PAC Learning with Bandit Feedback: Sharp Sample Complexity in the Realizable Setting

Abstract: We study the problem of multiclass PAC learning with bandit feedback in the realizable setting. In this framework, there is an unknown data distribution over an instance space $\mathcal{X}$ and a label space $\mathcal{Y}$, as in classical multiclass PAC learning, but the learner does not observe the labels of the i.i.d. training examples. Instead, in each round, it receives an unlabeled instance, predicts its label, and receives bandit feedback indicating only whether the prediction is correct. Despite this restriction, the goal remains the same as in classical PAC learning. We provide a general characterization of the optimal sample complexity of this problem, sharp for every concept class up to logarithmic factors. Our characterization is based on a new combinatorial dimension, termed the bandit $\mathrm{DS}$ dimension, defined via generalized combinatorial structures we call pseudo-boxes. These extend the pseudo-cubes underlying the $\mathrm{DS}$ dimension by allowing a different number of neighbors in each coordinate. In contrast to the $\mathrm{DS}$ dimension, which governs the full-information setting by counting the number of coordinates in the pseudo-cube, the bandit $\mathrm{DS}$ dimension aggregates the number of neighbors across coordinates, leading to a characterization in which the sample complexity scales with the total number of neighbors. We also propose a general learning algorithm achieving the upper bound, based on an algorithmic principle called ListCascade, which connects bandit learning to list learning and may be of independent interest.

• The paper's main contribution is the introduction of the bandit DS dimension, which sharpens sample complexity characterization for PAC learning with bandit feedback.
• The ListCascade algorithm efficiently narrows down candidate labels across epochs, achieving near-optimal sample usage proportional to the BDS dimension.
• The results resolve a longstanding gap in multiclass learning by precisely quantifying the cost difference between bandit and full-information feedback.

Sharp Sample Complexity for PAC Learning with Bandit Feedback in the Realizable Setting

Problem Setting and Motivation

The paper "PAC Learning with Bandit Feedback: Sharp Sample Complexity in the Realizable Setting" (2605.25678) addresses the fundamental problem of multiclass PAC learning under bandit feedback, formalizing the scenario where a learner receives only binary (correct/incorrect) feedback after predicting the label for each instance, rather than full label information. The label space is potentially large, and the underlying target function is assumed to be realizable by a fixed concept class $\mathcal{H} \subseteq \mathcal{Y}^{\mathcal{X}}$.

Prior results established upper and lower bounds on the sample complexity in this setting, exhibiting a multiplicative gap linear in the number of labels $K$ [daniely2011multiclass, daniely2015multiclass]. This gap motivates the search for a sharp, combinatorial characterization of sample complexity in bandit PAC learning analogous to the way VC, Natarajan, and DS dimensions govern other learning settings.

New Combinatorial Dimension: The Bandit DS Dimension

The central technical contribution of the paper is the introduction of the bandit DS (BDS) dimension, a new combinatorial parameter that exactly captures the sample complexity of multiclass PAC learning with bandit feedback up to logarithmic factors. The BDS dimension generalizes the classical DS dimension (used for the full-information case) by aggregating combinatorial local complexities across coordinates.

Formal Definition:

Given a concept class $\mathcal{H}$, the BDS dimension, $\operatorname{BDS}(\mathcal{H})$, is the maximum total of multiplicities (number of "neighborhoods") over coordinates in a pseudo-box shattered by $\mathcal{H}$. Pseudo-boxes generalize the combinatorial objects (pseudo-cubes) underlying the DS dimension by allowing for variable-sized neighborhoods in each coordinate. The key insight is that, under bandit feedback, the learning difficulty from each coordinate is additive rather than simply determined by the number of coordinates.

