This talk addresses the challenge of machine learning generalization under heavy-tailed distributions, providing a theoretical foundation for the stable improvement of models such as Reinforcement Learning from Human Feedback (RLHF). We propose a novel heavy-tailed information-theoretic framework for generalization bounds. Traditional information-theoretic bounds typically control the generalization error using KL divergence, which relies on the existence of the moment generating function (MGF) of the loss or reward, or on sub-Gaussian tail assumptions. However, in modern machine learning paradigms such as RLHF and stochastic optimization, losses and rewards often are heavy-tailed, where the MGF may not exist, rendering KL-based tools ineffective. Even when the KL mutual information is small, rare extreme events can dominate generalization performance. We show that under heavy-tailed rewards, KL divergence-based trust regions in RLHF can suffer from the "Catastrophic Goodhart effect" — where the proxy reward can blow up without bound even when the KL constraint is small. In contrast, RLHF with Rényi regularization effectively controls reward inflation under heavy-tailed sub-Weibull rewards. Furthermore, for the Best-of-n strategy, we show that Rényi-regularized alignment provides finite reward guarantees and ensures that best-of-N policies remain well-controlled. Finally, we successfully apply Rényi regularization to Direct Preference Optimization (DPO) for LLMs.