Bellman Unbiasedness: Toward Provably Efficient Distributional Reinforcement Learning with General Value Function Approximation

## Abstract

Abstract Distributional reinforcement learning improves performance by effectively capturing environmental stochasticity, but a comprehensive theoretical understanding of its effectiveness remains elusive.
In this paper, we present a regret analysis for distributional reinforcement learning with general value function approximation in a finite episodic Markov decision process setting.
We first introduce a key notion of Bellman unbiasedness for a tractable and exactly learnable update via statistical functional dynamic programming .
Our theoretical results show that approximating the infinite-dimensional return distribution with a finite number of moment functionals is the only method to learn the statistical information unbiasedly, including nonlinear statistical functionals.
Second, we propose a provably efficient algorithm, SF-LSVI , achieving a regret bound of O ~ ⁢ ( d E ⁢ H 3 2 ⁢ K ) ~ 𝑂 subscript 𝑑 𝐸 superscript 𝐻 3 2 𝐾 \tilde{O}(d_{E}H^{\frac{3}{2}}\sqrt{K}) over~ start_ARG italic_O end_ARG ( italic_d start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT divide start_ARG 3 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT square-root start_ARG italic_K end_ARG ) where H 𝐻 H italic_H is the horizon, K 𝐾 K italic_K is the number of episodes, and d E subscript 𝑑 𝐸 d_{E} italic_d start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT is the eluder dimension of a function class.

## 1 Introduction

Distributional reinforcement learning (DistRL) [, , , ] is an advanced approach to reinforcement learning (RL) that focuses on the entire probability distribution of returns rather than solely on the expected return.
By considering the full distribution of returns, distRL provides deeper insight into the uncertainty of each action, such as the mode or median.
This framework enables us to make safer and more effective decisions that account for various risks [, , , ], particularly in complex real-world situations, such as robotic manipulation [],
neural response [],
stratospheric balloon navigation [],
algorithm discovery [], and several game benchmarks [, ].

Although distRL has a solid theoretical basis and demonstrates effective empirical performance, its efficiency and tractability are still not fully understood.
In practice, distributions contain an infinite amount of information, and we should resort to approximations using a finite number of parameters or statistical functionals, such as categorical [] and quantile representations [].
However, not all statistical functionals can be learned through the Bellman operator, as the meaning of statistical functionals is not typically preserved after updates.
For example, the median is not preserved under the Bellman updates, as the median of a mixture of two distributions does not necessarily equate to the mixture of their medians.
Hence, a distinct analysis is required to determine whether there exists a corresponding Bellman operator for each statistical functional that ensures commutativity.
Rowland et al. [] defined Bellman closedness as a property of statistical functionals that can be exactly learned by the existence of a corresponding Bellman operator.

At this point, we take a closer look at the distributional Bellman update and revisit what additional properties of statistical functionals beyond Bellman closedness are needed to construct tractable algorithms.
In doing so, we identify and provide the following challenges that arise when statistical functionals are used to express the update process:

Figure: Figure 1: Venn-Diagram of Statistical Functional Classes. The diagram shows the category of statistical functional. (Yellow) Within the linear statistical functional class, Rowland et al. [] demonstrated that the solution of Bellman closedness is uniquely represented by polynomial exponential functionals. (Red) We extend this concept by introducing Bellman unbiasedness, which not only covers moment functionals but also central moment functionals from the broader class including nonlinear statistical functionals. (A) Maximum and minimum functionals are Bellman closed, while those are not unbiasedly estimatable. (B) Median and quantile functionals are neither Bellman closed and unbiased, highlighting that they are not proper to encode the distribution.
Refer to caption: x1.png

- •
Representing a mixture distribution with a finite, fixed number of parameters inevitably leads to approximation errors during the update process.
For example, when expressing the mixture of two distributions, each represented by $N$ parameters, compressing $2N$ into $N$ parameters in the mixture results in inevitable information loss.
- •
Due to the unknown nature of the transition probability $\mathbb{P}(\cdot|s,a)$, the target distribution is estimated by sampling the next state $s^{\prime}$.
For a tractable algorithm, it is crucial that the statistical functionals of the target distribution can be unbiasedly estimated using the statistical functionals from the sampled distribution.

In this paper, we describe the statistical functional and sketch of the distribution, and introduce the additional approximation errors that can occur with the distRL method.
We define the notion of Bellman unbiasedness, a property for representing the distribution tractably and prove that the moment functional is the only solution that satisfies this property, including nonlinear statistical functionals.
Based on this property, we propose a provably efficient statistical functional reinforcement learning algorithm with general value function approximation, denoted as SF-LSVI.
We will discuss the inherent issues of distributional Bellman completeness, a structural assumption previously defined in the literature [, ], and investigate the benefits of redesigning this concept using a sketch.

In summary, our main contributions are as follows:

- •
Introduce a key property of Bellman unbiasedness for collections of statistical functionals (sketch) and show that the moment functionals are the unique structure that can be unbiasedly estimated and exactly learned from distributions, including nonlinear statistical functionals.
- •
Revisit the existing concept of distributional Bellman Completeness through a statistical functional lens.
We enhance the tractability by using moment functional, that the agent can only access a finite-dimensional embedding space.
This removes the misspecification errors commonly encountered when discretizing infinite-dimensional distributions.
- •
Propose a tractable and provably efficient distributional RL algorithm called SF-LSVI, achieving a tightest regret upper bound $\tilde{O}(d_{E}H^{\frac{3}{2}}\sqrt{K})$ (^1^11We ignore poly-log terms in $H,S,A,K$ in the $\tilde{O}(\cdot)$ notation.) in distributional framework.
Our framework yields a better regret bound with a weaker structural assumption compared to prior results.

## 2 Related Work

#### Distributional RL.

In classical RL, the Bellman equation, which is based on expected returns, has a closed-form expression.
However, it is questionable whether any statistical functionals of return distribution always have their corresponding closed-form expressions.
Rowland et al. [] introduced the notion of Bellman Closedness for collections of statistical functionals that can be updated in a closed form via Bellman update.
They showed that the only Bellman-closed statistical functionals in the discounted setting are the moments $\mathbb{E}_{Z\sim\eta}[Z^{k}]$.
More recently, Marthe et al. [] proposed a general framework for distRL, where the agent plans to maximize their own utility functionals instead of expected return and formalized the property of Bellman Optimizability.
They proved that the only $W_{1}$-continuous and linear Bellman Optimizable statistical functionals are shown to be exponential utilities $\frac{1}{\lambda}\log\mathbb{E}_{Z\sim\eta}[\exp(\lambda Z)]$ in the undiscounted setting.

In practice, C51 [] and QR-DQN [] are notable distributional RL algorithms where the convergence guarantees of sampled-based algorithms are proved [, ].
Dabney et al. [] expanded the class of policies on arbitrary distortion risk measures by taking the based distribution non-uniformly and improve the sample efficiency from their implicit representation of the return distribution.
Cho et al. [] highlighted the drawbacks of optimistic exploration in distRL, introducing a randomized exploration method that perturbs the return distribution when the agent selects their next action.

**Table 1: Comparison for different methods under distributional RL framework. $\mathcal{H}$ represents a subspace of infinite-dimensional space $\mathcal{F}^{\infty}$. To bound the eluder dimesion $d_{E}$, Wang et al. [] and Chen et al. [] assumed the discretized reward MDP.**
| Algorithm | Regret | Eluder dimension $d_{E}$ | Bellman Completeness | MDP assumption | Tractability of<br>Distribution Estimation |
| --- | --- | --- | --- | --- | --- |
| O-DISCO<br>[] | $\tilde{\mathcal{O}}(\text{poly}(d_{E}H)\sqrt{K})$ | $\text{dim}_{E}(\mathcal{H},\epsilon)$ | distributional BC | discretized reward,<br>small-loss bound | ✗ |
| V-EST-LSR<br>[] | $\tilde{\mathcal{O}}(d_{E}H^{2}\sqrt{K})$ (^2^22In Chen et al. [], the regret bound is written as $\tilde{O}(d_{E}L_{\infty}(\rho)H\sqrt{K})$, where $L_{\infty}(\rho)$ represents the lipschitz constant of the risk measure $\rho$, i.e., $|\rho(Z)-\rho(Z^{\prime})|\leq L_{\infty}(\rho)\|F_{Z}-F_{Z^{\prime}}\|_{\infty}$. Since $L_{\infty}(\rho)\geq H$ in risk-neutral setting, we translate the regret bound into $\tilde{O}(d_{E}H^{2}\sqrt{K})$.) | $\text{dim}_{E}(\mathcal{H},\epsilon)$ | distributional BC | discretized reward,<br>lipschitz continuity | ✗ |
| [3pt]<br>SF-LSVI<br>[Ours] | $\tilde{\mathcal{O}}(d_{E}H^{\frac{3}{2}}\sqrt{K})$ | $\text{dim}_{E}(\mathcal{F}^{N},\epsilon)$ | statistical functional BC | none | ✓ |

#### RL with General Value Function Approximation.

Regret bounds have been studied for a long time in RL, across various domains such as bandit [, , ], tabular RL [, , , , ], and linear function approximation [, , ].
In recent years, deep RL has shown significant performance using deep neural networks as function approximators, and attempts have been made to analyze whether it is efficient in terms of general function approximation.
Wang et al. [] established a provably efficient RL algorithm with general value function approximation based on the eluder dimension $d_{E}$ [] and achieves a regret upper bound of $\tilde{O}(\text{poly}(d_{E}H)\sqrt{K})$.
To circumvent the intractability from computing the upper confidence bound, Ishfaq et al. [] injected the stochasticity on the training data and get the optimistic value function instead of upper confidence bound, enhancing computationally efficiency.
Beyond risk-neutral setting, several prior works have shown regret bounds under risk-sensitive objectives (e.g., entropic risk [, ], CVaR []), which align with our approach in that they are built on a distribution framework.
Liang and Luo [] achieved the regret upper bound of $\tilde{O}(\exp(H)\sqrt{|\mathcal{S}|^{2}|\mathcal{A}|H^{2}K})$ and the lower bound of $\Omega(\exp(H)\sqrt{|\mathcal{S}||\mathcal{A}|HK})$ in tabular setting.

#### DistRL with General Value Function Approximation.

Recently, only few efforts have aimed to bridge the gap between two fields.
Wang et al. [] proposed a distributional RL algorithm, O-DISCO, which enjoys small-loss bound by using a log-likelihood objective.
Similarly, Chen et al. [] provided a risk-sensitive reinforcement learning framework with static lipschitz risk measure.
While these studies analyze within a distributional framework, they do not address the intractability of implementation in infinite-dimensional space of distributions.
In contrast, our approach focuses on a statistical functional framework, providing a detailed comparison with other distRL methods as shown in Table 1.

## 3 Preliminaries

#### Episodic MDP.

We consider a episodic Markov decision process which is defined as a $\mathcal{M}=(\mathcal{S},\mathcal{A},H,\mathbb{P},r)$ characterized by state space $\mathcal{S}$, action space $\mathcal{A}$, horizon length $H$, transition kernels $\mathbb{P}=\{\mathbb{P}_{h}\}_{h\in[H]}$, and reward $r=\{r_{h}\}_{h\in[H]}$ at step $h\in[H]$.
The agent interacts with the environment across $K$ episodes. For each $k\in[K]$ and $h\in[H]$, $\mathbb{H}_{h}^{k}=(s_{1}^{1},a_{1}^{1},\dots,s_{H}^{1},a_{H}^{1},\dots,s_{h}^
{k},a_{h}^{k})$ represents the history up to step $h$ at episode $k$.
We assume the reward is bounded by $[0,1]$ and the agent always transit to terminal state $s_{\text{end}}$ at step $H+1$ with $r_{H+1}=0$.

#### Policy and Value Functions.

A (deterministic) policy $\pi$ is a collection of $H$ functions $\{\pi_{h}:\mathcal{S}\rightarrow\mathcal{A}\}_{h=1}^{H}$.
Given a policy $\pi$, a step $h\in[H]$, and a state-action pair $(s,a)\in\mathcal{S}\times\mathcal{A}$, the $Q$ and $V$-function are defined as
$Q_{h}^{\pi}(s,a)(:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R})\coloneqq
\mathbb{E}_{\pi}\left[\sum_{h^{\prime}=h}^{H}r_{h^{\prime}}(s_{h^{\prime}},a_{
h^{\prime}})\mid s_{h}=s,a_{h}=a\right]$ and
$V_{h}^{\pi}(s)(:\mathcal{S}\rightarrow\mathbb{R})\coloneqq\mathbb{E}_{\pi}
\left[\sum_{h^{\prime}=h}^{H}r_{h^{\prime}}(s_{h^{\prime}},a_{h^{\prime}})\mid
s
_{h}=s\right].$

#### Random Variables and Distributions.

For a sample space $\Omega$, we extend the definition of the $Q$-function into a random variable and its distribution,

$$ $\displaystyle Z_{h}^{\pi}(s,a)(:\mathcal{S}\times\mathcal{A}\times\Omega \rightarrow\mathbb{R})\coloneqq\sum_{h^{\prime}=h}^{H}r_{h^{\prime}}(s_{h^{ \prime}},a_{h^{\prime}})\mid s_{h}=s,a_{h}=a,a_{h^{\prime}}=\pi_{h^{\prime}}(s _{h^{\prime}}),$ $\displaystyle\eta_{h}^{\pi}(s,a)(:\mathcal{S}\times\mathcal{A}\rightarrow \mathscr{P}(\mathbb{R}))\coloneqq\text{law}(Z_{h}^{\pi}(s,a)).$ $$

Analogously, we extend the definition of $V$-function by introducing a bar notation.

$$ $\displaystyle\bar{Z}_{h}^{\pi}(s)(:\mathcal{S}\times\Omega\rightarrow\mathbb{R })\coloneqq\sum_{h^{\prime}=h}^{H}r_{h^{\prime}}(s_{h^{\prime}},a_{h^{\prime}} )\mid s_{h}=s,a_{h^{\prime}}=\pi_{h^{\prime}}(s_{h^{\prime}}),$ $\displaystyle\bar{\eta}_{h}^{\pi}(s)(:\mathcal{S}\rightarrow\mathscr{P}( \mathbb{R}))\coloneqq\text{law}(\bar{Z}_{h}^{\pi}(s)).$ $$

Note that $\bar{Z}^{\pi}_{h}(s)=Z^{\pi}_{h}(s,\pi(s))$ and $\bar{\eta}_{h}^{\pi}(s)=\eta_{h}^{\pi}(s,\pi(s))$.
We use $\pi^{\star}$ to denote an optimal policy
and denote $V_{h}^{\star}(s)=V_{h}^{\pi^{\star}}(s),Q_{h}^{\star}(s,a)=Q_{h}^{\pi^{\star}}
(s,a),\eta_{h}^{\star}(s,a)=\eta_{h}^{\pi^{\star}}(s,a)$, and $\bar{\eta}_{h}^{\star}(s)=\bar{\eta}_{h}^{\pi^{\star}}(s)$.
For notational simplicity, we denote the expectation over transition,
$[\mathbb{P}_{h}V_{h+1}^{\pi}](s,a)=\mathbb{E}_{s^{\prime}\sim\mathbb{P}_{h}(
\cdot|s,a)}V_{h+1}^{\pi}(s^{\prime})$,
$[\mathbb{P}_{h}\bar{Z}_{h+1}^{\pi}](s,a)=\mathbb{E}_{s^{\prime}\sim\mathbb{P}_
{h}(\cdot|s,a)}\bar{Z}_{h+1}^{\pi}(s^{\prime})$,
and $[\mathbb{P}_{h}\bar{\eta}_{h+1}^{\pi}](s,a)=\mathbb{E}_{s^{\prime}\sim\mathbb{
P}_{h}(\cdot|s,a)}\bar{\eta}_{h+1}^{\pi}(s^{\prime}).$
(^3^33Note that $\mathbb{E}_{s^{\prime}\sim\mathbb{P}_{h}(\cdot|s,a)}\bar{\eta}^{\pi}_{h+1}(s^{ \prime})$ is a mixture distribution.)
For brevity, we refer to $\bar{\eta}^{\pi}$ simply as $\bar{\eta}$.

In the episodic MDP, the agent aims to learn the optimal policy through a fixed number of interactions with the environment across a number of episodes.
At the beginning of each episode $k(\in[K])$, the agent starts at the initial state $s_{1}^{k}$ and choose a policy $\pi^{k}$.
In step $h(\in[H])$, the agent observes $s^{k}_{h}(\in\mathcal{S})$, takes an action $a^{k}_{h}(\in\mathcal{A})\sim\pi^{k}_{h}(\cdot|s^{k}_{h})$, receives a reward $r_{h}(s^{k}_{h},a^{k}_{h})$, and the environment transits to the next state $s^{k}_{h+1}\sim\mathbb{P}_{h}(\cdot|s^{k}_{h},a^{k}_{h}).$
Finally, we measure the suboptimality of an agent by its regret, which is the accumulated difference between the ground truth optimal and the return received from the interaction.
The regret after $K$ episodes is defined as
$\text{Reg}(K)=\sum_{k=1}^{K}V_{1}^{\star}(s_{1}^{k})-V_{1}^{\pi^{k}}(s_{1}^{k}).$

#### Distributional Bellman Optimality Equation.

Recall that $\eta^{\star}_{h}$ satisfies the distributional Bellman optimality equation:

$$ $\displaystyle\eta_{h}^{\star}(s,a)=(\mathcal{T}_{h}\eta_{h+1}^{\star})(s,a)$ $\displaystyle=\mathbb{E}_{s^{\prime}\sim\mathbb{P}_{h}(\cdot|s,a),a^{\prime} \sim\pi_{h}^{\star}(\cdot|s^{\prime})}[(\mathcal{B}_{r_{h}})_{\#}\eta_{h+1}^{ \star}(s^{\prime},a^{\prime})]$ $\displaystyle=(\mathcal{B}_{r_{h}})_{\#}[\mathbb{P}_{h}\eta_{h+1}^{\star}](s,a)$ $$

where $\mathcal{B}_{r}:\mathbb{R}\rightarrow\mathbb{R}$ is defined by $\mathcal{B}_{r}(x)=r+x$, and
$g_{\#}\eta\in\mathscr{P}(\mathbb{R})$ is the pushforward of the distribution $\eta$ through $g$, i.e., $g_{\#}\eta(A)=\eta(g^{-1}(A))$ for any Borel set $A\subseteq\mathbb{R}$.

