Dimensionality Reduction for Stationary Time Series via Stochastic Nonconvex Optimization
This addresses the problem of efficient stochastic optimization for dependent data in machine learning, though it appears incremental as it builds on existing methods like Oja's algorithm.
The paper tackled the challenge of performing dimensionality reduction via streaming PCA on stationary time series data with nonconvex objectives, proposing a variant of Oja's algorithm with downsampling that achieved near optimal asymptotic sample complexity.
Stochastic optimization naturally arises in machine learning. Efficient algorithms with provable guarantees, however, are still largely missing, when the objective function is nonconvex and the data points are dependent. This paper studies this fundamental challenge through a streaming PCA problem for stationary time series data. Specifically, our goal is to estimate the principle component of time series data with respect to the covariance matrix of the stationary distribution. Computationally, we propose a variant of Oja's algorithm combined with downsampling to control the bias of the stochastic gradient caused by the data dependency. Theoretically, we quantify the uncertainty of our proposed stochastic algorithm based on diffusion approximations. This allows us to prove the asymptotic rate of convergence and further implies near optimal asymptotic sample complexity. Numerical experiments are provided to support our analysis.