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Why Generalization in RL is Difficult: Epistemic POMDPs and Implicit Partial Observability

Dibya Ghosh, Jad Rahme, Aviral Kumar, Amy Zhang, Ryan P. Adams, Sergey Levine

arXiv:2107.06277v1cs.LGcs.AIstat.ML

TL;DR

RL generalization from finite training contexts is harder than supervised-learning generalization because epistemic uncertainty induces implicit partial observability. The paper reframes the problem as an epistemic POMDP, proposes approximate optimization with LEEP, and reports significant ProcGen generalization gains over standard RL methods. It also shows that generic stochasticity techniques can fail in this setting.

  • Problem

    Generalization to unseen contexts is a central RL challenge, and standard empirical-risk-minimization approaches can be sub-optimal even when test contexts share the training distribution.

  • Method

    The paper models finite-context RL generalization as an epistemic POMDP and proposes LEEP, an ensemble-based approximation to optimizing return in that POMDP.

  • Results

    LEEP achieves significant gains in test-time performance over standard RL methods on several ProcGen benchmark tasks.

  • Takeaways & Limitations

    Generalizing RL requires policies that account for epistemic uncertainty and can degrade gracefully when learned representations fail to generalize.

  • Takeaways & Limitations

    Generic stochasticity methods such as entropy regularization can fail catastrophically in epistemic POMDPs, and their expected return may be arbitrarily below Bayes-optimal behavior.

Abstract

from arXiv · show

Generalization is a central challenge for the deployment of reinforcement learning (RL) systems in the real world. In this paper, we show that the sequential structure of the RL problem necessitates new approaches to generalization beyond the well-studied techniques used in supervised learning. While supervised learning methods can generalize effectively without explicitly accounting for epistemic uncertainty, we show that, perhaps surprisingly, this is not the case in RL. We show that generalization to unseen test conditions from a limited number of training conditions induces implicit partial observability, effectively turning even fully-observed MDPs into POMDPs. Informed by this observation, we recast the problem of generalization in RL as solving the induced partially observed Markov decision process, which we call the epistemic POMDP. We demonstrate the failure modes of algorithms that do not appropriately handle this partial observability, and suggest a simple ensemble-based technique for approximately solving the partially observed problem. Empirically, we demonstrate that our simple algorithm derived from the epistemic POMDP achieves significant gains in generalization over current methods on the Procgen benchmark suite.

1 Introduction

RL generalization can fail even when test contexts share the training distribution, because finite training data induces epistemic uncertainty and implicit partial observability. The paper formalizes this as an epistemic POMDP and proposes LEEP, which improves test-time generalization on ProcGen tasks.

  • Motivation: Generalizing RL policies to unseen contexts remains difficult despite empirical-risk-minimization methods that often support supervised-learning generalization.The paper frames this as a deployment-relevant challenge beyond optimizing a fixed task.
  • Motivation: 95% success from peeking can outperform a training-optimal map classifier when classifier generalization falls below 95% in new zoos.The classifier is optimal in training zoos but can be less reliable than peeking at test time.
  • Core idea: Finite training contexts create epistemic uncertainty that makes even fully observed test environments implicitly partially observed.The agent cannot identify the true classifier or MDP parameters from limited training contexts.
  • Core idea: The paper recasts generalization as solving an epistemic POMDP induced by uncertainty about the test environment.This reframing connects test-time generalization to partially observed decision-making rather than ordinary MDP optimization.
  • Contribution: LEEP approximates epistemic-POMDP optimization with an ensemble of policies and achieves significant test-time gains over standard RL methods on ProcGen tasks.The paper also emphasizes policies that degrade gracefully when learned networks fail to generalize.

2 Related Work

Prior work studies RL generalization through representation and behavior changes, while Bayesian RL models optimal action under uncertainty over MDPs. This paper brings the Bayesian-RL POMDP perspective to fixed-context generalization.

