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Lipschitz Continuity in Model-based Reinforcement Learning
Kavosh Asadi, Dipendra Misra, Michael L. Littman
TL;DR
In model-based reinforcement learning, small model errors can compound during planning, making reliable multi-step prediction an important challenge. The paper studies Lipschitz models and finds that multi-step prediction accuracy relates to both one-step accuracy and the model’s Lipschitz constant.
Problem
Method
Results
Multi-step prediction accuracy is related to both one-step accuracy and the model’s Lipschitz constant, with this dependence extending to value prediction and policy quality.
Source: §1 · p. 1Takeaways & Limitations
The authors conclude that Lipschitz continuity plays a central role in multi-step prediction and value-estimation error, and report benefits from using Wasserstein for model-based reinforcement learning.
Source: §9 · p. 8The Lipschitz model class covers only stochastic transitions representable as a state-independent distribution over deterministic transitions.
Source: §3 · p. 2
Abstract
from arXiv · showhide
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 how Lipschitz continuity affects model-based reinforcement learning, showing that prediction and value-estimation quality depend on both one-step accuracy and model smoothness. It introduces a stochastic Lipschitz model class and an efficient neural-network regularization approach based on controlling the Lipschitz constant.
- Model errors can be magnified during planning in approximate settings, causing poor performance.The introduction contrasts this instability with the better-understood behavior of model-based algorithms in tabular environments.Source: §1 · p. 1
- Multi-step prediction accuracy depends on both one-step accuracy and the model’s Lipschitz constant, with that dependence extending to value prediction and policy quality.Source: §1 · p. 1
- For continuous states and stochastic transitions, the paper measures distributional distance with Wasserstein metrics and represents dynamics through deterministic components in a Lipschitz model class.It also provides an Expectation-Maximization algorithm to learn this class.Source: §1 · p. 1
- Controlling model Lipschitz constants is a regularization approach that can be applied efficiently to neural-network transition models.Source: §1 · p. 1
2. Background
The paper studies continuous-state MDPs with discrete actions and uses Lipschitz continuity and the Wasserstein metric to describe function smoothness and distances between transition distributions.
2. Background
- The MDP uses a continuous state space, discrete actions, reward and transition functions, and a discount rate; with one action, it is an MRP.Source: §2 · p. 1
2.1. Lipschitz Continuity
- A Lipschitz function has bounded variation relative to distances in its input and output spaces; constants of 1 and below define non-expansions and contractions, respectively.Source: §2.1 · pp. 1–2
2.2. Wasserstein Metric
- The Wasserstein metric is the minimum expected distance between paired points with specified distribution marginals, and Kantorovich-Rubinstein duality expresses it as a maximization over Lipschitz functions.The theory uses both the primal and dual definitions.Source: §2.2 · p. 2
3. Lipschitz Model Class
The paper represents stochastic transitions using action-conditioned distributions over deterministic state-transition functions. This model class captures state-independent mixtures of deterministic transitions and induces a distribution-to-distribution transition map measured with Wasserstein distance.
3. Lipschitz Model Class
- The model class represents stochastic MDP transitions as action-conditioned distributions over deterministic functions from states to states.The definition also specifies a Lipschitz model class, though the supplied passage cuts off before its formal condition.Source: §3 · p. 2
- The class captures state-independent mixtures of deterministic transitions, and any finite MDP transition probabilities can be decomposed this way.The paper cites an appendix proof for the finite-MDP decomposition.Source: §3 · p. 2
- In the gridworld example, four deterministic direction functions form a Lipschitz model class with KF = 2.For an action to move up, the intended transition has probability 0.8 and the neighboring directions each have probability 0.1; state distances are Manhattan distances.Source: Figure 2 · p. 3
- The generalized transition map takes a state distribution to a probability distribution, whose distances are quantified using Wasserstein.The paper focuses on the Lipschitz constant of this map.Source: §3 · pp. 2–3
4. On the Choice of Probability Metric
In a stochastic model-based example, the section compares KL divergence, Total Variation, and Wasserstein as metrics for the similarity of true and approximate transition outputs. It argues that Wasserstein best reflects their intuitive closeness.
