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Linear Convergence with Condition Number Independent Access of Full Gradients
Lijun Zhang · Mehrdad Mahdavi · Rong Jin

Thu Dec 05 07:00 PM -- 11:59 PM (PST) @ Harrah's Special Events Center, 2nd Floor
For smooth and strongly convex optimization, the optimal iteration complexity of the gradient-based algorithm is $O(\sqrt{\kappa}\log 1/\epsilon)$, where $\kappa$ is the conditional number. In the case that the optimization problem is ill-conditioned, we need to evaluate a larger number of full gradients, which could be computationally expensive. In this paper, we propose to reduce the number of full gradient required by allowing the algorithm to access the stochastic gradients of the objective function. To this end, we present a novel algorithm named Epoch Mixed Gradient Descent (EMGD) that is able to utilize two kinds of gradients. A distinctive step in EMGD is the mixed gradient descent, where we use an combination of the gradient and the stochastic gradient to update the intermediate solutions. By performing a fixed number of mixed gradient descents, we are able to improve the sub-optimality of the solution by a constant factor, and thus achieve a linear convergence rate. Theoretical analysis shows that EMGD is able to find an $\epsilon$-optimal solution by computing $O(\log 1/\epsilon)$ full gradients and $O(\kappa^2\log 1/\epsilon)$ stochastic gradients.

Author Information

Lijun Zhang (Nanjing University (NJU))
Mehrdad Mahdavi (Michigan State University (MSU))
Rong Jin (Michigan State University (MSU))

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