#### Additional Notations.

For a given $N$, we denote an $N-$dimensional function class
$\mathcal{F}^{N}\coloneqq\mathcal{F}^{(1)}\times\dots\times\mathcal{F}^{(N)}
\subseteq\Big{\{}f=[f^{(1)},\cdots,f^{(N)}]:\mathcal{S}\times\mathcal{A}
\rightarrow\mathbb{R}^{N}\Big{\}}.$
Given a dataset $\mathcal{D}=\{(s_{t},a_{t},[z_{t}^{(1)},\dots,z_{t}^{(N)}])\}_{t=1}^{|\mathcal
{D}|}\subseteq\mathcal{S}\times\mathcal{A}\times\mathbb{R}^{N}$,
a set of state-action pairs $\mathcal{Z}=\{(s_{t},a_{t})\}_{t=1}^{|\mathcal{Z}|}\subseteq\mathcal{S}\times
\mathcal{A}$
and
for a function $f:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}^{N}$, we define the norm $\|f^{(n)}\|_{\infty},\|f\|_{\infty,1},\|f\|_{\mathcal{D}},\|f\|_{\mathcal{Z}}$ as written in Appendix A.
For a set of (vector-valued) functions $\mathcal{F}^{N}\subseteq\{f:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}^
{N}\}$,
the width function of $(s,a)$ is defined as
$w^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(n)}}(\mathcal{F}^{N},s,
a)\coloneqq\max_{f,g\in\mathcal{F}^{N}}|f^{{\color[rgb]{0,0,0}\definecolor[
named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}
\pgfsys@color@gray@fill{0}(n)}}(s,a)-g^{{\color[rgb]{0,0,0}\definecolor[named]
{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}
\pgfsys@color@gray@fill{0}(n)}}(s,a)|$.

## 4 Statistical Functionals in Distributional RL

Figure: Figure 2: Illustrative representation of sketch-based Bellman updates for a mixture distribution. Instead of updating the distributions directly, each sampled distribution is embedded through a sketch $\psi$ (e.g., mean $\mu$, quantile $q_{i}$). The transformation $\phi_{\psi}$ aims to compress the mixture distribution into the same number of parameters, ensuring unbiasedness to prevent information loss.
Refer to caption: x2.png

### 4.1 Bellman Unbiasedness

We revisit the definitions of statistical functionals and illustrate the property of Bellman closedness described in Bellemare et al. [].
Then we introduce another pivotal property, Bellman unbiasedness, required for learning in terms of statistical functionals, rather than approximated distribution.

###### Definition 4.1 (Statistical functionals, Sketch; [ 9 ] ) .

A statistical functional is a mapping from a probability distribution to a real value $\psi:\mathscr{P}(\mathbb{R})\rightarrow\mathbb{R}$.
A sketch is a vector-valued function ${\psi}_{1:N}:\mathscr{P}(\mathbb{R})\rightarrow\mathbb{R}^{N}$ specified by an $N$-tuple where each component is a statistical functional,

$$ ${\psi}_{1:N}(\cdot)=(\psi_{1}(\cdot),\cdots,\psi_{N}(\cdot)).$ $$

We denote the domain of sketch as $\mathscr{P}_{\psi_{1:N}}(\mathbb{R})$ and its image as
$I_{\psi_{1:N}}=\{\psi(\bar{\eta}):\bar{\eta}\in\mathscr{P}_{\psi_{1:N}}(
\mathbb{R})\}.$
We further extend to state return distribution functions
$\psi_{1:N}(\bar{\eta})=\Big{(}\psi_{1:N}(\bar{\eta}(s)):s\in\mathcal{S}\Big{)}.$

###### Definition 4.2 (Bellman closedness; [ 41 ] ) .

A sketch ${\psi}_{1:N}$ is Bellman closed if there exists an operator $\mathcal{T}_{\psi_{1:N}}:I_{\psi_{1:N}}^{\mathcal{S}}\rightarrow I_{\psi_{1:N}
}^{\mathcal{S}}$ such that

$$ $\psi_{1:N}(\mathcal{T}\bar{\eta})=\mathcal{T}_{\psi_{1:N}}\psi_{1:N}(\bar{\eta })\quad\text{for all }\bar{\eta}\in\mathscr{P}(\mathbb{R})^{\mathcal{S}}$ $$

which is closed under the distributional Bellman operator $\mathcal{T}:\mathscr{P}(\mathbb{R})^{\mathcal{S}}\rightarrow\mathscr{P}(
\mathbb{R})^{\mathcal{S}}$.

Bellman closedness is the property that a sketch are exactly learnable when updates are performed from the infinite-dimensional distribution space to the finite-dimensional embedding space.
Notably, Rowland et al. [] showed that the only finite linear statistical functionals that are Bellman closed are given by the collections of statistical functionals where its linear span is equal to the set of exponential-polynomial functionals
where $\psi_{0}$ is the constant functional equal to $1$.

###### Remark 4.3 .

Rowland et al. [] discussed only linear statistical functionals when defining a sketch, leaving open questions about the Bellman closedness of nonlinear statistical functionals such as variance, central moments, or quantiles.
However, it is noteworthy that nonlinear statistical functionals, such as maximum or minimum, can also be Bellman closed.
In their proof, there might be an ambiguity regarding the assumption that the corresponding sketch Bellman operator for quantile is linear.
In this paper, we show the non-existence of sketch Bellman operator for quantile and discuss in detail in Appendix B.1.

###### Theorem 4.4 .

Quantile functional cannot be Bellman closed under any additional sketch.

We investigate whether there exists another sketch that is both nonlinear and Bellman closed, and introduce additional conditions under which the Bellman closed property might be missing.
During the implementation phase, it is important to note that the agent does not have access to the transition kernel.
Specifically, the agent can only access the empirical transition kernel $\hat{\mathbb{P}}$ which is derived from a finite number of sample estimates.
This limitation implies that the operator should be treated as an empirical operator $\hat{\mathcal{T}}_{\psi}$, rather than $\mathcal{T}_{\psi}$.
Therefore, we introduce a new notion of Bellman unbiasedness that allows us to unbiasedly estimate the expected distribution $(\mathcal{B}_{r})_{\#}\mathbb{E}_{s^{\prime}\sim\mathbb{P}(\cdot|s,a)}[\bar{
\eta}(s^{\prime})]$, which is a mixture by transitions, from the sample distribution $(\mathcal{B}_{r})_{\#}\bar{\eta}(s^{\prime})$.

Figure: Figure 3: (Left) Bellman Closedness. (Right) Bellman Unbiasedness. The above path represents an ideal distributional Bellman update. Due to the infinite-dimensionality, the update process should be represented by using a finite-dimensional embedding (sketch) $\psi$. Since the transition kernel $\mathbb{P}$ is unknown, the below path describes that the implementation should sample the next state and update by using $\hat{\mathcal{T}}_{\psi}$ with the empirical transition kernel $\hat{\mathbb{P}}$. A sketch $\psi$ is Bellman unbiased if $\hat{\mathcal{T}}_{\psi}\circ\psi$ can unbiasedly estimate $\psi\circ\mathcal{T}$ through some transformation $\phi_{\psi}$.

###### Definition 4.5 (Bellman unbiasedness) .

A sketch ${\psi}$ is Bellman unbiased if there exists an integer $k$ and a corresponding vector-valued estimator $\phi_{\psi}=\phi_{\psi}(\psi(x_{1}),\cdots,\psi(x_{k})):(I_{\psi}^{\mathcal{S}
})^{k}\rightarrow I_{\psi}^{\mathcal{S}}$ where the sketch of expected distribution $(\mathcal{B}_{r})_{\#}\mathbb{E}_{s^{\prime}\sim\mathbb{P}(\cdot|s,a)}[\bar{
\eta}(s^{\prime})]$ can be unbiasedly estimated by $\phi_{\psi}$ using the $k$ samples from the sketch of the sample distribution $(\mathcal{B}_{r})_{\#}\bar{\eta}(s^{\prime})$, i.e.,

$$ $\displaystyle\mathbb{E}_{s^{\prime}_{1},\cdots,s^{\prime}_{k}\sim\mathbb{P}( \cdot|s,a)}\Bigg{[}{\phi_{\psi}}\Bigg{(}\underbrace{\psi\Big{(}(\mathcal{B}_{r })_{\#}\bar{\eta}(s^{\prime}_{1})}_{\hat{\mathcal{T}}_{\psi}\psi(\eta)}\Big{)} ,\cdots,\psi\Big{(}(\mathcal{B}_{r})_{\#}\bar{\eta}(s_{k}^{\prime})\Big{)} \Bigg{)}\Bigg{]}$ $\displaystyle=\psi\Big{(}(\mathcal{B}_{r})_{\#}\mathbb{E}_{s^{\prime}\sim \mathbb{P}(\cdot|s,a)}[\bar{\eta}(s^{\prime})]\Big{)}.$ $$

Bellman unbiasedness is another natural definition, similar to Bellman closedness, which takes into account a finite number of samples for the transition.
As a simple example, median functional is not Bellman unbiased as the sample median is not an unbiased estimate of the median functional.
In general, there is no unbiased estimator for the median, implying that the median functional cannot be unbiasedly estimated within an embedding space represented by a finite number of statistical functionals.
On the other hand, variance functional is a nonlinear statistical functional but can be unbiasedly estimated.
Then, the following question naturally arises;

$$ “Which sketches are unbiasedly estimatable under the sketch-based Bellman update?” $$

The following lemma answers this question.

###### Lemma 4.6 .

Let $F_{\bar{\eta}}$ be a CDF of the probability distribution $\bar{\eta}\in\mathscr{P}_{\psi}(\mathbb{R})^{\mathcal{S}}$.
Then a sketch is Bellman unbiased if and only if the sketch is homogeneous over $\mathscr{P}_{\psi}(\mathbb{R})^{\mathcal{S}}$ of degree $k$, i.e.,
there exists some vector-valued function $h=h(x_{1},\cdots,x_{k}):\mathcal{X}^{k}\rightarrow\mathbb{R}^{N}$ such that

$$ $\psi(\bar{\eta})=\int\cdots\int h(x_{1},\cdots,x_{k})dF_{\bar{\eta}}(x_{1}) \cdots dF_{\bar{\eta}}(x_{k}).$ $$

Lemma 4.6 states that in statistical functional dynamic programming, the unbiasedly estimatable embedding of a distribution can only be structured in the form of functions that are homogeneous of finite degree.
(e.g., variance functional is a homogeneous of degree $2$ []).
Taking this concept further and combining it with the results on Bellman closedness, we prove that even when including a nonlinear statistical functional, the only sketch that can be unbiasedly estimated in a finite-dimensional embedding space is the moment functional.

###### Theorem 4.7 .

The only finite statistical functionals that are both Bellman unbiased and closed are given by the collections of $\psi_{1},\dots,\psi_{N}$ where its linear span $\{\sum_{n=0}^{N}\alpha_{n}\psi_{n}|\ \alpha_{n}\in\mathbb{R}\ ,\forall N\}$ is equal to the set of exponential polynomial functionals $\{\eta\rightarrow\mathbb{E}_{Z\sim\eta}[Z^{l}\exp{(\lambda Z)}]|\ l=0,1,\dots,
L,\lambda\in\mathbb{R}\}$, where $\psi_{0}$ is the constant functional equal to $1$.
In discount setting, it is equal to the linear span of the set of moment functionals $\{\eta\rightarrow\mathbb{E}_{Z\sim\eta}[Z^{l}]|\ l=0,1,\dots,L\}$ for some $L\leq N$.

Compared to Rowland et al. [], our results extend beyond linear statistical functionals to include unbiasedly estimatable nonlinear statistical functionals, showing the uniqueness of the moment functional.
Our theoretical results not only show that high-order central moments such as variance or skewness are exactly learnable, but also reveal that other nonlinear statistical functionals like median or quantiles inevitably involve approximation errors due to biased estimations.
In Section 4.2, we emphasize that unbiasedly estimable sketches play an important role in mitigating misspecification errors, thereby enabling us to achieve sublinear regret.

###### Definition 4.1 (Statistical functionals, Sketch; [ 9 ] ) .

###### Definition 4.2 (Bellman closedness; [ 41 ] ) .

###### Remark 4.3 .

###### Theorem 4.4 .

###### Definition 4.5 (Bellman unbiasedness) .

###### Lemma 4.6 .

###### Theorem 4.7 .

### 4.2 Statistical Functional Bellman Completeness

We consider distributional reinforcement learning with value function approximation.
For successful TD learning, reinforcement learning with function approximation commonly requires the assumption, Bellman Completeness, that after applying Bellman operator, the output lies in the function class $\mathcal{F}$ [, , ].
On the other hand, our approach receives a tuple of function class $\mathcal{F}^{N}\subseteq\{f:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}^
{N}\}$ as input to represent $N$-moment of distribution.
Building on this, we assume that for any $\bar{\eta}:\mathcal{S}\rightarrow\mathscr{P}([0,H])$, the sketch of target function that results from applying sketch Bellman operator lies in the function class $\mathcal{F}^{N}$.

In the seminal works, Wang et al. [] and Chen et al. [] assumed that the function class
$\mathcal{H}\subseteq\{\eta:\mathcal{S}\times\mathcal{A}\rightarrow\mathscr{P}(
[0,H])\}$ follows the distributional Bellman Completeness assumption, i.e.,
if $\eta\in\mathcal{H}$ for all $\pi,h\in[H]$, then $\mathcal{T}_{h}^{\pi}\eta\in\mathcal{H}$.
However, since the distributional Bellman operator mixes distributions for the next state transition, it implies that a function class $\mathcal{H}$ must be closed under mixture.
Due to this closedness under mixture,
both previous studies assumed a discretized reward MDP, which requires the prior knowledge that outcomes of return distribution are additively closed.
In general, representing a distribution with a fixed number of parameters inevitably increases the required number of parameters due to the emergence of a mixture distribution.
Consequently, trying to represent it with a limited number of parameters leads to approximating the learning outcome, thereby causing model misspecification
which generally introduces an additional linear regret.
Precisely, a regret lower bound with model misspecification error $\zeta$ is known as $\Omega(\zeta K)$ in a linear bandit setting [].
Therefore, assuming Bellman Completeness on the distribution space is invalid as it significantly constrains the MDP structure or causes linear regret.

To circumvent this issue, we revisit the assumption of distributional Bellman completeness through the statistical functional lens.
We propose a new framework that matches a finite number of all statistical functionals to the target function, rather than the entire distribution itself.

###### Assumption 4.8 (Statistical Functional Bellman Completeness) .

For any distribution $\bar{\eta}:\mathcal{S}\rightarrow\mathscr{P}([0,H])$ and $h\in[H]$, there exists $f_{\bar{\eta}}\in\mathcal{F}^{N}$ which satisfies

$$ $f_{\bar{\eta}}(s,a)=\psi_{1:N}\big{(}(\mathcal{B}_{r_{h}})_{\#}[\mathbb{P}_{h} \bar{\eta}](s,a)\big{)}\quad\forall(s,a)\in\mathcal{S}\times\mathcal{A}$ $$

###### Assumption 4.9 (Model Misspecification) .

There exists a set of function $\mathcal{F}^{N}$ and a real number $\zeta>0$, such that for any $\bar{\eta}:\mathcal{S}\rightarrow\mathscr{P}([0,H])$ and $h\in[H]$, there exists $f_{\bar{\eta}}\in\mathcal{F}^{N}$ which satisfies

$$ $\max_{(s,a)\in\mathcal{S}\times\mathcal{A}}\left\|f_{\bar{\eta}}(s,a)-\psi_{1: N}\big{(}(\mathcal{B}_{r_{h}})_{\#}[\mathbb{P}_{h}\bar{\eta}](s,a)\big{)} \right\|\leq\zeta\quad\forall(s,a)\in\mathcal{S}\times\mathcal{A}.$ $$

We call $\zeta$ the misspecification error.

###### Assumption 4.8 (Statistical Functional Bellman Completeness) .

###### Assumption 4.9 (Model Misspecification) .

## 5 Algorithms

In this section, we propose SF-LSVI for distributional RL framework with general value function approximation.
Leveraging the result from Theorem 4.7, we introduce a moment least square regression.
This allows us to capture a finite set of moment information from the distribution, which can be unbiasedly estimated, thereby leading to the truncated moment problem.
Unlike previous work [, ] that estimates in infinite-dimensional distribution spaces, our method enables tractable distribution estimation in finite-dimensional embedding spaces without approximation error.

