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Recent work across many machine learning disciplines has highlighted thatstandard descent methods, even without explicit regularization, do not merelyminimize the training error, but also exhibit an implicit bias. This bias istypically towards a certain regularized solution, and relies upon the detailsof the learning process, for instance the use of the cross-entropy loss.  In this work, we show that for empirical risk minimization over linearpredictors with arbitrary convex, strictly decreasing losses, if the risk doesnot attain its infimum, then the gradient-descent path and thealgorithm-independent regularization path converge to the same direction(whenever either converges to a direction). Using this result, we provide ajustification for the widely-used exponentially-tailed losses (such as theexponential loss or the logistic loss): while this convergence to a directionfor exponentially-tailed losses is necessarily to the maximum-margin direction,other losses such as polynomially-tailed losses may induce convergence to adirection with a poor margin.

Gradient descent follows the regularization path for general losses