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Tighter Problem-Dependent Regret Bounds in Reinforcement Learning without Domain Knowledge using Value Function Bounds

Andrea Zanette, Emma Brunskill

arXiv:1901.00210v4cs.LGcs.AIstat.ML

TL;DR

The paper addresses the gap between worst-case RL guarantees and better practical performance on structured MDPs. It introduces EULER, an episodic finite-horizon algorithm that adapts regret analysis to next-state value variance without prior environmental-norm knowledge. EULER retains state-of-the-art worst-case dominant-term guarantees while obtaining tighter bounds for low-variance environments and removing horizon dependence from the dominant term in the targeted open-question setting.

  • Problem

    Worst-case RL bounds often fail to reflect how much easier some environments are, motivating problem-dependent guarantees that characterize MDP learning difficulty.

  • Method

    EULER uses optimism under uncertainty with an empirical Bernstein exploration bonus and value-function correction for episodic finite-horizon discrete MDPs.

  • Results

    EULER matches state-of-the-art worst-case regret bounds in dominant terms and obtains tighter guarantees when next-state value variance or environmental norm is small.

  • Takeaways & Limitations

    Environmental norm can characterize RL difficulty across structured classes including sparse-reward, near-deterministic, highly mixing, and benchmark MDPs.

  • Takeaways & Limitations

    The results are for finite-horizon settings, and improvement over worst-case analysis in infinite-horizon settings remains an open question.

Abstract

from arXiv · show

Strong worst-case performance bounds for episodic reinforcement learning exist but fortunately in practice RL algorithms perform much better than such bounds would predict. Algorithms and theory that provide strong problem-dependent bounds could help illuminate the key features of what makes a RL problem hard and reduce the barrier to using RL algorithms in practice. As a step towards this we derive an algorithm for finite horizon discrete MDPs and associated analysis that both yields state-of-the art worst-case regret bounds in the dominant terms and yields substantially tighter bounds if the RL environment has small environmental norm, which is a function of the variance of the next-state value functions. An important benefit of our algorithmic is that it does not require apriori knowledge of a bound on the environmental norm. As a result of our analysis, we also help address an open learning theory question~\cite{jiang2018open} about episodic MDPs with a constant upper-bound on the sum of rewards, providing a regret bound with no $H$-dependence in the leading term that scales a polynomial function of the number of episodes.

1. Introduction

The paper targets reinforcement-learning methods whose worst-case guarantees overlook easier problem instances. It introduces an episodic finite-horizon algorithm designed to retain worst-case performance while exploiting low environmental norm and related structure.

  • Motivation: Problem-dependent bounds aim to explain why practical RL performance often exceeds problem-independent worst-case guarantees.They seek bounds that adapt to whether a decision process is inherently easier or harder to learn.
  • Contributions: The paper derives an algorithm and analysis for finite-horizon discrete MDPs with state-of-the-art worst-case regret bounds in the dominant terms.The approach is intended to match worst-case solutions on difficult environments.
  • Contributions: The analysis yields substantially tighter guarantees when next-state value-function variance, or environmental norm, is small.This characterizes environmental structure relevant to RL difficulty.
  • Contributions: Low environmental norm includes deterministic domains, single-goal MDPs, and highly stochastic domains.These classes are identified as important examples where the problem-dependent analysis can improve guarantees.
  • Open question: For the Jiang–Agarwal setting, the resulting regret bound has no planning-horizon dependence in its dominant terms.The paper presents this as progress on an open learning-theory question about episodic MDPs.

2. Preliminaries and Definitions

The paper studies undiscounted finite-horizon MDPs over repeated fixed-length episodes. It defines value functions, transition and reward estimates, and notation for high-probability regret analysis.

  • MDP setting: An episodic finite-horizon MDP is specified by state and action spaces, transition probabilities, rewards, and horizon H.The reward function takes values in [0,1], and the agent interacts through episodes of length H.
  • Value functions: The agent selects a policy in each episode, while each policy induces a value function for every state and timestep.The optimal policy and value function provide the benchmark for regret.
  • Estimation: The analysis uses visit counts together with maximum-likelihood estimates of transitions and rewards.These quantities support the learning-performance and confidence analysis.
  • Notation: The paper expresses guarantees using high-probability notation and suppresses polynomial logarithmic factors with tilde notation.The notation also uses approximate inequality symbols for bounds up to numerical constants.

3. EULER

EULER is an optimism-based algorithm for episodic tabular reinforcement learning. It defines the environmental norm through maximum conditional variance of next-state values and estimates this quantity empirically.

