Lexicographic and Depth-Sensitive Margins in Homogeneous and Non-Homogeneous Deep Models

With an eye toward understanding complexity control in deep learning, we study how infinitesimal regularization or gradient descent optimization lead to margin maximizing solutions in both homogeneous and non homogeneous models, extending previous work that focused on infinitesimal regularization only in homogeneous models. To this end we study the limit of loss minimization with a diverging norm constraint (the constrained path''), relate it to the limit of amargin path'' and characterize the resulting solution. For non-homogeneous models we show that this solution is biased toward the deepest part of the model, discarding the shallowest parts if they are unnecessary. For homogeneous models, we show convergence to a ``lexicographic max margin solution'', and provide conditions under which max margin solutions are also attained as the limit of unconstrained gradient descent.