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Sample-Adaptivity Tradeoff in On-Demand Sampling

Nika Haghtalab, Omar Montasser, Mingda Qiao

TL;DR

The paper analyzes the trade-off between adaptive rounds and sample complexity in on-demand sampling for Multi-Distribution Learning (MDL). It derives a tight realizable bound where the optimal sample complexity scales as $O\left(d k^{\Theta(1/r)} / \varepsilon\right)$ for $r$ rounds, and shows that roughly $\log k$ rounds are necessary to reach near-optimal performance; in the agnostic setting, it introduces the Optimization via On-Demand Sampling (OODS) framework to obtain near-optimal $\widetilde O((d+k)/\varepsilon^2)$ samples in $\widetilde O(\sqrt{k})$ rounds. The OODS framework abstracts the adaptivity-sample trade-off and yields both upper and lower bounds on round complexity, revealing inherent limitations of adaptive strategies that rely on restricted sample information. Together, these results illuminate how limited adaptivity can dramatically improve sample efficiency, while also identifying fundamental barriers to achieving sub-polynomial round complexity without new techniques. The work has implications for data-adaptive sampling in large-scale learning pipelines and motivates further study into combining adaptive strategies with polylogarithmic-round guarantees.

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 $dk^{Θ(1/r)} / ε$. For the general agnostic case, we present an algorithm that achieves near-optimal sample complexity of $\widetilde O((d + k) / ε^2)$ 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.

Sample-Adaptivity Tradeoff in On-Demand Sampling

TL;DR

The paper analyzes the trade-off between adaptive rounds and sample complexity in on-demand sampling for Multi-Distribution Learning (MDL). It derives a tight realizable bound where the optimal sample complexity scales as for rounds, and shows that roughly rounds are necessary to reach near-optimal performance; in the agnostic setting, it introduces the Optimization via On-Demand Sampling (OODS) framework to obtain near-optimal samples in rounds. The OODS framework abstracts the adaptivity-sample trade-off and yields both upper and lower bounds on round complexity, revealing inherent limitations of adaptive strategies that rely on restricted sample information. Together, these results illuminate how limited adaptivity can dramatically improve sample efficiency, while also identifying fundamental barriers to achieving sub-polynomial round complexity without new techniques. The work has implications for data-adaptive sampling in large-scale learning pipelines and motivates further study into combining adaptive strategies with polylogarithmic-round guarantees.

Abstract

We study the tradeoff between sample complexity and round complexity in on-demand sampling, where the learning algorithm adaptively samples from 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 -round algorithm scales approximately as . For the general agnostic case, we present an algorithm that achieves near-optimal sample complexity of within 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 -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.

Paper Structure

This paper contains 78 sections, 29 theorems, 130 equations, 1 table.

Table of Contents

  1. Introduction
  2. Our Contributions and Results
  3. Related Work
  4. Multi-Distribution Learning
  5. Power of Adaptivity in Learning and Beyond
  6. Preliminaries
  7. Multi-Distribution Learning (MDL)
  8. MDL via Game Dynamics
  9. Sample-Adaptivity Tradeoff for Realizable MDL
  10. Overview of Upper Bound
  11. Additional Notations
  12. Technical Overview
  13. Overview of Lower Bound
  14. Sample-Adaptivity Tradeoff for Agnostic MDL
  15. The MDL Algorithm of ZZCDL24
  16. ...and 63 more sections

Key Result

Theorem 1

alg:realizable is an $r$-round $(\varepsilon, \delta)$-PAC algorithm for realizable MDL with sample complexity $O(k^{2/r}\log k\cdot\frac{d}{\varepsilon} + \frac{k\log(k)\log(k/\delta)}{\varepsilon})$.

Theorems & Definitions (57)

  • Theorem 1: Informal version of thm:realizable-upper
  • Remark 1
  • Theorem 2: Informal version of thm:realizable-lower
  • Proposition 1: Corollaries \ref{['cor:MDL-box']} and \ref{['cor:MDL-ellipsoid']}
  • Proposition 2
  • proof : Proof sketch
  • Proposition 3
  • Lemma 3: Lemma 3 of ZZCDL24
  • proof : Proof sketch of prop:agnostic-ellipsoid-version
  • Definition 1: Optimization via On-Demand Sampling
  • ...and 47 more