Regularized Bundle Methods for Convex and Non-Convex Risks

Machine learning is most often cast as an optimization problem. Ideally, one expects a convex objective function to rely on efficient convex optimizers with nice guarantees such as no local optima. Yet, non-convexity is very frequent in practice and it may sometimes be inappropriate to look for convexity at any price. Alternatively one can decide not to limit a priori the modeling expressivity to models whose learning may be solved by convex optimization and rely on non-convex optimization algorithms. The main motivation of this work is to provide efficient and scalable algorithms for non-convex optimization. We focus on regularized unconstrained optimization problems which cover a large number of modern machine learning problems such as logistic regression, conditional random fields, large margin estimation, etc. We propose a novel algorithm for minimizing a regularized objective that is able to handle convex and non-convex, smooth and non-smooth risks. The algorithm is based on the cutting plane technique and on the idea of exploiting the regularization term in the objective function. It may be thought as a limited memory extension of convex regularized bundle methods for dealing with convex and non convex risks. In case the risk is convex the algorithm is proved to converge to a stationary solution with accuracy ε with a rate O(1/λε) where λ is the regularization parameter of the objective function under the assumption of a Lipschitz empirical risk. In case the risk is not convex getting such a proof is more difficult and requires a stronger and more disputable assumption. Yet we provide experimental results on artificial test problems, and on five standard and difficult machine learning problems that are cast as convex and non-convex optimization problems that show how our algorithm compares well in practice with state of the art optimization algorithms.