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Non-Asymptotic Gap-Dependent Regret Bounds for Tabular MDPs

Max Simchowitz, Kevin Jamieson

arXiv:1905.03814v2cs.LGmath.OCmath.STstat.ML

TL;DR

The paper addresses gap-dependent regret for episodic tabular MDPs without relying on diameter or ergodicity assumptions. It develops an approach extending to model-based optimistic algorithms and establishes logarithmic gap-dependent bounds, while leaving some horizon dependence and extensions unresolved.

  • Problem

    Previous gap-dependent analyses appeal to diameter or ergodicity assumptions to support reasoning about future-state value functions, while existing bounds can be overly pessimistic about problem complexity.

  • Method

    The paper develops a new approach for logarithmic gap-dependent bounds that extends to model-based optimistic algorithms, analyzing StrongEuler through optimistic value estimates and confidence bonuses.

  • Results

    The analysis gives StrongEuler a gap-dependent regret bound that scales logarithmically with 1/δ and applies to episodic tabular MDPs.

  • Takeaways & Limitations

    The approach provides logarithmic gap-dependent guarantees for a broad family of model-based optimistic algorithms without the prior analyses’ stated reliance on diameter or ergodicity assumptions.

  • Takeaways & Limitations

    The bounds retain dependence on the smallest nonzero gap and an almost gap-independent burn-in term, while optimal horizon dependence remains unresolved.

Abstract

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This paper establishes that optimistic algorithms attain gap-dependent and non-asymptotic logarithmic regret for episodic MDPs. In contrast to prior work, our bounds do not suffer a dependence on diameter-like quantities or ergodicity, and smoothly interpolate between the gap dependent logarithmic-regret, and the $\widetilde{\mathcal{O}}(\sqrt{HSAT})$-minimax rate. The key technique in our analysis is a novel "clipped" regret decomposition which applies to a broad family of recent optimistic algorithms for episodic MDPs.

1 Introduction

The paper seeks finite-time, gap-dependent regret bounds for tabular episodic MDPs that exploit favorable instances without relying on diameter-like or ergodicity quantities. It develops a clipped regret analysis for optimistic algorithms, yielding logarithmic regret that interpolates with minimax performance.

  • Motivation: Prior minimax bounds capture worst-case dependence on states, actions, and horizon but can be overly pessimistic about favorable problem structure.Earlier gap-dependent analyses improve to C log T but are asymptotic and may depend on conservative hitting-time or ergodicity quantities.
  • Contribution: The paper demonstrates that optimism-based algorithms can attain gap-dependent, problem-specific regret bounds in finite-time episodic MDPs.The analysis specializes to StrongEuler and extends more broadly to other optimistic algorithms.
  • Main results: StrongEuler has a high-probability bound of CM log(1/δ) for all T ≥1, with CM depending on sub-optimality gaps, horizon length, and an almost-gap-independent polynomial term.Choosing δ = 1/T yields a non-asymptotic expected regret bound scaling as CM log T.
  • Interpretation: The bounds avoid worst-case dependence on mixing times, hitting times, and ergodicity measures while exploiting benign problem instances.The approach establishes logarithmic regret after a burn-in polynomial in S, A, H, and the gaps, and smoothly connects to the minimax HSAT rate.
  • Main results: The regret interpolates between instance-dependent CM log T and minimax e O(√(HSAT)), with the minimax rate potentially sharper for smaller T.For benign instances, the dependence on H may also be refined.
  • Technical contribution: The clipped regret decomposition applies to a broad family of optimistic algorithms and supports analogous gap-dependent logarithmic-regret bounds beyond StrongEuler.The paper identifies this decomposition as its main technical contribution and notes that the subsequent analysis is tailored to model-based methods.

2 Main Results

The paper gives StrongEuler finite-time, gap-dependent regret guarantees for episodic MDPs, with an analysis template that extends to other optimistic algorithms. The results include logarithmic behavior, favorable horizon dependence in special settings, matching lower bounds for key terms, and interpolation toward gap-independent regimes.

