Linear Convergence of Accelerated Stochastic Gradient Descent for Nonconvex Nonsmooth Optimization
This provides a theoretical guarantee for faster convergence in stochastic optimization for nonconvex problems, which is incremental but addresses a known bottleneck in machine learning.
The paper tackles the problem of nonconvex nonsmooth optimization by proposing an accelerated stochastic gradient descent method that combines variance reduction and Nesterov's extrapolation, proving linear convergence to a stationary point with numerical validation.
In this paper, we study the stochastic gradient descent (SGD) method for the nonconvex nonsmooth optimization, and propose an accelerated SGD method by combining the variance reduction technique with Nesterov's extrapolation technique. Moreover, based on the local error bound condition, we establish the linear convergence of our method to obtain a stationary point of the nonconvex optimization. In particular, we prove that not only the sequence generated linearly converges to a stationary point of the problem, but also the corresponding sequence of objective values is linearly convergent. Finally, some numerical experiments demonstrate the effectiveness of our method. To the best of our knowledge, it is first proved that the accelerated SGD method converges linearly to the local minimum of the nonconvex optimization.