Hypothesis Selection: A High Probability Conundrum

In the hypothesis selection problem, we are given a finite set of candidate distributions (hypotheses), $\mathcal{H} = \{H_1, \ldots, H_n\}$, and samples from an unknown distribution $P$. Our goal is to find a hypothesis $H_i$ whose total variation distance to $P$ is comparable to that of the nearest hypothesis in $\mathcal{H}$. If the minimum distance is $\mathsf{OPT}$, we aim to output an $H_i$ such that, with probability at least $1-\delta$, its total variation distance to $P$ is at most $C \cdot \mathsf{OPT} + \varepsilon$. Despite decades of work, key aspects of this problem remain unresolved, including the optimal running time for algorithms that achieve the optimal sample complexity and best possible approximation factor of $C=3$. The previous state-of-the-art result [Aliakbarpour, Bun, Smith, NeurIPS 2024] provided a nearly linear in $n$ time algorithm but with a sub-optimal dependence on the other parameters, running in $\tilde{O}(n/(\delta^3\varepsilon^3))$ time. We improve this time complexity to $\tilde{O}(n/(\delta \varepsilon^2))$, significantly reducing the dependence on the confidence and error parameters. Furthermore, we study hypothesis selection in three alternative settings, resolving or making progress on several open questions from prior works. (1) We settle the optimal approximation factor when bounding the \textit{expected distance} of the output hypothesis, rather than its high-probability performance. (2) Assuming the numerical value of \textit{$\mathsf{OPT}$ is known} in advance, we present an algorithm obtaining $C=3$ and runtime $\tilde{O}(n/\varepsilon^2)$ with the optimal sample complexity and succeeding with high probability in $n$. (3) Allowing polynomial \textit{preprocessing} step on the hypothesis class $\mathcal{H}$ before observing samples, we present an algorithm with $C=3$ and subquadratic runtime which succeeds with high probability in $n$.