4. On the Choice of Probability Metric
- The KL-divergence comparison notes an exception when the two constants are exactly equal.Source: §4 · p. 3
- Total Variation gives the same comparison when distributions have disjoint supports, regardless of the distance between those supports.Source: §4 · p. 3
- Wasserstein best matches intuitive transition-model closeness because it is sensitive to how far apart the compared distributions are.The section illustrates this with a true transition that adds or subtracts c1 and an approximate model that adds or subtracts c2.Source: §4 · p. 3, Figure 3 · p. 3
- Wasserstein’s relevance is especially emphasized for high-dimensional spaces, where true distributions usually lie on low-dimensional manifolds.Source: §4 · p. 3
5. Understanding the Compounding Error Phenomenon
The section addresses compounding error in long-horizon model predictions by bounding multi-step Wasserstein error from single-step error and model smoothness. It also relates the result to prior bounds for discrete, continuous, and deterministic settings.
5. Understanding the Compounding Error Phenomenon
- Compounding error occurs when recursively feeding model predictions into later steps, causing poor long-horizon predictions and ineffective planning.The section formalizes model accuracy as ∆-accuracy of the induced transition function.Source: §5 · p. 3
- The generalized transition function is Lipschitz, so similar input distributions produce similar output distributions up to a KF factor.The proof of the multi-step bound uses this property together with the one-step accuracy definition.Source: §5 · p. 4
- Theorem 1 bounds n-step Wasserstein prediction error for a ∆-accurate model using the one-step error and min{KF, KT}.Here KF and KT are the Lipschitz constants of the model and transition function, respectively.Source: §5 · p. 4
- Prior results include the Simulation Lemma for discrete-state MDPs, a total-variation trajectory bound for continuous states, and a deterministic prediction-error bound generalized here to probabilistic models.The cited deterministic result is from Venkatraman et al. (2015); the Simulation Lemma is attributed to Kearns & Singh (2002) and Strehl et al. (2009), and the continuous-state result to Kakade et al. (2003).Source: §5 · p. 5
6. Value Error with Lipschitz Models
This section bounds state-value error between MRPs that differ only in their transition function, under Lipschitz model and reward assumptions. It also identifies conditions for tightness and notes that bounded rewards are not required.
6. Value Error with Lipschitz Models
- With a Δ-accurate Lipschitz model and a reward function with Lipschitz constant KR, the state-value error is bounded by γKRΔ/((1−γ)(1−γK̄)), where K̄ = min(KF, KT).The bound applies to the single-action MRP setup with state-dependent rewards.Source: §6 · p. 5
- The same value-error bound holds in the reverse direction because Wasserstein distance is symmetric.Source: §6 · p. 5
- The value-error bound is tight when the reward function is linear, and its proof does not require bounded rewards.Source: §6 · p. 5
- For comparison, Theorem 1’s bound is tight when the transition model is deterministic and linear.Source: §6 · p. 5
7. Lipschitz Generalized Value Iteration
This section shows that, given a Lipschitz transition model, Generalized Value Iteration yields a Lipschitz state-action value function. It derives implications for continuous-state PAC exploration and value-aware model learning.
7. Lipschitz Generalized Value Iteration
- The proof uses Lipschitz-function and operator lemmas to bound the Lipschitz constant across GVI updates and at convergence.The passage identifies the operators as Lipschitz and gives their constants, but the supplied text omits the details.Source: §7 · pp. 5–6
- With a Lipschitz transition model, GVI using any non-expansion backup operator from Lemma 4 computes a value function with a bounded Lipschitz constant.The theorem concerns the fixed point of the Bellman equations; the supplied passages do not include the bound’s formula.Source: §7 · pp. 5–6
- A Lipschitz model is sufficient for the Lipschitz-value-function assumption used in PAC exploration in continuous state spaces.The passage notes that confirming the model assumption may be easier.Source: §7 · p. 6
- Minimizing Wasserstein is equivalent to minimizing the VAML objective (Asadi et al., 2018).This implication follows from the theorem relating Lipschitz models to value functions.Source: §7 · p. 6
8. Experiments
Experiments show that controlling model Lipschitz constants can improve policy learning and value estimation, while Wasserstein error is most informative when nearby states have similar values. The paper also evaluates EM for learning stochastic Lipschitz model classes in supervised and reinforcement-learning settings.
