{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-broad-optimality-of-profile-maximum","title":"The Broad Optimality of Profile Maximum Likelihood","arxiv_id":"1906.03794","date":"2019-06-10","proceeding":"NeurIPS 2019 12","authors":["Yi Hao","Alon Orlitsky"],"abstract":"We study three fundamental statistical-learning problems: distribution estimation, property estimation, and property testing. We establish the profile maximum likelihood (PML) estimator as the first unified sample-optimal approach to a wide range of learning tasks. In particular, for every alphabet size $k$ and desired accuracy $\\varepsilon$: $\\textbf{Distribution estimation}$ Under $\\ell_1$ distance, PML yields optimal $\\Theta(k/(\\varepsilon^2\\log k))$ sample complexity for sorted-distribution estimation, and a PML-based estimator empirically outperforms the Good-Turing estimator on the actual distribution; $\\textbf{Additive property estimation}$ For a broad class of additive properties, the PML plug-in estimator uses just four times the sample size required by the best estimator to achieve roughly twice its error, with exponentially higher confidence; $\\boldsymbol{\\alpha}\\textbf{-R\\'enyi entropy estimation}$ For integer $\\alpha>1$, the PML plug-in estimator has optimal $k^{1-1/\\alpha}$ sample complexity; for non-integer $\\alpha>3/4$, the PML plug-in estimator has sample complexity lower than the state of the art; $\\textbf{Identity testing}$ In testing whether an unknown distribution is equal to or at least $\\varepsilon$ far from a given distribution in $\\ell_1$ distance, a PML-based tester achieves the optimal sample complexity up to logarithmic factors of $k$. Most of these results also hold for a near-linear-time computable variant of PML. Stronger results hold for a different and novel variant called truncated PML (TPML).","url_abs":"https://arxiv.org/abs/1906.03794v3","url_pdf":"https://arxiv.org/pdf/1906.03794v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"the-broad-optimality-of-profile-maximum","repo_url":"https://github.com/ucsdyi/PML","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1906.03794","atlas_url":"https://app.syntology.ai/?focus=1906.03794","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}