Figure: Algorithm 1 Statistical Functional Least Square Value Iteration (${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1} \pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0} \texttt{SF-LSVI}}(\delta)$)

#### Overview.

At the beginning of episode $k\in[K]$, we maintain all previous samples $\{(s^{\tau}_{h^{\prime}},a^{\tau}_{h^{\prime}},r^{\tau}_{h^{\prime}})\}_{(\tau
,h^{\prime})\in[k-1]\times[H]}$ and initialize a sketch $\psi_{1:N}(\bar{\eta}^{k}_{H+1}(\cdot))=\bm{0}^{N}$.
For each step $h=H,\dots,1$, we compute the normalized sample moments of target distribution $\{(\mathcal{B}_{r_{h^{\prime}}^{\tau}})_{\#}\bar{\eta}^{k}_{h+1}(s_{h^{\prime}
+1}^{\tau})\}_{h^{\prime}\in[H]}$ with the help of binomial theorem,

$$ $\displaystyle\psi_{n}\Big{(}(\mathcal{B}_{r_{h^{\prime}}^{\tau}})_{\#}\bar{ \eta}_{h}(s_{h^{\prime}+1}^{\tau})\Big{)}$ $\displaystyle=\mathbb{E}[(\bar{Z}_{h+1}^{k}(s_{h^{\prime}+1}^{\tau})+r_{h^{ \prime}}^{\tau})^{n}]{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}/H^{n-1}}$ $\displaystyle=\sum_{n^{\prime}=0}^{n}{\color[rgb]{0,0,0}\definecolor[named]{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill {0}H^{n^{\prime}}}\psi_{n^{\prime}}\Big{(}\bar{\eta}_{h}(s_{h^{\prime}+1}^{ \tau})\Big{)}(r_{h^{\prime}}^{\tau})^{n-n^{\prime}}{\color[rgb]{0,0,0} \definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0} \pgfsys@color@gray@fill{0}/H^{n-1}}$ $$

and iteratively solve the $N$-moment least square regression

$$ $\displaystyle\tilde{f}^{k}_{h,\bar{\eta}}\leftarrow\arg\min_{f\in\mathcal{F}} \sum_{\tau=1}^{k-1}\sum_{h^{\prime}=1}^{H}\Big{(}\sum_{n=1}^{N}f^{(n)}(s_{h^{ \prime}}^{\tau},a_{h^{\prime}}^{\tau})-\psi_{n}\Big{(}(\mathcal{B}_{r_{h^{ \prime}}^{\tau}})_{\#}\bar{\eta}_{h+1}^{k}(s_{h^{\prime}+1}^{\tau})\Big{)}\Big {)}^{2}$ $$

based on the dataset $\mathcal{D}^{k}_{h}$ which contains the sketch of temporal target $\psi_{1:N}\Big{(}(\mathcal{B}_{r_{h^{\prime}}^{\tau}})_{\#}\bar{\eta}_{h+1}^{k
}(s_{h^{\prime}+1}^{\tau})\Big{)}$.
Then we define $Q^{k}_{h}(\cdot,\cdot)=\min\{(\tilde{f}^{k}_{h,\bar{\eta}})^{(1)}(\cdot,\cdot)
+b^{k}_{h}(\cdot,\cdot),H\}$ and choose the greedy policy $\pi^{k}_{h}(\cdot)$ with respect to $Q^{k}_{h}$.
We repeat the procedure until all the $K$ episodes are completed.

###### Remark 5.1 .

For an optimistic planning, we define the bonus function as the width function $b^{k}_{h}(s,a)\coloneqq w^{k}_{h}((\mathcal{F}^{N})^{k}_{h},s,a)$ where $(\mathcal{F}^{N})^{k}_{h}$ denotes a confidence region at step $h$, episode $k$.
When $\mathcal{F}$ is a linear function class, the width function can be evaluated by simply computing the maximal distance of weight vector.
For a general function class $\mathcal{F}$, computing the width function requires to solve a set-constrained optimization problem, which is known as NP-hard [].
However, a width function is computed simply for optimistic exploration, and approximation errors are known to have a small effect on regret [].

###### Remark 5.1 .

## 6 Theoretical Analysis

In this section, we provide the theoretical guarantees for SF-LSVI under assumption 4.8.
We apply proof techniques from Wang et al. [] and extend the result to a statistical functional lens.
First, we generalize the concept of eluder dimension [] to the vector-valued function, which has been widely used in RL literatures [, , ] to measure the complexity of learning with the function approximators.

###### Definition 6.1 ( $\epsilon$ -dependent, $\epsilon$ -independent, Eluder dimension for vector-valued function) .

Let $\epsilon\geq 0$ and $\mathcal{Z}=\{(s_{i},a_{i})\}_{i=1}^{n}\subseteq\mathcal{S}\times\mathcal{A}$ be a sequence of state-action pairs.

- •
A state-action pair $(s,a)\in\mathcal{S}\times\mathcal{A}$ is $\bm{\epsilon}$-dependent on $\mathcal{Z}$ with respect to $\mathcal{F}^{N}$ if $\|f-g\|_{\mathcal{Z}}\leq\epsilon$ for any vector-valued function $f,g\in\mathcal{F}^{N}$, then $|f^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(1)}}(s,a)-g^{{\color[
rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(1)}}(s,a)|\leq\epsilon$.
- •
An $(s,a)$ is $\bm{\epsilon}$-independent on $\mathcal{Z}$ with respect to $\mathcal{F}^{N}$ if $(s,a)$ is not $\epsilon$-dependent on $\mathcal{Z}$.
- •
The $\bm{\epsilon}$-eluder dimension $\text{dim}_{E}(\mathcal{F}^{N},\epsilon)$ of a vector-valued function class $\mathcal{F}^{N}$ is the length of the longest sequence of elements in $\mathcal{S}\times\mathcal{A}$ such that, for some $\epsilon^{\prime}\geq\epsilon$, every element is $\epsilon^{\prime}$-independent on its predecessors.

We assume that the function class $\mathcal{F}^{N}$ and state-action space $\mathcal{S}\times\mathcal{A}$ have bounded covering numbers.

###### Assumption 6.2 (Covering number) .

For any $\epsilon>0$, the following holds:

- •
there exists an $\epsilon$-cover $\mathcal{C}(\mathcal{F}^{N},\epsilon)\subseteq\mathcal{F}^{N}$ with size $|\mathcal{C}(\mathcal{F}^{N},\epsilon)|\leq\mathcal{N}(\mathcal{F}^{N},\epsilon)$, such that for any $g\in\mathcal{F}^{N}$, there exists $g^{\prime}\in\mathcal{C}(\mathcal{F}^{N},\epsilon)$ with $\|g-g^{\prime}\|_{\infty,1}\leq\epsilon$.
- •
there exists an $\epsilon$-cover $\mathcal{C}(\mathcal{S}\times\mathcal{A},\epsilon)$ with size $|\mathcal{C}(\mathcal{S}\times\mathcal{A},\epsilon)|\leq\mathcal{N}(\mathcal{S
}\times\mathcal{A},\epsilon)$, such that for any $(s,a)\in\mathcal{S}\times\mathcal{A}$, there exists $(s^{\prime},a^{\prime})\in\mathcal{C}(\mathcal{S}\times\mathcal{A},\epsilon)$ with $\max_{f\in\mathcal{F}}|f(s,a)-f(s^{\prime},a^{\prime})|\leq\epsilon$

The following two lemmas give confidence bounds on the sum of the $l_{2}$ norms of all normalized moments.

###### Lemma 6.3 (Single Step Optimization Error) .

Consider a fixed $k\in[K]$ and a fixed $h\in[H]$. Let $\mathcal{Z}_{h}^{k}=\{(s_{h}^{\tau},a_{h}^{\tau})\}_{\tau\in[k-1]}$ and
$\mathcal{D}^{k}_{h,\bar{\eta}}=\Big{\{}\Big{(}s_{h}^{\tau},a_{h}^{\tau},{\psi_
{1:N}\Big{(}(\mathcal{B}_{r_{h^{\prime}}}^{\tau})_{\#}\bar{\eta}(s_{h^{\prime}
+1}^{\tau})\Big{)}}\Big{)}\Big{\}}_{\tau\in[k-1]}$ for any $\bar{\eta}:\mathcal{S}\rightarrow\mathscr{P}([0,H])$.
Define $\tilde{f}_{h,\bar{\eta}}^{k}=\arg\min_{f\in\mathcal{F}^{N}}\|f\|^{2}_{\mathcal
{D}^{k}_{h,\bar{\eta}}}$.
For any $\bar{\eta}$ and $\delta\in(0,1)$, there is an event $\mathcal{E}(\bar{\eta},\delta)$ such that conditioned on $\mathcal{E}(\bar{\eta},\delta)$, with probability at least $1-\delta$, for any $\bar{\eta}^{\prime}:\mathcal{S}\rightarrow\mathscr{P}([0,H])$ with
$\|\psi_{1:N}(\bar{\eta}^{\prime})-\psi_{1:N}(\bar{\eta})\|_{\infty,1}\leq 1/T$
,we have

$$ $\displaystyle\left\|\tilde{f}_{h,\bar{\eta}^{\prime}}(\cdot,\cdot)-\psi_{1:N} \Big{(}(\mathcal{B}_{r(\cdot,\cdot)})_{\#}[\mathbb{P}\bar{\eta}^{\prime}]( \cdot,\cdot)\Big{)}\right\|_{\mathcal{Z}_{h}^{k}}$ $\displaystyle\leq{c^{\prime}\left(N^{\frac{1}{2}}H\sqrt{\log(1/\delta)+\log \mathcal{N}(\mathcal{F}^{N},1/T)}\right)}$ $$

for some constant $c^{\prime}>0$.

We remark that using moment functional as a sketch is a unique structure that leads to a concentration results for the difference between the target and sampled sketches
$\psi\Big{(}(\mathcal{B}_{r})_{\#}[\mathbb{P}\bar{\eta}]\Big{)}-\psi\Big{(}(
\mathcal{B}_{r})_{\#}\bar{\eta}\Big{)}$, as the tranformation $\phi_{\psi}$ of moment functional is an identity.
Furthermore, using normalized moments minimizes the size of the confidence region from $O(H^{N})$ to $O(\sqrt{N})$.
This adjustment is akin to scaling the optimization function in multi-objective optimization to treat each objective equally, which effectively prevents the model from favoring objectives with larger scales.

###### Lemma 6.4 (Confidence Region) .

Let $(\mathcal{F}^{N})_{h}^{k}=\{f\in\mathcal{F}^{N}|\|f-\tilde{f}^{k}_{h,\bar{\eta
}}\|^{2}_{\mathcal{Z}^{k}_{h}}\leq\beta(\mathcal{F}^{N},\delta)\},$ where

$$ $\beta(\mathcal{F}^{N},\delta)\geq c^{\prime}\cdot NH^{2}(\log(T/\delta)+\log \mathcal{N}(\mathcal{F}^{N},1/T))$ $$

for some constant $c^{\prime}>0$.
Then with probability at least $1-\delta/2$, for all $k,h\in[K]\times[H]$, we have

$$ $\psi_{n}\Big{(}(\mathcal{B}_{r_{h}(\cdot,\cdot)})_{\#}[\mathbb{P}_{h}\bar{\eta }_{h+1}^{k}](\cdot,\cdot)\Big{)}\in(\mathcal{F}^{N})_{h}^{k}$ $$

Lemma 6.4 guarantees that the sequence of moments from the target distribution $\psi_{1:N}\Big{(}(\mathcal{B}_{r_{h}(\cdot,\cdot)})_{\#}[\mathbb{P}_{h}\bar{
\eta}_{h+1}^{k}](\cdot,\cdot)\Big{)}$ lies in the confidence region $(\mathcal{F}^{N})^{k}_{h}$ with high probability.
Supported by the aforementioned lemma, we can further guarantee that all $Q$-functions are optimistically estimated with high probability and derive our final result.

###### Theorem 6.5 .

Under Assumption 4.8, with probability at least $1-\delta$, SF-LSVI achieves a regret bound of

$$ $\text{Reg}(K)\leq 2H\text{dim}_{E}(\mathcal{F}^{N},1/T)+4H\sqrt{KH\log(1/ \delta)}$ $$

Under weaker structural assumptions, we show that SF-LSVI enjoys sublinear regret of order $\tilde{O}(d_{E}H^{\frac{3}{2}}\sqrt{K})$, which is $\sqrt{H}$ better than the state-of-the-art distributional framework algorithm V-EST-LSR [].
For the linear MDP setting, we have $d_{E}=\tilde{O}(d)$ and thus the regret bound of SF-LSVI is $\tilde{O}(\sqrt{d^{2}H^{3}K})$ which matches a lower bound $\Omega(\sqrt{d^{2}H^{3}K})$
[].
In our analysis, we highlight two main technical novelties which significantly reduces the complexity of regret;

- 1.
We refine previous lemma of Osband et al. [] and Wang et al. [] to remove the dependency of $\beta(\mathcal{F}^{N},1/\delta)$ (See Appendix C.4), ensuring that regret bound depends only on the defined function class, not on the number of moment extracted.
- 2.
As shown in Table 1, we define the eluder dimension $d_{E}$ in a finite-dimensional embedding space $\mathcal{F}^{N}$, while other methods rely on an infinite-dimensional distribution space $\mathcal{H}\subseteq\mathcal{F}^{\infty}$.

###### Definition 6.1 ( ϵ italic-ϵ \epsilon italic_ϵ -dependent, ϵ italic-ϵ \epsilon italic_ϵ -independent, Eluder dimension for vector-valued function) .

###### Assumption 6.2 (Covering number) .

###### Lemma 6.3 (Single Step Optimization Error) .

###### Lemma 6.4 (Confidence Region) .

###### Theorem 6.5 .

## 7 Conclusions

We describe the sources of approximation error inherent in distribution-based updates and introduce a pivotal concept of Bellman unbiasedness, which enables to exactly learn the information of distribution.
We also present a provably efficient distRL algorithm, SF-LSVI, with general value function approximation.
Notably, our algorithm achieves a near-optimal regret bound of $\tilde{O}(d_{E}H^{\frac{3}{2}}\sqrt{K})$, matching the tightest upper bound achieved by non-distributional framework [, ].
One interesting future direction would be to reformulate the definition of regret as discrepencies in moments rather than the expected return, and to show the sample-efficiency of distRL.
We hope that our work sheds some light on future research in analyzing the provable efficiency of distRL.

## Appendix

## Appendix A Notation

**Table 2: Table of notation**
| Notation | Description |
| --- | --- |
| $\mathcal{S}$ | state space of size $S$ |
| $\mathcal{A}$ | action space of size $A$ |
| $H$ | horizon length of one episode |
| $T$ | number of episodes |
| $r_{h}(s,a)$ | reward of $(s,a)$ at step $h$ |
| $\mathbb{P}_{h}(s^{\prime}|s,a)$ | probability transition of $(s,a)$ to $s^{\prime}$ at step $h$ |
| $\mathbb{H}^{k}_{h}$ | history up to step $h$, episode $k$ |
| $N$ | number of statistical functionals |
| $Q^{\pi}_{h}(s,a)$ | Q-function of a given policy $\pi$ at step $h$ |
| $V^{\pi}_{h}(s)$ | V-function of a given policy $\pi$ at step $h$ |
| $Z^{\pi}_{h}(s,a)$ | random variable of $Q$-function |
| $\bar{Z}^{\pi}_{h}(s)$ | random variable of $V$-function |
| $\eta^{\pi}_{h}(s,a)$ | probability distribution of $Q$-function |
| $\bar{\eta}^{\pi}_{h}(s)$ | probability distribution of $V$-function |
| $[\mathbb{P}_{h}(\cdot)]$ | expectation over transition $[\mathbb{P}_{h}(\cdot)]=\mathbb{E}_{s^{\prime}\sim\mathbb{P}_{h}}(\cdot)$ |
| $(\mathcal{B}_{r})_{\#}$ | pushforward of the distribution through $\mathcal{B}_{r}(x)\coloneqq r+x$ |
| $f^{(n)}$ | $n$-th element of $N$-dimensional vector $f$ |
| $\|f\|_{\infty}$ | max norm of $f:X\rightarrow\mathbb{R}$ defined as $\|f\|_{\infty}\coloneqq\max_{x\in X}|f^{(n)}(x)|$ |
| $\|f\|_{\infty,1}$ | $l_{1}$-norm of max norm of $f:X\rightarrow\mathbb{R}$ defined as $\|f\|_{\infty,1}\coloneqq\sum_{n=1}^{N}\max_{x\in X}|f^{(n)}(x)|$ |
| $\mathcal{F}^{N}$ | a function class of $N$-dimensional embedding space |
| $\mathcal{Z}$ | a set of state-action pairs $\mathcal{Z}\coloneqq\{(s_{t},a_{t})\}_{t=1}^{|\mathcal{Z}|}$ |
| $\mathcal{D}$ | a dataset $\mathcal{D}\coloneqq\{(s_{t},a_{t},[d_{t}^{(1)},\cdots,d_{t}^{(N)}])\}_{t=1}^{<br>|\mathcal{D}|}$ |
| $\|f\|_{\mathcal{Z}}^{2}$ | for $f:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}$,<br>define $\|f\|_{\mathcal{Z}}^{2}\coloneqq\sum_{n=1}^{N}\sum_{(s,a)\in\mathcal{Z}}(f^{(n<br>)}(s_{t},a_{t}))^{2}$ |
| $\|f\|_{\mathcal{D}}^{2}$ | for $f:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}$,<br>define $\|f\|_{\mathcal{D}}^{2}\coloneqq\sum_{n=1}^{N}\sum_{t=1}^{\mathcal{D}}(f^{(n)}<br>(s_{t},a_{t})-d_{t}^{(n)})^{2}$ |
| $w^{(n)}(\mathcal{F}^{N},s,a)$ | width function of $(s,a)$ defined as $w^{(n)}(\mathcal{F}^{N},s,a)\coloneqq\max_{f,g\in\mathcal{F}^{N}}|f^{(n)}(s,a)<br>-g^{(n)}(s,a)|$ |
| $\tilde{f}^{k}_{h,\bar{\eta}}$ | a solution of moment least squre regression, defined as $\tilde{f}^{k}_{h,\bar{\eta}}\coloneqq\arg\min_{f\in\mathcal{F}^{N}}\|f\|_{<br>\mathcal{D}^{k}_{h}}$ |
| $f_{\bar{\eta}}$ | a target sketch of distribution $\bar{\eta}$,<br>defined as $f_{\bar{\eta}}\coloneqq\psi_{1:N}((\mathcal{B}_{r})_{\#}[\mathbb{P}_{h}\bar{<br>\eta}])$ |
| $(\mathcal{F}^{N})^{k}_{h}$ | a confidence region at step $h$, episode $k$, defined as $(\mathcal{F}^{N})^{k}_{h}\coloneqq\{f\in\mathcal{F}^{N}|\ \|f-\tilde{f}^{k}_{h<br>,\bar{\eta}}\|^{2}_{\mathcal{Z}^{k}_{h}}\leq\beta(\mathcal{F}^{N},\delta)\}$ |
| $\psi(\bar{\eta})$ | a statistical functional $\mathscr{P}_{\psi}(\mathbb{R})^{\mathcal{S}}\rightarrow\mathbb{R}^{S}$ |
| $\psi_{1:N}(\bar{\eta})$ | a $N-$collection of statistical functional $\mathscr{P}_{\psi_{1:N}}(\mathbb{R})^{\mathcal{S}}\rightarrow\mathbb{R}^{N<br>\times S}$ |
| $\mathscr{P}_{\psi_{1:N}}(\mathbb{R})$ | a domain of sketch $\psi_{1:N}$ |
| $I_{\psi_{1:N}}$ | an image of sketch $\psi_{1:N}$ |
| $\mathcal{T}$ | distributional Bellman operator, defined as $\mathcal{T}\bar{\eta}\coloneqq(\mathcal{B}_{r})_{\#}[\mathbb{P}\bar{\eta}]$ |
| $\mathcal{T}_{\psi}$ | sketch Bellman operator w.r.t $\psi$, defined as<br>$\mathcal{T}_{\psi}\psi(\bar{\eta})\coloneqq\psi\Big{(}(\mathcal{B}_{r})_{\#}[<br>\mathbb{P}\bar{\eta}]\Big{)}$ |
| $\hat{\mathcal{T}}_{\psi}$ | empirical sketch Bellman operator w.r.t $\psi$, defined as<br>$\hat{\mathcal{T}}_{\psi}\psi(\bar{\eta})\coloneqq\psi\Big{(}(\mathcal{B}_{r})_<br>{\#}[\hat{\mathbb{P}}\bar{\eta}]\Big{)}$ |
| $\mathcal{N}(\mathcal{F}^{N},\epsilon)$ | covering number of $\mathcal{F}^{N}$ w.r.t the $\epsilon-$ball |
| $\text{dim}_{E}(\mathcal{F}^{N},\epsilon)$ | eluder dimension of $\mathcal{F}^{N}$ w.r.t $\epsilon$ |

