2019/10/10 by Adil Salim, Salim, Adil · 1 citation
Decision Sciences · Mathematics · #Approximation Theory and Sequence Spaces #FOS: Mathematics #Mathematical Approximation and Integration #Optimization and Control (math.OC) #Probability (math.PR) #Risk and Portfolio Optimization
paper · pdf · doi:10.48550/arxiv.1910.04405
openalex publication_date 2019/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Random monotone operators are stochastic versions of maximal monotone operators which play an important role in stochastic nonsmooth optimization. Several stochastic nonsmooth optimization algorithms have been shown to converge to a zero of a mean operator defined as the expectation, in the sense of the Aumann integral, of a random monotone operator. In this note, we prove a strong law of large numbers for random monotone operators where the limit is the mean operator. We apply this result to the empirical risk minimization problem appearing in machine learning. We show that if the empirical risk minimizers converge as the number of data points goes to infinity, then they converge to an expected risk minimizer.