Gradient Descent Averaging and Primal-dual Averaging for Strongly Convex Optimization
This work provides improved theoretical convergence guarantees for optimization algorithms, which is important for researchers and practitioners working with strongly convex problems in machine learning.
This paper addresses the suboptimal convergence rates of gradient descent methods for strongly convex optimization due to a logarithmic factor. The authors propose two new algorithms, Gradient Descent Averaging (GDA) and Primal-dual Averaging for Strongly Convex Cases (SC-PDA), which achieve optimal convergence rates for output averaging and individual convergence, respectively.
Averaging scheme has attracted extensive attention in deep learning as well as traditional machine learning. It achieves theoretically optimal convergence and also improves the empirical model performance. However, there is still a lack of sufficient convergence analysis for strongly convex optimization. Typically, the convergence about the last iterate of gradient descent methods, which is referred to as individual convergence, fails to attain its optimality due to the existence of logarithmic factor. In order to remove this factor, we first develop gradient descent averaging (GDA), which is a general projection-based dual averaging algorithm in the strongly convex setting. We further present primal-dual averaging for strongly convex cases (SC-PDA), where primal and dual averaging schemes are simultaneously utilized. We prove that GDA yields the optimal convergence rate in terms of output averaging, while SC-PDA derives the optimal individual convergence. Several experiments on SVMs and deep learning models validate the correctness of theoretical analysis and effectiveness of algorithms.