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We establish conditions under which gradient descent applied to fixed-widthdeep networks drives the logistic loss to zero, and prove bounds on the rate ofconvergence. Our analysis applies for smoothed approximations to the ReLU, suchas Swish and the Huberized ReLU, proposed in previous applied work. We providetwo sufficient conditions for convergence. The first is simply a bound on theloss at initialization. The second is a data separation condition used in prioranalyses.

When does gradient descent with logistic loss interpolate using deep networks with smoothed ReLU activations?