The Sample Complexity of Multiclass and Sparse Contextual Bandits

Researchers present optimal algorithms for sparse contextual bandits that achieve sample complexity of Õ((s/ε² + |A|/ε)log|Π|/δ), closing a gap from prior work that had exponential dependence on action set size. The results apply to multiclass classification and combinatorial semi-bandits through information-theoretic and algorithmic approaches.

This paper addresses a fundamental problem in online learning: how efficiently can algorithms learn optimal decision policies when receiving limited feedback. The authors tackle contextual bandits in a sparse reward regime where most actions yield zero reward for a given context—a realistic constraint in applications like recommendation systems and multiclass classification at scale.

The theoretical contribution eliminates an exponential term (Θ(|A|^9)) that plagued previous analyses by Erez et al., representing a major simplification. The sample complexity now scales linearly with the sparsity parameter s and inversely with squared error tolerance ε, matching information-theoretic lower bounds up to logarithmic factors. This completeness—proving matching upper and lower bounds—is rare in contextual bandit literature and indicates the theoretical landscape is now well-understood for this setting.

The dual algorithmic approach is noteworthy: the decision-estimation coefficient framework provides theoretical guarantees through complex optimization, while the low-variance exploration method offers practical, implementable algorithms. The extension to contextual combinatorial semi-bandits broadens applicability beyond simple classification to structured action spaces relevant in ranking, assortment, and combinatorial optimization problems.

While this is primarily a theoretical contribution with limited immediate commercial impact, the work establishes foundations for deploying bandit algorithms in high-dimensional action spaces efficiently. For companies developing recommendation engines, ranking systems, or adaptive learning platforms, these results validate that such systems can operate with sample complexity proportional to sparsity rather than action set size—critical for scalability in real-world deployments with thousands of potential actions.

• →Sample complexity bounds for sparse contextual bandits improved by removing Θ(|A|^9) exponential dependence from prior work
• →Matching upper and lower bounds (up to logs) prove the theoretical analysis is tight for this problem class
• →Decision-estimation coefficient framework and low-variance exploration provide complementary algorithmic and theoretical approaches
• →Results extend to contextual combinatorial semi-bandits with improved guarantees for bandit multiclass list classification
• →Scaling with sparsity parameter s rather than full action set size |A| enables practical deployment in high-dimensional settings