Generalizable Optimal Control with Transformers: One Policy Across Diverse Systems

In this paper, we develop a rigorous optimal control-theoretic approach to Transformer training that respects key structural constraints such as (i) realized-input-independence during execution, (ii) the ensemble control nature of the problem, and (iii) positional dependence. We model the Transformer architecture as a discrete-time controlled particle system with shared actions, exhibiting noise-free controlled McKean--Vlasov dynamics, and we show that lifting the dynamics to probability measures produces a fully-observed Markov decision process (MDP). Positional encodings are incorporated into the state space to preserve the sequence order under lifting. Using the dynamic programming principle, we establish the existence of globally optimal policies under mild assumptions of compactness. We further prove that closed-loop policies in the lifted problem is equivalent to an initial-distribution dependent open-loop policy, which are realized-input-independent and compatible with standard Transformer training. To train a Transformer, we propose a triply quantized training procedure for the lifted MDP by quantizing the state space, the space of probability measures, and the action space, and show that any optimal policy for the triply quantized model is near-optimal for the original training problem. Finally, we establish stability and empirical consistency properties of the lifted model by showing that the value function is continuous with respect to the perturbations of the initial empirical measures and convergence of policies as the data size increases. This approach provides a globally optimal and robust theoretical alternative to gradient-based training without requiring smoothness or convexity.

Decoder-only transformers compute the conditional probability of the next token from a sequence of past observations. This paper derives, from first principles, inference architectures that solve the same prediction problem - and in doing so, recovers transformer-like layer operations as a consequence of optimal control theory. The framework is developed for two model classes: a nonlinear model of discrete-valued processes, directly motivated by the transformer, and a linear Gaussian model as a tractable baseline. For both model classes, the prediction objective is reformulated as an optimal control problem whose solution yields an explicit inference algorithm, the dual filter, with a layer structure that mirrors the layer structure of a decoder-only transformer. Numerical experiments provide a comparison of the optimal control to attention weights from a trained transformer. These experiments reveal that when the embedding dimension is insufficient, the transformer implicitly exploits non-Markovian structure.

Transformers can implement reinforcement-learning algorithms in context, yet how off-policy control emerges within Transformers remains poorly understood. We study Q-learning, a foundational model-free control algorithm that combines off-policy bootstrapping with greedy policy improvement. We prove that a two-layer attention-only Transformer can implement a finite-temperature form of Q-learning, where the temperature $τ>0$ controls the approximation to greedy action selection, recovering window Q-learning as $τ\to0$. We train the structured Transformer on randomly generated MDP tasks using output-only supervision. Experiments show that the theoretically predicted parameter structure emerges during training, including the internal greedy action-selection mechanism, and that the trained Transformer achieves performance comparable to Q-learning on unseen tasks. Our theoretical analysis further shows that the equivalent realizations form a solution manifold and that, under suitable task-diversity conditions, the training iterates converge locally to this manifold at an exponential rate. For the random-task model analyzed here, the local worst-case number of gradient steps needed to reduce the distance to the manifold by a factor $ε$ is $Θ\left(τ^{-5}\log(1/\varepsilon)\right)$, quantifying the increasing optimization cost as the learned update approaches window Q-learning.