Convergence Analysis of Block Coordinate Algorithms with Determinantal Sampling

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

Sampling from a kk-DPP without looking at all items

Determinantal point processes (DPPs) are a useful probabilistic model for selecting a small diverse subset out of a large collection of items, with applications in summarization, stochastic optimization, active learning and more. Given a kernel function and a subset size $k$, our goal is to sample $k$ out of $n$ items with probability proportional to the determinant of the kernel matrix induced by the subset (a.k.a. $k$-DPP). Existing $k$-DPP sampling algorithms require an expensive preprocessing step which involves multiple passes over all $n$ items, making it infeasible for large datasets. A na\"ive heuristic addressing this problem is to uniformly subsample a fraction of the data and perform $k$-DPP sampling only on those items, however this method offers no guarantee that the produced sample will even approximately resemble the target distribution over the original dataset. In this paper, we develop an algorithm which adaptively builds a sufficiently large uniform sample of data that is then used to efficiently generate a smaller set of $k$ items, while ensuring that this set is drawn exactly from the target distribution defined on all $n$ items. We show empirically that our algorithm produces a $k$-DPP sample after observing only a small fraction of all elements, leading to several orders of magnitude faster performance compared to the state-of-the-art