Regularized Diffusion Adaptation via Conjugate Smoothing

--The purpose of this work is to develop and study a distributed strategy for Pareto optimization of an aggregate cost consisting of regularized risks. Each risk is modeled as the expectation of some loss function with unknown probability distribution while the regularizers are assumed deterministic, but are not required to be differentiable or even continuous. The individual, regularized, cost functions are distributed across a strongly-connected network of agents and the Pareto optimal solution is sought by appealing to a multi-agent diffusion strategy. T o this end, the regularizers are smoothed by means of infimal convolution and it is shown that the Pareto solution of the approximate, smooth problem can be made arbitrarily close to the solution of the original, non-smooth problem. Performance bounds are established under conditions that are weaker than assumed before in the literature, and hence applicable to a broader class of adaptation and learning problems. Index T erms --Distributed optimization, diffusion strategy, smoothing, proximal operator, non-smooth regularizer, proximal diffusion, regularized diffusion. The objective of distributed learning is the solution of global, stochastic optimization problems across networks of agents through localized interactions and without information about the statistical properties of the data. Using streaming data, the resulting strategies are adaptive in nature and able to track drifts in the location of the minimizers due to variations in the statistical properties of the data. Regularization is one useful technique to encourage or enforce structural properties on the sought after minimizer, such as sparsity or constraints. A substantial number of regularizers are inherently non-smooth, while many cost functions are differentiable.