Convergence Analysis of Block Coordinate Algorithms with Determinantal Sampling
This work addresses a theoretical bottleneck in optimization algorithms for researchers, offering incremental improvements in convergence analysis through a novel sampling technique.
The paper tackles the challenge of analyzing the convergence rate of a randomized Newton-like method for convex optimization by introducing determinantal sampling for coordinate blocks, showing that this approach yields a convergence rate dependent solely on the eigenvalue distribution of the Hessian-over-approximation matrix and provides analytical tractability.
We analyze the convergence rate of the randomized Newton-like method introduced by Qu et. al. (2016) for smooth and convex objectives, which uses random coordinate blocks of a Hessian-over-approximation matrix $\bM$ instead of the true Hessian. The convergence analysis of the algorithm is challenging because of its complex dependence on the structure of $\bM$. However, we show that when the coordinate blocks are sampled with probability proportional to their determinant, the convergence rate depends solely on the eigenvalue distribution of matrix $\bM$, and has an analytically tractable form. To do so, we derive a fundamental new expectation formula for determinantal point processes. We show that determinantal sampling allows us to reason about the optimal subset size of blocks in terms of the spectrum of $\bM$. Additionally, we provide a numerical evaluation of our analysis, demonstrating cases where determinantal sampling is superior or on par with uniform sampling.