  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.
