Weight-Decay Turns Transformer Loss Landscapes Villani: Functional-Analytic Foundations for Optimization and Generalization

Weight decay is widely used as a regularizer in large language models, yet its precise role in shaping Transformer loss landscapes remains theoretically underexplored. This paper provides the first rigorous functional-analytic characterization of the standard Transformer objective--cross-entropy loss with $L^2$ regularization--by proving it satisfies Villani's criteria for coercive energy functions. Specifically, we show that the regularized loss $\mathcal{F}$ is infinitely differentiable, grows at least quadratically, has Gaussian-integrable tails, and satisfies the differential growth condition $-\Delta\mathcal{F} + \tfrac{1}{s}\|\nabla\mathcal{F}\|^{2} \to \infty$ as $\|\theta\| \to \infty$ for all $s>0$. From this structure, we derive explicit log-Sobolev and Poincar\'e constants $C_{\mathrm{LS}} \leq \lambda^{-1} + d/\lambda^{2}$, linking the regularization strength $\lambda$ and model dimension $d$ to finite-time convergence guarantees for noisy stochastic gradient descent and PAC-Bayesian generalization bounds that tighten with increasing $\lambda$. To validate our theory, we introduce a scalable Villani diagnostic $\Psi_s(\theta) = -\Delta \mathcal{F} + s^{-1}\|\nabla \mathcal{F}\|^2$ and estimate it efficiently using Hutchinson trace probes in models with over 100M parameters. Experiments on GPT-Neo-125M across Penn Treebank and WikiText-103 confirm the predicted quadratic growth of $\Psi_s$, spectral inflation of the Hessian, and exponential convergence behavior consistent with our log-Sobolev analysis. These results demonstrate that weight decay not only improves generalization empirically but also establishes the mathematical conditions required for fast Langevin mixing and theoretically grounded curvature-aware optimization in deep learning.