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Linear regression in $ell_p$-norm is a canonical optimization problem that arises in several applications, including sparse recovery, semi-supervised learning, and signal processing. Generic convex optimization algorithms for solving $ell_p$-regression are slow in practice. Iteratively Reweighted Least Squares (IRLS) is an easy to implement family of algorithms for solving these problems that has been studied for over 50 years. However, these algorithms often diverge for p > 3, and since the work of Osborne (1985), it has been an open problem whether there is an IRLS algorithm that is guaranteed to converge rapidly for p > 3. We propose p-IRLS, the first IRLS algorithm that provably converges geometrically for any $p in [2,infty).$ Our algorithm is simple to implement and is guaranteed to find a $(1+varepsilon)$-approximate solution in $O(p^{3.5} m^{frac{p-2}{2(p-1)}} log frac{m}{varepsilon}) le O_p(sqrt{m} log frac{m}{varepsilon} )$ iterations. Our experiments demonstrate that it performs even better than our theoretical bounds, beats the standard Matlab/CVX implementation for solving these problems by 10--50x, and is the fastest among available implementations in the high-accuracy regime.
We give improved algorithms for the $ell_{p}$-regression problem, $min_{x} |x|_{p}$ such that $A x=b,$ for all $p in (1,2) cup (2,infty).$ Our algorithms obtain a high accuracy solution in $tilde{O}_{p}(m^{frac{|p-2|}{2p + |p-2|}}) le tilde{O}_{p}(m^
We give the first polynomial-time algorithm for robust regression in the list-decodable setting where an adversary can corrupt a greater than $1/2$ fraction of examples. For any $alpha < 1$, our algorithm takes as input a sample ${(x_i,y_i)}_{i leq
Diffusion is a fundamental graph procedure and has been a basic building block in a wide range of theoretical and empirical applications such as graph partitioning and semi-supervised learning on graphs. In this paper, we study computationally effici
We propose a provably convergent method, called Efficient Learned Descent Algorithm (ELDA), for low-dose CT (LDCT) reconstruction. ELDA is a highly interpretable neural network architecture with learned parameters and meanwhile retains convergence gu
In this paper, we develop a simple and fast online algorithm for solving a class of binary integer linear programs (LPs) arisen in general resource allocation problem. The algorithm requires only one single pass through the input data and is free of