Duality between subgradient and conditional gradient methods

Many problems in machine learning, statistics and signal processing may be cast as convex optimization problems. In large-scale situations, simple gradient-based algorithms with potentially many cheap iterations are often preferred over methods, such as Newton's method or interior-point methods, that rely on fewer but more expensive iterations. The choice of a first-order method depends on the structure of the problem, in particular (a) the smoothness and/or strong convexity of the objective function, and (b) the computational efficiency of certain operations related to the non-smooth parts of the objective function, when it is decomposable in a smooth and a non-smooth part. In this paper, we consider two classical algorithms, namely (a) subgradient descent and its mirror descent extension [29, 24, 4], and (b) conditional gradient algorithms, sometimes referred to as Frank-Wolfe algorithms [16, 13, 15, 14, 19]. Subgradient algorithms are adapted to non-smooth unstructured situations, and after t steps have a convergence rate of O(1/ t) in terms of objective values.