Generalization Bounds on Optimal Control for Transformer Training and Wasserstein Distributional Robustness

We derive finite-sample generalization bounds for Transformers trained with dynamic programming recursions. Building on the doubly lifted, measure-valued formulation of Transformer dynamics, we view data sets as probability laws on pairs of empirical input-output measures, allowing us to interpret the training problem as a finite-horizon Markovian control problem. We then analyze a quantized model, derived by quantizing the state, action, and measure-state spaces, and derive explicit finite-sample generalization bounds using concentration inequalities for empirical laws on finite metric spaces together with a Lipschitz stability estimate for the value function. These bounds are transferred to the base model at the cost of an explicit approximation error. Finally, we show that the same machinery yields a distributionally robust control formulation of the training problem, connecting Transformer generalization to Wasserstein distributionally robust optimization.

Sharp Stability Threshold and Certification for Designing Stable Residual Architectures

We propose \emph{the sublinear-growth principle} for deep residual architectures -- a sharp stability threshold on the input-magnitude exponent of every residual block's velocity field: $$\|v(x, t)\| \leq c\,\|x\|^q + b, \qquad q \in [0, 1].$$ The threshold $q = 1$ is established via two independent arguments. Classical ODE theory gives a global forward flow on $[0, T]$ at $q \le 1$ and exhibits divergent velocity fields at any $q > 1$. The optimal-control analysis, via the Hamilton-Jacobi-Bellman equation, sharpens this to a selection statement: the training optimum is bang-bang on the boundary of the admissible class, so the optimum at $q > 1$ blows up while the optimum at $q \le 1$ is safe by construction. The exponent criterion $q \le 1$ is thereby a necessary and sufficient condition for stable training. It clarifies architectural placements that ensure the stability of training and inference, explaining, for instance, the stabilizing role of layer normalization. The sublinear-growth velocity fields form \emph{the right function space} on which forward dynamics, adjoint sensitivity, and architectural composition are all well-controlled. An arithmetic of input-magnitude exponents under the five operations that build residual blocks enables efficient certification of $q_k \le 1$ at the level of architectural primitives, in place of ad hoc trial and error in the search for stable neural architectural designs. A parameter-free modification reduces the supercritical Mamba block from $q = 5$ to $q = 1$ without layer normalization, demonstrating this point. Experiments on Mamba and PatchTST confirm that the $q \le 1$ variants train stably: the criterion is the input-magnitude exponent, not the presence of a normalization layer.

Noise Schedule Design for Diffusion Models: An Optimal Control Perspective

We develop a principled framework for analyzing and designing noise schedules in diffusion models. We show that one can recast this design problem as an optimal control problem, whose state is the Fisher information of the diffusion process which evolves according to an ODE and the control input is the noise schedule. The objective of the optimal control problem is a functional involving the Fisher information, which is shown to be an upper bound on the Kullback-Leibler sampling error. By solving this optimal control problem, we obtain sufficient conditions on noise schedules under which state-of-the-art $\tilde{\mathcal{O}} (d/n)$ sampling error is achievable, where $d$ is the data dimension and $n$ is the number of discretization steps. While existing theoretical work also prove that $\tilde{\mathcal{O}}(d/n)$ sampling error bounds are achievable, these results hold for specific noise schedules, which do not include the schedules used in practice. Under a further parametric assumption on the data distribution, we show that one can obtain closed-form expressions for the noise schedules. These noise schedules generalize standard empirical schedules such as exponential and sigmoid schedules by allowing additional parameters that can be tuned. Systematically tuning the parameters of these schedules yields new schedules that achieve superior FID scores on image generation benchmarks.