Implicit MLE: Backpropagating Through Discrete Exponential Family Distributions (Paper Explained)

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#imle #backpropagation #discrete

Backpropagation is the workhorse of deep learning, but unfortunately, it only works for continuous functions that are amenable to the chain rule of differentiation. Since discrete algorithms have no continuous derivative, deep networks with such algorithms as part of them cannot be effectively trained using backpropagation. This paper presents a method to incorporate a large class of algorithms, formulated as discrete exponential family distributions, into deep networks and derives gradient estimates that can easily be used in end-to-end backpropagation. This enables things like combinatorial optimizers to be part of a network's forward propagation natively.

OUTLINE:

0:00 - Intro & Overview

4:25 - Sponsor: Weights & Biases

6:15 - Problem Setup & Contributions

8:50 - Recap: Straight-Through Estimator

13:25 - Encoding the discrete problem as an inner product

19:45 - From algorithm to distribution

23:15 - Substituting the gradient

26:50 - Defining a target distribution

38:30 - Approximating marginals via perturb-and-MAP

45:10 - Entire algorithm recap

56:45 - Github Page & Example

Paper: https://arxiv.org/abs/2106.01798

Code (TF): https://github.com/nec-research/tf-imle

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Abstract:

Combining discrete probability distributions and combinatorial optimization problems with neural network components has numerous applications but poses several challenges. We propose Implicit Maximum Likelihood Estimation (I-MLE), a framework for end-to-end learning of models combining discrete exponential family distributions and differentiable neural components. I-MLE is widely applicable as it only requires the ability to compute the most probable states and does not rely on smooth relaxations. The framework encompasses several approaches such as perturbation-based implicit differentiation and recent methods to differentiate through black-box combinatorial solvers. We introduce a novel class of noise distributions for approximating marginals via perturb-and-MAP. Moreover, we show that I-MLE simplifies to maximum likelihood estimation when used in some recently studied learning settings that involve combinatorial solvers. Experiments on several datasets suggest that I-MLE is competitive with and often outperforms existing approaches which rely on problem-specific relaxations.

Authors: Mathias Niepert, Pasquale Minervini, Luca Franceschi

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