Gradient descent follows the regularization path for general losses
This addresses the implicit bias in gradient descent for machine learning practitioners, offering theoretical insights into loss function selection, though it is incremental as it builds on prior work about regularization paths.
The paper shows that for linear predictors with convex, strictly decreasing losses, gradient descent paths converge to the same direction as regularization paths when the risk does not reach its infimum, providing a justification for exponentially-tailed losses like logistic loss, which lead to maximum-margin solutions, unlike polynomially-tailed losses that may result in poor margins.
Recent work across many machine learning disciplines has highlighted that standard descent methods, even without explicit regularization, do not merely minimize the training error, but also exhibit an implicit bias. This bias is typically towards a certain regularized solution, and relies upon the details of the learning process, for instance the use of the cross-entropy loss. In this work, we show that for empirical risk minimization over linear predictors with arbitrary convex, strictly decreasing losses, if the risk does not attain its infimum, then the gradient-descent path and the algorithm-independent regularization path converge to the same direction (whenever either converges to a direction). Using this result, we provide a justification for the widely-used exponentially-tailed losses (such as the exponential loss or the logistic loss): while this convergence to a direction for exponentially-tailed losses is necessarily to the maximum-margin direction, other losses such as polynomially-tailed losses may induce convergence to a direction with a poor margin.