Instance-Dependent Regret Bound in RL

• Instance-dependent regret bounds adapt the regret guarantee to problem-specific hardness by leveraging characteristics of the MDP and reward landscape.
• The introduction of the return gap refines traditional gap definitions by focusing on trajectory-dependent relevance, ensuring that only reachable state-action pairs impact regret.
• This framework sharpens exploration in reinforcement learning, yielding tighter upper and lower bounds that guide the design of efficient, risk-aware algorithms.

An instance-dependent regret bound is a regret guarantee for sequential decision-making algorithms—such as those in reinforcement learning and bandit settings—that adapts its scaling to problem-specific hardness parameters rather than worst-case quantities. Such bounds leverage characteristics of the underlying Markov decision process (MDP), reward landscape, or action structure, often resulting in significantly improved (smaller) regret on “easy” instances, and recover the minimax rates in the worst case.

1. Alternative Definitions of the Gap and Return Gap

Classically, instance-dependent regret bounds in Markov decision processes are formulated in terms of per-state-action value-function gaps, which quantify the suboptimality of choosing an action $a$ at state $s$:

$\gap(s, a) = V^*(s) - Q^*(s, a)$

where $V^*(s)$ denotes the optimal value function at state $s$, and $Q^*(s, a)$ is the optimal Q-value.

The innovation in "Beyond Value-Function Gaps: Improved Instance-Dependent Regret Bounds for Episodic Reinforcement Learning" (2107.01264) is the introduction of the return gap, defined in a way that accounts for the trajectory-dependent accessibility of state-action pairs. Formally,

$\mathrm{return\ gap}(s,a) := \gap(s,a) \vee \min_{\pi : \mathbb{P}_\pi(B(s,a)) > 0} \ \frac{1}{H} \mathbb{E}_\pi\left[\sum_{h=1}^{\kappa(s)} \gap(S_h, A_h) \mid B(s,a)\right]$

where $B(s,a)$ indicates the event along a trajectory that $(s,a)$ is reached after taking at least one suboptimal previous action, and $H$ is the episode length.

This path-aware gap considers the policy support—whether a state-action pair is actually visited under (near-)optimal policies—and thus only charges regret to reachable and relevant state-action pairs. Such refinement allows regret bounds to discount “irrelevant” parts of the state–action space (e.g., dead ends).

2. Main Theoretical Results: Upper and Lower Bounds

The main upper bound for any optimistic algorithm (e.g., StrongEuler, UCBVI) is given as:

$$R(K) \lessapprox \sum_{(s,a): \mathrm{return\ gap}(s,a) > 0} \frac{V^*(s,a) \log K}{\mathrm{return\ gap}(s,a)}$$

where $K$ is the number of episodes, and $V^*(s,a)$ is the optimal value achievable through $(s,a)$.

For deterministic MDPs:

$$R(K) \lessapprox \sum_{(s,a): \Pi_{s,a} \neq \emptyset} \frac{H \log K}{v^* - v^{*}_{s,a}}$$

where $v^*_{s,a}$ denotes the maximal return of any policy that must pass through $(s,a)$, and $\Pi_{s,a}$ the set of such policies.

On the lower bound side, the information-theoretic lower bound is framed as:

$$\min_{\eta(\pi) \ge 0} \sum_{\pi \in \Pi} \eta(\pi) (v^*_\theta - v^\pi_\theta)$$

subject to

$$\sum_{\pi \in \Pi} \eta(\pi) KL(P_\theta^\pi,P_\lambda^\pi) \geq 1, \quad \forall \lambda\in \Lambda(\theta)$$

In symmetric or irreducible MDPs, this yields:

$\liminf_{K\to\infty} \frac{\mathbb{E}[R_\theta(K)]}{\log K} \geq \sum_{(s,a): \gap_\theta(s,a) > 0} \frac{1}{\gap_\theta(s,a)}$

These bounds are adaptive to the instance difficulty: if certain action–state pairs are only accessible via highly suboptimal policies, the agent avoids unnecessary exploration and the bound depends only on the reachable and relevant portions of the MDP.

3. Implications for Algorithm Design and Analysis

The primary implication is that exploration (and thus regret) should be concentrated along or near optimal trajectories. The agent does not need to efficiently estimate Q-values in all state-actions—only in those plausibly visited under optimal or near-optimal policies. For “hard-to-reach” or “catastrophic” actions—reachable only after a long sequence of poor choices—the contribution to regret is dramatically reduced or eliminated.

Optimistic algorithms, which operate by maintaining upper confidence bounds over Q-values (OFU-style algorithms), cannot in general achieve the information-theoretic lower bound—especially when more than one optimal policy exists—since they still “pay” regret for each reachable tuple, regardless of its likelihood under optimal policies.

An important technical contribution is surplus clipping—refining the way confidence bonuses are aggregated across states and actions—which further enhances the instance sensitivity of these algorithms.

4. Structural and Practical Insights

These findings sharpen previous gap-dependent bounds (e.g., those of Simchowitz et al. 2019, Jin et al. 2020), which scale as $O(H \log K / \gap_{\min})$ over all state–actions and degrade when any gap is small (i.e., $\gap_{\min}\to0$).

Compared to these, return gaps detach the regret from irrelevant or unreachable portions of the state space, providing much tighter bounds for structured tasks—such as real-world navigation, robotic control, or safety-critical systems—where not all actions, paths or policies are relevant in practice.

In deterministic MDPs, where only a unique optimal policy exists, optimistic algorithms can fully match the lower bounds up to horizon or log factors. In general, however, even in deterministic systems, the more general information-theoretic lower bounds can be unattainable by standard optimistic methods unless policy structure is further exploited (e.g., policy elimination techniques).

5. Comparative Analysis and Open Directions

Bound Type	Value-function Gap	Return Gap (this work)
General Upper Bound	$O\left(\sum_{s,a} \frac{H\log K}{\gap(s,a)}\right)$	$$O\left(\sum_{s,a} \frac{\log K}{\mathrm{return\ gap}(s,a)}\right)$$
Deterministic MDP Lower Bound	$\Omega\left(\sum_{s,a} \frac{\log K}{\gap(s,a)}\right)$	$$\Omega\left(\sum_{s,a} \frac{\log K}{H\ \mathrm{return\ gap}(s,a)}\right)$$

The difference arises precisely in structured environments: the return gap can be much larger than the value-function gap (the denominator in the regret bound), leading to much smaller regret.

A remaining challenge is designing algorithms that fully attain the lower bound in stochastic MDPs with multiple optimal policies, possibly requiring fundamentally new algorithmic approaches—such as those analyzing return distributions at the policy level or incorporating “policy elimination” or “path-based” optimism.

6. Practical Deployment and Real-World Impacts

These advances offer substantial guidance for designing exploration schemes and confidence updating in applied RL. In domains such as robotics or safety-critical AI, focusing exploration and learning on reachable, meaningful actions and trajectories yields improved sample efficiency and reduces unnecessary risk or exploration of unsafe actions.

The formalism validates heuristic strategies, such as reward shaping, early termination upon deviation from optimal trajectories, or concentrating exploration on the “important” part of the state–action space.

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In summary, the notion of an instance-dependent regret bound—recast here in terms of return gaps—not only tightens theoretical understanding of the learning problem in RL but also prescribes more practical and efficient algorithmic approaches for structured real-world domains. Through both upper and lower bounds, the work clarifies the inherent limits and potential for adaptivity—shifting the focus from all-state–action exploration to trajectory and path-based learning, and demonstrating the statistical and computational advantages of such an approach over uniform, gap-blind regret guarantees.