  • Generalization in RL: Prior RL generalization methods improve function approximators through inductive biases or encourage behaviors that transfer to unseen contexts.Examples include environment-dynamics imitation, bisimulation, and other representational approaches.
  • Bayesian RL: Bayesian RL studies acting optimally under a belief distribution over MDPs, commonly reducing Bayes-optimal behavior to a POMDP or belief-state MDP.The epistemic POMDP is presented as a specific instantiation of this framework.
  • Bayesian RL: Exact Bayes-optimal policy learning is generally intractable, motivating relaxations such as value of perfect information and Thompson sampling.These methods can remain asymptotically optimal for learning in relevant Bayesian-RL settings.

3 Problem Setup

The paper studies contextual MDPs in which training covers only sampled contexts and evaluation uses unseen contexts. It also introduces sequential classification as an example where figure-grounded feedback creates a generalization challenge.

  • MDP formulation: The MDP is defined by state, action, reward, transition, initial-state distribution, and discount-factor components.The setup specifies a Markovian transition function and bounded rewards.
  • MDP formulation: Classical MDP results identify a deterministic Markovian policy as optimal among history-dependent policies for the standard objective.This property becomes problematic when the effective test-time problem is partially observed.
  • Contextual MDPs: A contextual MDP decomposes each state into a fixed context vector and a varying sub-state, with context-specific dynamics and rewards sharing structure.The agent trains only on a sampled subset of contexts and is evaluated across unseen contexts.

4 Warmup: A Sequential Classification RL Problem

Sequential classification exposes a mismatch between standard MDP solutions and RL generalization: feedback from incorrect guesses should change later actions, but deterministic policies repeat them. Simple structure-aware variants generalize better than DQN.

  • Problem structure: Although the image fully determines its label, a finite training set makes perfect generalization unavailable.The one-step and sequential tasks share the same deterministic optimal policy in the fully observed MDP.
  • Empirical result: DQN achieves lower test performance than simple variants that leverage the sequential structure of RL.The variants include random guessing after an initial error and process of elimination.
  • Problem structure: Sequential RL policies can exploit multiple attempts and incorrect-guess feedback by trying alternative labels or changing actions after failure.These options are additional structure unavailable in the one-step classification formulation.
  • Empirical result: All three agents have the same training performance, but DQN repeatedly selects the same incorrect label and generalizes more poorly.This behavior follows from modeling the empirical training dataset as an MDP with a deterministic Markovian optimum.

5 Modeling Generalization in RL as an Epistemic POMDP

The paper models generalization from limited training contexts as an epistemic POMDP, because uncertainty about the unseen environment creates implicit partial observability at test time. This framework explains why standard MDP-based policies can generalize poorly and why Bayes-optimal behavior may require memory, stochasticity, or actions that are sub-optimal in individual posterior MDPs.

  • Limited training evidence induces implicit partial observability at test time, even when the underlying domains are fully observed.
  • The epistemic POMDP samples an unknown MDP from the posterior at each episode while exposing the agent only its current state.
  • When the prior is well specified, performance in the epistemic POMDP equals expected test-time return, and its optimal policy is Bayes-optimal for generalization.
  • Bayes-optimal behavior can require memory or stochasticity because episode experience reveals information about the hidden MDP identity.
  • Policies optimized for individual posterior MDPs, including standard RL policies and generic entropy-regularized policies, may have arbitrarily low expected test-time return relative to Bayes-optimal behavior.

6 Learning Policies that Generalize Well Using the Epistemic POMDP

Because the exact epistemic POMDP is difficult to recover in high-dimensional settings, the paper derives an approximate ensemble method, LEEP, that learns policies on posterior samples and combines them into one policy. Its bootstrap construction and linked-policy objective provide a practical approximation to epistemic-POMDP optimization.