## Appendix B Related Work and Discussion

### B.1 Technical Clarifications on Linearity Assumption in Existing Results

#### Bellman Closedness and Linearity.

Rowland et al. [] proved that quantile functional is not Bellman closed by providing a specific counterexample.
However, their discussion based on counterexamples can be generalized as it assumes that the sketch Bellman operator for the quantile functional needs to be linear.

They consider an discounted MDP with initial state $s_{0}$ with single action $a$, which transits to one of two terminal states $s_{1},s_{2}$ with equal probability.
Letting no reward at state $s_{0}$, $\texttt{Unif}([0,1])$ at state $s_{1}$, and $\texttt{Unif}([1/K,1+1/K])$ at state $s_{2}$, the return distribution at state $s_{0}$ is computed as mixture $\frac{1}{2}\texttt{Unif}([0,\gamma])+\frac{1}{2}\texttt{Unif}([\gamma/K,\gamma
+\gamma/K])$.
Then the $\frac{1}{2K}-$quantile at state $s_{0}$ is $\frac{\gamma}{K}$.
They proposed a counterexample where each quantile distribution of state $s_{1},s_{2}$ is represented as $\frac{1}{K}\sum_{k=1}^{K}\delta_{\frac{2k-1}{K}}$ and $\frac{1}{K}\sum_{k=1}^{K}\delta_{\frac{2k+1}{K}}$ respectively, the $\frac{1}{2K}-$quantile of state $s_{0}$ is
$\psi_{q_{2K}}\Big{(}\frac{1}{2K}\sum_{k=1}^{K}\delta_{\frac{\gamma(2k-1)}{K}}+
\delta_{\frac{\gamma(2k+1)}{K}}\Big{)}=\frac{3\gamma}{2K}.$
However, this example does not consider that the mixture of quantiles is not a quantile of the mixture distribution
(i.e., $\psi_{q}(\lambda\eta_{1}+(1-\lambda)\eta_{2})\neq\lambda\psi_{q}(\eta_{1})+(1-
\lambda)\psi_{q}(\eta_{2})$), due to the nonlinearity of the quantile functional.
Therefore, this does not present a valid counterexample to prove that quantile functionals are not Bellman closed.

#### Bellman Optimizability and Linearity.

Marthe et al. [] proposed the notion of Bellman optimizable statistical functional which redefine the Bellman update by planning with respect to statistical functionals rather than expected returns.
They proved that $W_{1}$-continuous Bellman Optimizable statistical functionals are characterized by exponential utilities $\frac{1}{\lambda}\log\mathbb{E}_{Z\sim\eta}[\exp(\lambda Z)]$.
However, their proof requires some technical clarification regarding the assumption that such statistical functionals are linear.

To illustrate, they define a statistical functional $\psi_{f}$ and consider two probability distributions $\eta_{1}=\frac{1}{2}(\delta_{0}+\delta_{h})$ and $\eta_{2}=\delta_{\phi(h)}$ where $\phi(h)=f^{-1}\Big{(}\frac{1}{2}(f(0)+f(h))\Big{)}$.
Using the translation property, they lead $\psi_{f}(\eta_{1})=\psi_{f}(\eta_{2})$ to
$\frac{1}{2}(f(x)+f(x+h))=f(x+\phi(h))$ for all $x\in\mathbb{R}$.
However, this equality $\psi_{f}\Big{(}\frac{1}{2}(\delta_{x}+\delta_{x+h})\Big{)}=\frac{1}{2}(f(x)+f(
x+h))$ holds only if $\psi_{f}$ is linear, which is not necessarily a valid assumption for all statistical functionals.

### B.2 Existence of Nonlinear Bellman Closed Sketch.

The previous two examples may not have considered the possibility that the sketch Bellman operator might not necessarily be linear.
However, some statistical functionals are Bellman-closed even if they are nonlinear, so it is open question whether there is a nonlinear sketch Bellman operator that makes the quantile functional Bellman-closed.
In this section, we present examples of maximum and minimum functionals that are Bellman-closed, despite being nonlinear.

In a nutshell, consider the maximum of return distribution at state $s_{1},s_{2}$ is $\gamma,\gamma+\gamma/K$ respectively.
Beyond linearity, the maximum of return distribution at state $s_{0}$ can be computed by taking the maximum of these values;

$$ $\max(\max(\bar{\eta}(s_{1})),\max(\bar{\eta}(s_{2})))=\max(\gamma,\gamma+ \gamma/K)=\gamma+\gamma/K$ $$

which produces the desired result.
This implies the existence of a nonlinear sketch that is Bellman closed.
More precisely, by defining $\max_{s^{\prime}\sim\mathbb{P}(\cdot|s,a)}$ and $\min_{s^{\prime}\sim\mathbb{P}(\cdot|s,a)}$ as the maximum and minimum of the sampled sketch $\psi\big{(}(\mathcal{B}_{r})_{\#}\bar{\eta}(s^{\prime})\big{)}$ with the distribution $\mathbb{P}(\cdot|s,a)$, we can derive the sketch Bellman operator for maximum and minimum functionals as follows;

- •
$\mathcal{T}_{\psi_{\text{max}}}\Big{(}\psi_{\text{max}}(\bar{\eta}(s))\Big{)}=
\max_{s^{\prime}\sim\mathbb{P}(\cdot|s,a)}\Big{(}\psi_{\text{max}}\big{(}(
\mathcal{B}_{r})_{\#}\bar{\eta}(s^{\prime})\big{)}\Big{)}$
- •
$\mathcal{T}_{\psi_{\text{min}}}\Big{(}\psi_{\text{min}}(\bar{\eta}(s))\Big{)}=
\min_{s^{\prime}\sim\mathbb{P}(\cdot|s,a)}\Big{(}\psi_{\text{min}}\big{(}(
\mathcal{B}_{r})_{\#}\bar{\eta}(s^{\prime})\big{)}\Big{)}$.

### B.3 Non-existence of sketch Bellman operator for quantile functional

In this section, we prove that quantile functional cannot be Bellman closed under any additional sketch.
First we introduce the definition of mixture-consistent, which is the property that the sketch of a mixture can be computed using only the sketch of the distribution of each component.

###### Definition B.1 (mixture-consistent) .

A sketch $\psi$ is mixture-consistent if for any $\lambda\in[0,1]$ and any distributions $\eta_{1},\eta_{2}\in\mathscr{P}_{\psi}(\mathbb{R})$, there exists a corresponding function $h_{\psi}$ such that

$$ $\displaystyle\psi(\lambda\eta_{1}+(1-\lambda)\eta_{2})=h_{\psi}\Big{(}\psi( \eta_{1}),\psi(\eta_{2}),\lambda\Big{)}.$ $$

Next, we will provide some examples of determining whether a sketch is mixture-consistent or not.

###### Definition B.1 (mixture-consistent) .

#### Example 1.

Every moment or exponential polynomial functional is mixture-consistent.

###### Proof.

For any $n\in[N]$ and $\lambda\in\mathbb{C}$,

$$ $\displaystyle\mathbb{E}_{Z\sim\lambda\eta_{1}+(1-\lambda)\eta_{2}}[Z^{n}\exp( \lambda Z)]=\lambda\mathbb{E}_{Z\sim\eta_{1}}[Z^{n}\exp(\lambda Z)]+(1-\lambda )\mathbb{E}_{Z\sim\eta_{2}}[Z^{n}\exp(\lambda Z)].$ $$

∎

###### Proof.

#### Example 2.

Variance functional is not mixture-consistent.

###### Proof.

Let $\lambda=\frac{1}{2}$ and $Z,Y$ be the random variables where $Z\sim\frac{1}{2}\delta_{0}+\frac{1}{2}\delta_{2}$ and $Y\sim\frac{1}{2}\delta_{k}+\frac{1}{2}\delta_{k+2}$.
Then, $\text{Var}(Z)=\text{Var}(Y)=1$.
While RHS is constant for any $k$,
LHS is not a constant for any $k$, i.e.,

$$ $\displaystyle\text{Var}_{X\sim\frac{1}{2}(\frac{1}{2}\delta_{0}+\frac{1}{2} \delta_{2})+\frac{1}{2}(\frac{1}{2}\delta_{k}+\frac{1}{2}\delta_{k+2})}(X)= \frac{1}{4}(k^{2}+5).$ $$

∎

While variance functional is not mixture consistent by itself, it can be mixture consistent with another statistical functional, the mean.

###### Proof.

#### Example 3.

Variance functional is mixture-consistent under mean functional.

###### Proof.

Notice that mean functional is mixture-consistent. We need to show that variance functional is mixture-consistent under mean functional.

$$ $\displaystyle\text{Var}_{Z\sim\lambda\eta_{1}+(1-\lambda)\eta_{2}}[Z]$ $\displaystyle=\mathbb{E}_{Z\sim\lambda\eta_{1}+(1-\lambda)\eta_{2}}[Z^{2}]-( \mathbb{E}_{Z\sim\lambda\eta_{1}+(1-\lambda)\eta_{2}}[Z])^{2}$ $\displaystyle=\lambda\mathbb{E}_{Z\sim\eta_{1}}[Z^{2}]+(1-\lambda)\mathbb{E}_{ Z\sim\eta_{2}}[Z^{2}]-(\lambda\mathbb{E}_{Z\sim\eta_{1}}[Z]+(1-\lambda)\mathbb {E}_{Z\sim\eta_{2}}[Z])^{2}$ $\displaystyle=\lambda(\text{Var}_{Z\sim\eta_{1}}[Z]+(\mathbb{E}_{Z\sim\eta_{1} }[Z])^{2})+(1-\lambda)(\text{Var}_{Z\sim\eta_{2}}[Z]+(\mathbb{E}_{Z\sim\eta_{2 }}[Z])^{2})$ $\displaystyle\qquad-(\lambda\mathbb{E}_{Z\sim\eta_{1}}[Z]+(1-\lambda)\mathbb{E }_{Z\sim\eta_{2}}[Z])^{2}.$ $$

∎

This means that to determine whether it is mixture-consistent or not, we should check it on a per-sketch basis, rather than on a per-statistical functional basis.

###### Proof.

#### Example 4.

Maximum and minimum functional are both mixture-consistent.

###### Proof.

$$ $\displaystyle\max_{Z\sim\lambda\eta_{1}+(1-\lambda)\eta_{2}}[Z]=\max(\max_{Z \sim\eta_{1}}[Z],\max_{Z\sim\eta_{2}}[Z])$ $$

and

$$ $\displaystyle\min_{Z\sim\lambda\eta_{1}+(1-\lambda)\eta_{2}}[Z]=\min(\min_{Z \sim\eta_{1}}[Z],\min_{Z\sim\eta_{2}}[Z])$ $$

∎

Since maximum and minimum functionals are mixture consistent, we can construct a nonlinear sketch bellman operator like the one in section B.2.
This is possible because there is a nonlinear function $h_{\psi}$ that ensures the sketch is closed under mixture.

Before demonstrating that a quantile sketch cannot be mixture consistent under any additional sketch, we will first illustrate with the example of a median functional that is not mixture consistent.

###### Proof.

#### Example 5.

Median sketch is not mixture-consistent.

###### Proof.

Let $\lambda=\frac{1}{2}$ and $Z,Y$ be the random variables where $Z\sim 0.2\delta_{0}+0.8\delta_{1}$ and $Y\sim 0.6\delta_{0}+0.4\delta_{k}$ for some $0<k<1$.
Then $\psi_{\text{med}}(Z)=1$ and $\psi_{\text{med}}(Y)=0$.
However,

$$ $\displaystyle\text{med}_{X=\frac{Z+Y}{2}}[X]=\psi_{\text{med}}(0.4\delta_{0}+0 .2\delta_{k}+0.4\delta_{1})=k$ $$

which is dependent in $k$.
∎

###### Lemma B.2 .

Quantile sketch cannot be mixture-consistent, under any additional sketch.

###### Proof.

For a given integer $N>0$ and a quantile level $\alpha\in[0,1]$,
let $\lambda=\frac{1}{2}$ and a random variable $Y\sim p_{y_{0}}\delta_{0}+p_{y_{1}}\delta_{y_{1}}+\cdots+p_{y_{N}}\delta_{y_{N
}}(0<y_{1}<\cdots<y_{N}<1)$ where $p_{y_{0}}>\alpha$ so that $\psi_{q_{\alpha}}(Y)=0$.
Consider another random variable $Z\sim p_{z_{0}}\delta_{0}+p_{z_{1}}\delta_{1}$ where $p_{z_{0}}<\alpha$ so that $\psi_{q_{\alpha}}(Z)=1$.
Then the $\alpha-$quantile of the mixture $X=\frac{Y+Z}{2}$ is

$$ $\psi_{q_{\alpha}}[X]=y_{n}\text{ where }n=\min\left\{n\leq N\Big{|}\ \frac{1}{ 2}\sum_{n^{\prime}=0}^{n}p_{y_{n^{\prime}}}+\frac{1}{2}p_{z_{0}}>\alpha\right\}.$ $$

Letting $p_{z_{0}}=2\alpha-\sum_{n^{\prime}=0}^{n}p_{y_{n^{\prime}}}$, we can manipulate $\psi_{q_{\alpha}}[X]$ to be any value of $y_{n}$.
Hence, $\psi(q_{\alpha})[X]$ is a function of all possible outcomes of $Y$.

If there exists a finite number of statistical functionals which make quantile sketch mixture-consistent, then such sketch would uniquely determine the distribution for any $N$.
This results in a contradiction that infinite-dimensional distribution space can be represented by a finite number of statistical functional.
∎

###### Lemma B.3 .

If a sketch $\psi$ is Bellman closed, then it is mixture-consistent.

###### Proof.

Consider an MDP where initial state $s_{0}$ has no reward and transits to two state $s_{1},s_{2}$ with probability $\lambda,1-\lambda$ and reward distribution $\bar{\eta}_{1},\bar{\eta}_{2}$.
Since $\psi$ is Bellman closed, $\psi(\bar{\eta}(s_{0}))$ is a function of $\psi(\bar{\eta}(s_{1}))$ and $\psi(\bar{\eta}(s_{2}))$, (i.e., $\psi(\bar{\eta}(s_{0}))$ = $g_{\psi}(\psi(\bar{\eta}(s_{1}))$, $\psi(\bar{\eta}(s_{2})))$ for some $g_{\psi}$).
Since $\psi(\bar{\eta}(s_{0}))=\psi(\lambda\bar{\eta}(s_{1})+(1-\lambda)\bar{\eta}(s_
{2}))$, it implies that $\psi$ is mixture-consistent.
∎

Combining the results of Lemma B.2 and Lemma B.3, we prove that a quantile sketch cannot be Bellman closed, no matter what additional sketches are provided.

###### Proof.

###### Lemma B.2 .

###### Proof.

###### Lemma B.3 .

###### Proof.

## Appendix C Proof

###### Theorem ( 4.4 ) .

Quantile functional cannot be Bellman closed under any additional sketch.