This combinatorial viewpoint yields:

$$\mathbf{m}^B_{\mathcal{H}}(\epsilon, \delta) \in \widetilde{\Theta} \left( \frac{\operatorname{BDS}(\mathcal{H})}{\epsilon} \right)$$

where $\mathbf{m}^B_{\mathcal{H}}$ is the sample complexity of realizable PAC bandit learning and the $\widetilde{\Theta}$ notation omits polylogarithmic factors in $K$, $1/\epsilon$, and $K$0.

Algorithmic Framework: ListCascade

The sample complexity upper bound is achieved with a novel framework named ListCascade. The central idea is to iteratively reduce the candidate label set using a sequence of list learners:

• The process begins with the full label set of size $K$1.
• In each epoch, the learner predicts uniformly over the current candidate list for each instance, collecting examples with positive bandit feedback (i.e., when the prediction is correct).
• These retained examples are used to train a smaller list predictor, progressively narrowing the list size (typically halving it each epoch) until a final list of size one is learned.
• The sample complexity per epoch scales with the size of the current list, and the overall complexity accumulates across epochs.

ListCascade leverages recent advances in list learning theory, specifically the one-inclusion list algorithm [charikar2023characterization], and sharp bounds provided by the optimal $K$2-ary Sauer lemma (Hanneke et al., 14 Apr 2026). The careful combination of combinatorial and algorithmic ideas ensures that the cost per epoch is asymptotically optimal with respect to the BDS dimension.

Sharp Lower Bound Construction

The matching lower bound is developed by constructing an adversarial distribution using a pseudo-box that witnesses the BDS dimension of $K$3. Mass is concentrated on an "anchor" point, with the remaining mass distributed among coordinates according to their locality multiplicities. If the learner fails to adequately sample each important coordinate a number of times proportional to its neighborhood size, error cannot be reduced below $K$4. This construction forces any learner to require at least $K$5 samples.

Implications and Connections

Theoretical Implications

• Resolution of Open Problem: The results close the sample complexity gap from previous works [daniely2011multiclass, daniely2015multiclass], providing a tight combinatorial characterization for the realizable setting.
• Bandit versus Full-Information: The distinction between the DS and BDS dimensions precisely quantifies the extra cost of bandit feedback. The additive aggregation in BDS contrasts with the cardinality-based regime of DS, highlighting fundamental differences between learning with full and restricted (bandit) feedback.
• List Learning as a Bridge: The ListCascade framework demonstrates that list learning is an essential algorithmic and analytic tool for reducing bandit feedback problems to manageable subproblems. The refined analysis of list sample complexity—eliminating an extraneous list-size factor via the $K$6-ary Sauer lemma (Hanneke et al., 14 Apr 2026)—is critical to optimality.

Practical Implications

• Label Efficiency: In settings where label querying is expensive or practically restricted to yes/no (bandit-style) queries, the BDS dimension provides actionable guidance on the minimal requirements for sample allocation. This is highly relevant in medical trial design and large-scale interactive systems.
• Generalization to Other Bandit Problems: The approach separates multiclass classification with bandit feedback from general contextual bandits by exploiting the sparsity and structure inherent in classification reward functions. The characterization does not (yet) extend directly to agnostic or general bandit settings where such structure may be absent or less exploitable.

Open Directions

The paper leaves several important questions open, notably:

• Whether the remaining logarithmic factors in the upper bound can be eliminated.
• Characterizing the sample complexity in the agnostic (non-realizable) regime.
• Extending combinatorial approaches to broader or more structured bandit problems.

Conclusion

This work introduces the bandit DS dimension as the key combinatorial complexity governing PAC learning with bandit feedback in the realizable multiclass setting (2605.25678). The authors establish tight upper and lower bounds on sample complexity, reveal the power of list learning as an algorithmic primitive via the ListCascade framework, and connect these results to foundational learning theory via optimal combinatorial bounds. The implications reach both theory—by resolving open complexity characterizations—and practice, by informing algorithm design in feedback-limited environments. The extension of these ideas to agnostic learning and more general bandit models remains a promising direction for future research.