Estimating the principal eigenvector of a stochastic matrix: Mirror Descent Algorithms via game approach with application to PageRank problem

The problem of estimating the principal eigenvector related to the largest eigenvalue of a given (left) stochastic matrix A has many applications in ranking search results, multi-agent consensus, networked control and data mining. The well-known power method is a typical tool, but it modifies matrix and, therefore, the related solution. We propose and study both deterministic and randomized game algorithms based on Mirror Descent (MD) method which are intended for bounding the Euclidean norm residual ∥Ax-x∥₂ on the standard simplex in ℝ^N. We prove the explicit uniform upper bounds of type O(√ln(N)/n) with arbitrary horizon n ≥ 1. They improve the similar earlier results with respect to n which have been proved for the squared norm residual, i.e. ∥Ax-x∥₂². Numerical results for N = 100 illustrate the general decrease of the norm residual ∥Ax̂_t-x̂_t∥₂ in time t and corroborate theoretical results.