The Alternating Direction Method of Multipliers (ADMM) has been studied for years. The traditional ADMM algorithm needs to compute, at each iteration, an (empirical) expected loss function on all training examples, resulting in a computational complexity proportional to the number of training examples. To reduce the time complexity, stochastic ADMM algorithms were proposed to replace the expected function with a random loss function associated with one uniformly drawn example plus a Bregman divergence. The Bregman divergence, however, is derived from a simple second order proximal function, the half squared norm, which could be a suboptimal choice. In this paper, we present a new family of stochastic ADMM algorithms with optimal second order proximal functions, which produce a new family of adaptive subgradient methods. We theoretically prove that their regret bounds are as good as the bounds which could be achieved by the best proximal function that can be chosen in hindsight. Encouraging empirical results on a variety of real-world datasets confirm the effectiveness and efficiency of the proposed algorithms.
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