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Unifying PAC and Regret: Uniform PAC Bounds for Episodic Reinforcement Learning

Christoph Dann, Tor Lattimore, Emma Brunskill

arXiv:1703.07710v3cs.LGcs.AIstat.ML

TL;DR

RL needs guarantees that remain reliable across domains, especially in high-stakes applications, but classical PAC and regret criteria leave complementary gaps. The paper introduces Uniform-PAC and shows that UBEV achieves Uniform-PAC guarantees with near-optimal PAC and high-probability regret bounds for episodic MDPs.

  • Problem

    Classical PAC and regret guarantees provide incomplete performance assessments, motivating a unified criterion for RL algorithms in high-stakes settings.

  • Method

    The paper defines Uniform-PAC, requiring simultaneous high-probability ε-optimality control for all ε > 0, and develops the optimistic episodic-RL algorithm UBEV using time-uniform LIL confidence bounds.

  • Results

    UBEV is Uniform-PAC, with PAC bounds never worse than ˜O(S2AH4/ε2), improving on MBIE by H2 and tightening the horizon dependence over MoRMax for small ε.

  • Takeaways & Limitations

    Uniform-PAC bridges PAC and high-probability regret criteria, making UBEV the first algorithm reported to satisfy PAC and sublinear, nearly optimal regret guarantees.

  • Takeaways & Limitations

    Regret alone can permit infinitely many mistakes, while PAC alone can permit errors larger than ε/2 on every episode.

Abstract

from arXiv · show

Statistical performance bounds for reinforcement learning (RL) algorithms can be critical for high-stakes applications like healthcare. This paper introduces a new framework for theoretically measuring the performance of such algorithms called Uniform-PAC, which is a strengthening of the classical Probably Approximately Correct (PAC) framework. In contrast to the PAC framework, the uniform version may be used to derive high probability regret guarantees and so forms a bridge between the two setups that has been missing in the literature. We demonstrate the benefits of the new framework for finite-state episodic MDPs with a new algorithm that is Uniform-PAC and simultaneously achieves optimal regret and PAC guarantees except for a factor of the horizon.

1 Introduction

The paper introduces Uniform-PAC, a performance framework that simultaneously controls ε-errors for every accuracy level and connects PAC guarantees with high-probability regret. It proposes UBEV, an episodic RL algorithm using time-uniform concentration bounds to obtain near-optimal PAC and regret guarantees.

  • Contribution: Uniform-PAC simultaneously bounds the number of ε-errors for every ε > 0, unlike conventional PAC or regret guarantees.The framework is intended to support convergence to an optimal policy with high probability while yielding both PAC and high-probability regret bounds.
  • Motivation: PAC and regret guarantees alone need not ensure high-probability convergence to an optimal policy.The paper notes that small regret can coexist with infinitely many maximally suboptimal episodes.
  • Algorithm: UBEV is a new optimistic episodic RL algorithm that is Uniform-PAC and uses law-of-iterated-logarithm confidence intervals uniform over time.These intervals avoid horizon-dependent confidence levels and more tightly control failure events in which the algorithm behaves poorly.
  • Guarantees: UBEV achieves near-optimal dependence on the episode horizon and optimal dependence on state-space size, action-space size, and episode count.Its PAC dependence is linear in the state space, while its regret dependence is square-root in the state space.
  • Setting: The setting is finite-state episodic MDPs with time-dependent dynamics, where each episode contains H time steps and policies may depend on within-episode time.The paper defines performance using expected return and episode-level optimality gaps and regret.

2 Uniform PAC and Existing Learning Frameworks

Uniform-PAC strengthens PAC by controlling ε-errors simultaneously, providing a bridge to high-probability regret while avoiding limitations of either framework alone. The paper formalizes these relationships and shows that Uniform-PAC yields convergence and strong regret guarantees.

