Lipschitz Continuity in Model-based Reinforcement Learning

Kavosh Asadi; Dipendra Misra; Michael L. Littman

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Lipschitz Continuity in Model-based Reinforcement Learning

Kavosh Asadi, Dipendra Misra, Michael L. Littman

arXiv:1804.07193v3Apr 19, 2018cs.LGcs.AIstat.ML

TL;DR

Model-based reinforcement learning can suffer when small model errors compound during long-horizon planning. This paper develops Lipschitz model classes and Wasserstein-based bounds for multi-step prediction and value estimation, and reports benefits from controlling neural-network model smoothness.

Problem
Approximate model-based RL can magnify small model errors through planning, motivating analysis of how model smoothness affects long-horizon prediction and policy quality.
Method
The paper represents stochastic transitions as distributions over deterministic Lipschitz components, uses Wasserstein distance for distributional error, and provides an EM learning algorithm for continuous state spaces.
Results
The paper derives bounds linking multi-step prediction and value-estimation error to one-step accuracy and Lipschitz constants, with experiments showing benefits from controlling neural-network model smoothness.
Takeaways & Limitations
Lipschitz continuity and Wasserstein distance provide tools for analyzing and improving model-based RL with function approximation.
Takeaways & Limitations
The Lipschitz model-class representation is a subset of stochastic transitions characterized by a state-independent distribution over deterministic functions, and larger problems remain future work.

Abstract

from arXiv · show

We examine the impact of learning Lipschitz continuous models in the context of model-based reinforcement learning. We provide a novel bound on multi-step prediction error of Lipschitz models where we quantify the error using the Wasserstein metric. We go on to prove an error bound for the value-function estimate arising from Lipschitz models and show that the estimated value function is itself Lipschitz. We conclude with empirical results that show the benefits of controlling the Lipschitz constant of neural-network models.

1. Introduction

The paper studies model-based reinforcement learning through Lipschitz continuity, linking model smoothness to multi-step prediction and value-estimation accuracy. It proposes Lipschitz model learning, including an EM algorithm for continuous state spaces, and examines Lipschitz regularization for neural-network transition models.

Approximate model-based RL can be unstable because small model errors may be magnified by planning, producing poor performance.
Multi-step prediction accuracy depends on both one-step model accuracy and the model’s Lipschitz constant.
Value-prediction error also depends on the model’s Lipschitz constant and influences the quality of policies found by planning.
The paper represents stochastic dynamics through deterministic components and provides an Expectation-Maximization algorithm for learning this class in continuous state spaces.
Controlling model complexity through the Lipschitz constant is examined as an efficiently applicable regularization technique for neural-network transition models.

2. Background

This section introduces the MDP framework, Lipschitz continuity as a smoothness condition, and Wasserstein distance for comparing probability distributions. Kantorovich-Rubinstein duality connects Wasserstein distance to optimization over Lipschitz functions.

The MDP is defined by state space S, action set A, reward function R, transition dynamics T, and discount rate γ; one-action MDPs are MRPs.
Lipschitz continuity bounds how much a function’s output distance can change relative to its input distance.
A Lipschitz function is a non-expansion when its constant equals 1 and a contraction when its constant is less than 1.
The Wasserstein metric measures distributional distance through the minimum expected distance between paired points under couplings matching the two marginals.
Kantorovich-Rubinstein duality computes Wasserstein distance by maximizing over functions with Lipschitz constant at most one.

3. Lipschitz Model Class

The paper models stochastic transitions as distributions over deterministic component functions whose smoothness defines a Lipschitz model class. This representation supports generalized distribution-valued transitions and connects their Lipschitz behavior to Wasserstein distance.

