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

The best-arm identification (BAI) problem is one of the most fundamental problems in interactive machine learning, which has two flavors: the fixed-budget setting (FB) and the fixed-confidence setting (FC). For $K$-armed bandits with the unique best arm, the optimal sample complexities for both settings have been settled down, and they match up to logarithmic factors. This prompts an interesting research question about the generic, potentially structured BAI problems: Is FB harder than FC or the other way around? In this paper, we show that FB is no harder than FC up to logarithmic factors. We do this constructively: we propose a novel algorithm called FC2FB (fixed confidence to fixed budget), which is a meta algorithm that takes in an FC algorithm $\mathcal{A}$ and turn it into an FB algorithm. We prove that this FC2FB enjoys a sample complexity that matches, up to logarithmic factors, that of the sample complexity of $\mathcal{A}$. This means that the optimal FC sample complexity is an upper bound of the optimal FB sample complexity up to logarithmic factors. Our result not only reveals a fundamental relationship between FB and FC, but also has a significant implication: FC2FB, combined with existing state-of-the-art FC algorithms, leads to improved sample complexity for a number of FB problems.

💡 Research Summary

The paper investigates the relationship between the two canonical formulations of best‑arm identification (BAI): the fixed‑budget (FB) setting, where a total sampling budget B is given, and the fixed‑confidence (FC) setting, where a target error probability δ is prescribed and the algorithm stops as soon as it can certify the answer with confidence 1‑δ. While for the classic unstructured K‑armed bandit the optimal sample complexities of FB and FC are known to match up to logarithmic factors, it has remained unclear whether this equivalence extends to more general, possibly structured BAI problems.

The authors first formalize a class of “strong” fixed‑confidence algorithms. A strong FC algorithm is δ‑correct for any δ∈(0,1) and enjoys a high‑probability sample‑complexity bound of the form T*_δ = A·ln(1/δ)+C, where A and C are problem‑dependent constants that are typically unknown to the practitioner. The central contribution is a meta‑algorithm called FC2FB (Fixed‑Confidence to Fixed‑Budget). FC2FB takes any strong FC algorithm as a black‑box and, without knowledge of A or C, converts it into an FB algorithm whose error probability decays exponentially in the allocated budget.

The conversion works by partitioning the total budget B into R stages, each receiving roughly B′ = ⌊B/R⌋ samples. In stage r the algorithm runs the FC subroutine with a failure rate δ_r = δ₀·2^{-(R−r)} (δ₀ is a constant, e.g., 1/e). If the FC subroutine terminates before exhausting its stage budget, its recommended arm is output and the whole procedure stops; otherwise the stage is discarded and the next, more permissive δ_r+1 is tried. By exponentially increasing the allowed failure probability across stages, the method guarantees that the first stage that actually terminates provides a recommendation whose error probability is at most O(exp(−B/(4Q)·ln(1/δ₀))) where Q is a user‑chosen parameter controlling the minimal per‑stage sample size (often set to the intrinsic dimension of the problem, such as K for unstructured bandits or d for linear bandits).

Theorem 3.2 formalizes this guarantee: for a budget B satisfying
 B ≥ 2A·ln(1/δ₀) + (C+1)·ln(2A·Q·ln(1/δ₀)) + 2(C+1)·Q,
the output arm ˆJ satisfies
 P(ˆJ ≠ optimal) ≤ 3·exp