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Boundary-aware Reinforcement Learning for Hypercube State Spaces via Deterministic Policy Gradient

Lijun Bo, Yijie Huang, Chenhao Lu

arXiv:2610.09712v1math.OCcs.LG

TL;DR

Continuous-time RL for reflected state dynamics needs to account for Neumann boundary conditions, while stochastic policies can require action sampling and integration. This paper develops a deterministic policy-gradient framework and CT-DDPG-R with soft or hard boundary treatments, and reports reduced boundary residuals and improved training stability in reservoir problems.

  • Problem

    Continuous-time RL must address reflected state boundaries, whose Neumann conditions shape the Bellman equation, while stochastic policies can require action sampling and action-distribution integration.

    Source: §1 · p. 3

  • Method

    The paper derives a Neumann Bellman equation, deterministic policy-gradient and martingale results, then develops CT-DDPG-R with soft boundary penalties or a hard cosine-Fourier architecture.

    Source: §1 · p. 4

  • Results

    Boundary-aware methods substantially reduce boundary residuals and improve training stability on one-, two-, and five-dimensional reservoir regulation problems, with advantages more pronounced as dimension increases.

    Source: §1 · p. 4

  • Takeaways & Limitations

    The discrepancy estimate supports using piecewise-constant exploratory actions in implementation as the time grid is refined and exploration noise vanishes.

    Source: §1 · p. 4, §5.2.4 · p. 20

    In the one-dimensional experiment, imposing the boundary condition can drive the policy toward the boundary and leave it trapped there for a long training period, especially under the hard constraint.

    Source: §6.1 · p. 23

Abstract

from arXiv · show

We develop a continuous-time deterministic policy gradient framework for reinforcement learning with reflected state dynamics, where the state process is governed by a controlled reflected stochastic differential equation on a hypercube. Under suitable regularity assumptions, we establish the connection between the value function and the Neumann Bellman equation, introduce an advantage-rate function that yields a deterministic policy gradient formula, and prove the martingale characterization theorem. Motivated by these theoretical results, we propose a continuous-time deep deterministic policy gradient algorithm for reflected stochastic systems, in which the Neumann boundary condition is imposed via either soft penalization or hard architectural constraint. We further quantify the discrepancy between the ideal continuous-time dynamics and the discretely sampled exploratory dynamics executed in practice, showing that the error decays as the time grid is refined and exploration noise vanishes. Our experiments on reservoir control problems illustrate the effectiveness of the RL framework, highlighting that boundary-aware methods substantially reduce Neumann boundary residuals and enhance learning stability.

1 Introduction

The introduction motivates continuous-time RL for systems whose states evolve continuously, then focuses on state constraints and reflected dynamics as an underexplored issue. It develops a boundary-aware deterministic actor-critic framework, its theoretical basis, and an analysis connecting ideal policies to discretely executed exploratory actions.

1 Introduction

  • Continuous-time RL is motivated by systems with continuously evolving states, while discrete-time methods can be sensitive to time discretization near the continuous-time limit.The introduction argues that time discretization should be an implementation choice rather than the foundation of the RL formulation.Source: §1 · p. 2
  • Stochastic or relaxed continuous-time policies can require high-frequency action sampling and action-distribution integration, motivating deterministic feedback policies.The introduction notes that Monte Carlo action integration may increase policy-update costs, especially in high-dimensional action spaces.Source: §1 · pp. 2–3
  • Reflected state constraints make the Bellman equation a Neumann boundary-value problem, shaping critic behavior, regulator terms, and value approximation near the boundary.The paper identifies state boundaries as a comparatively underexplored issue in continuous-time RL.Source: §1 · p. 3
  • For fixed deterministic policies, the value function satisfies a Neumann Bellman equation, and the deterministic policy gradient excludes regulator terms through that boundary condition.The paper also introduces an advantage-rate function and martingale characterizations of both it and the value function.Source: §1 · pp. 3–4
  • CT-DDPG-R combines martingale-based critic learning and deterministic actor updates with soft Neumann penalties or a hard cosine-Fourier architecture.On one-, two-, and five-dimensional reservoir problems, boundary-aware methods reduce boundary residuals and improve training stability, with advantages more pronounced at higher dimensions.Source: §1 · p. 4
  • The paper quantifies the discrepancy between ideal continuous-time feedback and discretely sampled exploratory actions, providing theoretical support for the algorithm’s data-collection procedure.The estimate concerns piecewise-constant exploratory actions; the introduction also previews its use in analyzing real-environment state-process error.Source: §1 · p. 4

