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In this paper, we consider unregularized online learning algorithms in aReproducing Kernel Hilbert Spaces (RKHS). Firstly, we derive explicitconvergence rates of the unregularized online learning algorithms forclassification associated with a general gamma-activating loss (see Definition1 in the paper). Our results extend and refine the results in Ying and Pontil(2008) for the least-square loss and the recent result in Bach and Moulines(2011) for the loss function with a Lipschitz-continuous gradient. Moreover, weestablish a very general condition on the step sizes which guarantees theconvergence of the last iterate of such algorithms. Secondly, we establish, forthe first time, the convergence of the unregularized pairwise learningalgorithm with a general loss function and derive explicit rates under theassumption of polynomially decaying step sizes. Concrete examples are used toillustrate our main results. The main techniques are tools from convexanalysis, refined inequalities of Gaussian averages, and an induction approach.

Unregularized Online Learning Algorithms with General Loss Functions