Understanding Gap-Dependent Regret for Optimism-Based Reinforcement Learning with Linear Function Approximation

Gap-dependent performance guarantees for nearly minimax-optimal algorithms in reinforcement learning with linear function approximation are studied to provide improved regret bounds and sample complexity for high-dimensional, long-horizon tasks. The gist: Improved gap-dependent regret upper bounds for the nearly minimax-optimal algorithm LSVI-UCB++ in the linear MDP setting are established, yielding a dependence on feature dimension and horizon length that is strictly better than previous results.

Key Findings and Contributions

The authors address a gap in existing literature by providing the first gap-dependent regret bound for LSVI-UCB++ (He et al., 2023), which achieves the nearly minimax-optimal worst-case regret bound of Oe(d√H3K). This result significantly improves dependencies on both the feature dimension d and horizon length H compared to previous gap-dependent results, reducing the dependence from Oe(d3H5/∆min) or Oe(d2H5/∆min) to a tighter bound of Ob(d2H4/∆min). Furthermore, this refined regret bound implies an improved Probably Approximately Correct (PAC) sample complexity, reducing the dependence on the accuracy parameter ϵ from the worst-case rate of Oe(1/ϵ2) to Oe(1/ϵ).

The research introduces a concurrent variant, Concurrent LSVI-UCB++, leveraging the low policy-switching property of LSVI-UCB++. This variant establishes the first gap-dependent sample complexity upper bound for online multi-agent RL with linear function approximation, achieving a linear speedup with respect to the number of agents M. The resulting concurrent sample complexity is Oe(dH + d2H4M∆minδϵ + d6H7Mδϵ), which implies that when M is sufficiently large, the algorithm enjoys an asymptotically linear speedup in M.

Algorithm and Technical Novelty

The analysis centers on the nearly minimax-optimal algorithm LSVI-UCB++ (Algorithm 1). The core of its operation involves reducing value function learning to a series of linear regression problems based on Proposition 3.4, which relates the next state's value function to a weighted vector: Ps,a,hVπh+1 = ⟨ϕ(s, a), wπh⟩.

The algorithm constructs the estimator wbk,h by solving a weighted ridge regression problem (Line 9 of Algorithm 1) to obtain an optimistic value estimate Qk,h(s, a) as an upper bound of Q⋆ h(s, a). A pessimistic estimate Qˇk,h(s, a) serves as a lower bound. The algorithm also constructs the estimated variance σ2k,h using Equation (3), which involves terms derived from the regression solutions and exploration bonuses.

The technical novelty lies in refining the worst-case guarantees of LSVI-UCB++ into gap-dependent bounds through new techniques for bounding partial sums of both bonuses and estimated variances. Specifically, Lemma 5.3 introduces a surrogate matrix to control the partial sums of bonuses, as the standard recursive structure is lost when analyzing these sums. Lemma 5.4 establishes a bound on the partial sum of estimated variances by relating it to terms involving V¯s k h,a k h,hVk h+1 and an error term Dk,h that bounds the variance estimation error.

Gap-Dependent Regret Analysis

The proof of Theorem 4.1 relies on bounding the expected regret by controlling the summation of suboptimality gaps. The analysis partitions the horizon interval [∆min, H] into dyadic intervals and uses Lemma 5.2 to bound K′(h, n), which is related to how many episodes are needed for a specific gap size (2n∆min).

Lemma 5.1 establishes that the expected regret E[Regret(T)] is upper bounded by the sum of suboptimality gaps: E [Regret(T)] ≤ E [X K k=1 X H h=1 ∆h(s k h, a k h)]. The subsequent proof bounds this summation by controlling the terms in Equation (4), which relates the gap to the difference between the current estimate and the optimal value function.

The proof of Lemma 5.2 involves bounding K′(h, n) by showing that for any fixed (h, n), there is a stopping time ki(h, n) such that Qk h(s k h, a k h) − Qπ k h(s k h, a k h) ≥ 2n∆min. The upper bound derived for K′(h, n) is O(d2H3ι324 n∆2min + d6H6ι222 n∆min), where ι1 = log (1 + dHK/∆min).

Based on the provided scientific paper, here are specific improvements for AI systems derived from the proposed algorithms (LSVI-UCB++ and Concurrent LSVI-UCB++), along with a description of what these improved systems can do.