###### Proof.

See Lemma B.2 and Lemma B.3.
∎

###### Lemma ( 4.6 ) .

Let $F_{\bar{\eta}}$ be a CDF of the probability distribution $\bar{\eta}\in\mathscr{P}(\mathbb{R})^{\mathcal{S}}$.
Then a sketch is Bellman unbiased if and only if the sketch is a homogeneous of degree $k$, i.e.,
there exists some vector-valued function $h=h(x_{1},\cdots,x_{k}):\mathcal{X}^{k}\rightarrow\mathbb{R}^{N}$ such that

$$ $\psi(\bar{\eta})=\int\cdots\int h(x_{1},\cdots,x_{k})dF_{\bar{\eta}}(x_{1}) \cdots dF_{\bar{\eta}}(x_{k}).$ $$

###### Proof.

($\Rightarrow$)
Consider an two-stage MDP with a single action $a$, and an initial state $s_{0}$ which transits to one of terminal state $\{s_{1},\cdots,s_{K}\}$ with transition kernel $\mathbb{P}(\cdot|s_{0},a)$. Assume that the reward $r(s_{0})=0$.
Then $\bar{\eta}(s_{0})=\sum_{k=1}^{K}\mathbb{P}(s_{k})\delta_{r(s_{k})}$.
Note that $s^{\prime}_{1},\cdots,s^{\prime}_{k}$ are independent and identically distributed random variable in distribution $\mathbb{P}(\cdot|s,a)$.

$$ $\displaystyle\mathbb{E}_{s^{\prime}\sim\mathbb{P}(\cdot|s_{0},a)}\left[\phi_{ \psi}\left(\psi\Big{(}(\mathcal{B}_{r})_{\#}\bar{\eta}({s^{\prime}_{1}})\Big{) },\cdots,\psi\Big{(}(\mathcal{B}_{r})_{\#}\bar{\eta}({s^{\prime}_{k}})\Big{)} \right)\right]=\psi_{1:N}\Big{(}(\mathcal{B}_{r})_{\#}\mathbb{E}_{s^{\prime} \sim\mathbb{P}(\cdot|s_{0},a)}[\bar{\eta}(s^{\prime})]\Big{)}$ $\displaystyle\Longrightarrow\mathbb{E}_{s^{\prime}\sim\mathbb{P}(\cdot|s_{0},a )}\left[\phi_{\psi}\left(\psi\Big{(}\delta_{r({s^{\prime}_{1}})}\Big{)},\cdots ,\psi\Big{(}\delta_{r({s^{\prime}_{k}})}\Big{)}\right)\right]=\psi\Big{(} \mathbb{E}_{s^{\prime}\sim\mathbb{P}(\cdot|s_{0},a)}[\delta_{r(s^{\prime})}] \Big{)}$ $\displaystyle\Longrightarrow\mathbb{E}_{s^{\prime}\sim\mathbb{P}(\cdot|s_{0},a )}\left[\phi_{\psi}\Big{(}g({s^{\prime}_{1}}),\cdots,g({s^{\prime}_{k}})\Big{) }\right]=\psi\Big{(}{\bar{\eta}}({s_{0}})\Big{)}$ $\displaystyle\Longrightarrow\int\cdots\int h(s^{\prime}_{1},\cdots,s^{\prime}_ {k})dF_{\bar{\eta}}(s^{\prime}_{1})\cdots dF_{\bar{\eta}}(s^{\prime}_{k})=\psi \Big{(}{\bar{\eta}}({s_{0}})\Big{)}.$ $$

($\Leftarrow$)

$$ $\displaystyle\psi\Big{(}(\mathcal{B}_{r})_{\#}\mathbb{E}_{s^{\prime}\sim \mathbb{P}(\cdot|s,a)}[\bar{\eta}(s^{\prime})]\Big{)}$ $\displaystyle=\int\cdots\int h(x_{1},\cdots,x_{k})dF_{(\mathcal{B}_{r})_{\#} \mathbb{E}_{s^{\prime}\sim\mathbb{P}(\cdot|s,a)}[\bar{\eta}(s^{\prime})]}(x_{1 }),\cdots,dF_{(\mathcal{B}_{r})_{\#}\mathbb{E}_{s^{\prime}\sim\mathbb{P}(\cdot |s,a)}[\bar{\eta}(s^{\prime})](x_{k})}$ $\displaystyle=\int\cdots\int h(x_{1}+r,\cdots,x_{k}+r)d\Big{(}\mathbb{E}_{s^{ \prime}\sim\mathbb{P}(\cdot|s,a)}F_{\bar{\eta}(s^{\prime})}(x_{1})\Big{)}, \cdots,d\Big{(}\mathbb{E}_{s^{\prime}\sim\mathbb{P}(\cdot|s,a)}F_{\bar{\eta}(s ^{\prime})}(x_{k})\Big{)}$ $\displaystyle=\mathbb{E}_{s^{\prime}\sim\mathbb{P}(\cdot|s,a)}\left[\int\cdots \int h(x_{1}+r,\cdots,x_{k}+r)dF_{\bar{\eta}(s^{\prime})}(x_{1})\cdots dF_{ \bar{\eta}(s^{\prime})}(x_{k})\right]$ $\displaystyle=\mathbb{E}_{s^{\prime}\sim\mathbb{P}(\cdot|s,a)}\left[\psi\Big{( }(\mathcal{B}_{r})_{\#}[\bar{\eta}(s^{\prime})]\Big{)}\right]$ $$

∎

###### Theorem ( 4.7 ) .

The only finite statistical functionals that are Bellman unbiased and closed are given by the collections of $\psi_{1},\dots,\psi_{N}$ where its linear span $\{\sum_{n=0}^{N}\alpha_{n}\psi_{n}|\ \alpha_{n}\in\mathbb{R}\ ,\forall N\}$ is equal to the set of exponential polynomial functionals $\{\eta\rightarrow\mathbb{E}_{Z\sim\eta}[Z^{l}\exp{(\lambda Z)}]|\ l=0,1,\dots,
L,\lambda\in\mathbb{R}\}$, where $\psi_{0}$ is the constant functional equal to $1$.
In discount setting, it is equal to the linear span of the set of moment functionals $\{\eta\rightarrow\mathbb{E}_{Z\sim\eta}[Z^{l}]|\ l=0,1,\dots,L\}$ for some $L\leq N$.

###### Proof.

Our proof is mainly based on the proof techniques of Rowland et al. [] and we describe in an extended form.
Since their proof also considers the discounted setting, we will define $\mathcal{B}_{r,\gamma}(x)=r+\gamma x$ for discount factor $\gamma\in[0,1)$.
By assumption of Bellman closedness,
$\psi_{n}\Big{(}(\mathcal{B}_{r,\gamma})_{\#}\bar{\eta}(s^{\prime})\Big{)}$ will be written as $g(r,\gamma,\psi_{1:N}(\bar{\eta}(s^{\prime}))$ for some $g$.
By assumption of Bellman unbiasedness and Lemma 4.6,
both $\psi_{1:N}(\bar{\eta}(s^{\prime}))$ and $\psi_{n}\Big{(}(\mathcal{B}_{r,\gamma})_{\#}\bar{\eta}(s^{\prime})\Big{)}$ are affine as functions of the distribution $\bar{\eta}(s^{\prime})$,

$$ $\displaystyle\psi_{1:N}(\alpha\bar{\eta}_{1}(s^{\prime})+(1-\alpha)\bar{\eta}_ {2}(s^{\prime}))$ $\displaystyle=\mathbb{E}_{Z_{i}\sim\alpha\bar{\eta}_{1}(s^{\prime})+(1-\alpha) \bar{\eta}_{2}(s^{\prime})}[h_{1:N}(\bar{Z}_{1},\cdots,\bar{Z}_{k})]$ $\displaystyle=\alpha\mathbb{E}_{\bar{Z}_{i}\sim\bar{\eta}_{1}(s^{\prime})}[h_{ 1:N}(\bar{Z}_{1},\cdots,\bar{Z}_{k})]+(1-\alpha)\mathbb{E}_{\bar{Z}_{i}\sim \bar{\eta}_{2}(s^{\prime})}[h_{1:N}(\bar{Z}_{1},\cdots,\bar{Z}_{k})]$ $\displaystyle=\alpha\psi_{1:N}(\bar{\eta}_{1}(s^{\prime}))+(1-\alpha)\psi_{1:N }(\bar{\eta}_{2}(s^{\prime}))$ $$

and

$$ $\displaystyle\psi_{n}\Big{(}(\mathcal{B}_{r,\gamma})_{\#}(\alpha\bar{\eta}_{1} (s^{\prime})+(1-\alpha)\bar{\eta}_{2}(s^{\prime}))\Big{)}$ $\displaystyle=\mathbb{E}_{Z_{i}\sim\alpha\bar{\eta}_{1}(s^{\prime})+(1-\alpha) \bar{\eta}_{2}(s^{\prime})}[h_{n}(r+\gamma\bar{Z}_{1},\cdots,r+\gamma\bar{Z}_{ k})]$ $\displaystyle=\alpha\mathbb{E}_{\bar{Z}_{i}\sim\bar{\eta}_{1}(s^{\prime})}[h_{ n}(r+\gamma\bar{Z}_{1},\cdots,r+\gamma\bar{Z}_{k})]+(1-\alpha)\mathbb{E}_{\bar {Z}_{i}\sim\bar{\eta}_{2}(s^{\prime})}[h_{n}(r+\gamma\bar{Z}_{1},\cdots,r+ \gamma\bar{Z}_{k})]$ $\displaystyle=\alpha\psi_{n}\Big{(}(\mathcal{B}_{r,\gamma})_{\#}\bar{\eta}_{1} (s^{\prime})\Big{)}+(1-\alpha)\psi_{n}\Big{(}(\mathcal{B}_{r,\gamma})_{\#}\bar {\eta}_{2}(s^{\prime})\Big{)}$ $$

Therefore, $g(r,\gamma,\cdot)$ is also affine on the convex codomain of $\psi_{1:N}$.
Thus, we have

$$ $\mathbb{E}_{\bar{Z}_{i}\sim\bar{\eta}}[\phi_{\psi_{n}}(r+\gamma\bar{Z}_{1}, \cdots,r+\gamma\bar{Z}_{k})]=a_{0}(r,\gamma)+\sum_{n^{\prime}=1}^{N}a_{n^{ \prime}}(r,\gamma)\mathbb{E}_{\bar{Z}_{i}\sim\bar{\eta}}[\phi_{\psi_{n^{\prime }}}(\bar{Z}_{1},\cdots,\bar{Z}_{k})]$ $$

for some function $a_{0:N}:\mathbb{R}\times[0,1]\rightarrow\mathbb{R}$.
By taking $\bar{\eta}(s^{\prime})=\delta_{x}$, we obtain

$$ $\phi_{\psi_{n}}(r+\gamma x,\cdots,r+\gamma x)=a_{0}(r,\gamma)+\sum_{n^{\prime} =1}^{N}a_{n^{\prime}}(r,\gamma)\phi_{\psi_{n^{\prime}}}(x,\cdots,x).$ $$

According to Engert [], for any translation invariant finite-dimensional space is spanned by a set of function of the form

$$ $\{x\mapsto x^{l}\exp(\lambda_{j}x)|\ j\in[J],0\leq l\leq L\}$ $$

for some finite subset $\{\lambda_{1},\cdots,\lambda_{J}\}$ of $\mathbb{C}$.
Hence, each function $x\mapsto\phi_{\psi_{n}}(x,\cdots,x)$ is expressed as linear combination of exponential polynomial functions.
In addition, the linear combination of $\phi_{\psi_{n}}$ should be closed under composition with for any discount factor $\gamma\in[0,1]$, all $\lambda_{j}$ should be zero.
Hence, the linear combination of $\phi_{\psi_{1}},\cdots,\phi_{\psi_{N}}$ must be equal to the span of $\{x\mapsto x^{l}|\ 0\leq l\leq L\}$ for some $L\in\mathbb{N}$.

∎

###### Lemma ( 6.3 ) .

Consider a fixed $k\in[K]$ and a fixed $h\in[H]$. Let $\mathcal{Z}_{h}^{k}=\{(s_{h}^{\tau},a_{h}^{\tau})\}_{\tau\in[k-1]}$ and
$\mathcal{D}^{k}_{h,\bar{\eta}}=\Big{\{}\Big{(}s_{h}^{\tau},a_{h}^{\tau},{\psi_
{1:N}\Big{(}(\mathcal{B}_{r_{h^{\prime}}}^{\tau})_{\#}\bar{\eta}(s_{h^{\prime}
+1}^{\tau})\Big{)}}\Big{)}\Big{\}}_{\tau\in[k-1]}$ for any $\bar{\eta}:\mathcal{S}\rightarrow\mathscr{P}([0,H])$.
Define $\tilde{f}_{h,\bar{\eta}}^{k}=\arg\min_{f\in\mathcal{F}^{N}}\|f\|^{2}_{\mathcal
{D}^{k}_{h,\bar{\eta}}}$.
For any $\bar{\eta}$ and $\delta\in(0,1)$, there is an event $\mathcal{E}(\bar{\eta},\delta)$ such that conditioned on $\mathcal{E}(\bar{\eta},\delta)$, with probability at least $1-\delta$, for any $\bar{\eta}^{\prime}:\mathcal{S}\rightarrow\mathscr{P}([0,H])$ with
$\|\psi_{1:N}(\bar{\eta}^{\prime})-\psi_{1:N}(\bar{\eta})\|_{\infty,1}\leq 1/T$ or $\sum_{n=1}^{N}\|\psi_{n}(\bar{\eta}^{\prime})-\psi_{n}(\bar{\eta})\|_{\infty}
\leq 1/T$, we have

$$ $\displaystyle\left\|\tilde{f}_{h,\bar{\eta}^{\prime}}(\cdot,\cdot)-\psi_{1:N} \Big{(}(\mathcal{B}_{r(\cdot,\cdot)})_{\#}[\mathbb{P}\bar{\eta}^{\prime}]( \cdot,\cdot)\Big{)}\right\|_{\mathcal{Z}_{h}^{k}}$ $\displaystyle\leq{c^{\prime}\left(N^{\frac{1}{2}}H\sqrt{\log(1/\delta)+\log \mathcal{N}(\mathcal{F}^{N},1/T)}\right)}$ $$

for some constant $c^{\prime}>0$.

###### Proof.

Define the sketch of target $f_{\bar{\eta}}:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}^{N}$,

$$ $\displaystyle f_{\bar{\eta}}(\cdot,\cdot)\coloneqq\psi_{1:N}\Big{(}(\mathcal{B }_{r(\cdot,\cdot)})_{\#}[\mathbb{P}\bar{\eta}](\cdot,\cdot)\Big{)}$ $$

for all $i\in[N].$

For any $f\in\mathcal{F}$,

$$ $\displaystyle\|f\|^{2}_{\mathcal{D}^{k}_{h,\bar{\eta}^{\prime}}}-\|f_{\bar{ \eta}^{\prime}}\|^{2}_{\mathcal{D}^{k}_{h,\bar{\eta}^{\prime}}}$ $\displaystyle=\sum_{n=1}^{N}\sum_{s_{h}^{\tau},a_{h}^{\tau}\in{\mathcal{Z}^{k} _{h,\bar{\eta}^{\prime}}}}\Big{(}f^{(n)}(s_{h}^{\tau},a_{h}^{\tau})-\psi_{n} \Big{(}(\mathcal{B}_{r_{h}^{\tau}})_{\#}\bar{\eta}^{\prime}(s_{h+1}^{\tau}) \Big{)}\Big{)}^{2}-\Big{(}f_{\bar{\eta}^{\prime}}^{(n)}(s_{h}^{\tau},a_{h}^{ \tau})-\psi_{n}\Big{(}(\mathcal{B}_{r_{h}^{\tau}})_{\#}\bar{\eta}^{\prime}(s_{ h+1}^{\tau})\Big{)}\Big{)}^{2}$ $\displaystyle=\sum_{n=1}^{N}\sum_{s_{h}^{\tau},a_{h}^{\tau}\in{\mathcal{Z}^{k} _{h,\bar{\eta}^{\prime}}}}(f^{(n)}(s_{h}^{\tau},a_{h}^{\tau})-f_{\bar{\eta}^{ \prime}}^{(n)}(s_{h}^{\tau},a_{h}^{\tau}))^{2}$ $\displaystyle\qquad+2(f^{(n)}(s_{h}^{\tau},a_{h}^{\tau})-f_{\bar{\eta}^{\prime }}^{(n)}(s_{h}^{\tau},a_{h}^{\tau}))\Big{(}f^{(n)}_{\bar{\eta}^{\prime}}(s_{h} ^{\tau},a_{h}^{\tau})-\psi_{n}\Big{(}(\mathcal{B}_{r_{h}^{\tau}})_{\#}\bar{ \eta}^{\prime}(s_{h+1}^{\tau})\Big{)}\Big{)}$ $\displaystyle\geq\|f-f_{\bar{\eta}^{\prime}}\|^{2}_{\mathcal{Z}_{h}^{k}}{-4 \sum_{n=1}^{N}\|f_{\bar{\eta}}^{(n)}-f_{\bar{\eta}^{\prime}}^{(n)}\|_{\infty}( H+1)|\mathcal{Z}_{h}^{k}|}$ $\displaystyle\qquad+\sum_{n=1}^{N}\sum_{s_{h}^{\tau},a_{h}^{\tau}\in{\mathcal{ Z}^{k}_{h,\bar{\eta}^{\prime}}}}\Big{[}\underbrace{2(f^{(n)}(s_{h}^{\tau},a_{h }^{\tau})-f_{\bar{\eta}}^{(n)}(s_{h}^{\tau},a_{h}^{\tau}))\Big{(}f^{(n)}_{\bar {\eta}}(s_{h}^{\tau},a_{h}^{\tau})-\psi_{n}\Big{(}(\mathcal{B}_{r_{h}^{\tau}}) _{\#}\bar{\eta}(s_{h+1}^{\tau})\Big{)}\Big{)}}_{\chi_{h}^{\tau}(f^{(n)})}\Big{]}$ $\displaystyle\geq\|f-f_{\bar{\eta}^{\prime}}\|^{2}_{\mathcal{Z}_{h}^{k}}{ \color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-4N(H+1)}-\Big{|}\sum_{n =1}^{N}\sum_{s_{h}^{\tau},a_{h}^{\tau}\in\mathcal{Z}^{k}_{h,\bar{\eta}^{\prime }}}\chi^{\tau}_{h}(f^{(n)})\Big{|}.$ $$

