability results for this representation: first, they prove that using it for value function learning via least-squares policy evaluation will converge to the fixed point. Second, they show that it is "Bellman complete," an alternative notion of stability. Finally, the authors propose an algorithm for learning this representation from data and show that it is useful for stable value function learning in practice both in tabular MDP environments and in DeepMind Control Suite and D4RL tasks.
weaknesses: The scope of the contributions may be a little narrow. I believe the figures could use significant improvement. They are generally hard to read, using only color to differentiate the different methods and thin lines that are difficult to distinguish when the confidence interval is 0. While the learning traces in figure 3 are interesting, I wonder if there might be alternative ways to show these results that are easier to parse, such as summary statistics that capture the degree of instability (similar to figure 4 but focusing on the value error over the course of training in the different environments). For example, in figure 3b the plot is dominated by what looks like a single unstable seed for BCRL-na which makes it hard to tell what "good" vs "bad" seeds look like for each method.

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confidence: 5
contribution: 2
initial_rating: 3
presentation: 3
questions: Few questions related to the weaknesses : 

1. Can the authors compare theoertically, why bisimulation with the kernel representations can learn the true underlying latent dynamics? How do we know whether inverse or forward dynamics objectives are better/worse compared to KROPE? What does the algorithm mean in context of learning the underlying latent dynamics of the world? 

2. Can authors compare to past works on bisimulation based representations, both theoretically and empirically - and show how well does KROPE perform and scale in complex control tasks in light of these past objectives? 

3. I would expect to see some more experimental results in more challenging benchmarks - I think the current suite of tasks are not enough to show effectiveness of the proposed KROPE objective. 

4. Zang et al for example also studies bisimulation in context of noise (similar to Amy Zhang et al’s past work on bisimulation for control) - can the authors comment on what would happen with KROPE, given reliance on kernels, in presence of noise in the environment?

5. All experiments are done on raw state-action spaces; there are no tasks with pixel based observations? I can imagine kernel representations would fail on more complex pixelized environments. Can authors comment on the trade-offs here?

6. I would like to understand the Bellman completeness claim much better in context of KROPE, compared to past works. Can authors comment on this? Or in short - what is the theoretical guarantee of this algorithm, and how well can we tell that KROPE learns good representations?

7. Is the claim here to learn true underlying latent dynamics, or more on the stability with the KROPE representations? Each claim would have it own questions to answer, depending on how the authors would like to move ahead - what’s the goal here?
soundness: 2
strengths: The paper is well written and proposes a kernel based representation objective through first principles in RL; cares about why policy evaluation is an important measure/metric to look into for analyse stability of value function learning - and thoroughly discusses prior works in this widely studied area of learning good representations for online/offline RL. Furthermore, experiments analyse stability-metrics that are important to look into, when proposing new representation objectives; Figure 4 is a good example of that. It draws insights from prior works and discusses theoretical properties such as the Bellman completeness, that are worth looking into when analysing the strength of a new objective.
summary: This paper proposes a new algorithm for representation learning in offline RL; with the goal to make value function learning more stable and stabilize policy optimization through learning better representations. It builds up on prior works on bisimulation based representations, and proposes a kernel based objective to learn representations for similar state-action pairs. Experimental results are demonstrated on a few tasks to demonstrate why the proposed KROPE algorithm can do stable and accurate policy evaluation.
weaknesses: There are several key weaknesses that I find concerning in this paper : 

1. The proposed objective adds to the line of work around representation learning for RL; in addition to several past works that recently also studied bisimulation based objectives for control and offline RL. While the paper refers to past works, little comparisons are made to those prior works, either theoretically or experimentally - and it seems the authors start with policy evaluartion - discusses metrics related to stability for it - and claims to show expeirmentally why policy evaluation can be done well with the KROPE objective. However, several prior works have done this already, whether implicitly or explicitly showing the policy evaluation measure; and have demonstrated significantly empirically why bisimulation can be a good measure fior learning representations for offline RL. 

2. Following from the above comment - I do not understand the exact benefits the kernel based approach buys for this; whether we do bisimulartion based representation or not (such as looking into inverse or forward dynamics models for learning representations) - the benefits of kernels seem to be not explained too well here. Are there new techniques in deep kernel learning literature that can be exploited here? What happens when we go to large state action spaces, with complex tasks - how well would the kernel based approach scale here? 

3. Experiments are done on simple tasks where kernel based representations can assumably do similar to other objectives. However, as you scale to larger state-action spaces, I do not think the paper demonstrates well how well their algorithm would scale in this context. I would experiment more thorough comparisons with a large body of prior works for comparison, either empirically or theoretically.

4. Recent works (e.g Lamb et al.,) have significantly studied inverse dynamics models and claimed how these objectives can learn the true underlying latent dynamics in an unknown environment. In addition, Zang et al, studied and compared some of these inverse dynamics objectives with the bisimulation based objectives in offline RL - and demonstrated effectiveness of each approach on several empirical benchmarks. How well does the proposed KROPE representation do, in light of all these past works? In all these works, there is also the implicit or explicit demonstration of good enough policy evaluations to then do control. I do not think the discussion of policy evalaution and stability here is enough to show that KROPE can learn good policies for control.

Meta-review: This paper proposes learning a stable representation for least-squares value iteration by leveraging a bisimulation-style representation. However, my primary concern lies with the definition of the krope kernel introduced in the paper, which seems fundamentally flawed. The main issue stems from the definition of $k_2$ (Lines 231--232): since $(s_1', a_1')$ and $(s_2', a_2')$ are drawn independently, this metric fails to effectively capture the similarity between the transitions of $(s_1, a_1)$ and $(s_2, a_2)$. In fact, it is possible for $k_2((s_1, a_1), (s_2, a_2))$ to be larger than $k_2((s_1, a_1), (s_1, a_1))$, even when $(s_2, a_2)$ and $(s_1, a_1)$ have very different transitions. For example, suppose $(s_1, a_1)$ transitions to a reward-one state with probability 0.6 and a reward-zero one otherwise, $(s_2, a_2)$ transitions to the same reward-one state with probability 1. In this case, the proposed kernel can return a higher similarity between $(s_1,a_1)$ and $(s_2,a_2)$ than $(s_1,a_1)$ and itself.

To put it plainly, the krope kernel can assign higher similarity scores to state-action pairs that are very different than to pairs that are identical. This undermines its validity as a bisimulation metric from the outset, and, as a result, the subsequent results and claims built upon it seem unsubstantiated.

Decision: Reject