We examine gradient descent on unregularized logistic regression problems, with homogeneous linear predictors on linearly separable datasets. We show the predictor converges to the direction of the max-margin (hard margin SVM) solution. The result also generalizes to other monotone decreasing loss functions with an infimum at infinity, to multi-class problems, and to training a weight layer in a deep network in a certain restricted setting. Furthermore, we show this convergence is very slow, and only logarithmic in the convergence of the loss itself. This can help explain the benefit of continuing to optimize the logistic or cross-entropy loss even after the training error is zero and the training loss is extremely small, and, as we show, even if the validation loss increases. Our methodology can also aid in understanding implicit regularization n more complex models and with other optimization methods.
- Pub Date:
- October 2017
- Statistics - Machine Learning;
- Computer Science - Machine Learning
- Final JMLR version, with improved discussions over v3. Main improvements in journal version over conference version (v2 appeared in ICLR): We proved the measure zero case for main theorem (with implications for the rates), and the multi-class case