Incremental Randomized Sketching for Online Kernel Learning

Randomized sketching has been used in offline kernel learning, but it cannot be applied directly to online kernel learning due to the lack of incremental maintenances for randomized sketches with regret guarantees. To address these issues, we propose a novel incremental randomized sketching approach for online kernel learning, which has efficient incremental maintenances with theoretical guarantees. We construct two incremental randomized sketches using the sparse transform matrix and the sampling matrix for kernel matrix approximation, update the incremental randomized sketches using rank-$1$ modifications, and construct an time-varying explicit feature mapping for online kernel learning. We prove that the proposed incremental randomized sketching is statistically unbiased for the matrix product approximation, obtains a $1 + \epsilon$ relative-error bound for the kernel matrix approximation, enjoys a sublinear regret bound for online kernel learning, and has constant time and space complexities at each round for incremental maintenances. Experimental results demonstrate that the incremental randomized sketching achieves a better learning performance in terms of accuracy and efficiency even in adversarial environments.