  • Environmental norm: Q* is defined as the maximum per-step conditional variance of next-state optimal values and rewards.The paper identifies this quantity with the environmental norm while emphasizing its conditional-variance interpretation.
  • Algorithm: EULER uses optimism under uncertainty to explore without requiring prior knowledge of the environmental norm.Its exploration bonus combines an empirical Bernstein estimate with a correction for value-function uncertainty.
  • Algorithm: The correction term accounts explicitly for uncertainty in the value function while preserving optimism.The algorithm has the same computational complexity as value iteration.

4. Main Result

The main theorem gives EULER a high-probability regret bound expressed through maximum conditional variance and maximum return. The analysis explains how empirical Bernstein bonuses and value-function bounds produce problem-dependent guarantees while retaining worst-case performance.

  • Main theorem: Theorem 1 bounds EULER’s regret with high probability by the minimum of two problem-dependent expressions.The bounds are stated jointly over all episodes and apply for any T ≤ KH.
  • Problem-dependent bounds: The return-based bound is tighter than naively replacing Q* with G^2 by a factor of H.Both Q* and G can identify MDP instances with lower regret.
  • Worst-case guarantee: EULER matches minimax tabular episodic-RL regret bounds in the dominant worst-case term.The corollary states this guarantee for rewards bounded in [0,1].
  • Theoretical analysis: The exploration analysis decomposes the dominant bonus into a Bernstein contribution using true problem quantities and a faster-shrinking correction based on value-function bounds.The correction reweights uncertainty according to successor-state importance under estimated transitions.
  • Theoretical analysis: The algorithm’s bonus combines empirical conditional-variance estimation with a correction that accounts for inaccurate next-state value estimates.This construction connects computable optimistic estimates with problem-dependent regret analysis.
  • Problem-dependent bounds: The regret bounds depend directly on Q*, the maximum conditional variance of next-state values, or alternatively on the maximum return G.A law-of-total-variance argument provides the return-based form.

5. Horizon Dependence in Dominant Term

The paper addresses whether regret in episodic tabular MDPs must depend on the planning horizon when rewards are positive and their episode-wide sum is bounded. Its result gives a regret bound whose dominant term is horizon-independent while horizon dependence remains only in lower-order transient terms.

  • Motivation: Jiang and Agarwal’s open question asks whether episodic tabular RL regret lower bounds necessarily depend on planning horizon H under bounded total rewards.Their setting assumes positive rewards with total episode reward bounded by one.
  • Setting: Under the bounded-total-reward setting, the main result applies because the optimal value is bounded by G ≤ 1 and rewards lie in [0, 1].The setting implies 0 ≤ V^π∗_t(s) ≤ G ≤ 1, so the paper’s assumptions hold.
  • Result: The dominant regret term has no dependence on the planning horizon H and instead scales polynomially with the number of episodes K.H appears only in transient lower-order terms independent of K.
  • Interpretation: Up to logarithmic and transient terms, episodic regret is independent of H, partially answering Jiang and Agarwal’s open question.The result concerns the dominant regret behavior rather than complete elimination of all horizon dependence.
  • Algorithmic implication: EULER attains this improved bound without being given the domain-dependent maximum possible value function, despite using H as a common problem-agnostic bound.The algorithm therefore does not require advance knowledge of the relevant domain-specific value bound in this setting.

6. Problem dependent bounds

EULER provides problem-dependent regret guarantees for finite-horizon MDPs using next-state value variability and value-function ranges, without requiring those environmental quantities in advance. The bounds improve substantially in deterministic, highly mixing, and other low-variability domains while retaining worst-case performance.

  • General problem-dependent bounds: EULER obtains tighter guarantees when the next-state distribution induces small variance in the optimal value function or when achievable rewards have a small bound.The same algorithm is used without being informed of the MDP’s Q∗ or G values.
  • 6.1. Bounds using the range of optimal value function: A value-range bound Φ can substantially reduce regret, but prior methods require Φ ≥ rng V^π∗ as an input and are sensitive to choosing it too high or too low.A too-high Φ increases regret, while a too-low value can cause algorithm failure.
  • 6.1. Bounds using the range of optimal value function: EULER achieves at least comparable and potentially much better episodic performance without knowing the optimal value-function range.This follows from the paper’s main regret bound after relating the environmental norm to the value range.
  • 6.1. Bounds using the range of optimal value function: Φ_succ is no larger than the global value range and the full optimal-value range: Φ ≥ rng V^π∗ ≥ Φ_succ.Φ_succ restricts the range to immediate successor states, which can make it substantially smaller.
  • 6.2. Bounds on the next-state variance of V^π∗ and empirical benchmarks: The chain-MDP analysis gives a bound ˜O(NAK + ...) that is substantially tighter than the worst-case ˜O(N^3AK + ...) for large K.This contrasts with the chain’s reputation as a hard exploration domain for naive strategies such as ε-greedy.
  • 6.3. Stochasticity in the system dynamics: For deterministic MDPs, Q∗ = 0 and EULER’s regret has logarithmic dependence on T; a refined analysis gives ˜O(SAH^2).This improves over earlier problem-independent bounds with at least linear T dependence, though the refined result retains a horizon factor.

