An optimal randomized incremental gradient method
This provides a more efficient algorithm for large-scale machine learning and data analysis problems, though it is incremental as it builds on existing primal-dual frameworks.
The paper tackles finite-sum convex optimization by developing a randomized primal-dual gradient method that reduces gradient evaluations by a factor of O(√m) compared to deterministic methods, achieving optimal complexity with a new lower bound proof.
In this paper, we consider a class of finite-sum convex optimization problems whose objective function is given by the summation of $m$ ($\ge 1$) smooth components together with some other relatively simple terms. We first introduce a deterministic primal-dual gradient (PDG) method that can achieve the optimal black-box iteration complexity for solving these composite optimization problems using a primal-dual termination criterion. Our major contribution is to develop a randomized primal-dual gradient (RPDG) method, which needs to compute the gradient of only one randomly selected smooth component at each iteration, but can possibly achieve better complexity than PDG in terms of the total number of gradient evaluations. More specifically, we show that the total number of gradient evaluations performed by RPDG can be ${\cal O} (\sqrt{m})$ times smaller, both in expectation and with high probability, than those performed by deterministic optimal first-order methods under favorable situations. We also show that the complexity of the RPDG method is not improvable by developing a new lower complexity bound for a general class of randomized methods for solving large-scale finite-sum convex optimization problems. Moreover, through the development of PDG and RPDG, we introduce a novel game-theoretic interpretation for these optimal methods for convex optimization.