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Escaping the Gravitational Pull of Softmax

Jincheng Mei
Chenjun Xiao
Lihong Li
Dale Schuurmans
Advances in Neural Information Processing Systems 33 (NeurIPS 2020)


The softmax is the standard transformation used in machine learning to map real-valued vectors to categorical distributions. Unfortunately, the softmax poses serious drawbacks for gradient descent optimization. We establish two negative results for this transform: (1) optimizing any expectation with respect to the softmax must exhibit extreme sensitivity to parameter initialization (``the 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 Lojasiewicz inequality. To circumvent these shortcomings we investigate an alternative transformation, the escort (p-norm) mapping, that demonstrates better optimization properties. 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.

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