A Constrained Optimization Perspective of Unrolled Transformers

Understanding whether deep neural networks are effectively optimized remains challenging, as training occurs in highly nonconvex landscapes and standard metrics provide limited visibility into layer-wise learning quality. This challenge is particularly acute for transformer-based language models, where training is expensive, models are often reused in frozen form, and poorly optimized layers can silently degrade performance. We propose a layer-wise peeling framework for monitoring training dynamics, in which each transformer layer is locally optimized against intermediate representations of the trained model. By constructing lightweight, layer-specific reference solutions and projecting layers onto multiple intermediate outputs via different permutations, we obtain achievable baselines that enable fine-grained diagnosis of under-optimized layers. Experiments on decoder-only transformer models show that these layer-wise reference bounds can match or even surpass the trained model at various stages of training, exposing inefficiencies that remain hidden in aggregate loss curves. We further demonstrate that this analysis remains effective under binarization and quantized settings, where training dynamics are particularly fragile. Across all numerical results, the proposed bounds consistently separate apparent convergence from effective optimality, highlighting optimization opportunities that are invisible when relying on training loss alone.

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.

Pre-trained transformers have demonstrated remarkable generalization abilities, at times extending beyond the scope of their training data. Yet, real-world deployments often face unexpected or adversarial data that diverges from training data distributions. Without explicit mechanisms for handling such shifts, model reliability and safety degrade, urging more disciplined study of out-of-distribution (OOD) settings for transformers. By systematic experiments, we present a mechanistic framework for delineating the precise contours of transformer model robustness. We find that OOD inputs, including subtle typos and jailbreak prompts, drive language models to operate on an increased number of fallacious concepts in their internals. We leverage this device to quantify and understand the degree of distributional shift in prompts, enabling a mechanistically grounded fine-tuning strategy to robustify LLMs. Expanding the very notion of OOD from input data to a model's private computational processes, a new transformer diagnostic at inference time is a critical step toward making AI systems safe for deployment across science, business, and government.