Escaping the Gravitational Pull of Softmax

The softmax is the standard transformation used in machine learning to map realvalued vectors to categorical distributions. Unfortunately, this transform poses serious drawbacks for gradient descent (ascent) optimization. We reveal this difficulty by establishing two negative results: (1) optimizing any expectation with respect to the softmax must exhibit sensitivity to parameter initialization ("softmax gravity well"), and (2) optimizing log-probabilities under the softmax must exhibit slow convergence ("softmax damping"). Both findings are based on an analysis of convergence rates using the Non-uniform Łojasiewicz (NŁ) inequalities. To circumvent these shortcomings we investigate an alternative transformation, the escort mapping, that demonstrates better optimization properties. The disadvantages of the softmax and the effectiveness of the escort transformation are further explained using the concept of NŁ coefficient. In addition to proving bounds on convergence rates to firmly establish these results, we also provide experimental evidence for the superiority of the escort transformation.