From Oja's Algorithm to the Multiplicative Weights Update Method with Applications
This provides a theoretical link between two algorithms, potentially aiding analysis in optimization contexts, but appears incremental as it builds on known methods.
The paper connects Oja's algorithm to the multiplicative weights update method, showing that its regret can be bounded in terms of the latter when applied to sequences of symmetric matrices with common eigenvectors, and discusses applications to optimization with quadratic forms over the unit sphere.
Oja's algorithm is a well known online algorithm studied mainly in the context of stochastic principal component analysis. We make a simple observation, yet to the best of our knowledge a novel one, that when applied to a any (not necessarily stochastic) sequence of symmetric matrices which share common eigenvectors, the regret of Oja's algorithm could be directly bounded in terms of the regret of the well known multiplicative weights update method for the problem of prediction with expert advice. Several applications to optimization with quadratic forms over the unit sphere in $\reals^n$ are discussed.