We consider stochastic gradient descent and its averaging variant for binary
classification problems in a reproducing kernel Hilbert space. In the
traditional analysis using a consistency property of loss functions, it is
known that the expected classification error converges more slowly than the
expected risk even when assuming a low-noise condition on the conditional label
probabilities. Consequently, the resulting rate is sublinear. Therefore, it is
important to consider whether much faster convergence of the expected
classification error can be achieved. In recent research, an exponential
convergence rate for stochastic gradient descent was shown under a strong
low-noise condition but provided theoretical analysis was limited to the
squared loss function, which is somewhat inadequate for binary classification
tasks. In this paper, we show an exponential convergence of the expected
classification error in the final phase of the stochastic gradient descent for
a wide class of differentiable convex loss functions under similar assumptions.
As for the averaged stochastic gradient descent, we show that the same
convergence rate holds from the early phase of training. In experiments, we
verify our analyses on the L2​-regularized logistic regression.Comment: 15 pages, 2 figure