  • The exact epistemic POMDP is difficult to approximate because coherent MDP models and posterior distributions are hard to maintain in high-dimensional domains.
  • LEEP approximates the posterior with finitely many sampled MDPs, learns one policy per sample, and combines them into a single policy.
  • A lower-bound result supports high empirical-epistemic-POMDP return when each component policy performs well in its sampled MDP and can be imitated by the combined policy.
  • LEEP bootstrap-samples training contexts to form overlapping context sets, trains an ensemble with PPO, and updates each policy using reward plus a disagreement penalty against the combined policy.
  • The max_j π_j link function is treated as an inductive bias for combining policies, reflecting the need to approximate optimal behavior in the empirical epistemic POMDP.

7 Experiments

The Procgen experiments test whether epistemic-POMDP-inspired policies improve generalization across unseen procedural levels, including settings with few training contexts and component ablations. LEEP generally outperforms PPO and avoids degradation when training data are scarce.

  • The study evaluates LEEP on Procgen games with image observations, 200 training levels, 15 actions, and an ensemble of 4 policies.
  • On BigFish, LEEP matches PPO’s test return while exhibiting lower variance across the five random seeds.
  • LEEP outperforms PPO by a significant margin on Maze, Heist, and Dodgeball, while reducing the training-test generalization gap in all four evaluated environments.
  • With only 50 Maze training levels, PPO’s test return declines below the starting random policy, whereas LEEP avoids this degradation.
  • Ablations indicate that unregularized ensembles improve over PPO but do not match LEEP, while the optimistic link outperforms averaging and Distral improves training-context learning without improving generalization.

8 Discussion

The paper argues that RL generalization creates an epistemic POMDP because uncertainty about the test-time MDP induces partial observability, and presents LEEP as an approximate solution with improved generalization.

  • 8 Discussion: Generalization in RL creates a partially observed problem because epistemic uncertainty obscures the underlying MDP, unlike standard MDP solution settings.The paper calls this induced setting the epistemic POMDP.
  • 8 Discussion: LEEP optimizes a bound for performance in an approximation of the epistemic POMDP.
  • 8 Discussion: LEEP achieves significant generalization improvements over RL methods that do not incorporate epistemic uncertainty into policy optimization.
  • 8 Discussion: The approach optimizes a crude epistemic-POMDP approximation with a small number of posterior samples.

A.1 Implementation details

The image-classification analysis models RL classification as sequential guessing: discounted rewards favor trying labels in descending posterior probability, while γ = 0 recovers ordinary classification.

  • A.1 Implementation details: The RL image-classification environment samples an image-label episode and penalizes incorrect guesses until the correct label is selected.Correct classification ends the episode with reward 0, while an incorrect action gives reward −1.
  • A.1 Implementation details: The analysis restricts attention to deterministic memory-based policies and excludes repeated actions because repetition cannot improve the episode return.
  • A.1 Implementation details: An optimal policy tries labels sequentially from highest to lowest probability under the belief distribution.
  • A.1 Implementation details: For γ > 0, optimal memoryless behavior is obtained through a constrained optimization problem, with an explicit form available when γ = 1.
  • A.1 Implementation details: For γ = 0, the objective becomes supervised classification, and the optimal memoryless policy deterministically selects the most probable label.

B Theoretical Results

The theoretical results establish that the epistemic POMDP captures expected test-time generalization and that MDP-optimal policies can be poor or incompatible guides for Bayes-optimal behavior.

  • B Theoretical Results: The epistemic POMDP’s expected return equals a policy’s expected test-time return under the posterior over unknown MDPs.
  • B Theoretical Results: The epistemic-POMDP optimal policy is Bayes-optimal for generalization to the unknown MDP.
  • B Theoretical Results: A deterministic Markov policy can be optimal with probability at least 1 − ϵ yet be outperformed by a uniformly random policy in the epistemic POMDP.
  • B Theoretical Results: MDP-optimal policies can have action support disjoint from the Bayes-optimal policy, so combining posterior MDP optima cannot always produce Bayes-optimal behavior.
  • B Theoretical Results: For a suitable prior, the maximum-entropy solution in a surrogate bandit MDP is optimal in the corresponding epistemic POMDP.
  • B Theoretical Results: The goal-action construction extends to goal-state-reaching settings when time-varying reward functions are allowed.