For the first inequality, we change the second term from $\bar{\eta}^{\prime}$ to $\bar{\eta}$ which are the $\epsilon$-covers.
Notice that $AC-BC^{\prime}\geq-|AC-BC^{\prime}|\geq-|(A-B)C|-|(A-B)C^{\prime}|\geq-2|A-B||
\max(C,C^{\prime})|$.

$$ $\displaystyle(f^{(n)}(s_{h}^{\tau},a_{h}^{\tau})-f_{\bar{\eta}^{\prime}}^{(n)} (s_{h}^{\tau},a_{h}^{\tau}))\Big{(}f^{(n)}_{\bar{\eta}^{\prime}}(s_{h}^{\tau}, a_{h}^{\tau})-\psi_{n}\Big{(}(\mathcal{B}_{r_{h}^{\tau}})_{\#}\bar{\eta}^{ \prime}(s_{h+1}^{\tau})\Big{)}\Big{)}$ $\displaystyle\qquad-(f^{(n)}(s_{h}^{\tau},a_{h}^{\tau})-f_{\bar{\eta}}^{(n)}(s _{h}^{\tau},a_{h}^{\tau}))\Big{(}f^{(n)}_{\bar{\eta}}(s_{h}^{\tau},a_{h}^{\tau })-\psi_{n}\Big{(}(\mathcal{B}_{r_{h}^{\tau}})_{\#}\bar{\eta}(s_{h+1}^{\tau}) \Big{)}\Big{)}$ $\displaystyle\geq-2\|f_{\bar{\eta}^{\prime}}^{(n)}(s_{h}^{\tau},a_{h}^{\tau})- f_{\bar{\eta}}^{(n)}(s_{h}^{\tau},a_{h}^{\tau})\|$ $\displaystyle\qquad\times\max\Big{(}\Big{|}f^{(n)}_{\bar{\eta}^{\prime}}(s_{h} ^{\tau},a_{h}^{\tau})-\psi_{n}\Big{(}(\mathcal{B}_{r_{h}^{\tau}})_{\#}\bar{ \eta}^{\prime}(s_{h+1}^{\tau})\Big{)}\Big{|},\Big{|}f^{(n)}_{\bar{\eta}}(s_{h} ^{\tau},a_{h}^{\tau})-\psi_{n}\Big{(}(\mathcal{B}_{r_{h}^{\tau}})_{\#}\bar{ \eta}(s_{h+1}^{\tau})\Big{)}\Big{|}\Big{)}$ $\displaystyle\geq-2\|f_{\bar{\eta}^{\prime}}^{(n)}(s_{h}^{\tau},a_{h}^{\tau})- f_{\bar{\eta}}^{(n)}(s_{h}^{\tau},a_{h}^{\tau})\|(H+1)$ $$

For the second inequality, consider $\bar{\eta}^{\prime}:\mathcal{S}\rightarrow\mathscr{P}([0,H])$ with $\sum_{n=1}^{N}\|\psi_{n}(\bar{\eta}^{\prime})-\psi_{n}(\bar{\eta})\|_{\infty}
\leq 1/T$.
We have

$$ $\displaystyle\|f^{(n)}_{\bar{\eta}}-f^{(n)}_{\bar{\eta}^{\prime}}\|_{\infty}$ $\displaystyle=\max_{s,a}\Big{|}\sum_{n^{\prime}=1}^{n}{\color[rgb]{0,0,0} \definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0} \pgfsys@color@gray@fill{0}H^{n^{\prime}}}[\psi_{n^{\prime}}([\mathbb{P}\bar{ \eta}](s,a))-\psi_{n^{\prime}}([\mathbb{P}\bar{\eta}^{\prime}](s,a))]r^{n-n^{ \prime}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}/H^{n-1}}\Big{|}$ $\displaystyle\leq\sum_{n^{\prime}=1}^{n}\max_{s^{\prime}}\Big{|}\psi_{n^{ \prime}}(\bar{\eta}(s^{\prime}))-\psi_{n^{\prime}}(\bar{\eta}^{\prime}(s^{ \prime}))\Big{|}$ $\displaystyle\leq 1/T.$ $$

Defining $\mathbb{F}_{h}^{k}$ as the filtration induced by the sequence
$\{(s_{h^{\prime}}^{\tau},a_{h^{\prime}}^{\tau})\}_{\tau,h^{\prime}\in[k-1]
\times[H]}\cup\{(s_{1}^{k},a_{1}^{k}),(s_{2}^{k},a_{2}^{k}),\dots,(s_{h}^{k},a
_{h}^{k})\}$,
notice that

$$ $\displaystyle\mathbb{E}\Big{[}\sum_{n=1}^{N}\chi_{h}^{\tau}(f^{(n)})\Big{|} \mathbb{F}_{h}^{\tau}\Big{]}$ $\displaystyle=\sum_{n=1}^{N}2(f^{(n)}(s_{h}^{\tau},a_{h}^{\tau})-f_{\bar{\eta} }^{(n)}(s_{h}^{\tau},a_{h}^{\tau}))(f^{(n)}_{\bar{\eta}}(s_{h}^{\tau},a_{h}^{ \tau})-\mathbb{E}\Big{[}\psi_{n}\Big{(}(\mathcal{B}_{r_{h}^{\tau}})_{\#}{\bar{ \eta}}(s_{h+1}^{\tau})\Big{)}\Big{|}\mathbb{F}_{h}^{\tau}\Big{]})$ $\displaystyle=\sum_{n=1}^{N}2(f^{(n)}(s_{h}^{\tau},a_{h}^{\tau})-f_{\bar{\eta} }^{(n)}(s_{h}^{\tau},a_{h}^{\tau}))(f^{(n)}_{\bar{\eta}}(s_{h}^{\tau},a_{h}^{ \tau})-\mathbb{E}_{s_{h+1}^{\tau}\sim\mathbb{P}_{h}(\cdot|s_{h}^{\tau},a_{h}^{ \tau})}\Big{[}\psi_{n}\Big{(}(\mathcal{B}_{r_{h}^{\tau}})_{\#}{\bar{\eta}}(s_{ h+1}^{\tau})\Big{)}\Big{]})$ $\displaystyle=\sum_{n=1}^{N}2(f^{(n)}(s_{h}^{\tau},a_{h}^{\tau})-f_{\bar{\eta} }^{(n)}(s_{h}^{\tau},a_{h}^{\tau}))(f^{(n)}_{\bar{\eta}}(s_{h}^{\tau},a_{h}^{ \tau})-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\psi_{n}\Big{(}(\mathcal {B}_{r_{h}^{\tau}})_{\#}\mathbb{E}_{s_{h+1}^{\tau}\sim\mathbb{P}_{h}(\cdot|s_{ h}^{\tau},a_{h}^{\tau})}[{\bar{\eta}}(s_{h+1}^{\tau})]\Big{)})}$ $\displaystyle=0$ $$

and

$$ $\displaystyle\Big{|}\sum_{n=1}^{N}\chi_{h}^{\tau}(f^{(n)})\Big{|}$ $\displaystyle=\Big{|}\sum_{n=1}^{N}2(f^{(n)}(s_{h}^{\tau},a_{h}^{\tau})-f_{ \bar{\eta}}^{(n)}(s_{h}^{\tau},a_{h}^{\tau}))(f_{\bar{\eta}}^{(n)}(s_{h}^{\tau },a_{h}^{\tau})-\psi_{n}\Big{(}(\mathcal{B}_{r_{h}^{\tau}})_{\#}{\bar{\eta}}(s _{h+1}^{\tau})\Big{)})\Big{|}$ $\displaystyle\leq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\max_{n\in[N]}\Big {\{}2(f_{\bar{\eta}}^{(n)}(s_{h}^{\tau},a_{h}^{\tau})-\psi_{n}\Big{(}(\mathcal {B}_{r_{h}^{\tau}})_{\#}{\bar{\eta}}(s_{h+1}^{\tau})\Big{)})\Big{\}}}\sum_{n=1 }^{N}\Big{|}f^{(n)}(s_{h}^{\tau},a_{h}^{\tau})-f_{\bar{\eta}}^{(n)}(s_{h}^{ \tau},a_{h}^{\tau})\Big{|}$ $\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0 }\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\leq 2(H+1)\sum_{n=1}^{ N}\Big{|}f^{(n)}(s_{h}^{\tau},a_{h}^{\tau})-f_{\bar{\eta}}^{(n)}(s_{h}^{\tau}, a_{h}^{\tau})\Big{|}}$ $$

In third equality, we emphasize that only Bellman unbiased sketch can derive the martingale difference sequence which induce the concentration result.
Since every moment functional is commutable with mixing operation, the transformation $\phi_{\psi_{n}}$ in Definition 4.5 is identity for all $n\in[N]$.
Hence, we choose the sketch as moment which already knows $\phi_{\psi}$.

By Azuma-Hoeffding inequality,

$$ $\displaystyle\mathbb{P}\Big{[}\Big{|}\sum_{(\tau,h)\in[k-1]\times[H]}\sum_{n=1 }^{N}\chi_{h}^{\tau}(f^{(n)})\Big{|}\geq\epsilon\Big{]}$ $\displaystyle\leq 2\exp\Big{(}-\frac{\epsilon^{2}}{2(2(H+1))^{2}\sum_{(\tau,h) \in[k-1]\times[H]}\Big{(}\sum_{n=1}^{N}|f^{(n)}-f_{\bar{\eta}}^{(n)}|\Big{)}^{ 2}}\Big{)}$ $\displaystyle\leq 2\exp\Big{(}-\frac{\epsilon^{2}}{2(2(H+1))^{2}\sum_{(\tau,h) \in[k-1]\times[H]}\Big{(}N\sum_{n=1}^{N}|f^{(n)}-f_{\bar{\eta}}^{(n)}|^{2}\Big {)}}\Big{)}$ $\displaystyle=2\exp\Big{(}-\frac{\epsilon^{2}}{2N(2(H+1))^{2}\|f-f_{\bar{\eta} }\|_{\mathcal{Z}_{h}^{k}}^{2}}\Big{)}$ $$

where the second inequality follows from the Cauchy-Schwartz inequality.

We set

$$ $\displaystyle\epsilon=\sqrt{8N(H+1)^{2}\|f-f_{\bar{\eta}}\|^{2}_{\mathcal{Z}_{ h}^{k}}\log\Big{(}\frac{\mathcal{N}(\mathcal{F}^{N},1/T)}{\delta}\Big{)}}$ $$

With union bound for all $f\in\mathcal{C}(\mathcal{F}^{N},1/T)$, with probability at least $1-\delta$,

$$ $\displaystyle\Big{|}\sum_{(\tau,h)\in[k-1]\times[H]}\sum_{n=1}^{N}\chi_{h}^{ \tau}(f^{(n)})\Big{|}\leq c^{\prime}N^{\frac{1}{2}}(H+1)\|f-f_{\bar{\eta}}\|_{ \mathcal{Z}_{h}^{k}}\sqrt{\log\Big{(}\frac{\mathcal{N}(\mathcal{F}^{N},1/T)}{ \delta}\Big{)}}$ $$

for some constant $c^{\prime}>0$.

For all $f\in\mathcal{F}^{N}$, there exists $g\in\mathcal{C}(\mathcal{F}^{N},1/T)$, such that $\|f-g\|_{\infty,1}\leq 1/T$ or $\sum_{n=1}^{N}\|f^{(n)}-g^{(n)}\|_{\infty}\leq 1/T$ for all $n\in[N]$,

$$ $\displaystyle\Big{|}\sum_{(\tau,h)\in[k-1]\times[H]}\sum_{n=1}^{N}\chi_{h}^{ \tau}(f^{(n)})\Big{|}$ $\displaystyle\leq\Big{|}\sum_{(\tau,h)\in[k-1]\times[H]}\sum_{n=1}^{N}\chi_{h} ^{\tau}(g^{(n)})\Big{|}+2(H+1)|\mathcal{Z}^{k}_{h}|\sum_{n=1}^{N}\frac{1}{T}$ $\displaystyle\leq c^{\prime}N^{\frac{1}{2}}(H+1)\|g-f_{\bar{\eta}}\|_{\mathcal {Z}^{k}_{h}}\sqrt{\log\Big{(}\frac{\mathcal{N}(\mathcal{F}^{N},1/T)}{\delta} \Big{)}}+2N(H+1)$ $\displaystyle\leq c^{\prime}N^{\frac{1}{2}}(H+1)(\|f-f_{\bar{\eta}}\|_{ \mathcal{Z}^{k}_{h}}+1)\sqrt{\log\Big{(}\frac{\mathcal{N}(\mathcal{F}^{N},1/T) }{\delta}\Big{)}}+2N(H+1)$ $\displaystyle\leq c^{\prime}N^{\frac{1}{2}}(H+1)(\|f-f_{\bar{\eta}^{\prime}}\| _{\mathcal{Z}^{k}_{h}}+2)\sqrt{\log\Big{(}\frac{\mathcal{N}(\mathcal{F}^{N},1/ T)}{\delta}\Big{)}}+2N(H+1)$ $$

where the third inequality follows from,

$$ $\displaystyle\|f-g\|_{\mathcal{Z}_{h}^{k}}^{2}$ $\displaystyle\leq\sum_{n=1}^{N}\sum_{(\tau,h)\in[k-1]\times[H]}|f^{(n)}(s_{h}^ {\tau},a_{h}^{\tau})-g^{(n)}(s_{h}^{\tau},a_{h}^{\tau})|^{2}$ $\displaystyle\leq NT\Big{(}\frac{1}{T}\Big{)}^{2}$ $\displaystyle\leq 1.$ $$

Recall that $\tilde{f}^{k}_{h,\eta^{\prime}}=\arg\min_{f\in\mathcal{F}}\|f\|^{2}_{\mathcal{
D}^{k}_{h,\eta^{\prime}}}$.
We have $\|\tilde{f}^{k}_{h,\eta^{\prime}}\|^{2}_{\mathcal{D}^{k}_{h,\eta^{\prime}}}-\|
{f_{\bar{\eta}^{\prime}}}\|^{2}_{\mathcal{D}^{k}_{h,\eta^{\prime}}}\leq 0$, which implies,

$$ $\displaystyle 0$ $\displaystyle\geq\|\tilde{f}^{k}_{h,\bar{\eta}^{\prime}}\|^{2}_{\mathcal{D}^{k }_{h,\bar{\eta}^{\prime}}}-\|f_{\bar{\eta}^{\prime}}\|^{2}_{\mathcal{D}^{k}_{h ,\bar{\eta}^{\prime}}}$ $\displaystyle=\|\tilde{f}^{k}_{h,\bar{\eta}^{\prime}}-f_{\bar{\eta}^{\prime}} \|^{2}_{\mathcal{Z}^{k}_{h}}$ $\displaystyle\qquad+2\sum_{n=1}^{N}\sum_{(\tau,h)\in[k-1]\times[H]}\Big{[}(({ \tilde{f}^{k}_{h,\bar{\eta}^{\prime}}})^{(n)}(s_{h}^{\tau},a_{h}^{\tau})-f^{(n )}_{\bar{\eta}^{\prime}}(s_{h}^{\tau},a_{h}^{\tau}))((f_{\bar{\eta}^{\prime}}^ {(n)}(s_{h}^{\tau},a_{h}^{\tau})-\psi_{n}\Big{(}(\mathcal{B}_{r_{h}^{\tau}})_{ \#}\bar{\eta}^{\prime}(s^{\tau}_{h+1})\Big{)})\Big{]}$ $\displaystyle\geq\|\tilde{f}^{k}_{h,\bar{\eta}^{\prime}}-f_{\bar{\eta}^{\prime }}\|^{2}_{\mathcal{Z}^{k}_{h}}-c^{\prime}N^{\frac{1}{2}}(H+1)(\|\hat{f}^{k}_{h ,\bar{\eta}^{\prime}}-f_{\bar{\eta}^{\prime}}\|_{\mathcal{Z}^{k}_{h}}+2)\sqrt{ \log(2/\delta)+\log\mathcal{N}(\mathcal{F}^{N},1/T)}-6N(H+1).$ $$

Recall that if $x^{2}-2ax-b\leq 0$ holds for constant $a,b>0$, then $x\leq a+\sqrt{a^{2}+b}\leq c^{\prime}\cdot a$ for some constant $c^{\prime}>0$.