  • Horizon dependence: For contextual bandits and bounded-reward MDPs, the regret bound improves its dependence on the horizon H, with contextual bandits eliminating H from the first term.For contextual bandits, the first term coincides with known asymptotically optimal instance-specific regret.
  • StrongEuler guarantees: The gap-dependent term sums contributions from suboptimal state-action pairs, and a lower bound shows this dependence is unavoidable in general.The lower bound applies when the total number of rounds is large and constructs an instance for every possible choice of gaps.
  • Horizon dependence: The optimal horizon dependence remains unresolved: H^2 is expected from infinite-horizon results, but H^3 may be unavoidable for non-asymptotic bounds.The paper leaves resolution of the optimal H dependence to future work.
  • Algorithmic scope: The gap-clipping analysis extends beyond StrongEuler to other recent optimistic algorithms, although some require modified confidence intervals to obtain logarithmic rather than log^2 T regret.The analysis is specialized to StrongEuler for sharper H-dependence but is presented as broadly applicable.
  • Interpolation and lower bounds: The main bound interpolates between gap-dependent logarithmic regret and gap-independent regimes, while lower bounds show an |S|/gapmin dependence can be unavoidable for optimistic algorithms.An explicit construction demonstrates over-exploration from optimism, with one small-gap state and all other states having gaps at least 1/2.

3 Gap-Dependent bounds via ‘clipping’

The paper introduces a clipped regret decomposition that combines surplus-based minimax analysis with gap-dependent control, yielding finite-time logarithmic bounds for optimistic episodic-MDP algorithms. The technique handles future surpluses recursively, applies even at optimal state-action pairs, and avoids exponential horizon dependence.

  • Optimistic algorithms: Strong optimism requires E_k,h(x, a) ≥ 0 for every episode, state-action pair, and stage, and supports sharper horizon dependence in contextual bandits.The surplus framework is intended to extend beyond StrongEuler once an analogous surplus bound is established.
  • Clipping trick: The clipping trick merges surplus-based minimax analysis with gap-dependent control of suboptimal-action selections.It clips current surpluses at gap-dependent thresholds and transfers remaining contribution to future surpluses.
  • Bounds: The resulting analysis gives gap-dependent logarithmic regret and interpolates with minimax behavior, with improved horizon dependence in benign settings.For contextual bandits, the variance and effective-horizon factors can be replaced by 1; bounded cumulative rewards also improve the variance dependence.
  • Clipping trick: A suboptimal action is selected only when its current surplus or expected future surplus exceeds a gap-dependent threshold.This separates insufficient visitation of the current pair from insufficient visitation of future state-action pairs.
  • Clipping trick: The recursive clipping argument incurs a factor of at most (1 + O(...))^H ≲ 1, avoiding exponential dependence on H.The displayed expression is partially fragmented, but the stated conclusion is that the horizon factor remains controlled.
  • Clipping trick: Half-clipping enables surplus clipping at state-action pairs whose actions are optimal, a step required for the main clipped regret decomposition.The proof first controls suboptimality and then subtracts gapmin/(2H) from surplus terms, at most halving the suboptimality.
  • Limitations: The half-clipping step is coarse and produces an S/gapmin term that cannot be significantly improved for existing optimistic algorithms in general.The paper attributes this dependence to a lower-bound limitation for the analyzed algorithm class.

4 Conclusion

The paper presents logarithmic gap-dependent regret bounds for episodic, non-generative tabular MDPs and extends the analysis to a broad class of model-based optimistic algorithms. It identifies unresolved limitations involving model-free methods, horizon dependence, and smallest-gap dependence.

  • Conclusion: The paper proposes logarithmic, gap-dependent bounds for tabular MDPs in the episodic, non-generative setting.The approach is intended for a common class of optimism-based algorithms.
  • Conclusion: The clipped regret decomposition provides a modular recipe for analogous guarantees across existing model-based optimistic algorithms.The appendix describes the tools as modular and reusable for related settings.
  • Conclusion: Extending the bounds to model-free Q-learning approaches and resolving the optimal horizon dependence remain future work.These are explicitly identified as open directions in the conclusion.
  • Conclusion: Existing optimistic algorithms necessarily incur an additional S/gapmin dependence on worst-case instances.The paper suggests that removing this dependence will require new algorithmic ideas.
  • Conclusion: The paper also proposes extending logarithmic bounds to infinite-arm settings and adaptive discretization with embedded state-action pairs.These directions are presented as possible applications of the techniques.

B Precise Statement and Rigorous Proof Sketch of Main Regret Bounds

The appendix develops precise regret statements and a proof architecture that combines a granular clipping bound with StrongEuler surplus bounds. Its modular tools are intended to support analogous analyses for other model-based optimistic algorithms.

  • Proof strategy: The appendix refines the main regret bounds using problem-dependent quantities and organizes their complete proofs.It introduces the refined statements before presenting the technical lemmas and algorithm-specific surplus analysis.
  • Proof strategy: The proof combines Theorem B.3’s granular clipping bound with Proposition B.4’s StrongEuler surplus bound.Lemma B.6 combines these ingredients into a form used for the upper-bound proof.
  • Proof strategy: The resulting arguments establish both the logarithmic-regret result and its interpolation with the minimax regime.The appendix modifies the proof to obtain the interpolation theorem after proving the logarithmic bound.
  • Scope: The section’s tools are presented as a general recipe for similar regret bounds for existing model-based optimistic algorithms.The authors describe the proof components as modular enough to be borrowed in related settings.

