The Sample Complexity of Simple Binary Hypothesis Testing

Ankit Pensia, Varun Jog, Po-Ling Loh
Proceedings of Thirty Seventh Conference on Learning Theory, PMLR 247:4205-4206, 2024.

Abstract

The sample complexity of simple binary hypothesis testing is the smallest number of i.i.d. samples required to distinguish between two distributions $p$ and $q$ in either: (i) the prior-free setting, with type-I error at most $\alpha$ and type-II error at most $\beta$; or (ii) the Bayesian setting, with Bayes error at most $\delta$ and prior distribution $(\alpha, 1-\alpha)$. This problem has only been studied when $\alpha = \beta$ (prior-free) or $\alpha = 1/2$ (Bayesian), and the sample complexity is known to be characterized by the Hellinger divergence between $p$ and $q$, up to multiplicative constants. In this paper, we derive a formula that characterizes the sample complexity (up to multiplicative constants that are independent of $p$, $q$, and all error parameters) for: (i) all $0 \le \alpha, \beta \le 1/8$ in the prior-free setting; and (ii) all $\delta \le \alpha/4$ in the Bayesian setting. In particular, the formula admits equivalent expressions in terms of certain divergences from the Jensen–Shannon and Hellinger families. The main technical result concerns an $f$-divergence inequality between members of the Jensen–Shannon and Hellinger families, which is proved by a combination of information-theoretic tools and case-by-case analyses. We explore applications of our results to robust and distributed (locally-private and communication-constrained) hypothesis testing.

Cite this Paper

BibTeX
@InProceedings{pmlr-v247-pensia24a, title = {The Sample Complexity of Simple Binary Hypothesis Testing}, author = {Pensia, Ankit and Jog, Varun and Loh, Po-Ling}, booktitle = {Proceedings of Thirty Seventh Conference on Learning Theory}, pages = {4205--4206}, year = {2024}, editor = {Agrawal, Shipra and Roth, Aaron}, volume = {247}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--03 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v247/pensia24a/pensia24a.pdf}, url = {https://proceedings.mlr.press/v247/pensia24a.html}, abstract = {The sample complexity of simple binary hypothesis testing is the smallest number of i.i.d. samples required to distinguish between two distributions $p$ and $q$ in either: (i) the prior-free setting, with type-I error at most $\alpha$ and type-II error at most $\beta$; or (ii) the Bayesian setting, with Bayes error at most $\delta$ and prior distribution $(\alpha, 1-\alpha)$. This problem has only been studied when $\alpha = \beta$ (prior-free) or $\alpha = 1/2$ (Bayesian), and the sample complexity is known to be characterized by the Hellinger divergence between $p$ and $q$, up to multiplicative constants. In this paper, we derive a formula that characterizes the sample complexity (up to multiplicative constants that are independent of $p$, $q$, and all error parameters) for: (i) all $0 \le \alpha, \beta \le 1/8$ in the prior-free setting; and (ii) all $\delta \le \alpha/4$ in the Bayesian setting. In particular, the formula admits equivalent expressions in terms of certain divergences from the Jensen–Shannon and Hellinger families. The main technical result concerns an $f$-divergence inequality between members of the Jensen–Shannon and Hellinger families, which is proved by a combination of information-theoretic tools and case-by-case analyses. We explore applications of our results to robust and distributed (locally-private and communication-constrained) hypothesis testing.} }
Endnote
%0 Conference Paper %T The Sample Complexity of Simple Binary Hypothesis Testing %A Ankit Pensia %A Varun Jog %A Po-Ling Loh %B Proceedings of Thirty Seventh Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2024 %E Shipra Agrawal %E Aaron Roth %F pmlr-v247-pensia24a %I PMLR %P 4205--4206 %U https://proceedings.mlr.press/v247/pensia24a.html %V 247 %X The sample complexity of simple binary hypothesis testing is the smallest number of i.i.d. samples required to distinguish between two distributions $p$ and $q$ in either: (i) the prior-free setting, with type-I error at most $\alpha$ and type-II error at most $\beta$; or (ii) the Bayesian setting, with Bayes error at most $\delta$ and prior distribution $(\alpha, 1-\alpha)$. This problem has only been studied when $\alpha = \beta$ (prior-free) or $\alpha = 1/2$ (Bayesian), and the sample complexity is known to be characterized by the Hellinger divergence between $p$ and $q$, up to multiplicative constants. In this paper, we derive a formula that characterizes the sample complexity (up to multiplicative constants that are independent of $p$, $q$, and all error parameters) for: (i) all $0 \le \alpha, \beta \le 1/8$ in the prior-free setting; and (ii) all $\delta \le \alpha/4$ in the Bayesian setting. In particular, the formula admits equivalent expressions in terms of certain divergences from the Jensen–Shannon and Hellinger families. The main technical result concerns an $f$-divergence inequality between members of the Jensen–Shannon and Hellinger families, which is proved by a combination of information-theoretic tools and case-by-case analyses. We explore applications of our results to robust and distributed (locally-private and communication-constrained) hypothesis testing.
APA
Pensia, A., Jog, V. & Loh, P.. (2024). The Sample Complexity of Simple Binary Hypothesis Testing. Proceedings of Thirty Seventh Conference on Learning Theory, in Proceedings of Machine Learning Research 247:4205-4206 Available from https://proceedings.mlr.press/v247/pensia24a.html.

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