  • Uniform-PAC framework: Uniform-PAC bounds control the number of ε-errors simultaneously for every ε > 0, unlike conventional PAC bounds.They are defined through a polynomial bound on the number of errors at all accuracy levels.
  • Limitations of existing frameworks: Regret bounds constrain cumulative error but may permit infinitely many severe mistakes, including infinitely many worst-possible treatments.This limitation matters in high-stakes settings such as drug-treatment optimization.
  • Limitations of existing frameworks: PAC bounds constrain errors at one fixed ε but may permit persistent errors above ε/2 after learning stops.The same issue affects TCE and KWIK guarantees.
  • Uniform-PAC consequences: Uniform-PAC implies PAC for every ε and, in episodic MDPs, supports high-probability regret and convergence to optimal policies.Theorem 3 gives convergence with probability at least 1 − δ and a regret form ˜O(√(C2T) + max{C1, C2}).
  • Relationships between guarantees: A Θ(1/ε^2) PAC guarantee converts only to a high-probability regret bound with rate Ω(T^2/3) in the worst case.The paper also gives an MDP and algorithm attaining regret Ω(T^2/3) while satisfying that PAC bound.
  • Relationships between guarantees: High-probability regret does not generally imply finite PAC guarantees, and uniform high-probability regret can coexist with infinitely many small-accuracy mistakes.The paper also separates uniform high-probability regret from expected regret through a linear-regret construction.

3 The UBEV Algorithm

UBEV is an optimistic episodic-RL algorithm built around time-uniform LIL confidence intervals and value-aware concentration. These choices support faster learning while yielding Uniform-PAC guarantees with near-optimal PAC and regret dependence.

  • Algorithm design: UBEV computes an optimistic policy by backward induction using confidence intervals for transition probabilities and value-related quantities.Its optimistic Q estimates combine empirical next-state values and rewards with confidence bonuses.
  • Algorithm design: UBEV incorporates the value function directly into concentration analysis instead of bounding only the transition kernel.This saves a factor of S in sample complexity, at the cost of a more difficult analysis.
  • Algorithm design: UBEV updates its policy after every episode, allowing it to use new observations immediately rather than updating only when samples double.The comparison concerns algorithms that otherwise update less frequently and only finitely often.
  • Confidence bounds: UBEV uses time-uniform LIL confidence intervals, whose width scales as (log log n)/n rather than the log(k)/n rate used by several existing methods.The LIL intervals avoid horizon-dependent confidence levels and hold uniformly over time.

4 Uniform PAC Analysis

UBEV combines Uniform-PAC guarantees with near-optimal PAC and high-probability regret bounds for episodic reinforcement learning. Its analysis uses time-uniform LIL confidence intervals and optimism to control errors simultaneously across all accuracy levels and episodes.

  • Uniform-PAC guarantee: UBEV is Uniform-PAC and, with probability at least 1 −δ, converges to optimal policies while supporting bounds for all episodes T.The Uniform-PAC guarantee holds jointly for every ε > 0, and the convergence and regret results follow from it.
  • Confidence bounds: LIL confidence intervals shrink at the rate 1/n ln ln n, improving on 1/n ln n and 1/n ln T bounds while holding uniformly over time.These bounds avoid horizon-dependent confidence levels and are used to control infinitely many potential failure events.
  • Proof sketch: The proof selects an optimistic MDP, establishes that the true MDP remains in the feasible model set on the complement of a failure event, and bounds the resulting optimality gap.The analysis decomposes errors using visitation probabilities, empirical reward and transition estimates, and state-action pairs with non-negligible visitation.
  • Proof sketch: Except for at most O(AS2H4/ε2 polylog(A, S, H, 1/ε, 1/δ)) episodes, UBEV's optimality gap is at most ε on the complement of the failure event.A pigeonhole argument combines bounds on nice episodes and visitation-based error terms.
  • Discussion of UBEV bound: UBEV's (Uniform-)PAC bound is never worse than ˜O(S2AH4/ε2), improving on MBIE by H2 and on MoRMax's ˜O(SAH6/ε2) in the stated regime.For ε < 1/(S2A), the bound has linear dependence on the state-space size and H4 dependence on the horizon.
  • Discussion of UBEV bound: UBEV matches Azar et al.'s regret bound in S, A, and T terms but not H, while Azar et al.'s algorithm is not Uniform-PAC.The episode-based formulation contributes an H factor relative to time-step accounting.

5 Conclusion

The Uniform-PAC framework unifies PAC and high-probability regret criteria for episodic RL, while UBEV achieves both PAC and nearly optimal sublinear regret. The framework currently excludes infinite-horizon RL and unrealizable function approximation.

