Fixed Budget is No Harder Than Fixed Confidence in Best-Arm Identification up to Logarithmic Factors

Researchers prove that fixed-budget best-arm identification in bandit problems is no harder than fixed-confidence approaches up to logarithmic factors, introducing FC2FB—a meta-algorithm that converts fixed-confidence algorithms to fixed-budget ones while maintaining optimal sample complexity. This fundamental result establishes a previously unclear relationship between two core machine learning paradigms and enables improved algorithms across multiple problem classes.

This theoretical computer science result addresses a foundational question in interactive machine learning that has remained open despite decades of bandit algorithm research. The best-arm identification problem, which seeks to identify the optimal choice among multiple options with minimal sampling, appears in countless applications from clinical trials to recommendation systems. The paper's contribution is establishing that fixed-budget constraints (knowing your sampling limit upfront) are not fundamentally harder than fixed-confidence constraints (knowing your desired confidence level upfront)—they're equivalent up to logarithmic factors, a relatively small gap in complexity theory.

The constructive proof through the FC2FB algorithm is particularly significant because it provides a practical conversion mechanism rather than merely theoretical insight. By taking existing fixed-confidence algorithms and transforming them into fixed-budget variants, researchers can immediately leverage decades of accumulated optimization work in one setting to improve the other. This represents efficient knowledge transfer across algorithmic paradigms.

For the machine learning community, this work simplifies algorithm design and analysis. Instead of developing separate optimal solutions for both settings, practitioners can focus on one and convert as needed. The logarithmic gap suggests the two formulations capture similar fundamental difficulty, making the field's resource allocation more efficient. The paper's demonstration of improved sample complexity for multiple specific problems validates the practical utility beyond theoretical elegance, indicating that FC2FB with state-of-the-art fixed-confidence algorithms produces concrete performance gains that researchers and practitioners can immediately deploy.

• →FC2FB meta-algorithm converts fixed-confidence bandit algorithms to fixed-budget variants while preserving optimal sample complexity up to logarithmic factors
• →Fixed-budget and fixed-confidence best-arm identification have equivalent fundamental difficulty, resolving a longstanding open question in bandit theory
• →The constructive proof enables immediate practical improvements by combining FC2FB with existing optimized fixed-confidence algorithms
• →Logarithmic factors represent the complexity gap between the two settings, indicating they capture similar underlying hardness
• →Result has implications for diverse applications including clinical trials, online learning, and recommendation systems