Differentiable Dynamic Programming for Structured Prediction and Attention

TL;DR

This paper introduces a differentiable approach to dynamic programming by smoothing the max operator, enabling its integration into neural networks for structured prediction and attention mechanisms.

Contribution

It proposes a novel smoothing technique for DP algorithms, making them differentiable and applicable within neural network training via backpropagation.

Findings

Smoothed Viterbi algorithm for sequence prediction.

Smoothed DTW algorithm for time-series alignment.

Effective structured prediction and attention in neural machine translation.

Abstract

Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.