An Optimal Reduction of TV-Denoising to Adaptive Online Learning

Dheeraj Baby, Xuandong Zhao, Yu-Xiang Wang
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:2899-2907, 2021.

Abstract

We consider the problem of estimating a function from n noisy samples whose discrete Total Variation (TV) is bounded by Cn. We reveal a deep connection to the seemingly disparate problem of \emph{Strongly Adaptive} online learning [Daniely et al 2015] and provide an O(nlogn) time algorithm that attains the near minimax optimal rate of ˜O(n1/3C2/3n) under squared error loss. The resulting algorithm runs online and optimally \emph{adapts} to the \emph{unknown} smoothness parameter Cn. This leads to a new and more versatile alternative to wavelets-based methods for (1) adaptively estimating TV bounded functions; (2) online forecasting of TV bounded trends in time series.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-baby21a, title = { An Optimal Reduction of TV-Denoising to Adaptive Online Learning }, author = {Baby, Dheeraj and Zhao, Xuandong and Wang, Yu-Xiang}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {2899--2907}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/baby21a/baby21a.pdf}, url = {https://proceedings.mlr.press/v130/baby21a.html}, abstract = { We consider the problem of estimating a function from $n$ noisy samples whose discrete Total Variation (TV) is bounded by $C_n$. We reveal a deep connection to the seemingly disparate problem of \emph{Strongly Adaptive} online learning [Daniely et al 2015] and provide an $O(n \log n)$ time algorithm that attains the near minimax optimal rate of $\tilde O (n^{1/3}C_n^{2/3})$ under squared error loss. The resulting algorithm runs online and optimally \emph{adapts} to the \emph{unknown} smoothness parameter $C_n$. This leads to a new and more versatile alternative to wavelets-based methods for (1) adaptively estimating TV bounded functions; (2) online forecasting of TV bounded trends in time series. } }
Endnote
%0 Conference Paper %T An Optimal Reduction of TV-Denoising to Adaptive Online Learning %A Dheeraj Baby %A Xuandong Zhao %A Yu-Xiang Wang %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-baby21a %I PMLR %P 2899--2907 %U https://proceedings.mlr.press/v130/baby21a.html %V 130 %X We consider the problem of estimating a function from $n$ noisy samples whose discrete Total Variation (TV) is bounded by $C_n$. We reveal a deep connection to the seemingly disparate problem of \emph{Strongly Adaptive} online learning [Daniely et al 2015] and provide an $O(n \log n)$ time algorithm that attains the near minimax optimal rate of $\tilde O (n^{1/3}C_n^{2/3})$ under squared error loss. The resulting algorithm runs online and optimally \emph{adapts} to the \emph{unknown} smoothness parameter $C_n$. This leads to a new and more versatile alternative to wavelets-based methods for (1) adaptively estimating TV bounded functions; (2) online forecasting of TV bounded trends in time series.
APA
Baby, D., Zhao, X. & Wang, Y.. (2021). An Optimal Reduction of TV-Denoising to Adaptive Online Learning . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:2899-2907 Available from https://proceedings.mlr.press/v130/baby21a.html.

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