  • Uniform-PAC strengthens and unifies PAC and high-probability regret criteria for episodic reinforcement learning.
  • UBEV is Uniform-PAC and is the first algorithm that is both PAC and has sublinear, nearly optimal regret.
  • The framework applies directly to multi-armed and contextual bandits but not to infinite-horizon reinforcement learning.
  • Extending UBEV to infinite-horizon RL with linear state-space sample complexity remains an open research direction.
  • Practical large-scale RL theory must address unrealizable function approximation beyond the tabular setting.

Appendices of Unifying PAC and Regret:

The appendix section is identified only by the paper title and does not provide substantive content.

  • The supplied passage contains the paper title rather than substantive appendix content.

A.1 Proof of Theorem 1

Theorem 1 uses two hard-to-distinguish one-state, horizon-one MDPs to show that sublinear regret requires infinitely many suboptimal plays in one MDP, ruling out any finite PAC bound at a smaller accuracy.

  • The proof constructs two one-state, horizon-one MDPs with two actions and differing Bernoulli rewards for action 2.
  • Assuming action 2 is played at most N times in M1 with probability at least β establishes the proof’s finite-mistake premise.
  • The two MDPs share action-selection behavior and action-1 reward distributions conditional on observed histories.
  • When action 2 is chosen at most N times, the likelihood ratio of the full observation sequence is bounded by (1 + 2α)^N.
  • For sufficiently large T, M2 incurs Ω(T) expected regret under the finite-mistake premise, so sublinear regret requires infinitely many suboptimal plays in M1 and prevents any finite PAC bound for ε < α/2.

A.2 Proof of Theorem 2

Theorem 2 constructs algorithms showing that PAC and regret guarantees do not generally imply one another’s stronger forms. These constructions yield linear regret or infinitely many errors despite the other guarantee.

  • A PAC bound scaling as 1/ε^2 implies at best a high-probability regret bound of order O(T^2/3), tight only when T = Θ(1/ε^3).
  • For any fixed ε, Equation (8) permits an algorithm with uniform high-probability regret of Ω(T).
  • Figure 3 interprets regret as the area of shaded error regions accumulated through T episodes.
  • The PAC-to-uniform-regret construction incurs regret R(T) = NH + (T − N)ε = Ω(T) for T ≥ N.
  • A uniform high-probability regret guarantee can coexist with infinitely many ε/2-mistakes with probability 1.
  • A uniform high-probability regret guarantee can coexist with expected regret ER(T) = δεT = Ω(T) when the algorithm always chooses a suboptimal policy with probability δ.

A.3 Proof of Theorem 3

The proof converts a Uniform-PAC mistake bound into a high-probability regret bound by selecting a suitable accuracy threshold and summing regret across episodes. The resulting bound scales as the square root of the episode count up to logarithmic factors.

  • Regret conversion: The proof uses a decreasing mistake-bound function to choose ε_min so the relevant bound fits within T episodes.For sufficiently large T, an ε_min ∈ (0, H] is selected such that g(ε) ≤ T for ε > ε_min.
  • Regret conversion: The regret calculation assumes the worst case: maximal-regret mistakes occur first, followed by progressively less severe mistakes.The authors direct readers to Figure 3 for intuition about this ordering.
  • Regret conversion: The proof bounds regret term by term after substituting the selected ε_min and applying logarithmic inequalities.It uses ln(a + b) ≤ ln(a) + ln(b) = ln(ab) for a, b ≥ 2.
  • High-probability regret bound: R(T) ≤ (√C2T + C1) polylog(T, C3, C1, H) = O(√C2T polylog(T, C3, C1, H)).This is the resulting high-probability regret bound on the event where the Uniform-PAC conditions hold.

B Experimental Details

The experiments evaluate episodic RL algorithms on randomly generated finite-state MDPs with sparse deterministic rewards and non-deterministic transitions. All algorithms are compared under the shared assumption that immediate rewards are known.

