Logit Dynamics in Softmax Policy Gradient Methods

We analyzes the logit dynamics of softmax policy gradient methods. We derive the exact formula for the L2 norm of the logit update vector: $$\|\Delta \mathbf{z}\|_2 \propto \sqrt{1-2P_c + C(P)}$$ This equation demonstrates that update magnitudes are determined by the chosen action's probability ($P_c$) and the policy's collision probability ($C(P)$), a measure of concentration inversely related to entropy. Our analysis reveals an inherent self-regulation mechanism where learning vigor is automatically modulated by policy confidence, providing a foundational insight into the stability and convergence of these methods.