Accelerating Stochastic Alternating Direction Method of Multipliers with Adaptive Subgradient

The Alternating Direction Method of Multipliers (ADMM) has been studied for years, since it can be applied to many large-scale and data-distributed machine learning tasks. The traditional ADMM algorithm needs to compute an (empirical) expected loss function on all the training examples for each iteration, which results in a computational complexity propositional to the number of training examples. To reduce the time complexity, stochastic ADMM algorithm is proposed to replace the expected loss function by a random loss function associated with one single uniformly drawn example and Bregman divergence for a second order proximal function. The Bregman divergence in the original stochastic ADMM algorithm is derived from half squared norm, which could be a suboptimal choice. In this paper, we present a new stochastic ADMM algorithm, using Bregman divergence derived from second order proximal functions associated with iteratively updated matrices. Our new stochastic ADMM produces a new family of adaptive subgradient methods. We theoretically prove that their regret bounds are as good as the bounds achieved by the best proximal functions that can be chosen in hindsight. Encouraging results confirm the effectiveness and efficiency of the proposed algorithms.