Accelerated and Sparse Algorithms for Approximate Personalized PageRank and Beyond

TL;DR

This paper introduces accelerated and conjugate direction algorithms for sparse approximate personalized PageRank, improving convergence speed and efficiency while maintaining sparsity, applicable to broader quadratic optimization problems.

Contribution

It presents the first accelerated sparse algorithm for personalized PageRank and a conjugate directions method for exact solutions, addressing open questions and extending to general quadratic objectives.

Findings

Accelerated algorithm reduces dependence on condition number.

Conjugate directions method achieves exact solutions efficiently.

Both methods outperform previous approaches in certain regimes.

Abstract

It has recently been shown that ISTA, an unaccelerated optimization method, presents sparse updates for the $\ell_1$-regularized personalized PageRank problem, leading to cheap iteration complexity and providing the same guarantees as the approximate personalized PageRank algorithm (APPR) [FRS+19]. In this work, we design an accelerated optimization algorithm for this problem that also performs sparse updates, providing an affirmative answer to the COLT 2022 open question of [FY22]. Acceleration provides a reduced dependence on the condition number, while the dependence on the sparsity in our updates differs from the ISTA approach. Further, we design another algorithm by using conjugate directions to achieve an exact solution while exploiting sparsity. Both algorithms lead to faster convergence for certain parameter regimes. Our findings apply beyond PageRank and work for any quadratic objective whose Hessian is a positive-definite $M$-matrix.