A Lipschitz model class represents stochastic MDP transitions as a distribution over deterministic functions f:S→S, conditioned on action through g(f|a).
The class captures state-independent mixtures of deterministic transitions, and any finite MDP transition probabilities can be decomposed in this form.
The generalized transition function maps a state distribution and action to a probability distribution, so its Lipschitz constant is measured using Wasserstein distance.
In the gridworld example, an action attempts its corresponding direction with probability 0.8 and neighboring directions with probability 0.1 each.
For the gridworld component functions, Manhattan state distances yield a Lipschitz constant KF=2 for the model class.

4. On the Choice of Probability Metric

The paper argues that Wasserstein distance best captures how closely stochastic transition distributions approximate one another because it reflects the distance between their supports. In contrast, likelihood and total variation can miss meaningful differences in separated distributions.

Wasserstein distance is presented as a suitable metric for comparing stochastic transition distributions in model-based reinforcement learning.The paper motivates this choice through an example involving an environment and an approximate transition model.
A uniform state distribution is shifted by ±c1 in the environment and by ±c2 in the approximate model, producing distributions whose similarity depends on the relationship between c1 and c2.The example compares the resulting state distributions after one transition.
Likelihood-based comparison distinguishes the distributions unless c1 and c2 are exactly equal, even when their shifts may be close.
Total variation treats distributions with disjoint supports similarly regardless of how far apart those supports are.
Wasserstein distance is sensitive to the distance between the constants, matching the example’s intuitive notion of approximation quality.This sensitivity is especially relevant in high-dimensional spaces where distributions may lie on low-dimensional manifolds.

5. Understanding the Compounding Error Phenomenon

The paper analyzes compounding prediction error when approximate transition models are recursively applied over multiple steps. It bounds this error using one-step accuracy and the Lipschitz constant of the model and environment transitions.

Multi-step prediction feeds each model output into the next step, so small one-step errors can produce poor long-horizon predictions and ineffective planning.This phenomenon is identified as compounding error.
A Lipschitz model class is called Δ-accurate when its induced transition function satisfies the paper’s one-step Wasserstein accuracy condition.
The multi-step Wasserstein error is bounded in terms of the single-step Wasserstein error and the Lipschitz constant of the transition function.The analysis considers a start-state distribution and a fixed action sequence over n recursive model applications.
The generalized transition function induced by a Lipschitz model class is itself Lipschitz, with its constant bounded in terms of the model-class constant KF.This property supports recursive control of prediction error.
The composition lemma bounds the Lipschitz constant of composed functions by the product of their individual constants, enabling the recursive error analysis.Summation likewise preserves Lipschitz continuity with a constant bounded by the sum of component constants.
Theorem 1 combines one-step accuracy with the smaller of the model and true-transition Lipschitz constants, and the proof proceeds by induction.

6. Value Error with Lipschitz Models

The paper studies value-function error when two Markov reward processes differ only in their transition functions. Under Lipschitz assumptions, it bounds the resulting value error and notes conditions under which the bound is tight.

The value-error analysis compares two Markov reward processes with identical rewards and discounting but different transition functions, T and bT.The setting uses a single action and a reward function depending only on state.
Theorem 2 assumes a Lipschitz model class, a Δ-accurate model, and a Lipschitz reward function with constant KR.
The value-function error is bounded by γKRΔ / ((1−γ)(1−γK̄)) under the theorem’s assumptions.Here K̄ is the minimum of the model-class and true-transition Lipschitz constants.
The same bound applies with the value functions reversed because Wasserstein distance is symmetric.
Theorem 2’s bound is tight when the reward function is linear, while Theorem 1 is tight for deterministic linear transition models.The proof does not require a bounded reward function.

7. Lipschitz Generalized Value Iteration

Lipschitz Generalized Value Iteration shows that fixed points of suitable Bellman equations yield Lipschitz state-action value functions. The result supports PAC exploration with a Lipschitz model assumption and connects Wasserstein minimization to value-aware model learning.

