Finite-Time Regret Guarantees for Neural RL in Finite-Memory POMDPs

Summary:This work studies reinforcement learning in partially observable Markov decision processes, where long-range dependence makes learning difficult. For environments satisfying filter stability, it approximates the POMDP with a finite-memory superstate MDP and derives a new pointwise approximation-error bound. The authors then analyze temporal-difference learning with nonlinear neural value functions on correlated, POMDP-generated transitions, and combine the resulting finite-time critic bound with

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Core Contributions

This work addresses a fundamental tension in reinforcement learning: a POMDP's true state is hidden, so an agent must infer it from history, but longer histories increase both representational and sample complexity.

Under a filter-stability assumption, the authors approximate the POMDP with a finite-memory superstate MDP. A superstate is constructed from the most recent window of observations, turning the partially observable task into a finite-state MDP surrogate. The paper establishes three main results:

1. A new pointwise approximation-error bound for the finite-state MDP surrogate;
2. A finite-time critic-error bound for temporal-difference learning with nonlinear neural value functions, using correlated transitions generated by POMDP trajectories;
3. A finite-time sublinear regret guarantee relative to the optimal POMDP value, obtained by combining the critic bound with policy mirror descent.

According to the authors, this is the first finite-time sublinear regret result with neural function approximation for finite-memory POMDPs.

From History to Superstates

Let the hidden state, observation, and action be (s_t), (o_t), and (a_t). A finite-memory agent uses a window

[ h_t^m=(o_{t-m+1},\ldots,o_t) ]

as an expanded state. Filter stability is the condition that makes this construction useful: as the window grows, the posterior over hidden states should concentrate, allowing recent history to act as an approximate sufficient statistic.

The tradeoff is fundamental. Increasing (m) may reduce information loss, but the effective state space also grows, making data collection and function approximation harder. The new pointwise bound connects approximation quality to learning guarantees. It is this link that allows guarantees for an algorithm trained on the surrogate to be translated into a regret guarantee for the original POMDP.

Filter stability is not an implementation detail that can be assumed away. If hidden states change rapidly, observation noise accumulates, or the posterior remains multimodal over long periods, finite history may fail to recover an adequate sufficient statistic. In that case, the theoretical guarantees may cease to apply regardless of neural-network capacity.

Why Correlated Samples Matter

Many reinforcement-learning analyses assume independent and identically distributed samples. POMDP trajectories violate that assumption: consecutive transitions share histories, actions, and latent states, creating temporal correlation and distribution shift.

This paper applies TD learning directly to superstate transitions generated by POMDP trajectories and derives a finite-time critic bound under these correlated samples. This is a stronger analysis setting than treating the trajectory data as ordinary MDP samples.

The neural function-approximation result also narrows the gap between theory and modern deep RL. TD learning may use tabular, linear, or neural critics, but the paper's guarantee applies to nonlinear neural value functions. A separate implementation question remains: how faithfully specific optimizers, replay buffers, target-network updates, and exploration procedures map onto the algorithm assumed by the proof.

Policy Optimization and Regret

The critic estimate is combined with policy mirror descent, which updates policy weights through an exponential tilt relative to a reference distribution rather than taking an unconstrained gradient step. This geometry allows approximation error in the critic to be transferred into a policy-quality bound.

When the critic error decays sublinearly with the number of samples, the resulting policy regret can also be sublinear. The result suggests that neural critics need not be excluded from finite-sample guarantees in finite-memory POMDPs.

This is not an unconditional claim about every partially observable task. Window length, filter stability, function-approximation error, observation noise, and discount factor can all enter the regret constants. In high-dimensional, long-horizon environments, theoretical sublinear scaling does not automatically imply an practically achievable sample budget or policy quality.

Engineering Trade-offs

Memory Length versus Sample Efficiency

A longer history window can reduce information loss, but it also enlarges the effective state space and may make trajectory coverage sparse. The practical design problem is to select the smallest window that preserves the relevant state information.

Neural Networks versus Tractable Analysis

Linear approximation is easier to analyze but less expressive. Neural networks better reflect modern value-function architectures but introduce optimization and representation error. The paper narrows this gap without making results independent of architecture and hyperparameters.

Correlated On-policy Data

Using genuine POMDP trajectories avoids a mismatch between the data distribution and the environment, but correlated transitions can make gradient estimates noisier. Sequence-aware optimization, trajectory batching, conservative off-policy correction, or variance-control techniques may be needed in implementation.

Limits of Numerical Validation

The abstract states that experiments support the theory, but it does not disclose environments, window sizes, network sizes, regret slopes, or baseline configurations. Agreement in simulation does not establish robustness in unstable POMDPs, long-horizon tasks, or high-dimensional observation spaces.

Practical Guidance

Before applying the method to a real system:

1. Test filter stability. Examine posterior entropy, state-prediction error, and performance across several history-window sizes.
2. Separate information and capacity effects. Use ablations over both memory length and model size so that gains are not misattributed to a larger network.
3. Evaluate whole trajectories. Report return per episode, reward per step, policy regret, and confidence intervals rather than relying only on one-step prediction error.
4. Stress-test assumption violations. Include observation dropouts, action delay, and nonstationary task phases, which may invalidate the stability assumption.
5. Measure environment interactions. Count the real-environment samples needed to reach a target return, not merely gradient steps.

For research teams, the paper is best treated as a theoretical foundation for finite-memory POMDP algorithms: a framework for reasoning about critic error, policy optimization, and memory length, rather than a turnkey production recipe.

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Source: ArXiv Machine Learning (cs.LG) (2026-11-06)

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Tags: #POMDP Follow #Neural Reinforcement Learning Follow #Finite-Time Regret Follow #Temporal-Difference Learning Follow #Policy Mirror Descent Follow