Relu and softplus neural nets as zero-sum turn-based games
This provides a novel game-theoretic framework for analyzing and training neural networks, which could benefit researchers in machine learning theory and optimization, though it appears incremental as it builds on existing game theory concepts.
The authors tackled the problem of interpreting neural network outputs by showing that ReLU and Softplus networks can be represented as zero-sum turn-based games, where network evaluation corresponds to game value computation. This representation enables deriving path-integral formulas, bounding outputs, verifying robustness, and framing training as an inverse game problem.
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.