Rethinking GSPO: The Perplexity-Entropy Equivalence

Abstract: We provide a new perspective on GSPO's length-normalized importance ratios by establishing their connection to information-theoretic quantities. We show that GSPO's sequence-level weight $s(\theta) = (\pi_\theta/\pi_{\theta_{\text{old}}})^{1/|y|}$ can be equivalently expressed as the inverse perplexity ratio $\text{PPL}{\theta{\text{old}}}/\text{PPL}_\theta$ and as the exponential cross-entropy change $\exp(\Delta H)$. While the perplexity-entropy relationship follows from standard definitions, this observation provides a useful lens for understanding GSPO: the algorithm weights policy gradient updates by perplexity ratios, offering an information-theoretic interpretation of the importance weights. This perspective helps explain GSPO's empirical properties, including log-domain variance reduction through geometric averaging and stability in training mixture-of-experts models. We validate the mathematical equivalences and variance predictions through controlled experiments on mathematical reasoning tasks.