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Escaping the Gravitational Pull of Softmax
Jincheng Mei · Chenjun Xiao · Bo Dai · Lihong Li · Csaba Szepesvari · Dale Schuurmans

Tue Dec 08 09:00 AM -- 11:00 AM (PST) @ Poster Session 1 #251

The softmax is the standard transformation used in machine learning to map real-valued 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 \L{}ojasiewicz (N\L{}) inequalities. To circumvent these shortcomings we investigate an alternative transformation, the \emph{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 N\L{} 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.

Author Information

Jincheng Mei (University of Alberta / Google Brain)
Chenjun Xiao (University of Alberta)
Bo Dai (Google Brain)
Lihong Li (Amazon)
Csaba Szepesvari (DeepMind / University of Alberta)
Dale Schuurmans (Google Brain & University of Alberta)

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