Monotone polygon

Separable space

Discrete mathematics

Statistics

Descent direction

Applied mathematics

Regularization (linguistics)

Convergence (economics)

Gradient descent

Economics

Computer science

Mathematical optimization

Regression

Mathematics

Mathematical analysis

Artificial intelligence

Geometry

Infimum and supremum

Logarithm

Economic growth

Artificial neural network

Margin (machine learning)

Machine learning

Journal of Machine Learning Research

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 in more complex models and with other optimization methods.

The implicit bias of gradient descent on separable data