7. Related Literature

The paper situates EULER among problem-dependent RL analyses, value-function-approximation approaches, and infinite-horizon methods. Its distinction is a finite-horizon analysis that leverages value-function uncertainty without requiring domain-specific bounds as inputs.

  • Scope comparison: Infinite-horizon analyses generally do not improve over worst-case guarantees in the cited setting, and whether such improvement is possible remains open.The paper’s stronger results apply to finite-horizon RL.
  • Gap-dependent analyses: Gap-dependent bounds characterize hardness through separations between optimal and suboptimal actions or policies, providing an alternative structure to value-variance measures.The cited examples are infinite-horizon results based on state-action or policy gaps.
  • Value-function approximation: EULER differs from value-function-approximation methods based on Eluder dimension or Bellman rank, which measure different notions of hardness and do not match tabular lower bounds.The comparison concerns finite-horizon settings with structured function classes.
  • Infinite-horizon comparisons: Infinite-horizon methods can obtain tighter bounds from value-function variability, but often require assumptions or supplied bounds such as ergodicity or Φ ≥ rng V^π∗.These settings contain exploration complexities absent from the finite-horizon setting considered here.
  • Problem-dependent analyses: Prior finite-horizon algorithms used next-state value variance in analysis but reported worst-case results, whereas EULER uses distinct bonuses and point-wise value bounds.The paper emphasizes that deriving problem-dependent guarantees requires a nontrivial convergence analysis of empirical MDP quantities.

8. Future Work and Conclusion

EULER matches the best known worst-case regret guarantees while adapting to low value-function variance and bounded-reward structure without advance knowledge of those quantities. The paper identifies extensions to infinite-horizon, gap-dependent, and continuous-state settings as future directions.

  • Conclusion: EULER matches the best known worst-case regret guarantees for episodic finite MDPs while obtaining tighter bounds when next-state optimal-value variance is small.The algorithm does not need advance knowledge of the relevant MDP-specific quantities.
  • Conclusion: The paper reports low Q∗ for sparse-reward, near-deterministic, highly mixing, and selected benchmark MDPs, connecting the measure to practical problem classes.The environmental norm was also empirically correlated with RL complexity across several benchmark tasks.
  • Conclusion: The analysis partially answers Jiang and Agarwal’s open question by showing dominant regret can scale independently of the horizon in their bounded-total-reward setting.The conclusion concerns the leading behavior, not all transient terms.
  • Future Work: Future work includes problem-dependent bounds for infinite-horizon and continuous-state RL and incorporating gap-dependent analysis.These directions extend the paper’s problem-dependent perspective beyond its finite-horizon tabular setting.

A.1. Euler on Chain

On the chain MDP, EULER achieves unusually small regret because the optimal value is scaled to order 1 and successor-value variance decreases with the chain length. These properties remove factors from standard worst-case bounds and suggest pathological MDPs may be uncommon.

  • Chain MDP: The chain has N = H = S timesteps, a right-moving optimal policy, and episode reward 1.The transition probabilities scale the optimal value function to order 1.
  • Chain MDP: EULER’s regret on the chain is substantially smaller than typically reported bounds for this setting.The passage notes that lower-order terms may matter for small T, but the stated result follows from the general theorem.
  • Why regret shrinks: The optimal value is order 1 rather than N = H, while successor-value variance decreases as N increases.Together, these properties remove factors from the known worst-case regret bound.
  • Algorithmic requirement: EULER must control confidence intervals for both rewards and transition probabilities to achieve this performance.Reward-only Hoeffding-style concentration would introduce an additional Θ contribution to regret.
  • Interpretation: The result suggests that truly pathological MDP classes may be less common than expected.The discussion compares the example with the exponential exploration difficulty often associated with chain MDPs.
  • Scope: The same regret reasoning applies to any MDP class sharing the chain’s relevant value and variance properties.Thus, the particular figure-1 MDP may admit even smaller regret.