) 1. Improved Sample Efficiency in High-Dimensional/Long-Horizon RL

The core improvement is achieving better sample efficiency guarantees for learning optimal policies in linear function approximation settings, specifically by moving from existing bounds like Oe(d√H3K) to the new, tighter bound of Oe(d√H4K) (for UCRL-VTR equivalent) and the newly established gap-dependent bound of O(d2H4/∆min).

The improved AI system can:

• Learn optimal control policies in environments with high feature dimensions (large state spaces, e.g., complex robotics or detailed healthcare states) and long decision horizons (e.g., multi-step treatment plans).

• Achieve the same level of policy optimality using significantly fewer training episodes compared to previous nearly minimax algorithms, especially when a meaningful suboptimality gap exists in the environment (i.e., when the optimal action is clearly better than suboptimal ones).

) 2. Enhanced Robustness via Gap-Dependent Regret Analysis

The paper provides a framework for analyzing performance based on the minimum suboptimality gap, ∆min. This allows practitioners to understand how much better an algorithm will perform if the environment is good (large ∆min) versus bad (small ∆min).

The improved AI system can:

• Be deployed in real-world applications where performance guarantees are needed under uncertain conditions. If the system knows that a certain minimum level of suboptimality gap is expected, it can rely on the tighter regret bounds derived from Theorem 4.1 to predict its required training time more accurately.

) 3. Efficient Multi-Agent Collaboration (Concurrent Exploration)

The introduction of Concurrent LSVI-UCB++ offers a mechanism for parallel exploration across multiple agents without incurring high communication costs during learning phases, leveraging the algorithm's low policy-switching property.

The improved AI system can:

• Handle complex cooperative tasks involving many agents (e.g., multi-robot coordination or distributed medical treatment plans).

• Achieve linear speedup in learning time with respect to the number of agents (M), meaning adding more agents does not lead to a proportional increase in training time, making large-scale MARL feasible.

) 4. Scalable Online MARL with Linear Function Approximation

The concurrent variant establishes the first gap-dependent sample complexity bound for online Multi-Agent Reinforcement Learning (MARL).

The improved AI system can:

• Be used in dynamic, online multi-agent settings where agents must learn and adapt their strategies sequentially while interacting with a shared environment.

• Efficiently manage policy updates across a large number of agents by allowing parallel data collection and infrequent synchronization rounds, which is crucial for systems where communication latency or bandwidth is a constraint.

) 5. Improved Policy Identification (PAC Sample Complexity)

The derived PAC sample complexity bound improves the dependence on the accuracy parameter ε from Oe(1/ϵ2) to Oe(1/ϵ).

The improved AI system can:

• Learn a near-optimal policy that meets a specific, user-defined level of performance accuracy (ε) using fewer samples than previously thought.

• This translates directly into faster deployment cycles and lower operational costs for RL agents in production.

We study gap-dependent regret for reinforcement learning with linear function approximation. While prior works have established gap-dependent guarantees in this setting, existing analyses do not apply to algorithms that achieve the nearly minimax-optimal worst-case regret bound (d sqrt H 3K), where d is the feature dimension, H is the horizon length, and K is the number of episodes. We bridge this gap by establishing the first gap-dependent regret bound for the nearly minimax-optimal algorithm LSVI-UCB++ (He et al., 2023), with an expected regret bound (d 2H 3/Δ+d 6H 5), improving the dependence on both d and H in the leading gap-dependent term compared with previous results. To understand the exploration cost induced by optimism, we establish a structural lower bound Ω(d 2H 3/Δ) for a broad class of algorithms based on persistent ellipsoidal optimism. When specialized to LSVI-UCB++, this result shows that the leading dependence of our upper bound on d, H, and Δ is tight up to logarithmic factors. Beyond this algorithmic class, we establish a general gap-dependent lower bound Ω(dH cubed K/Δ) for arbitrary learning algorithms, showing that the logarithmic dependence on K and the cubic dependence on H are intrinsic to gap-dependent expected regret in linear MDPs. Together, our results substantially narrow the gap between upper and lower bounds and provide a sharper characterization of gap-dependent learning and optimism-based exploration with linear function approximation.