B.1 More Precise Statement of Regret Bound Theorem 2.4

The precise results bound StrongEuler’s regret with problem-dependent variance, gap, transition-suboptimality, and effective-horizon factors. They yield logarithmic behavior in favorable instances and interpolate toward minimax rates, while retaining explicit smallest-gap and burn-in limitations.

  • Main bounds: StrongEuler’s refined bounds depend on variance terms, minimal gaps, optimal and suboptimal state-action sets, and an effective horizon.The theorem statements use these quantities to express instance-sensitive regret guarantees.
  • Benign settings: Contextual-bandit instances replace Var and H_T by 1, while bounded cumulative rewards can also replace Var by 1.These substitutions yield sharper horizon dependence in the corresponding benign settings.
  • Interpolation: Theorem B.2 interpolates between logarithmic-regret and square-root-T regimes under transition-suboptimality assumptions.The bound applies when each tuple is α-transition suboptimal, a condition describing multiplicative closeness of transitions to an optimal action’s transition distribution.
  • Problem dependence: The number of optimal state-action pairs can be as large as H|S|, but is often bounded by |S| in structured environments.This quantity directly affects the optimal-pair contribution to the regret bound.
  • Limitations: The bounds retain a smallest-gap dependence and an almost gap-independent burn-in term, limiting their behavior on difficult instances.The paper identifies the S/gapmin dependence as unavoidable for existing optimistic algorithms in worst-case settings.

B.2 Rigorous proof of upper bounds: Preliminaries

This section develops the formal ingredients for proving StrongEuler’s regret bounds, including surplus control, strong optimism, and clipped regret decomposition. The analysis tracks lead and future terms, with variance dependence supporting the minimax behavior.

  • Clipped regret decomposition: The formal proof assumes a common transition-suboptimality parameter α for simplicity, although the bound can be refined to allow variation across tuples.The contextual-bandit case specializes to zero transition suboptimality.
  • Surplus and optimism: Proposition B.4 establishes a high-probability surplus bound for StrongEuler on event Aconc.The event has probability at least 1 − δ/2 and the bound holds for all states, actions, stages, and episodes.
  • Surplus and optimism: The surplus analysis separates a lead term from future terms generated by rollouts of πk.The lead term drives the dominant regret factors, while future terms contribute to the lower-order regret component.
  • Surplus and optimism: The lead term decays more slowly than the future terms and therefore governs the nearly gap-free portion of regret.The comparison is stated in terms of their respective decay rates.
  • Variance quantities: Policy-dependent variance quantities are crucial for obtaining the correct minimax dependence.The lead term depends on the minimum variance of the optimal value function, while the analysis also tracks policy-dependent variance.
  • Clipped regret decomposition: The clipped regret decomposition bounds regret by separately clipping lead and future contributions at gap-dependent thresholds.A technical lemma distributes clipping across sums, and the resulting bound applies to strongly optimistic algorithms, with α replaced by 1 for merely optimistic algorithms.

B.3 Proof of Corollary B.1: A proof via integration

The proof of Corollary B.1 converts local clipped regret terms into discrete integrals over visitation counts. Concentration relates empirical counts to visitation probabilities, and integration computations complete the logarithmic bound.

  • From local terms to integrals: Lemma B.6 expresses regret as a sum of local lead and future terms indexed by state-action visits.The terms are later analyzed as functions of cumulative sample counts.
  • Sampling concentration: A sampling event ensures empirical counts are sufficiently large relative to their expected visitation counts with total probability 1 − δ.The concentration event Aconc and sampling event Esamp jointly provide the stated probability guarantee.
  • From local terms to integrals: The proof views the regret sums as discrete integrals of non-increasing lead and future functions of visitation counts.The relation nk(x,a)/4 = Σj≤k ωj,h(x,a) connects cumulative counts to visitation probabilities.
  • Integral conversion: Lemma B.8 converts sums of non-increasing functions of counts into integral bounds under the sampling event.The conversion applies to functions bounded above by fmax and includes clipping thresholds εx,a,h.
  • Integral evaluation: Lemma B.9 evaluates the resulting integrals for clipped, logarithmically decaying functions and related cases.The computations accommodate both lead and future terms and remain valid under constant rescaling of the integration argument.
  • Anytime bounds: Using confidence intervals union-bounded over all T incurs log T · log log T dependence and therefore does not yield strictly O(log T) regret.This is contrasted with anytime estimates used in the preferred analysis.
  • Conclusion: Using these integral evaluations in the lead and future terms completes the proof of Corollary B.1.The paper explicitly concludes the proof after substituting the two bounds into the regret decomposition.