2 Problem Formulation

The problem formulation sets up controlled reflected diffusion on a hypercube and defines policy performance for deterministic Markov controls. Reflection acts at the domain’s faces, while the value function is the objective to optimize over policy parameters.

2 Problem Formulation

  • The state follows a controlled reflected diffusion on a hypercube, with componentwise nondecreasing regulators acting when coordinates reach their corresponding lower or upper faces.The setup specifies Brownian noise and drift and diffusion coefficients for the reflected dynamics.Source: §2 · p. 5
  • The control is a deterministic feedback law, parameterized as a family of policies indexed by parameters that are optimized to maximize the value function.The policy maps time and state to an action in the action set.Source: §2 · p. 5
  • Policy performance is defined by the expected cumulative return for the reflected state process, using running and terminal rewards with a positive discount factor.The value function is associated with the reflected process under the parameterized policy.Source: §2 · p. 5

3 Neumann Bellman Equation on Hypercube

Under regularity assumptions, the value function for a deterministic feedback policy solves a Neumann Bellman boundary-value problem on the hypercube, and classical solutions recover its probabilistic representation.

3 Neumann Bellman Equation on Hypercube

  • The paper assumes value-function regularity in the interior and continuity up to the boundary, rather than C1,2 regularity on the closed time-space domain.The authors note that terminal and Neumann compatibility conditions may fail, so full classical regularity cannot generally be guaranteed.Source: §3 · p. 6
  • Under Assumptions 3.1–3.4, the value function satisfies the interior generator–reward equation, zero normal derivatives on the boundary, and terminal condition g.The result also states that a classical solution of this boundary-value problem has the probabilistic representation (2.2).Source: §3 · p. 7
  • The proof derives the interior equation using local stopping, Itô’s formula, and the dynamic programming principle.It then obtains the boundary condition by contradiction, using positive probability of regulator increase when the uniformly nondegenerate diffusion starts at a boundary face.Source: §3 · pp. 7–8

4 Policy Gradient and Martingales on Hypercube

This section establishes a deterministic policy gradient and a martingale characterization for reflected diffusion control on a hypercube. These results provide the policy-improvement and policy-evaluation components of the actor-critic algorithm.

4 Policy Gradient and Martingales on Hypercube

  • The section’s two theoretical foundations are a deterministic policy gradient for reflected diffusions and a martingale characterization.The gradient guides actor updates, while the martingale characterization supplies model-free learning conditions for the value and advantage-rate functions.Source: §4 · pp. 8–9

4.1 Deterministic policy gradient on the hypercube

  • The value function is Gâteaux differentiable in the policy parameter, and the Neumann condition removes reflection terms from the final gradient formula.The formula uses the advantage-rate function; its boundary-regulator contributions vanish under the Neumann boundary condition.Source: §4.1 · pp. 9, 10, 12

4.2 Martingale characterization on the hypercube

  • Under the stated boundary, normalization, and local-control conditions, the martingale characterization identifies the candidate value and advantage-rate functions with their policy-specific counterparts.The martingale condition is required under locally perturbed controls, yielding an off-policy learning scheme with exploration.Source: §4.2 · pp. 12–13

5 Model-free CT-DDPG-R Algorithm

The paper develops a model-free continuous-time actor-critic algorithm for reflected diffusion control, imposing the Neumann boundary condition through soft penalties or a hard architectural constraint. It also analyzes how discrete action updates and exploration affect the ideal reflected dynamics and objective.

