TURF: Two-Factor, Universal, Robust, Fast Distribution Learning Algorithm

Yi Hao, Ayush Jain, Alon Orlitsky, Vaishakh Ravindrakumar
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:8427-8445, 2022.

Abstract

Approximating distributions from their samples is a canonical statistical-learning problem. One of its most powerful and successful modalities approximates every distribution to an ℓ1 distance essentially at most a constant times larger than its closest t-piece degree-d polynomial, where t≥1 and d≥0. Letting ct,d denote the smallest such factor, clearly c1,0=1, and it can be shown that ct,d≥2 for all other t and d. Yet current computationally efficient algorithms show only ct,1≤2.25 and the bound rises quickly to ct,d≤3 for d≥9. We derive a near-linear-time and essentially sample-optimal estimator that establishes ct,d=2 for all (t,d)≠(1,0). Additionally, for many practical distributions, the lowest approximation distance is achieved by polynomials with vastly varying number of pieces. We provide a method that estimates this number near-optimally, hence helps approach the best possible approximation. Experiments combining the two techniques confirm improved performance over existing methodologies.

Cite this Paper

BibTeX
@InProceedings{pmlr-v162-hao22a, title = {{TURF}: Two-Factor, Universal, Robust, Fast Distribution Learning Algorithm}, author = {Hao, Yi and Jain, Ayush and Orlitsky, Alon and Ravindrakumar, Vaishakh}, booktitle = {Proceedings of the 39th International Conference on Machine Learning}, pages = {8427--8445}, year = {2022}, editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan}, volume = {162}, series = {Proceedings of Machine Learning Research}, month = {17--23 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v162/hao22a/hao22a.pdf}, url = {https://proceedings.mlr.press/v162/hao22a.html}, abstract = {Approximating distributions from their samples is a canonical statistical-learning problem. One of its most powerful and successful modalities approximates every distribution to an $\ell_1$ distance essentially at most a constant times larger than its closest $t$-piece degree-$d$ polynomial, where $t\ge1$ and $d\ge0$. Letting $c_{t,d}$ denote the smallest such factor, clearly $c_{1,0}=1$, and it can be shown that $c_{t,d}\ge 2$ for all other $t$ and $d$. Yet current computationally efficient algorithms show only $c_{t,1}\le 2.25$ and the bound rises quickly to $c_{t,d}\le 3$ for $d\ge 9$. We derive a near-linear-time and essentially sample-optimal estimator that establishes $c_{t,d}=2$ for all $(t,d)\ne(1,0)$. Additionally, for many practical distributions, the lowest approximation distance is achieved by polynomials with vastly varying number of pieces. We provide a method that estimates this number near-optimally, hence helps approach the best possible approximation. Experiments combining the two techniques confirm improved performance over existing methodologies.} }
Endnote
%0 Conference Paper %T TURF: Two-Factor, Universal, Robust, Fast Distribution Learning Algorithm %A Yi Hao %A Ayush Jain %A Alon Orlitsky %A Vaishakh Ravindrakumar %B Proceedings of the 39th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Kamalika Chaudhuri %E Stefanie Jegelka %E Le Song %E Csaba Szepesvari %E Gang Niu %E Sivan Sabato %F pmlr-v162-hao22a %I PMLR %P 8427--8445 %U https://proceedings.mlr.press/v162/hao22a.html %V 162 %X Approximating distributions from their samples is a canonical statistical-learning problem. One of its most powerful and successful modalities approximates every distribution to an $\ell_1$ distance essentially at most a constant times larger than its closest $t$-piece degree-$d$ polynomial, where $t\ge1$ and $d\ge0$. Letting $c_{t,d}$ denote the smallest such factor, clearly $c_{1,0}=1$, and it can be shown that $c_{t,d}\ge 2$ for all other $t$ and $d$. Yet current computationally efficient algorithms show only $c_{t,1}\le 2.25$ and the bound rises quickly to $c_{t,d}\le 3$ for $d\ge 9$. We derive a near-linear-time and essentially sample-optimal estimator that establishes $c_{t,d}=2$ for all $(t,d)\ne(1,0)$. Additionally, for many practical distributions, the lowest approximation distance is achieved by polynomials with vastly varying number of pieces. We provide a method that estimates this number near-optimally, hence helps approach the best possible approximation. Experiments combining the two techniques confirm improved performance over existing methodologies.
APA
Hao, Y., Jain, A., Orlitsky, A. & Ravindrakumar, V.. (2022). TURF: Two-Factor, Universal, Robust, Fast Distribution Learning Algorithm. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:8427-8445 Available from https://proceedings.mlr.press/v162/hao22a.html.

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