Hence,

$$ $\displaystyle\|\tilde{f}^{k}_{h,\eta^{\prime}}-{f_{\bar{\eta}^{\prime}}}\|_{ \mathcal{Z}^{k}_{h}}\leq c^{\prime}(N^{\frac{1}{2}}H\sqrt{\log(1/\delta)+\log \mathcal{N}(\mathcal{F}^{N},1/T)})$ $$

for some constant $c^{\prime}>0$.
∎

###### Lemma ( 6.4 ) .

Let $(\mathcal{F}^{N})_{h}^{k}=\{f\in\mathcal{F}^{N}|\|f-\tilde{f}^{k}_{h,\bar{\eta
}}\|^{2}_{\mathcal{Z}^{k}_{h}}\leq\beta(\mathcal{F}^{N},\delta)\}$, where

$$ $\beta(\mathcal{F}^{N},\delta)\geq c^{\prime}\cdot NH^{2}(\log(T/\delta)+\log \mathcal{N}(\mathcal{F}^{N},1/T))$ $$

for some constant $c^{\prime}>0$.
Then with probability at least $1-\delta/2$, for all $k,h\in[K]\times[H]$, we have

$$ $\psi_{n}\Big{(}(\mathcal{B}_{r_{h}(\cdot,\cdot)})_{\#}[\mathbb{P}_{h}\bar{\eta }_{h+1}^{k}](\cdot,\cdot)\Big{)}\in(\mathcal{F}^{N})_{h}^{k}$ $$

###### Proof.

For all $(k,h)\in[K]\times[H]$,

$$ $\mathbf{S}\coloneqq\begin{cases}\Big{\{}\Big{(}\min\{f^{(1)}(\cdot,\cdot)+b^{k }_{h+1}(\cdot,\cdot),H\}\Big{)}\Big{|}\ f\in\mathcal{C}(\mathcal{F}^{N},1/T) \Big{\}}\cup\Big{\{}0\Big{\}}&n=1\\ \Big{\{}\Big{(}\min\{f^{(n)}(\cdot,\cdot),H\}\Big{)}\Big{|}\ f\in\mathcal{C}( \mathcal{F}^{N},1/T)\Big{\}}\cup\Big{\{}0\Big{\}}&2\leq n\leq N\end{cases}$ $$

is a $(1/T)$-cover of $\psi_{1:N}({\eta}^{k}_{h+1}(\cdot,\cdot))$ where

$$ $\psi_{1:N}(\eta^{k}_{h+1}(\cdot,\cdot))=\begin{cases}\min\{(f^{k}_{h+1})^{(1)} (\cdot,\cdot)+b^{k}_{h+1}(\cdot,\cdot),H\}&n=1\text{ and }h<H\\ \min\{(f^{k}_{h+1})^{(n)}(\cdot,\cdot),H\}&2\leq n\leq N\text{ and }h<H\\ \bm{0}^{N}&h=H\end{cases},$ $$

i.e., there exists $\psi_{1:N}({\eta})\in\mathbf{S}$ such that $\|\psi_{1:N}({\eta})-\psi_{1:N}({\eta}^{k}_{h+1})\|_{\infty,1}\leq 1/T$.
This implies

$$ $\bar{\mathbf{S}}\coloneqq\Big{\{}\psi_{1:N}\Big{(}{\eta}(\cdot,\arg\max_{a\in \mathcal{A}}\psi_{1}({\eta}(\cdot,a)))\Big{)}\mid\psi_{1:N}({\eta})\in\mathbf{ S}\Big{\}}$ $$

is a $(1/T)$-cover of $\psi_{1:N}(\bar{\eta}^{k}_{h+1})$ with $\log(|\bar{\mathbf{S}}|)\leq\log\mathcal{N}(\mathcal{F}^{N},1/T)$.

For each $\psi_{1:N}(\bar{\eta})\in\bar{\mathbf{S}}$, let $\mathcal{E}(\bar{\eta},\delta/2|\bar{\mathbf{S}}|T)$ be the event defined in Lemma 6.3.
By union bound for all $\psi_{1:N}(\bar{\eta})\in\bar{\mathbf{S}}$, we have $\text{Pr}[\bigcap_{\psi_{1:N}(\bar{\eta})\in\bar{\mathbf{S}}}\mathcal{E}(\bar{
\eta},\delta/2|\bar{\mathbf{S}}|T)]\geq 1-\delta/2T$.

Let $\psi_{1:N}(\bar{\eta})\in\bar{\mathbf{S}}$ such that $\|\psi_{1:N}(\bar{\eta})-\psi_{1:N}(\bar{\eta}_{h+1}^{k})\|_{\infty,1}\leq 1/T$.
Conditioned on $\bigcap_{s_{N}(\bar{\eta})\in\bar{\mathbf{S}}}\mathcal{E}(\bar{\eta},\delta/2|
\bar{\mathbf{S}}|T)$ and by Lemma 6.3, we have

$$ $\displaystyle\Big{\|}\tilde{f}^{k}_{h,\bar{\eta}}(\cdot,\cdot)-\psi_{1:N}\Big{ (}(\mathcal{B}_{r_{h}(\cdot,\cdot)})_{\#}[\mathbb{P}_{h}\bar{\eta}_{h+1}^{k}]( \cdot,\cdot)\Big{)}\Big{\|}^{2}_{\mathcal{Z}^{k}_{h}}\leq c^{\prime}\Big{(}NH^ {2}(\log(T/\delta)+\log\mathcal{N}(\mathcal{F}^{N},1/T))\Big{)}$ $$

for some constant $c^{\prime}>0$.

By union bound for all $(k,h)\in[K]\times[H]$, we have $\psi_{1:N}\Big{(}(\mathcal{B}_{r_{h}(\cdot,\cdot)})_{\#}[\mathbb{P}_{h}\bar{
\eta}_{h+1}^{k}](\cdot,\cdot)\Big{)}\in(\mathcal{F}^{N})_{h}^{k}$ with probability $1-\delta/2$.
∎

###### Lemma C.1 .

Let $Q^{k}_{h}(s,a)\coloneqq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{
rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\min\{H,
\tilde{f}^{k}_{h}(s,a)+b^{k}_{h}(s,a)\}}$ for some bonus function $b^{k}_{h}(s,a)$ for all $(s,a)\in\mathcal{S}\times\mathcal{A}$.
If $b^{k}_{h}(s,a)\geq w^{(1)}((\mathcal{F}^{N})^{k}_{h},s,a)$ ,
then with probability at least $1-\delta/2$,

$$ $Q^{*}_{h}(s,a)\leq Q^{k}_{h}(s,a)\text{ and }V^{*}_{h}(s)\leq V^{k}_{h}(s)$ $$

for all $(k,h)\in[K]\times[H]$, for all $(s,a)\in\mathcal{S}\times\mathcal{A}.$

###### Proof.

We use induction on $h$ from $h=H$ to $1$ to prove the statement.
Let $\mathcal{E}$ be the event that for $(k,h)\in[K]\times[H]$, $\psi_{1:N}\Big{(}(\mathcal{B}_{r_{h}(\cdot,\cdot)})_{\#}[\mathbb{P}_{h}\bar{
\eta}^{k}_{h+1}](\cdot,\cdot)\Big{)}\in(\mathcal{F}^{N})^{k}_{h}$.
By Lemma 6.4, $\text{Pr}|\mathcal{E}|\geq 1-\delta/2$.
In the rest of the proof, we condition on $\mathcal{E}$.

When $h=H+1$, the desired inequality holds as $Q^{*}_{H+1}(s,a)=V^{*}_{H+1}(s)=Q^{k}_{H+1}(s,a)=V^{k}_{H+1}(s)=0$.
Now, assume that $Q^{*}_{h+1}(s,a)\leq Q^{k}_{h+1}(s,a)$ and $V^{*}_{h+1}(s)\leq V^{k}_{h+1}(s)$ for some $h\in[H]$.
Then, for all $(s,a)\in\mathcal{S}\times\mathcal{A}$,

$$ $\displaystyle Q^{*}_{h}(s,a)$ $\displaystyle=\min\{H,r_{h}(s,a)+[\mathbb{P}_{h}V^{*}_{h+1}](s,a)\}$ $\displaystyle\leq\min\{H,r_{h}(s,a)+[\mathbb{P}_{h}V^{k}_{h+1}](s,a)\}$ $\displaystyle\leq\min\{H,\tilde{f}^{k}_{h}(s,a)+w^{(1)}(\mathcal{F}^{k}_{h},s, a)\}$ $\displaystyle=\min\{H,Q^{k}_{h}(s,a)-b^{k}_{h}(s,a)+w^{(1)}(\mathcal{F}^{k}_{h },s,a)\}$ $\displaystyle\leq Q^{k}_{h}(s,a)$ $$

∎

###### Lemma C.2 (Regret decomposition) .

With probability at least $1-\delta/4$, we have

$$ $\text{Reg}(K)\leq\sum_{k=1}^{K}\sum_{h=1}^{H}(2b^{k}_{h}(s^{k}_{h},a^{k}_{h})+ \xi^{k}_{h}),$ $$

where $\xi^{k}_{h}=[\mathbb{P}_{h}(V^{k}_{h+1}-V^{\pi^{k}}_{h+1})](s^{k}_{h},a^{k}_{h
})-(V^{k}_{h+1}(s^{k}_{h+1})-V^{\pi^{k}}_{h+1}(s^{k}_{h+1}))$ is a martingale difference sequence with respect to the filtration $\mathbb{F}^{k}_{h}$ induced by the history $\mathbb{H}^{k}_{h}$.

###### Proof.

We condition on the above event $\mathcal{E}$ in the rest of the proof.
For all $(k,h)\in[K]\times[H]$, we have

$$ $\displaystyle\Big{\|}\tilde{f}^{k}_{h,\bar{\eta}}(\cdot,\cdot)-\psi_{1:N}\Big{ (}(\mathcal{B}_{r_{h}(\cdot,\cdot)})_{\#}[\mathbb{P}_{h}\bar{\eta}_{h+1}^{k}]( \cdot,\cdot)\Big{)}\Big{\|}^{2}_{\mathcal{Z}^{k}_{h}}\leq\beta(\mathcal{F}^{N} ,\delta).$ $$

Recall that $(\mathcal{F}^{N})_{h}^{k}=\{f\in\mathcal{F}^{N}\mid\|f-\tilde{f}^{k}_{h,\bar{
\eta}}\|^{2}_{\mathcal{Z}^{k}_{h}}\leq\beta(\mathcal{F}^{N},\delta)\}$ is the confidence region.
Since $\psi_{1:N}\Big{(}(\mathcal{B}_{r_{h}(\cdot,\cdot)})_{\#}[\mathbb{P}_{h}\bar{
\eta}_{h+1}^{k}](\cdot,\cdot)\Big{)}\in(\mathcal{F}^{N})^{k}_{h}$, then by the definition of width function $w^{(1)}(\mathcal{F}^{k}_{h},s,a)$, for $(k,h)\in[K]\times[H]$, we have

$$ $\displaystyle{w^{(1)}(\mathcal{F}^{k}_{h},s,a)}$ $\displaystyle\geq\Big{|}\psi_{1}\Big{(}(\mathcal{B}_{r_{h}(s,a)})_{\#}[\mathbb {P}_{h}\bar{\eta}_{h+1}^{k}](s,a)\Big{)}-(\tilde{f}^{k}_{h,\bar{\eta}})^{(1)}( s,a)\Big{|}$ $\displaystyle=\Big{|}r_{h}(s,a)+[\mathbb{P}_{h}V^{k}_{h+1}](s,a)-(\tilde{f}^{k }_{h,\bar{\eta}})^{(1)}(s,a)\Big{|}.$ $$

Recall that $Q^{*}_{h}(\cdot,\cdot)\leq Q^{k}_{h}(\cdot,\cdot)$.

$$ $\displaystyle\text{Reg}(K)$ $\displaystyle=\sum_{k=1}^{K}V_{1}^{\star}(s_{1}^{k})-V_{1}^{\pi^{k}}(s_{1}^{k})$ $\displaystyle{\leq}\sum_{k=1}^{K}V_{1}^{k}(s_{1}^{k})-V_{1}^{\pi^{k}}(s_{1}^{k})$ $\displaystyle=\sum_{k=1}^{K}Q_{1}^{k}(s_{1}^{k},a_{1}^{k})-Q_{1}^{\pi^{k}}(s_{ 1}^{k},a_{1}^{k})$ $\displaystyle=\sum_{k=1}^{K}Q_{1}^{k}(s_{1}^{k},a_{1}^{k})-(r_{1}(s_{1}^{k},a_ {1}^{k})+[\mathbb{P}_{1}V_{2}^{k}](s_{1}^{k},a_{1}^{k}))+(r_{1}(s_{1}^{k},a_{1 }^{k})+[\mathbb{P}_{1}V_{2}^{k}](s_{1}^{k},a_{1}^{k}))$ $\displaystyle\qquad-Q_{1}^{\pi^{k}}(s_{1}^{k},a_{1}^{k})$ $\displaystyle\leq\sum_{k=1}^{K}w^{(1)}((\mathcal{F}^{N})^{k}_{1},s_{1}^{k},a_{ 1}^{k}){+b^{k}_{1}(s_{1}^{k},a_{1}^{k})}+[\mathbb{P}_{1}(V_{2}^{k}-V_{2}^{\pi^ {k}})](s_{1}^{k},a_{1}^{k})$ $\displaystyle\leq\sum_{k=1}^{K}w^{(1)}((\mathcal{F}^{N})^{k}_{1},s_{1}^{k},a_{ 1}^{k}){+b^{k}_{1}(s_{1}^{k},a_{1}^{k})}+(V_{2}^{k}(s_{2}^{k})-V_{2}^{\pi^{k}} (s_{2}^{k}))+\xi_{1}^{k}$ $\displaystyle\qquad\vdots$ $\displaystyle\leq\sum_{k=1}^{K}\sum_{h=1}^{H}(w^{(1)}((\mathcal{F}^{N})^{k}_{h },s_{h}^{k},a_{h}^{k}){+b^{k}_{h}(s_{h}^{k},a_{h}^{k})}+\xi_{h}^{k})$ $\displaystyle\leq\sum_{k=1}^{K}\sum_{h=1}^{H}(2b^{k}_{h}(s_{h}^{k},a_{h}^{k})+ \xi_{h}^{k})$ $$

∎

It remains to bound $\sum_{k=1}^{K}\sum_{h=1}^{H}b^{k}_{h}(s^{k}_{h},a^{k}_{h})$, for which we will exploit fact that $\mathcal{F}^{N}$ has bounded eluder dimension.

###### Lemma C.3 .

If $b^{k}_{h}(s,a)\geq w^{(1)}((\mathcal{F}^{N})^{k}_{h},s,a)$ for all $(s,a)\in\mathcal{S}\times\mathcal{A}$ and $k\in[K]$ where

$$ $(\mathcal{F}^{N})^{k}_{h}=\{f\in\mathcal{F}^{N}|\ \|f-\tilde{f}^{k}_{h,\bar{ \eta}}\|^{2}_{\mathcal{Z}^{k}_{h}}\leq\beta(\mathcal{F}^{N},\delta)\},$ $$

then

$$ $\sum_{k=1}^{K}\sum_{h=1}^{H}\bm{1}\{{\color[rgb]{0,0,0}\definecolor[named]{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill {0}b^{k}_{h}(s^{k}_{h},a^{k}_{h})}>\epsilon\}\leq\left(\frac{4\beta(\mathcal{F }^{N},\delta)}{\epsilon^{2}}+1\right)\text{dim}_{E}(\mathcal{F}^{N},\epsilon)$ $$

for some constant $c>0$.

###### Proof.

We first want to show that for any sequence $\{(s_{1},a_{1}),\ldots,(s_{\kappa},a_{\kappa})\}\subseteq\mathcal{S}\times
\mathcal{A}$,
there exists $j\in[\kappa]$ such that $(s_{j},a_{j})$ is $\epsilon$-dependent on at least $L=\lceil(\kappa-1)/\text{dim}_{E}(\mathcal{F}^{N},\epsilon)\rceil$ disjoint subsequences in $\{(s_{1},a_{1}),\ldots,(s_{j-1},a_{j-1})\}$ with respect to $\mathcal{F}^{N}$.
We demonstrate this by using the following procedure.
Start with $L$ disjoint subsequences of $\{(s_{1},a_{1}),\ldots,(s_{j-1},a_{j-1})\}$, $\mathcal{B}_{1},\mathcal{B}_{2},\ldots,\mathcal{B}_{L}$, which are initially empty.
For each $j$, if $(s_{j},a_{j})$ is $\epsilon$-dependent on every $\mathcal{B}_{1},\ldots,\mathcal{B}_{L}$, we achieve our goal so we stop the process.
Else, we choose $i\in[L]$ such that $(s_{j},a_{j})$ is $\epsilon$-independent on $\mathcal{B}_{i}$ and update $\mathcal{B}_{i}\leftarrow\mathcal{B}_{i}\cup\{(s_{j},a_{j})\}$, $j\leftarrow j+1$.
Since every element of $\mathcal{B}_{i}$ is $\epsilon$-independent on its predecessors, $|\mathcal{B}_{i}|$ cannot get bigger than $\text{dim}_{E}(\mathcal{F}^{N},\epsilon)$ at any point in this process.
Therefore, the process stops at most step $j=L\text{dim}_{E}(\mathcal{F}^{N},\epsilon)+1\leq\kappa$.

