Online Learning from Finite Training Sets and Robustness to Input Bias

We analyze online gradient descent learning from finite training sets at noninfinitesimal learning rates eta. Exact results are obtained for the time-dependent generalization error of a simple model system: a linear network with a large number of weights N, trained on p = alphaN examples. This allows us to study in detail the effects of finite training set size alpha on, for example, the optimal choice of learning rate eta. We also compare online and offline learning, for respective optimal settings of eta at given final learning time. Online learning turns out to be much more robust to input bias and actually outperforms offline learning when such bias is present; for unbiased inputs, online and offline learning perform almost equally well.