Sample-Adaptivity Tradeoff in On-Demand Sampling

TL;DR

This paper investigates the balance between sample and round complexity in adaptive on-demand sampling for multi-distribution learning, introducing new algorithms and a framework that elucidate fundamental tradeoffs and bounds.

Contribution

It provides the first tight bounds on sample and round complexity tradeoffs in on-demand sampling, introduces the OODS framework, and develops near-optimal algorithms for MDL.

Findings

Optimal sample complexity scales as $dk^{ heta(1/r)} / $ in $r$ rounds.

Achieves near-optimal $ ilde{O}((d + k) / ^2)$ sample complexity in $ ilde{O}(\u221A k)$ rounds.

Establishes nearly tight bounds on round complexity within the OODS framework.

Abstract

We study the tradeoff between sample complexity and round complexity in on-demand sampling, where the learning algorithm adaptively samples from $k$ distributions over a limited number of rounds. In the realizable setting of Multi-Distribution Learning (MDL), we show that the optimal sample complexity of an $r$-round algorithm scales approximately as $dk^{\Theta(1/r)} / \epsilon$. For the general agnostic case, we present an algorithm that achieves near-optimal sample complexity of $\widetilde O((d + k) / \epsilon^2)$ within $\widetilde O(\sqrt{k})$ rounds. Of independent interest, we introduce a new framework, Optimization via On-Demand Sampling (OODS), which abstracts the sample-adaptivity tradeoff and captures most existing MDL algorithms. We establish nearly tight bounds on the round complexity in the OODS setting. The upper bounds directly yield the $\widetilde O(\sqrt{k})$-round algorithm for agnostic MDL, while the lower bounds imply that achieving sub-polynomial round complexity would require fundamentally new techniques that bypass the inherent hardness of OODS.