Relu and softplus neural nets as zero-sum turn-based games

TL;DR

This paper interprets ReLU and Softplus neural networks as zero-sum turn-based games, providing a game-theoretic framework for understanding, analyzing, and verifying neural network outputs and robustness properties.

Contribution

It introduces a novel game-theoretic interpretation of ReLU and Softplus neural networks, connecting network evaluation to backward recursion in a zero-sum game and enabling robustness analysis.

Findings

Network outputs can be represented as values of a zero-sum game.

The approach provides bounds on network outputs based on input bounds.

Training can be viewed as an inverse game problem.

Abstract

We show that the output of a ReLU neural network can be interpreted as the value of a zero-sum, turn-based, stopping game, which we call the ReLU net game. The game runs in the direction opposite to that of the network, and the input of the network serves as the terminal reward of the game. In fact, evaluating the network is the same as running the Shapley-Bellman backward recursion for the value of the game. Using the expression of the value of the game as an expected total payoff with respect to the path measure induced by the transition probabilities and a pair of optimal policies, we derive a discrete Feynman-Kac-type path-integral formula for the network output. This game-theoretic representation can be used to derive bounds on the output from bounds on the input, leveraging the monotonicity of Shapley operators, and to verify robustness properties using policies as certificates. Moreover, training the neural network becomes an inverse game problem: given pairs of terminal rewards and corresponding values, one seeks transition probabilities and rewards of a game that reproduces them. Finally, we show that a similar approach applies to neural networks with Softplus activation functions, where the ReLU net game is replaced by its entropic regularization.