Model Consistency for Learning with Mirror-Stratifiable Regularizers

Low-complexity non-smooth convex regularizers are routinely used to impose some structure (such as sparsity or low-rank) on the coefficients for linear predictors in supervised learning. Model consistency consists then in selecting the correct structure (for instance support or rank) by regularized empirical risk minimization. It is known that model consistency holds under appropriate non-degeneracy conditions. However such conditions typically fail for highly correlated designs and it is observed that regularization methods tend to select larger models. In this work, we provide the theoretical underpinning of this behavior using the notion of mirror-stratifiable regularizers. This class of regularizers encompasses the most well-known in the literature, including the $\ell_1$ or trace norms. It brings into play a pair of primal-dual models, which in turn allows one to locate the structure of the solution using a specific dual certificate. We also show how this analysis is applicable to optimal solutions of the learning problem, and also to the iterates computed by a certain class of stochastic proximal-gradient algorithms.Comment: 14 pages, 4 figure

A Variance-Reduced and Stabilized Proximal Stochastic Gradient Method with Support Identification Guarantees for Structured Optimization

This paper introduces a new proximal stochastic gradient method with variance reduction and stabilization for minimizing the sum of a convex stochastic function and a group sparsity-inducing regularization function. Since the method may be viewed as a stabilized version of the recently proposed algorithm PStorm, we call our algorithm S-PStorm. Our analysis shows that S-PStorm has strong convergence results. In particular, we prove an upper bound on the number of iterations required by S-PStorm before its iterates correctly identify (with high probability) an optimal support (i.e., the zero and nonzero structure of an optimal solution). Most algorithms in the literature with such a support identification property use variance reduction techniques that require either periodically evaluating an exact gradient or storing a history of stochastic gradients. Unlike these methods, S-PStorm achieves variance reduction without requiring either of these, which is advantageous. Moreover, our support-identification result for S-PStorm shows that, with high probability, an optimal support will be identified correctly in all iterations with the index above a threshold. We believe that this type of result is new to the literature since the few existing other results prove that the optimal support is identified with high probability at each iteration with a sufficiently large index (meaning that the optimal support might be identified in some iterations, but not in others). Numerical experiments on regularized logistic loss problems show that S-PStorm outperforms existing methods in various metrics that measure how efficiently and robustly iterates of an algorithm identify an optimal support.Comment: The work is accepted for presentation at AISTATS 2023. This is a technical report versio

Screening for Sparse Online Learning

Sparsity promoting regularizers are widely used to impose low-complexity structure (e.g. l1-norm for sparsity) to the regression coefficients of supervised learning. In the realm of deterministic optimization, the sequence generated by iterative algorithms (such as proximal gradient descent) exhibit "finite activity identification", namely, they can identify the low-complexity structure in a finite number of iterations. However, most online algorithms (such as proximal stochastic gradient descent) do not have the property owing to the vanishing step-size and non-vanishing variance. In this paper, by combining with a screening rule, we show how to eliminate useless features of the iterates generated by online algorithms, and thereby enforce finite activity identification. One consequence is that when combined with any convergent online algorithm, sparsity properties imposed by the regularizer can be exploited for computational gains. Numerically, significant acceleration can be obtained