5.1 Algorithm design

  • CT-DDPG-R uses martingale TD residuals to train the critic and the deterministic policy gradient to guide actor updates without requiring prior knowledge of the dynamics or rewards.The method uses trajectory data collected through interaction with the environment.Source: §5 · p. 13, §5.1 · pp. 14–15
  • The Neumann condition is enforced either by penalizing squared normal derivatives or by constructing the value network with cosine Fourier features.The soft approach leaves the network architecture unchanged; the hard approach imposes the condition exactly for the hypercube domain.Source: §5 · p. 13, §5.1 · p. 14
  • Exploratory actions add Gaussian noise to the current deterministic policy, while actions are held piecewise constant between grid points.The exploration scale controls noise around the policy action, and the resulting transitions are stored for learning.Source: §5.1 · p. 15, §5.2.2 · p. 17

5.2.1 Extension of the probability space

  • The analysis places the independent exploration noise on a product probability space, where the Brownian motion remains Brownian under the enlarged filtration.Source: §5.2.1 · p. 17

5.2.2 Discretely executed exploratory reflected dynamics

  • For every time grid and positive exploration scale, the discretely executed exploratory reflected SDE has a unique strong solution under the stated assumptions.The piecewise-constant action construction mirrors the algorithm’s regular action updates.Source: §5.2.2 · p. 17

5.2.3 State process error

  • The state-process error theorem gives a bound with a constant independent of the grid and exploration scale, and the discrepancy vanishes as the grid is refined and exploration noise vanishes.The estimate compares the ideal target-policy process with the discretely sampled exploratory process.Source: §5.2.3 · p. 18

5.2.4 Objective error

  • The objective error bound implies that discretely executed exploratory-policy performance converges to the ideal deterministic-policy objective as the grid is refined and exploration noise vanishes.The analysis identifies exploration noise as introducing objective bias relative to the ideal deterministic policy.Source: §5.2.4 · p. 20

6 Numerical Experiments

Across one-, two-, and five-dimensional reflected reservoir tasks, the experiments compare soft and hard Neumann treatments with an unconstrained value approximation. Boundary-aware methods generally stabilize learning and better match reference behavior.

6 Numerical Experiments

  • The reservoir experiments compare soft and hard Neumann treatments with an unconstrained value approximation across one-, two-, and five-dimensional settings.The reservoir states occupy bounded regions imposed by storage capacities and safety limits.Source: §6 · p. 21

6.1 One-dimensional reservoir with an explicit optimal solution

  • The one-dimensional test has an explicit optimal value function and policy, enabling the learned approximations to be checked against ground truth.The value function is given by an explicit series representation.Source: §6.1 · pp. 21–22
  • In one dimension, both boundary-aware treatments reduce Neumann residuals and stabilize learning, while unconstrained networks recover the value function and bang-bang policy with occasional oscillations.The analytical value function and optimal policy provide ground truth for this comparison.Source: §6.1 · pp. 22–23
  • In the one-dimensional experiment, imposing the boundary condition can trap learning near the boundary for a long time, especially with the hard constraint; later log-penalty tasks do not show this issue.The paper reports that the benefits of Neumann constraints become more visible as state dimension and value or policy complexity increase.Source: §6.1 · p. 23

6.2 Two-dimensional reservoir cascade regulation

  • In two dimensions, boundary-aware learned policies agree more closely with the finite-difference reference, despite oscillations in that grid-based approximation.No closed-form solution is available, so the policies are assessed qualitatively.Source: §6.2 · pp. 24–25

6.3 Five-dimensional reservoir cascade regulation

  • In five dimensions, soft and hard Neumann treatments converge faster and more stably than the unconstrained baseline, with slightly better final performance and smoother, more physically plausible policies.The reported benefits are more pronounced as state dimension and problem complexity increase.Source: §6.3 · p. 27
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