Accelerating Frank Wolfe using Jacobi polynomials

Robin Francis

Constrained optimization provides a general framework in which a wide variety of design criteria, specifications and limitations on available resources can be imposed on the desired solution. An real life example for constraint optimization are recommender systems in Netflix, Amazon and YouTube, aimed at suggesting relevant items to the user. Frank-Wolfe (FW) method has gained much attention recently due to a significant reduction in computational complexity as compared to projected gradient descent methods. The reduction in computational complexity of FW is because it minimizes an affine function as compared to projecting onto the constraint set. The FW algorithm can significantly reduce the computational complexity but suffers from sublinear convergence of order O(1/k). The sublinear convergence of the FW algorithm can be attributed to the zig-zagging nature of the FW updates as we approach the optimal solution due to absence of an regularizer in the linear minimization step. The momentum technique can improve the convergence to sublinear of order O(1/k2 ) for a subclass of problems. On unrolling the FW iterates leads to an polynomial recursion and the Jacobi polynomials are seen to be the optimal polynomials that can ensure that at each update is close to the optimal path connecting the initial point and the optimal point. For smooth convex and strongly convex functions the Jacobi FW algorithm has convergence of order O(1/k2 ). The Jacobi FW method is able achieve linear rates of convergences for some smooth, strongly convex functions over convex and compact constrained set with minimizer in the interior and for strongly convex and compact constrained sets. I am truely honoured to have been awarded with the CISCO CNI fellowship for the term August 2020 to June 2021. I’m continuing for my PhD degree under Sundeep Prabhakar Chepuri and we are working on this project to submit it to ICASSP 2022 and in Transactions on Signal Processing.