Fully Implicit Online Learning

Regularized online learning is widely used in machine learning applications. In this paper we analyze a class of regularized online algorithms without linearizing the loss function or the regularizer, which we call \emph{fully implicit online learning} (FIOL). We show that the FIOL algorithm admits a better regret than the linearization approximate algorithm if each iteration in FIOL can be solved exactly. Then we show that by exploring the structure of a large class of loss functions and regularizers, the computational complexity of FIOL in each iteration is comparable to its linearized part, even if no closed-form solution exists. Experiments validate the proposed approaches.