Optimization in Machine Learning: a Distribution-Space Approach

We present the viewpoint that optimization problems encountered in machine learning can often be interpreted as minimizing a convex functional over a function space, but with a non-convex constraint set introduced by model parameterization. This observation allows us to repose such problems via a suitable relaxation as convex optimization problems in the space of distributions over the training parameters. We derive some simple relationships between the distribution-space problem and the original problem, e.g., a distribution-space solution is at least as good as a solution in the original space. Moreover, we develop a numerical algorithm based on mixture distributions to perform approximate optimization directly in the distribution space. Consistency of this approximation is established and the numerical efficacy of the proposed algorithm is illustrated in simple examples. In both theory and practice, this formulation provides an alternative approach to large-scale optimization in machine learning.