Iterative Hard Thresholding with Adaptive Regularization: Sparser Solutions Without Sacrificing Runtime

• URL: http://arxiv.org/abs/2204.08274v1
• Date: Mon, 11 Apr 2022 19:33:15 GMT
• Title: Iterative Hard Thresholding with Adaptive Regularization: Sparser Solutions Without Sacrificing Runtime
• Authors: Kyriakos Axiotis and Maxim Sviridenko
• Abstract summary: We propose a simple modification to the iterative hard thresholding algorithm, which recoversally sparser solutions as a function of the condition number. Our proposed algorithm, regularized IHT, returns a solution with sparsity $O(skappa)$. Our algorithm significantly improves over ARHT which also finds a solution of sparsity $O(skappa)$.
• Score: 17.60502131429094
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose a simple modification to the iterative hard thresholding (IHT) algorithm, which recovers asymptotically sparser solutions as a function of the condition number. When aiming to minimize a convex function $f(x)$ with condition number $\kappa$ subject to $x$ being an $s$-sparse vector, the standard IHT guarantee is a solution with relaxed sparsity $O(s\kappa^2)$, while our proposed algorithm, regularized IHT, returns a solution with sparsity $O(s\kappa)$. Our algorithm significantly improves over ARHT which also finds a solution of sparsity $O(s\kappa)$, as it does not require re-optimization in each iteration (and so is much faster), is deterministic, and does not require knowledge of the optimal solution value $f(x^*)$ or the optimal sparsity level $s$. Our main technical tool is an adaptive regularization framework, in which the algorithm progressively learns the weights of an $\ell_2$ regularization term that will allow convergence to sparser solutions. We also apply this framework to low rank optimization, where we achieve a similar improvement of the best known condition number dependence from $\kappa^2$ to $\kappa$.

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