Error metrics
- Wasserstein model error better indicates value-prediction error when state-space proximity implies similar rewards, but provides no advantage with random rewards.The comparison used finite MRPs with 10 states and γ = 0.95, across 105 trials.Source: Figure 4 · p. 6, Figure 5 · p. 6, §8 · p. 6
Lipschitz enforcement
- The experiments enforce neural-network Lipschitz continuity by bounding constants of network operations, whose bounds combine to bound the full network.Operation bounds are provided for different p-norms.Source: §8 · p. 7, Table 1 · p. 7
Cart Pole and Pendulum
- An intermediate Lipschitz constant yields the best performance in Cart Pole and Pendulum, consistent with controlling compounding prediction errors and improving value estimation.Models were trained on 15 * 10^3 Cart Pole tuples and 10^4 Pendulum samples.Source: §8 · p. 7, Figure 6 · p. 7
Stochastic model learning
- For stochastic settings, EM jointly learns a set of Lipschitz functions and a distribution over them, treating each sample’s generating function as a latent variable.The samples contain state, action, and next-state tuples but not the identity of the function that generated each sample.Source: §8 · p. 7
Stochastic supervised learning
- EM effectively fits stochastic functions in a supervised domain, and controlling the Lipschitz constant is useful; final Wasserstein loss has a U-shape with respect to k.The experiment generated 30 points from five functions and trained five neural networks.Source: §8 · p. 8, Figure 8 · p. 8
Gridworld
- In gridworld, value iteration with a Lipschitz model class outperforms tabular and expected-model baselines, which fail due to lack of generalization and unhelpful expected ghost motion.The evaluation used 1,000 environmental time-steps and compared policy quality and model test accuracy.Source: §8 · p. 8, Figure 9 · p. 8
9. Conclusion
The paper concludes that Lipschitz continuity of estimated models plays a central role in multi-step prediction and value-estimation error, and that Wasserstein metrics benefit model-based RL. Applying these ideas to larger problems is identified as future work.
- Lipschitz continuity of an estimated model plays a central role in multi-step prediction error and value-estimation error.Source: §9 · p. 8
- The authors report benefits from employing Wasserstein for model-based RL.Source: §9 · p. 8
- Applying these ideas to larger problems is identified as important future work.Source: §9 · p. 8
Appendix
The appendix supports the paper’s bounds with proofs and auxiliary results on finite-MDP transition representations, tightness, and Lipschitz continuity. It also derives basic function constants and describes the supervised-learning data setup.
Deterministic-function representation
- A finite MDP’s transition probabilities can be reconstructed as a distribution over a finite set of deterministic transition functions.The proof constructs the functions from cumulative transition probabilities and defines a distribution over them.Source: Appendix · p. 11
Tightness of the bounds
- For a linear transition and linear reward, the Theorem 2 bound is tight: it holds at n = 2, with equality shown more generally after n compositions of the models.Source: Appendix · pp. 11–12
Lipschitz transition function
- The generalized transition function induced by a Lipschitz model class is Lipschitz, as shown through a direct argument and Kantorovich–Rubinstein duality.The appendix also invokes the composition of Lipschitz functions to bound the relevant constant.Source: Appendix · pp. 12–13
Lipschitz lemmas and operators
- The appendix states the composition lemma: composing Lipschitz functions yields a Lipschitz constant no greater than the product of their constants.It also establishes Lipschitz constants for the operators discussed by Asadi & Littman (2017).Source: Appendix · pp. 13–14
Function constants
- ReLU has Lipschitz constant 1, and vector addition by a fixed vector has constant 1 for p = 0, 1, and ∞.The appendix also derives norm-dependent constants for matrix multiplication.Source: Appendix · pp. 15–16
Supervised-learning domain
- For the supervised-learning domain, the dataset was generated from five functions, each sampled 30 times using inputs drawn uniformly from [−2, 2].Source: Appendix · p. 16