B.4 Proof of Theorem B.2

Theorem B.2 is proved by specializing the integration framework to variance-based and H_T-based bounds. The resulting analysis preserves the intended interpolation across regret regimes.

  • Interpolation: The proof of Theorem B.2 combines the preceding clipped decomposition and integration machinery to derive an interpolating regret bound.The section explicitly presents the theorem as interpolating between regimes.
  • Gap regimes: The proof separately tracks state-action pairs with small gaps through Zsub(ε) and the quantity H_T.The decomposition is used to organize the T and log T regimes.
  • Conclusion: The final technical inequalities bound both clipped lead and future contributions after discarding clipping, and summing over episodes completes the argument.The same form is used for both inequalities before the episode-wise summation.
  • Variance bound: The variance-based bound controls the integrated lead terms through a Cauchy–Schwarz argument and a logarithmic factor.The displayed intermediate bound contains Var, nK(x,a), and log(TM/δ).
  • H_T bound: The H_T-based bound requires a separate Cauchy–Schwarz integration argument rather than directly reusing the Corollary B.1 recipe.The proof introduces f(u) = 1/max{u,1} and applies a dedicated integration lemma.
  • H_T bound: The H_T analysis absorbs the initial sampling contribution into terms already present in the final bound.The initial contribution is bounded using |Z0|Hsamplefmax and is stated to be of order SAH^2 log(M/δ).

C Proof of Technical Lemmas

This section supplies the technical concentration, integration, inversion, and clipping arguments used by the regret proofs. It converts visitation-weighted sums into tractable bounds for clipped functions and verifies the required sampling events.

  • Clipped surplus control: The proof uses strong optimism to apply the clipped surplus bound to lead and future terms.The resulting expressions contain clipped surpluses evaluated at transition-dependent gap thresholds.
  • Future-term expansion: Future contributions are expanded over successor state-action pairs weighted by conditional visitation probabilities.The number of such terms is bounded by SAH before applying the clipping-distribution lemma.
  • Future-term expansion: Clipping is moved through weighted sums using α ≤ 1 and probability weights bounded by one.These inequalities preserve the required upper bound on the expanded future terms.
  • Sampling event: The sampling event is obtained from a martingale concentration result and implies the event used by the integral conversion argument.The construction sets Hsample using H, S, A, and δ and establishes the needed lower bound on counts after τ(x,a).
  • Integral conversion: An integration lemma bounds sums of visitation probabilities multiplied by non-increasing functions of cumulative counts.The proof constructs a step function whose integral tracks the count process and uses monotonicity.
  • Integral evaluation: The technical bounds cover both single-logarithmic and two-factor forms, with constants controlled through assumptions on log C and log(CC′).The special case with C′ = 1 yields the form g(u) ≲ C log(Mu/δ)/u.
  • Integral evaluation: The general integration computations handle functions bounded by clipped non-increasing envelopes, including the ε = 0 case.The proof uses an inversion lemma for eL(u)=log(Mu/δ) to locate the effective endpoint of the clipped region.

D Proof of ‘clipping’ bound: Proposition 3.1 / Theorem B.3

The proof establishes a clipped regret decomposition by showing that positive gaps force either sufficiently large clipped surpluses or compensating value-function differences. Half-clipping and gap clipping then yield the desired bound for optimistic algorithms, with α=1 when strong optimism is unavailable.

  • Fundamental gap bound: The central lemma shows that a positive gap implies either a large surplus or a large expected next-stage value-function difference.This dichotomy drives the clipped analysis.
  • Half-clipping: The half-clipped surplus clips Ek,h(x, a) at ϵclip = gapmin/(2H) and defines a corresponding half-clipped MDP.After clipping, πk need not remain optimal, so the proof compares clipped and optimistic value functions.
  • Gap clipping: Gap clipping further clips bonuses for suboptimal actions and bounds the resulting fully-clipped surpluses.The bound applies under strong optimism with transition-suboptimality parameters αx,a,h, and with αx,a,h = 1 for merely optimistic algorithms.
  • Conclusion: The final argument combines the clipped-surplus bounds to establish Theorem B.3, including the non-strongly-optimistic case with α = 1.The proof concludes by summing visitation-weighted clipped surpluses.
  • Trajectory decomposition: The proof partitions trajectories according to the first stage at which the policy selects a suboptimal action and bounds the resulting suboptimality gap through clipped surpluses.The events distinguish whether the policy disagrees with an optimal action at each stage.