Lipschitz Generalized Value Iteration: A Lipschitz transition model yields a Lipschitz state-action value function when solving the fixed point of a class of Bellman equations.The analysis is framed through Generalized Value Iteration, which extends Value Iteration to arbitrary backup operators.
Lipschitz Generalized Value Iteration: Generalized Value Iteration computes a value function with a bounded Lipschitz constant for any non-expansion backup operator identified by the paper.
Implications: The result makes a Lipschitz model sufficient for PAC exploration in continuous state spaces, rather than requiring a separately assumed Lipschitz value function.The paper describes the model assumption as potentially easier to confirm.
Implications: Minimizing Wasserstein distance is equivalent to minimizing the value-aware model learning objective under the theorem’s conditions.

8. Experiments

The experiments compare error metrics and evaluate Lipschitz-controlled models across supervised and reinforcement-learning settings. Results indicate that Wasserstein error is especially informative when state similarity tracks value similarity, while intermediate Lipschitz constants perform best and Lipschitz model classes outperform baselines in gridworld.

Metric comparison: 105 trials compare TV, KL, and Wasserstein error metrics under random rewards and state-index rewards, where the latter makes Euclidean state proximity informative about value similarity.The experiments use finite MRPs with 10 states and γ = 0.95, varying γ for correlation analyses.
Metric comparison: When rewards equal state indices, Wasserstein error correlates highly with value-prediction error; with random rewards, it provides no advantage.The result supports Wasserstein as a useful model-error metric when state-space proximity indicates similar values.
Continuous-control experiments: An intermediate Lipschitz constant yields the best performance in both Cart Pole and Pendulum policy-learning experiments.The learned models are used for long-horizon stochastic actor-critic rollouts and policies are evaluated in the actual environments.
Stochastic models: The EM algorithm effectively trains Lipschitz model classes on a supervised-learning domain, where controlling the Lipschitz constant is useful.The dataset contains 30 points from five functions, and Figure 8 reports a U-shaped final Wasserstein loss with respect to k.
Gridworld: In gridworld, tabular and expected models perform poorly, whereas value iteration with a Lipschitz model class outperforms these baselines.Tabular models fail through lack of generalization, while expected models inadequately represent the ghosts’ stochastic movement.

9. Conclusion

The paper argues that Lipschitz continuity is central to multi-step prediction and value-estimation error in model-based reinforcement learning. It also reports benefits from Wasserstein metrics and identifies larger problems as future work.

Main conclusions: Lipschitz continuity of an estimated model plays a central role in multi-step prediction error and value-estimation error.The conclusion frames this as a step toward understanding model-based RL with function approximation.
Main conclusions: The paper shows benefits of employing Wasserstein for model-based reinforcement learning.The conclusion presents Wasserstein as one of the paper’s supported findings.
Future work: Applying these ideas to larger problems is identified as important future work.This is the explicit scope boundary stated in the conclusion.

Appendix

The appendix supplies supporting claims, constructions, tightness examples, Lipschitz lemmas, and neural-network operator bounds. It also details the supervised-learning data used to evaluate the EM procedure.

Stochastic transition representation: Finite MDP transition probabilities can be represented by a finite set of deterministic functions together with a distribution over those functions.This representation underlies the appendix’s stochastic transition construction.
Tightness: The bounds in Theorems 1 and 2 are tight for deterministic linear transitions paired with a linear reward signal.The appendix provides a specific linear construction and verifies that the resulting value prediction matches the Theorem 2 bound.
Lipschitz transition analysis: A generalized transition function induced by a Lipschitz model class is itself Lipschitz, with Wasserstein used to measure distances between input and output distributions.The appendix also invokes the composition lemma and related Lipschitz results to establish these properties.
Neural-network bounds: The appendix derives Lipschitz constants for neural-network components including ReLU, matrix multiplication, and vector addition.These operator bounds support controlling the Lipschitz constant of an entire neural network.
Evaluation setup: The supervised-learning evaluation samples each of five functions 30 times with inputs uniformly drawn from [−2, 2].These samples form the dataset used to evaluate the EM algorithm.