BEAR: Sketching BFGS Algorithm for Ultra-High Dimensional Feature Selection in Sublinear Memory

We consider feature selection for applications in machine learning where the dimensionality of the data is so large that it exceeds the working memory of the (local) computing machine. Unfortunately, current large-scale sketching algorithms show poor memory-accuracy trade-off in selecting features in high dimensions due to the irreversible collision and accumulation of the stochastic gradient noise in the sketched domain. Here, we develop a second-order feature selection algorithm, called BEAR, which avoids the extra collisions by efficiently storing the second-order stochastic gradients of the celebrated Broyden-Fletcher-Goldfarb-Shannon (BFGS) algorithm in Count Sketch, using a memory cost that grows sublinearly with the size of the feature vector. BEAR reveals an unexplored advantage of second-order optimization for memory-constrained high-dimensional gradient sketching. Our extensive experiments on several real-world data sets from genomics to language processing demonstrate that BEAR requires up to three orders of magnitude less memory space to achieve the same classification accuracy compared to the first-order sketching algorithms with a comparable run time. Our theoretical analysis further proves the global convergence of BEAR with O(1/t) rate in t iterations of the sketched algorithm.