Now we want to show that if for some $j\in[\kappa]$ such that
$b^{k}_{h}(s_{j},a_{j})>\epsilon$,
then $(s_{j},a_{j})$ is $\epsilon$-dependent on at most $4\beta(\mathcal{F}^{N},\delta)/\epsilon^{2}$ disjoint subsequences in $\{(s_{1},a_{1}),\ldots,(s_{j-1},a_{j-1})\}$ with respect to $\mathcal{F}^{N}$.
If
$b^{k}_{h}(s_{j},a_{j})>\epsilon$
and $(s_{j},a_{j})$ is $\epsilon$-dependent on a subsequence of $\{(s^{\prime}_{1},a^{\prime}_{1}),\ldots,(s^{\prime}_{l},a^{\prime}_{l})\}
\subseteq\{(s_{1},a_{1}),\ldots,(s_{\kappa},a_{\kappa})\}$,
it implies that there exists $f,g\in\mathcal{F}^{N}$ with $\|f-\tilde{f}^{k}_{h,\bar{\eta}}\|^{2}_{\mathcal{Z}^{k}_{h}}\leq\beta(\mathcal
{F}^{N},\delta)$ and $\|g-\tilde{f}^{k}_{h,\bar{\eta}}\|^{2}_{\mathcal{Z}^{k}_{h}}\leq\beta(\mathcal
{F}^{N},\delta)$ such that
$f^{(1)}(s^{\prime}_{t},a^{\prime}_{t})-g^{(1)}(s^{\prime}_{t},a^{\prime}_{t})\geq\epsilon$.
By triangle inequality, $\|f-g\|^{2}_{\mathcal{Z}^{k}_{h}}\leq 4\beta(\mathcal{F}^{N},\delta)$.
On the other hand, if $(s_{j},a_{j})$ is $\epsilon$-dependent on $L$ disjoint subsequences in $\{(s_{1},a_{1}),\ldots,(s_{\kappa},a_{\kappa})\}$, then

$$ $\displaystyle 4\beta(\mathcal{F}^{N},\delta)\geq\|f-g\|^{2}_{\mathcal{Z}^{k}}{ \color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\geq\|f^{(1)}-g^{(1)}\|^ {2}_{\mathcal{Z}^{k}}}\geq L\epsilon^{2}$ $$

resulting in $L\leq 4\beta(\mathcal{F}^{N},\delta)/\epsilon^{2}$.
Therefore, we have $(\kappa/\text{dim}_{E}(\mathcal{F}^{N},\epsilon))-1\leq 4\beta(\mathcal{F}^{N}
,\delta)/\epsilon^{2}$ which results in

$$ $\kappa\leq\left(\frac{4\beta(\mathcal{F},\delta)}{\epsilon^{2}}+1\right)\text{ dim}_{E}(\mathcal{F}^{N},\epsilon)$ $$

∎

###### Lemma C.4 (Refined version of Lemma 10 in Wang et al. [ 50 ] ) .

If $b^{k}_{h}(s,a)\geq w^{(1)}((\mathcal{F}^{N})^{k}_{h},s,a)$ for all $(s,a)\in\mathcal{S}\times\mathcal{A}$ and $k\in[K]$, then

$$ $\sum_{k=1}^{K}\sum_{h=1}^{H}b^{k}_{h}(s^{k}_{h},a^{k}_{h})\leq H\textnormal{ dim}_{E}(\mathcal{F}^{N},1/T).$ $$

###### Proof.

We first sort the sequence $\{b^{k}_{h}(s^{k}_{h},a^{k}_{h})\}_{(k,h)\in[K]\times[H]}$ in a decreasing order and denote it by $\{e_{1},\ldots,e_{T}\}(e_{1}\geq e_{2}\geq\dots\geq e_{T})$.
By Lemma C.3, for any constant $M>0$ and $e_{t}\geq 1/\sqrt{M}T$, we have

$$ $\displaystyle t\leq\Big{(}\frac{4\beta(\mathcal{F}^{N},\delta)}{Me_{t}^{2}}+1 \Big{)}\text{dim}_{E}(\mathcal{F}^{N},\sqrt{M}e_{t})\leq\Big{(}\frac{4\beta( \mathcal{F}^{N},\delta)}{Me_{t}^{2}}+1\Big{)}\text{dim}_{E}(\mathcal{F}^{N},1/T)$ $$

which implies

$$ $\displaystyle e_{t}\leq\Big{(}\frac{t}{\text{dim}_{E}(\mathcal{F}^{N},1/T)}-1 \Big{)}^{-1/2}\sqrt{\frac{4\beta(\mathcal{F}^{N},\delta)}{M}},$ $$

for $t\geq\text{dim}_{E}(\mathcal{F}^{N},1/T)$.
Since we have $e_{t}\leq H$,

$$ $\displaystyle\sum_{t=1}^{T}e_{t}$ $\displaystyle=\sum_{t=1}^{T}e_{t}\bm{1}\{e_{t}<1/\sqrt{M}T\}+\sum_{t=1}^{T}e_{ t}\bm{1}\{e_{t}\geq 1/\sqrt{M}T,t<\text{dim}_{E}(\mathcal{F}^{N},1/T)\}$ $\displaystyle\quad+\sum_{t=1}^{T}e_{t}\bm{1}\{e_{t}\geq 1/\sqrt{M}T,t\geq\text {dim}_{E}(\mathcal{F}^{N},1/T)\}$ $\displaystyle\leq\frac{1}{\sqrt{M}}+H\text{dim}_{E}(\mathcal{F}^{N},1/T)+\sum_ {\text{dim}_{E}(\mathcal{F}^{N},1/T)\leq t\leq T}\Big{(}\frac{t}{\text{dim}_{E }(\mathcal{F}^{N},1/T)}-1\Big{)}^{-1/2}\sqrt{\frac{4\beta(\mathcal{F}^{N}, \delta)}{M}}$ $\displaystyle\leq\frac{1}{\sqrt{M}}+H\text{dim}_{E}(\mathcal{F}^{N},1/T)+2\Big {(}{\frac{T}{\text{dim}_{E}(\mathcal{F}^{N},1/T)}-1}\Big{)}^{1/2}\text{dim}_{E }(\mathcal{F}^{N},1/T)\sqrt{\frac{4\beta(\mathcal{F}^{N},\delta)}{M}}$ $\displaystyle=\frac{1}{\sqrt{M}}+H\text{dim}_{E}(\mathcal{F}^{N},1/T)+\sqrt{16 \cdot\text{dim}_{E}(\mathcal{F}^{N},1/T)\cdot T\cdot\beta(\mathcal{F}^{N}, \delta)/M}.$ $$

Taking $M\rightarrow\infty$,

$$ $\sum_{k=1}^{K}\sum_{h=1}^{H}b^{k}_{h}(s^{k}_{h},a^{k}_{h})\leq H\text{dim}_{E} (\mathcal{F}^{N},1/T).$ $$

∎

###### Theorem ( 6.5 ) .

Under Assumption 4.8, with probability at least $1-\delta$, SF-LSVI achieves a regret bound of

$$ $\text{Reg}(K)\leq 2H\textnormal{dim}_{E}(\mathcal{F}^{N},1/T)+4H\sqrt{KH\log(2 /\delta)}.$ $$

###### Proof.

Recall that $\xi_{h}^{k}=[\mathbb{P}_{h}(V^{k}_{h+1}-V^{\pi^{k}}_{h+1})](s^{k}_{h},a^{k}_{h
})-(V^{k}_{h+1}(s^{k}_{h+1})-V^{\pi^{k}}_{h+1}(s^{k}_{h+1}))$ is a martingale difference sequence where $\mathbb{E}[\xi^{k}_{h}|\mathbb{F}^{k}_{h}]=0$ and $|\xi^{k}_{h}|\leq 2H$.
By Azuma-Hoeffding’s inequality, with probability at least $1-\delta/2$,

$$ $\displaystyle\sum_{k=1}^{K}\sum_{h=1}^{H}\xi^{k}_{h}\leq 4H\sqrt{KH\log(2/ \delta)}.$ $$

Conditioning on the above event and Lemma C.4, we have

$$ $\displaystyle\text{Reg}(K)$ $\displaystyle\leq 2\sum_{k=1}^{K}\sum_{h=1}^{H}b^{k}_{h}(s^{k}_{h},a^{k}_{h})+ \sum_{k=1}^{K}\sum_{h=1}^{H}\xi^{k}_{h}$ $\displaystyle\leq 2H\text{dim}_{E}(\mathcal{F}^{N},1/T)+4H\sqrt{KH\log(2/ \delta)}$ $$

∎

###### Theorem ( 4.4 ) .

###### Proof.

###### Lemma ( 4.6 ) .

###### Proof.

###### Theorem ( 4.7 ) .

###### Proof.

###### Lemma ( 6.3 ) .

###### Proof.

###### Lemma ( 6.4 ) .

###### Proof.

###### Lemma C.1 .

###### Proof.

###### Lemma C.2 (Regret decomposition) .

###### Proof.

###### Lemma C.3 .

###### Proof.

###### Lemma C.4 (Refined version of Lemma 10 in Wang et al. [ 50 ] ) .

###### Proof.

###### Theorem ( 6.5 ) .

###### Proof.

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initial_rating: 3
presentation: 3
questions: -	The justification for Bellman unbiasedness is confusing to me. Why is Bellman unbiasedness necessary in the finite sample regime? Is it a fundamental requirement (in other words, there are some statistical lower bounds if Bellman unbiasedness does not hold) or just a technical assumption required by the current analysis?
soundness: 4
strengths: -	As one of the major technical contributions, this paper studies the distributional RL problem with general reward functions, whereas prior works mostly focus on discretized rewards. This paper also identifies two key assumptions, Bellman closedness and Bellman unbiasedness, that allow statistically efficient learning algorithms. 
-	The choice of the particular choice of the sketch function, namely, the momentums, is theoretically justified by Theorem 4.6, which proves that “the only finite statistical functionals that are both Bellman unbiased and closed … is equal to the linear span of the set of moment functionals”.
summary: This paper designs a distributional reinforcement learning algorithm with general function approximation, called SF-LSVI, under the assumption that the Eluder dimension of the function class is small. In particular, the SF-LSVI algorithm estimates the momentums of the expected return in addition to its mean (which is the standard Q-function). This paper proves that the SF-LSVI algorithm achieves a near-optimal regret bound.
weaknesses: -	While this paper studies distributional RL, the final metric for the performance of the algorithm is still the standard regret. In the main theorem (Theorem 6.5), there is no guarantee of whether the estimated momentum is close to the ground truth. In fact, estimating the momentums is purely an independent component in the algorithm, and is independent of learning the optimal policy.

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confidence: 4
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. What is the notation "law" means in Lines 181 and 186.
soundness: 3
strengths: 1. This paper involve a novel framework "Bellman unbiasedness" which strictly contains the existing Bellman completness assumption, and addressing the infinite-dimensionality issue in DistRL.
2. The paper proposed novel theoretical analysis towards DistRL within more general assumptions, which is a nice contribution to the RL theory community.
summary: This paper improves the existing distributional reinforcement learning algorithm by introducing a new concept called "Bellman unbiasedness" which is  and revisit the framework through a statistical functional lens. The paper present a new algorithm, SF-LSVI, which is provably efficient and achieves the a tight regret upper bound $\tilde{O}(d_E H^{3/2} \sqrt{K})$.
weaknesses: 1. I think it is better to discuss more about the technical contribution,e.g., detailed introduce the technical intuition of removing the dependency $\beta$ and why the dimension term is seperate from $\sqrt{K}$. 
2. It is not really clear that the relationship between the new "Bellman unbiasedness" assumption and the exist assumptions. Could you please present more comparison and examples?

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confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. Can the authors provide empirical validation to demonstrate SF-LSVI’s practical performance and potential advantages over existing methods?
2. How restrictive is the Statistical Functional Bellman Completeness assumption in practical settings? Could the authors discuss specific environments where this assumption may or may not hold?
3. Are there any potential limitations or specific scenarios where Bellman unbiasedness may not provide advantages or could introduce new challenges?
soundness: 4
strengths: - **Originality**: The paper introduces Bellman unbiasedness, a novel concept that builds on Bellman closedness to allow unbiased estimation in finite-dimensional spaces. This approach opens new directions for efficient DistRL without requiring assumptions that are often infeasible in practical settings.
- **Theoretical Rigor**: The proofs and derivations are thorough and well-structured, lending strong theoretical support to the proposed SF-LSVI algorithm. The regret bound, $\tilde{O}(d_E H^{3/2} \sqrt{K})$, represents a competitive improvement within DistRL frameworks.
- **Clarity**: The paper is generally well-written, with each section logically building on the previous one. The background provided on limitations of prior approaches and the relevance of Bellman unbiasedness is helpful for contextualizing the contribution.
- **Significance**: This work could have broad implications in various applications requiring robust policy learning, such as robotics and finance, where DistRL has shown potential.
summary: The paper introduces a novel concept in Distributional Reinforcement Learning (DistRL) called **Bellman unbiasedness**, which enables accurate and unbiased updates for statistical functionals in finite-dimensional spaces. This approach addresses the challenge of infinite-dimensional return distributions without requiring strict assumptions, such as distributional Bellman completeness. The authors propose **SF-LSVI**, an algorithm designed to achieve efficient learning with a favorable regret bound of $\tilde{O}(d_E H^{3/2} \sqrt{K})$, offering improvements over existing DistRL methods in terms of theoretical efficiency and robustness.
weaknesses: - **Lack of Empirical Validation**: The theoretical claims would be strengthened by empirical results on DistRL benchmarks, which could provide evidence of SF-LSVI’s practical effectiveness and robustness. Without this, the impact on real-world tasks remains speculative.
- **Completeness Assumption**: While the Statistical Functional Bellman Completeness assumption is less restrictive than full distributional completeness, it may still be challenging to meet in some environments. Further discussion on how widely this assumption holds in practical applications would be beneficial.
- **Accessibility**: Some sections, particularly the detailed theoretical derivations, may be challenging for readers not specialized in reinforcement learning. Simplifying these explanations or adding illustrative examples could make the work more accessible.

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confidence: 3
contribution: 2
initial_rating: 6
presentation: 2
questions: - How to choose $N$? How does $N$ affects the sample efficieny and computational efficiency?
- The author claims the provable efficiency of the proposed algorithm. How does the computational complexity of the proposed algorithm scale with the size of the state and action spaces, and $N$?

minor questions / comments
- Fig 1 Yellow should be Blue? why is categorical unbiased and not closed?
- Defiition 4.4, what is x_k?
- the example below Definition 4.4 seems to not involve state transition
soundness: 3
strengths: - Introducing Bellman unbiasedness provides a new perspective on maintaining efficiency in DistRL, specifically addressing the challenge of high-dimensional distributions.
- Theoretical rigor is high, with comprehensive proofs and well-justified assumptions.
- Key terms like Bellman closedness and unbiasedness are well-defined, and the authors provide visual aids to clarify functional relationships.
summary: This paper introduces the concept of Bellman unbiasedness to address limitations in distributional reinforcement learning  regarding infinite-dimensionality and model misspecification. The authors propose an algorithm, Statistical Functional Least Squares Value Iteration, that operates within a finite-dimensional space, achieving tighter regret bounds under a weaker assumption called Statistical Functional Bellman Completeness. The work leverages moment functionals as finite approximations for the distribution of returns, demonstrating that only these functionals can maintain both Bellman closedness and unbiasedness.
weaknesses: -  The dense theoretical sections, particularly around statistical functionals and Bellman properties, may be challenging for readers not well-versed in DistRL. Additional illustrations or simplified explanations might improve comprehension.
- The motivation of using DistRL algorithm for RL with GVFA is not strong enough. The authors provide a regret bound for the proposed algorithm, but the benifits of DistRL methods over non-DistRL methods are not shown. In addition, the comparsion to V-EST-LSR (Chen et al.,2024) may not suffice to justify the effectiveness of DistRL, given that  V-EST-LSR is designed to solve risk-sensitive tasks.
- While the paper includes theoretical analysis and basic examples, empirical validation would better demonstrate SF-LSVI’s practical utility.

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confidence: 3
contribution: 2
initial_rating: 6
presentation: 2
questions: The term “Bellman closedness” generally refers to the standard RL setting (i.e., not the distributional RL setting). A brief clarification on this would be helpful.

References:

Error Bounds for Approximate Policy Iteration

Finite-Time Bounds for Fitted Value Iteration

Finite-Time Bounds for Sampling-Based Fitted Value Iteration

Information-Theoretic Considerations in Batch Reinforcement Learning

The Bellman unbiasedness concept is new to me. Does a similar concept exist in the standard RL setting? If so, is it meaningful within that context?

It seems that this paper assumes a deterministic reward. If so, can the Bellman unbiasedness concept and theoretical guarantees be extended to a stochastic reward scenario? In standard RL, extending to deterministic rewards is relatively straightforward, but I’m uncertain about how this would apply in the distributional RL setting.

In Table 1, the terms "finitely representable" and "exactly learnable" are a bit unclear. It seems these properties might limit the applicability of SF-LSVI, which could be seen as a drawback of the algorithm.

The abstract mentions "Our theoretical results demonstrate that the only way to exactly capture statistical information, including nonlinear statistical functionals, is to represent the infinite-dimensional return distribution with a finite number of moment functionals." I think it refers to Definition 4.5 and Theorem 4.6. Could you clarify its meaning?

The term "law" is unclear.

"Additional sketch" seems ambiguous.

In Definition 4.7, why is the infinity norm used? Would it be possible to use the L2 norm instead?

H' in Lemma 6.3 appears to be a typo.

What is the Dcal-norm? It is defined in the appendix but not mentioned in the main text, and it’s unclear what d represents in the appendix.
soundness: 3
strengths: The paper is generally easy to follow.

The concept of Bellman unbiasedness is novel and interesting.

Theoretical results are achievable without assumptions of discretized rewards, small-loss bounds, or Lipschitz continuity. Compared to previous works, the regret bound is tighter, as illustrated in Table 1.
summary: This paper presents a regret analysis for distributional reinforcement learning (RL) with general value function approximation in finite episodic Markov decision processes (MDPs), using statistical functional dynamic programming.

Initially, it introduces the concept of Bellman unbiasedness, proving that the moment functional is the unique structure within a class that includes nonlinear statistical functionals.

The paper then addresses a new challenge related to the inherent difficulty of managing the infinite dimensionality of a distribution, offering a theoretical analysis of how hidden approximation errors hinder the development of provably efficient algorithms. The authors also revisit the distributional Bellman Completeness assumption.

Lastly, it proposes a provably efficient distributional RL algorithm named SF-LSVI, which achieves a regret bound of O(d_E H^{3/2} sqrt K) . The results are tighter and rely on weaker assumptions.
weaknesses: The purpose behind introducing the Bellman unbiasedness concept is unclear.

Assumption 4.8 appears to lack sufficient motivation. Although Bellman unbiasedness is discussed earlier in the paper, there seems to be a disconnect between it and Assumption 4.8.

It is also unclear how the model misspecification term, zeta, would enter the bounds if Assumption 4.8 does not hold. It would be worthwhile to investigate whether this would lead to a polynomial or exponential blowup.

Meta-review: This paper studies distributional reinforcement learning with general value function approximation. The authors design a distributional RL algorithm with general function approximation, called SF-LSVI, and prove that SF-LSVI  achieves a near-optimal regret bound under the assumption that the Eluder dimension of the function class is small. 

The main weakness raised by the reviewers is the theoretical formulation of the distributional RL problem. In this paper, the goal is to achieve a small regret, which is independent of the distributional RL framework. Therefore, the problem formulation might not reflect the real use case of distributional RL. 

Given the high standards of ICLR and the weakness mentioned above, I would recommend rejecting this paper.

Decision: Reject