Mini-Batch Stochastic ADMMs for Nonconvex Nonsmooth Optimization
This addresses optimization challenges for machine learning practitioners dealing with complex data, but it is incremental as it extends existing stochastic ADMM methods to mini-batch settings.
The paper tackles the problem of large-scale nonconvex nonsmooth optimization in machine learning by proposing mini-batch stochastic ADMMs, proving convergence rates of O(1/T) for stationary points and demonstrating effectiveness through experiments on simulated and real-world data.
With the large rising of complex data, the nonconvex models such as nonconvex loss function and nonconvex regularizer are widely used in machine learning and pattern recognition. In this paper, we propose a class of mini-batch stochastic ADMMs (alternating direction method of multipliers) for solving large-scale nonconvex nonsmooth problems. We prove that, given an appropriate mini-batch size, the mini-batch stochastic ADMM without variance reduction (VR) technique is convergent and reaches a convergence rate of $O(1/T)$ to obtain a stationary point of the nonconvex optimization, where $T$ denotes the number of iterations. Moreover, we extend the mini-batch stochastic gradient method to both the nonconvex SVRG-ADMM and SAGA-ADMM proposed in our initial manuscript \cite{huang2016stochastic}, and prove these mini-batch stochastic ADMMs also reaches the convergence rate of $O(1/T)$ without condition on the mini-batch size. In particular, we provide a specific parameter selection for step size $η$ of stochastic gradients and penalty parameter $ρ$ of augmented Lagrangian function. Finally, extensive experimental results on both simulated and real-world data demonstrate the effectiveness of the proposed algorithms.