On the Properties of the Softmax Function with Application in Game Theory and Reinforcement Learning
This provides incremental theoretical insights for researchers in machine learning and game theory, enhancing understanding of softmax behavior.
The paper derived new mathematical properties of the softmax function, showing it is the monotone gradient map of log-sum-exp and linking the inverse temperature parameter to Lipschitz and co-coercivity properties, with an application in game-theoretic reinforcement learning.
In this paper, we utilize results from convex analysis and monotone operator theory to derive additional properties of the softmax function that have not yet been covered in the existing literature. In particular, we show that the softmax function is the monotone gradient map of the log-sum-exp function. By exploiting this connection, we show that the inverse temperature parameter determines the Lipschitz and co-coercivity properties of the softmax function. We then demonstrate the usefulness of these properties through an application in game-theoretic reinforcement learning.