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A priori Estimates for Deep Residual Network in Continuous-time Reinforcement Learning

Shuyu Yin, Qixuan Zhou, Fei Wen, Tao Luo

TL;DR

The paper develops a priori generalization bounds for continuous-time reinforcement learning with discretized transitions by exploiting semi-group and Lipschitz properties of the dynamics. It introduces two loss-transformations and a max-operator decomposition to bound the Bellman optimal loss directly, without relying on boundedness assumptions. The main result provides a bound that scales polynomially with the action space size and neural-network width, and explicitly depends on the discretization step $\Delta t$ and sample size $n$, enabling principled discretization choices. The work leverages residual networks in Barron spaces and employs Rademacher complexity to derive both approximation and generalization terms, offering practically meaningful guarantees for continuous-time control problems.

Abstract

Deep reinforcement learning excels in numerous large-scale practical applications. However, existing performance analyses ignores the unique characteristics of continuous-time control problems, is unable to directly estimate the generalization error of the Bellman optimal loss and require a boundedness assumption. Our work focuses on continuous-time control problems and proposes a method that is applicable to all such problems where the transition function satisfies semi-group and Lipschitz properties. Under this method, we can directly analyze the \emph{a priori} generalization error of the Bellman optimal loss. The core of this method lies in two transformations of the loss function. To complete the transformation, we propose a decomposition method for the maximum operator. Additionally, this analysis method does not require a boundedness assumption. Finally, we obtain an \emph{a priori} generalization error without the curse of dimensionality.

A priori Estimates for Deep Residual Network in Continuous-time Reinforcement Learning

TL;DR

The paper develops a priori generalization bounds for continuous-time reinforcement learning with discretized transitions by exploiting semi-group and Lipschitz properties of the dynamics. It introduces two loss-transformations and a max-operator decomposition to bound the Bellman optimal loss directly, without relying on boundedness assumptions. The main result provides a bound that scales polynomially with the action space size and neural-network width, and explicitly depends on the discretization step and sample size , enabling principled discretization choices. The work leverages residual networks in Barron spaces and employs Rademacher complexity to derive both approximation and generalization terms, offering practically meaningful guarantees for continuous-time control problems.

Abstract

Deep reinforcement learning excels in numerous large-scale practical applications. However, existing performance analyses ignores the unique characteristics of continuous-time control problems, is unable to directly estimate the generalization error of the Bellman optimal loss and require a boundedness assumption. Our work focuses on continuous-time control problems and proposes a method that is applicable to all such problems where the transition function satisfies semi-group and Lipschitz properties. Under this method, we can directly analyze the \emph{a priori} generalization error of the Bellman optimal loss. The core of this method lies in two transformations of the loss function. To complete the transformation, we propose a decomposition method for the maximum operator. Additionally, this analysis method does not require a boundedness assumption. Finally, we obtain an \emph{a priori} generalization error without the curse of dimensionality.
Paper Structure (17 sections, 15 theorems, 116 equations, 2 figures)

This paper contains 17 sections, 15 theorems, 116 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related works
  4. Preliminaries
  5. Continuous-time reinforcement learning and discretization
  6. Residual network and Barron space
  7. Rademacher complexity
  8. Bellman optimal loss minimization with neural network approximation
  9. Bellman effective loss minimization problem
  10. Main results
  11. Main theorem
  12. Proof sketch
  13. Proof of Theorems
  14. Lemmas for the operations of residual network
  15. Approximation error for the Bellman effective loss minimization problem
  16. ...and 2 more sections

Key Result

Lemma 2.8

Suppose that $\psi_i: \mathbb{R} \rightarrow \mathbb{R}$ is a $C$-Lipschitz function for each $i \in \{1,\ldots,n\}$. For any $\boldsymbol{y} \in \mathbb{R}^n$, let $\psi(\boldsymbol{y})=\left(\psi_1\left(y_1\right), \cdots, \psi_n\left(y_n\right)\right)^{\top}$. For an arbitrary set of vector funct

Figures (2)

  • Figure 1: Sketch of proof for Theorem \ref{['thm::AprioriGenErrBoun']}
  • Figure 2: The relation between $\pi_j^{(k)}$, which is a binary tree.

Theorems & Definitions (39)

  • Remark 2.1
  • Definition 2.2: weighted path norm
  • Definition 2.3: weighted path norm for vector-valued function
  • Definition 2.4: Barron space
  • Definition 2.5: Barron space for vector-valued function
  • Remark 2.6
  • Definition 2.7: Rademacher complexity of a function class $\mathcal{F}$
  • Lemma 2.8: contraction lemma ma2019priori
  • Theorem 2.9: two-sided Rademacher complexity and generalization gap shalev2014understanding
  • Theorem 3.2: a priori generalization error bound for Bellman optimal loss
  • ...and 29 more