Generalizing Complex Hypotheses on Product Distributions: Auctions, Prophet Inequalities, and Pandora’s Problem

Chenghao Guo, Zhiyi Huang, Zhihao Gavin Tang, Xinzhi Zhang
Proceedings of Thirty Fourth Conference on Learning Theory, PMLR 134:2248-2288, 2021.

Abstract

This paper explores a theory of generalization for learning problems on product distributions, complementing the existing learning theories in the sense that it does not rely on any complexity measures of the hypothesis classes. The main contributions are two general sample complexity bounds: (1) $\tilde{O} \big( \frac{nk}{\epsilon^2} \big)$ samples are sufficient and necessary for learning an $\epsilon$-optimal hypothesis in \emph{any problem} on an $n$-dimensional product distribution, whose marginals have finite supports of sizes at most $k$; (2) $\tilde{O} \big( \frac{n}{\epsilon^2} \big)$ samples are sufficient and necessary for any problem on $n$-dimensional product distributions if it satisfies a notion of strong monotonicity from the algorithmic game theory literature. As applications of these theories, we match the optimal sample complexity for single-parameter revenue maximization (Guo et al., STOC 2019), improve the state-of-the-art for multi-parameter revenue maximization (Gonczarowski and Weinberg, FOCS 2018) and prophet inequality (Correa et al., EC 2019; Rubinstein et al., ITCS 2020), and provide the first and tight sample complexity bound for Pandora’s problem.

Cite this Paper

BibTeX
@InProceedings{pmlr-v134-guo21a, title = {Generalizing Complex Hypotheses on Product Distributions: Auctions, Prophet Inequalities, and Pandora’s Problem}, author = {Guo, Chenghao and Huang, Zhiyi and Tang, Zhihao Gavin and Zhang, Xinzhi}, booktitle = {Proceedings of Thirty Fourth Conference on Learning Theory}, pages = {2248--2288}, year = {2021}, editor = {Belkin, Mikhail and Kpotufe, Samory}, volume = {134}, series = {Proceedings of Machine Learning Research}, month = {15--19 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v134/guo21a/guo21a.pdf}, url = {https://proceedings.mlr.press/v134/guo21a.html}, abstract = {This paper explores a theory of generalization for learning problems on product distributions, complementing the existing learning theories in the sense that it does not rely on any complexity measures of the hypothesis classes. The main contributions are two general sample complexity bounds: (1) $\tilde{O} \big( \frac{nk}{\epsilon^2} \big)$ samples are sufficient and necessary for learning an $\epsilon$-optimal hypothesis in \emph{any problem} on an $n$-dimensional product distribution, whose marginals have finite supports of sizes at most $k$; (2) $\tilde{O} \big( \frac{n}{\epsilon^2} \big)$ samples are sufficient and necessary for any problem on $n$-dimensional product distributions if it satisfies a notion of strong monotonicity from the algorithmic game theory literature. As applications of these theories, we match the optimal sample complexity for single-parameter revenue maximization (Guo et al., STOC 2019), improve the state-of-the-art for multi-parameter revenue maximization (Gonczarowski and Weinberg, FOCS 2018) and prophet inequality (Correa et al., EC 2019; Rubinstein et al., ITCS 2020), and provide the first and tight sample complexity bound for Pandora’s problem.} }
Endnote
%0 Conference Paper %T Generalizing Complex Hypotheses on Product Distributions: Auctions, Prophet Inequalities, and Pandora’s Problem %A Chenghao Guo %A Zhiyi Huang %A Zhihao Gavin Tang %A Xinzhi Zhang %B Proceedings of Thirty Fourth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2021 %E Mikhail Belkin %E Samory Kpotufe %F pmlr-v134-guo21a %I PMLR %P 2248--2288 %U https://proceedings.mlr.press/v134/guo21a.html %V 134 %X This paper explores a theory of generalization for learning problems on product distributions, complementing the existing learning theories in the sense that it does not rely on any complexity measures of the hypothesis classes. The main contributions are two general sample complexity bounds: (1) $\tilde{O} \big( \frac{nk}{\epsilon^2} \big)$ samples are sufficient and necessary for learning an $\epsilon$-optimal hypothesis in \emph{any problem} on an $n$-dimensional product distribution, whose marginals have finite supports of sizes at most $k$; (2) $\tilde{O} \big( \frac{n}{\epsilon^2} \big)$ samples are sufficient and necessary for any problem on $n$-dimensional product distributions if it satisfies a notion of strong monotonicity from the algorithmic game theory literature. As applications of these theories, we match the optimal sample complexity for single-parameter revenue maximization (Guo et al., STOC 2019), improve the state-of-the-art for multi-parameter revenue maximization (Gonczarowski and Weinberg, FOCS 2018) and prophet inequality (Correa et al., EC 2019; Rubinstein et al., ITCS 2020), and provide the first and tight sample complexity bound for Pandora’s problem.
APA
Guo, C., Huang, Z., Tang, Z.G. & Zhang, X.. (2021). Generalizing Complex Hypotheses on Product Distributions: Auctions, Prophet Inequalities, and Pandora’s Problem. Proceedings of Thirty Fourth Conference on Learning Theory, in Proceedings of Machine Learning Research 134:2248-2288 Available from https://proceedings.mlr.press/v134/guo21a.html.

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