Greedy methods, randomization approaches, and multiarm bandit algorithms for efficient sparsity-constrained optimization

Alain Rakotomamonjy, Sokol Koço, Liva Ralaivola

Abstract: Several sparsity-constrained algorithms, such as orthogonal matching pursuit (OMP) or the Frank-Wolfe (FW) algorithm, with sparsity constraints work by iteratively selecting a novel atom to add to the current nonzero set of variables. This selection step is usually performed by computing the gradient and then by looking for the gradient component with maximal absolute entry. This step can be computationally expensive especially for large-scale and high-dimensional data. In this paper, we aim at accelerating these sparsity-constrained optimization algorithms by exploiting the key observation that, for these algorithms to work, one only needs the coordinate of the gradient's top entry. Hence, we introduce algorithms based on greedy methods and randomization approaches that aim at cheaply estimating the gradient and its top entry. Another of our contribution is to cast the problem of finding the best gradient entry as …