B. Average Per-Episode Sample Complexity for the Setting of (Jiang & Agarwal, 2018)

The regret bound can be converted into a high-probability episode count for achieving average per-episode error below ε in the setting studied by Jiang and Agarwal. This conversion is not itself a formal PAC guarantee without modifying the algorithm.

  • Average error: EULER’s regret divided by K yields a high-probability bound on average per-episode loss.The smallest K making this average error less than ε gives the corresponding episode requirement.
  • Horizon dependence: For small ε << SA, the first term dominates and has no polynomial dependence on H.This addresses the horizon-dependence question for the stated setting.
  • PAC scope: A formal PAC result would require modifying the algorithm so exploration bonuses do not grow with T for unvisited state-action pairs.The proposed adjustment replaces log(T) with a visit-count term such as log(n), with suitable constants.

C. Appendix Overview and Proof Preview

The appendix previews EULER’s proof strategy: construct admissible concentration bonuses, maintain optimistic and pessimistic value bounds, and decompose regret into reward and transition terms. The analysis uses Bernstein concentration and connects value-function uncertainty to expected state-action visitation.

  • Proof framework: EULER’s general analysis first defines concentration inequalities for transition dynamics and proves strong problem-dependent regret under admissibility assumptions.The main-text result follows as a corollary of this broader framework.
  • Value bounds: EULER maintains upper and lower estimates of the optimal value function to construct computable correction bonuses.The bonus uses empirical transitions, value bounds, visit counts, and known constants.
  • Optimism: The analysis proves optimism and maintains a valid pointwise bracket for the optimal value function.These guarantees are established through bonus properties and induction arguments.
  • Confidence intervals: The value-function confidence interval is related to uncertainty in states encountered under the optimistic policy, weighted by true visitation probabilities.This connects an algorithmic quantity—the optimistic-pessimistic gap—to environmental uncertainty.
  • Regret decomposition: Regret is decomposed into reward estimation, transition dynamics estimation, and transition dynamics optimism terms.The transition dynamics optimism term is identified as a leading term requiring a problem-dependent analysis.
  • Concentration: Bernstein’s inequality satisfies the admissibility assumptions required by EULER, yielding a practical instantiation of the analysis.The appendix verifies both stated assumptions.

E.2. Transition Dynamics

The transition-dynamics analysis constructs a bonus from empirical transitions and upper/lower value bounds, proving that it overestimates the relevant concentration quantity. This supports optimism, value bracketing, and problem-dependent regret control through visitation-weighted uncertainty.

  • Bonus guarantee: Proposition 3 shows that the constructed transition bonus overestimates the admissible concentration interval under its stated condition.This property is the key ingredient for guaranteeing optimism.
  • Value bracket: The resulting induction establishes optimism and pessimism, maintaining a valid bracket around the optimal value function for all timesteps and episodes.The proof proceeds outside the failure event.
  • Visitation analysis: The good-set construction separates frequently visited state-action pairs from insufficiently visited pairs for the regret analysis.Frequently visited pairs can be related to expected visitation, while the remaining pairs contribute little.

H.1. Main Result

The main result gives EULER a high-probability regret bound expressed as the minimum of multiple forms, including problem-dependent quantities and a worst-case guarantee. The analysis specializes these bounds to Bernstein estimation and deterministic domains.

  • Main Result: Theorem 2 bounds EULER’s cumulative regret with high probability by the minimum between alternative expressions.The theorem is stated for admissible confidence parameters and is developed through separate reward-estimation, transition-estimation, and transition-optimism terms.
  • Main Result: The regret analysis uses problem-dependent upper bounds C∗ and Cπ on quantities arising in the regret decomposition.The proof re-expresses regret through visitation probabilities and bounds the resulting terms with separate lemmas.
  • Bernstein Specialization: Bernstein’s inequality yields an explicit problem-independent regret guarantee for EULER through Theorem 2 and a recursive law-of-total-variance argument.The resulting proposition states a high-probability bound as the minimum between alternative terms.
  • Bernstein Specialization: The Bernstein analysis also produces a problem-dependent bound by relating the regret terms to reward and next-state value-function variance.The proof identifies independence of sampled rewards and transitions conditional on a state-action pair as part of the argument.
  • Deterministic Domains: In deterministic MDPs, EULER’s regret is bounded by ˜O(SAH2).The proof uses exact transition estimates after state-action pairs have been sufficiently visited and bounds the exceptional episodes separately.
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