B.4 Failure of MaxEnt RL and Uncertainty-Agnostic Regularizations

The paper constructs binary-tree epistemic POMDPs showing that uncertainty-agnostic stochasticity can perform arbitrarily poorly, while both stochastic and deterministic memoryless policies can fail relative to the Bayes-optimal policy. It also gives an ensemble-based optimization whose solution is optimal for the empirical epistemic POMDP under stated conditions.

  • Failure of uncertainty-agnostic stochasticity: Everywhere-stochastic memoryless policies can have performance arbitrarily close to the uniformly random policy, despite the Bayes-optimal memoryless policy being stochastic.The construction uses two binary-tree MDPs with different terminal states and a posterior over them.
  • Binary-tree construction: A policy reaching the correct terminal state with probability p achieves return 1/(1 + (1−γ)/(pγ)), linking performance directly to terminal-state success probability.The expression follows from the geometrically distributed number of visits to the initial state.
  • Failure of uncertainty-agnostic stochasticity: As tree depth n increases, the performance ratio between an everywhere-stochastic policy and the Bayes-optimal policy can become arbitrarily small.The relevant asymptotic ratio scales as (1 − β)^(n−1).
  • Deterministic-policy failure: All deterministic memoryless policies also fail to visit both possible goal states, whereas the Bayes-optimal policy randomizes only at the top level and then acts deterministically.A deterministic policy repeatedly follows one path and receives reward in only one of the two MDPs.
  • Ensemble-based solution: The ensemble optimization is optimal for the empirical epistemic POMDP when its constituent policies are combined through a function f satisfying f(π, …, π) = π.The result formalizes a condition under which jointly optimized policies produce an optimal aggregate policy.

C Procgen Implementation and Experimental Setup

The Procgen experiments train LEEP with four linked policies using the recommended fixed-level training setup and compare it with PPO under matched update budgets. The implementation is parallelized across four GPUs, with larger ensembles limited by memory and runtime constraints.

  • Training and evaluation setup: Agents train on fixed levels and are tested on the full level distribution using Procgen’s easy difficulty and recommended 200 training levels.This setting is used because prior work reports a substantial train-test gap even at easy difficulty.
  • LEEP implementation: LEEP maintains four policies, each using the prescribed ResNet architecture, and optimizes entropy-regularized PPO objectives with one-step KL penalties linking policies.Gradients are not taken through the linked policy.
  • Training budget: LEEP and PPO receive the same number of PPO updates by reducing LEEP’s environment steps per iteration from 16384 to 4096.LEEP uses four policies under the same environment budget as a single PPO policy.
  • Computational constraints: Running LEEP on one Procgen environment for 50 million steps takes approximately 5 hours on four Tesla T4 GPUs, and larger ensembles were infeasible without significant slowdown.GPU memory constraints limited the practical number of ensemble members.

D Procgen Results

Across the Procgen comparisons, LEEP matches or exceeds PPO’s training returns while reducing the train-test generalization gap. With fewer training levels, LEEP avoids the overfitting observed for PPO.

  • LEEP versus PPO: LEEP achieves equal or higher training return than PPO while producing a lower generalization gap across four Procgen environments.Figure 7 averages results across five random seeds and reports training returns on top and test returns on the bottom.
  • Varying training levels: As the number of training levels varies, LEEP generally maintains a smaller train-test performance gap than PPO.The comparison examines performance as the supplied training levels change.
  • Varying training levels: With fewer training levels, PPO’s learned performance can fall below a random policy, whereas LEEP avoids this overfitting.This behavior is reported for the Maze task with only 50 levels.
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