Sparse Convex Optimization via Adaptively Regularized Hard Thresholding

Kyriakos Axiotis, Maxim Sviridenko
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:452-462, 2020.

Abstract

The goal of Sparse Convex Optimization is to optimize a convex function $f$ under a sparsity constraint $s\leq s^*\gamma$, where $s^*$ is the target number of non-zero entries in a feasible solution (sparsity) and $\gamma\geq 1$ is an approximation factor. There has been a lot of work to analyze the sparsity guarantees of various algorithms (LASSO, Orthogonal Matching Pursuit (OMP), Iterative Hard Thresholding (IHT)) in terms of the Restricted Condition Number $\kappa$. The best known algorithms guarantee to find an approximate solution of value $f(x^*)+\epsilon$ with the sparsity bound of $\gamma = O\left(\kappa\min\left\{\log \frac{f(x^0)-f(x^*)}{\epsilon}, \kappa\right\}\right)$, where $x^*$ is the target solution. We present a new Adaptively Regularized Hard Thresholding (ARHT) algorithm that makes significant progress on this problem by bringing the bound down to $\gamma=O(\kappa)$, which has been shown to be tight for a general class of algorithms including LASSO, OMP, and IHT. This is achieved without significant sacrifice in the runtime efficiency compared to the fastest known algorithms. We also provide a new analysis of OMP with Replacement (OMPR) for general $f$, under the condition $s > s^* \frac{\kappa^2}{4}$, which yields Compressed Sensing bounds under the Restricted Isometry Property (RIP). When compared to other Compressed Sensing approaches, it has the advantage of providing a strong tradeoff between the RIP condition and the solution sparsity, while working for any general function $f$ that meets the RIP condition.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-axiotis20a, title = {Sparse Convex Optimization via Adaptively Regularized Hard Thresholding}, author = {Axiotis, Kyriakos and Sviridenko, Maxim}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {452--462}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/axiotis20a/axiotis20a.pdf}, url = {https://proceedings.mlr.press/v119/axiotis20a.html}, abstract = {The goal of Sparse Convex Optimization is to optimize a convex function $f$ under a sparsity constraint $s\leq s^*\gamma$, where $s^*$ is the target number of non-zero entries in a feasible solution (sparsity) and $\gamma\geq 1$ is an approximation factor. There has been a lot of work to analyze the sparsity guarantees of various algorithms (LASSO, Orthogonal Matching Pursuit (OMP), Iterative Hard Thresholding (IHT)) in terms of the Restricted Condition Number $\kappa$. The best known algorithms guarantee to find an approximate solution of value $f(x^*)+\epsilon$ with the sparsity bound of $\gamma = O\left(\kappa\min\left\{\log \frac{f(x^0)-f(x^*)}{\epsilon}, \kappa\right\}\right)$, where $x^*$ is the target solution. We present a new Adaptively Regularized Hard Thresholding (ARHT) algorithm that makes significant progress on this problem by bringing the bound down to $\gamma=O(\kappa)$, which has been shown to be tight for a general class of algorithms including LASSO, OMP, and IHT. This is achieved without significant sacrifice in the runtime efficiency compared to the fastest known algorithms. We also provide a new analysis of OMP with Replacement (OMPR) for general $f$, under the condition $s > s^* \frac{\kappa^2}{4}$, which yields Compressed Sensing bounds under the Restricted Isometry Property (RIP). When compared to other Compressed Sensing approaches, it has the advantage of providing a strong tradeoff between the RIP condition and the solution sparsity, while working for any general function $f$ that meets the RIP condition.} }
Endnote
%0 Conference Paper %T Sparse Convex Optimization via Adaptively Regularized Hard Thresholding %A Kyriakos Axiotis %A Maxim Sviridenko %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-axiotis20a %I PMLR %P 452--462 %U https://proceedings.mlr.press/v119/axiotis20a.html %V 119 %X The goal of Sparse Convex Optimization is to optimize a convex function $f$ under a sparsity constraint $s\leq s^*\gamma$, where $s^*$ is the target number of non-zero entries in a feasible solution (sparsity) and $\gamma\geq 1$ is an approximation factor. There has been a lot of work to analyze the sparsity guarantees of various algorithms (LASSO, Orthogonal Matching Pursuit (OMP), Iterative Hard Thresholding (IHT)) in terms of the Restricted Condition Number $\kappa$. The best known algorithms guarantee to find an approximate solution of value $f(x^*)+\epsilon$ with the sparsity bound of $\gamma = O\left(\kappa\min\left\{\log \frac{f(x^0)-f(x^*)}{\epsilon}, \kappa\right\}\right)$, where $x^*$ is the target solution. We present a new Adaptively Regularized Hard Thresholding (ARHT) algorithm that makes significant progress on this problem by bringing the bound down to $\gamma=O(\kappa)$, which has been shown to be tight for a general class of algorithms including LASSO, OMP, and IHT. This is achieved without significant sacrifice in the runtime efficiency compared to the fastest known algorithms. We also provide a new analysis of OMP with Replacement (OMPR) for general $f$, under the condition $s > s^* \frac{\kappa^2}{4}$, which yields Compressed Sensing bounds under the Restricted Isometry Property (RIP). When compared to other Compressed Sensing approaches, it has the advantage of providing a strong tradeoff between the RIP condition and the solution sparsity, while working for any general function $f$ that meets the RIP condition.
APA
Axiotis, K. & Sviridenko, M.. (2020). Sparse Convex Optimization via Adaptively Regularized Hard Thresholding. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:452-462 Available from https://proceedings.mlr.press/v119/axiotis20a.html.

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