E The StrongEuler Algorithm

StrongEuler modifies EULER with count-dependent confidence logarithms and strong optimism. These changes support logarithmic gap-dependent regret while preserving sharper behavior in settings such as contextual bandits.

  • Algorithm design: StrongEuler modifies EULER in two ways: count-dependent logarithmic factors and strong optimism.The algorithm is presented as a refinement of EULER rather than a wholly separate optimistic-value-iteration framework.
  • Algorithm design: Its bonus logarithms depend on nk(x, a), using L(nk(x, a)) proportional to log(Mnk(x, a)/δ), rather than the total time T.The passage identifies this refinement as necessary for optimal log T regret.
  • Algorithm design: Strong optimism is not required for gap-dependent bounds, but can sharpen bounds when transition dynamics are simple, including contextual bandits.EULER and related algorithms also obtain gap-dependent bounds under the clipping analysis.
  • Algorithm operation: Each rollout records reward, transition, count, and variance information, which is then used to construct exploration bonuses.The rollout and bonus procedures implement optimistic value iteration with variance-aware quantities.
  • Analysis setup: The algorithm assumes a good concentration event and analyzes strong optimism through lower and upper bounds on the surplus.The section separates establishing nonnegative surpluses from controlling their complexity.

F.1 Proof of Optimism

The optimism proof shows, on a high-probability concentration event, that StrongEuler’s Q-values dominate the optimal Q-values and its surpluses are nonnegative. The argument proceeds by backward induction using reward, transition, variance, and value-function confidence bounds.

  • Optimism guarantee: Under Aconc, StrongEuler is optimistic: Qk,h(x, a) ≥ Q⋆h(x, a), and its policy selects an action maximizing Qk,h.The proposition states optimism across all stages.
  • Proof strategy: The proof establishes these properties by backward induction on h, using the terminal equality at h = H + 1 and confidence bounds for rewards and transitions.The induction compares empirical and true transition terms against optimistic value functions.
  • Confidence analysis: Variance-sensitive bonuses control uncertainty in rewards, empirical transitions, and value-function differences.The analysis uses norm and variance inequalities to relate empirical and true transition quantities.
  • Confidence analysis: The concentration event Aconc is an intersection of reward, transition, value, and variance events occurring with probability at least 1 − δ/2.The constituent bounds are obtained using Bernstein, Hoeffding, and variance-concentration arguments with count-dependent logarithms.

G.4.1 Proof of Proposition G.3

The lower-bound construction shows that optimistic algorithms can incur substantial regret even when only one state has a small gap and the remaining gaps are large. The proof reduces the behavior to repeated two-arm bandit problems and uses adaptive-sampling arguments to establish the result.

  • Proof reduction: Conditioned on visitation counts, the proof decomposes the MDP into S independent two-arm bandit games with nk(x) rounds.This reduction provides the probabilistic structure used to lower-bound optimistic estimates.
  • Proof reduction: The analysis handles adaptive action sampling by representing rewards as a pre-sampled i.i.d. sequence and controlling empirical means through several high-probability events.The argument combines count bounds, maximal inequalities, and a symmetry event.

H.3 Proof of Equation (30)

The proof lower-bounds Equation (30) using a change-of-measure argument between two episodic MDPs that differ only at one state-action pair. It controls the resulting KL divergence by relating the transition laws to Bernoulli distributions and then applies uniform-goodness assumptions to obtain the required bound.

  • Change of measure: The proof follows standard information-theoretic lower-bound steps and introduces an MDP analogue of a binary-KL change-of-measure lemma.The lemma compares two MDPs through the laws of next-state and reward observations and the filtration generated by rollouts.
  • MDP construction: For a fixed state-action pair, the alternate MDP M′ differs from M only in its observation law at that pair.This construction isolates the contribution of the selected pair to the divergence between the two MDP-induced histories.
  • KL control: A universal-constant KL bound controls the divergence between the local observation laws of M and M′.The proof states this as Claim H.2 before applying the change-of-measure argument.
  • KL control: The transition laws at the selected pair reduce to Bernoulli distributions with parameters 3/4 − ∆x,a and 3/4 + η, respectively.The proof then bounds their binary KL divergence by a constant multiple of ∆x,a^2 using a Taylor expansion under the stated parameter range.
  • Conclusion: For sufficiently large K, uniform goodness yields bounds on the expectation of the constructed random variable under M′ and M, completing the lower-bound argument.The proof also verifies that the constructed variable lies in [0,1] with probability one and uses the selected pair’s visitation probabilities.
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