Diffusion stochastic optimization with non-smooth regularizers

We develop an effective distributed strategy for seeking the Pareto solution of an aggregate cost consisting of regularized risks. The focus is on stochastic optimization problems where each risk function is expressed as the expectation of some loss function and the probability distribution of the data is unknown. We assume each risk function is regularized and allow the regularizer to be non-smooth. Under conditions that are weaker than assumed earlier in the literature and, hence, applicable to a broader class of adaptation and learning problems, we show how the regularizers can be smoothed and how the Pareto solution can be sought by appealing to a multi-agent diffusion strategy. The formulation is general enough and includes, for example, a multi-agent proximal strategy as a special case.