  • MDP generation: The study generates MDPs with S = 5, 50, 200 states, A = 3 actions, and H = 10 timesteps.Transition probabilities are sampled independently from Dirichlet distributions.
  • MDP generation: Rewards are deterministic, equal to zero with probability 85%, and otherwise sampled uniformly from [0, 1].This produces concentrated but non-deterministic transitions and sparse rewards.
  • Evaluation assumptions: All algorithms are evaluated assuming the immediate rewards r(s, a, t) are known.The implementations are adapted accordingly, including replacing UBEV’s reward-confidence term with the true rewards.
  • Algorithm adaptations: Competitor algorithms are adapted to episodic MDPs with time-dependent transition dynamics using time-dependent learning and finite-horizon planning.The authors state that confidence intervals were adapted without re-deriving each algorithm’s constants.
  • Evaluation protocol: Figure 2 reports expected return averaged over 1000 episodes for a single randomly generated MDP.Repeated runs with different random seeds produced qualitatively similar results.
  • Reproducibility: Source code for the experiments is publicly available in the FiniteEpisodicRL.jl repository.The passage provides the repository URL.

C PAC Lower Bound

The lower-bound section constructs difficult episodic MDPs showing that PAC learning requires substantial sample complexity. Because Uniform-PAC is stronger than fixed-ε PAC, the lower bound also applies to Uniform-PAC guarantees.

  • Lower-bound statement: For sufficiently small ε, there exist time-dependent episodic MDPs where any algorithm fails to return an ε-optimal policy with probability at least 1 − δ0 when n ≤ cASH3.The construction uses positive constants c, δ0, and ε0, with S ≥ 4 and A ≥ 2.
  • Implication for Uniform-PAC: The lower bound applies to arbitrary fixed-ε PAC guarantees and therefore immediately extends to the stronger Uniform-PAC guarantees.This is because Uniform-PAC requires guarantees simultaneously across all ε > 0.
  • Construction: The hard instances are built by layering difficult multi-armed bandit problems across the episodic horizon.The proof follows the standard construction attributed to Jiang et al., with additional horizon steps accumulating reward.
  • Comparison with UBEV: For small ε, UBEV’s sample-complexity bound is optimal except for a factor of H and logarithmic terms.This comparison is the section’s stated consequence of the lower bound.

E.1 Proof of Theorem 4

The proof establishes UBEV’s Uniform-PAC guarantee by controlling a global failure event with uniform concentration arguments and then bounding non-optimal episodes outside that event. It concludes that all but polynomially many episodes are ε-optimal.

  • Episode classification: Outside the failure event, Lemma E.2 bounds the number of episodes that are not friendly by 48A2S3H4/ε times logarithmic factors.The supplied passage reports the polynomial dependence and logarithmic factor in abbreviated form.
  • PAC guarantee: All but at most 9216/ε polylog(A, S, H, 1/ε, 1/δ) friendly episodes are ε-optimal.This follows from Lemma E.8.
  • Failure probability: The proof defines a failure event F and shows that its probability is at most δ using uniform concentration statements.The argument combines concentration results through a union bound.
  • Concentration analysis: The proof controls transition and reward-related concentration events by analyzing sampled trajectories with filtrations and martingale-difference sequences.The Markov property supports the martingale arguments, followed by union bounds over state, action, and time indices.

E.3 Nice and Friendly Episodes

The analysis separates episodes into nice and stronger friendly classes, establishing properties that control uncertainty and optimality gaps. Rate lemmas then bound exceptional episodes and yield UBEV’s sample-complexity guarantees.

  • Sample-complexity consequences: A refined analysis of friendly episodes yields sample complexity linear-polylogarithmic in S, while the nice-episode gap bound controls ε-suboptimal episodes.Together with the non-nice episode bound, these results produce the sample complexity stated in Theorem 4.
  • Episode classifications: Nice episodes satisfy two probability-and-uncertainty conditions for every state, action, and time step.Friendly episodes additionally satisfy a corresponding condition for predecessor state-action pairs and earlier times.
  • Episode classifications: On the good event, nice and friendly episodes satisfy the stated confidence-related properties uniformly over their relevant state-action-time tuples.These properties support subsequent bounds on the optimality gap.
  • Counting exceptional episodes: The number of non-friendly episodes and the number of non-nice episodes are separately bounded, enabling control of exceptional episodes.The non-friendly count is further analyzed through accumulated probability terms and thresholds such as wmin and w′.
  • Rate bounds: Rate lemmas bound how often weighted uncertainty terms can exceed thresholds, using sequence inequalities and Hölder’s inequality.These bounds are applied to episodes whose optimality gap exceeds ε′.
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