Hyperparameter Trajectory Inference with Conditional Lagrangian Optimal Transport

Neural networks (NNs) often have critical behavioural trade-offs that are set at design time with hyperparameters—such as reward weighting in reinforcement learning or quantile targets in regression. Post-deployment, however, user preferences can evolve, making initially optimal settings undesirable, necessitating expensive retraining. To circumvent this, we introduce the task of Hyperparameter Trajectory Inference (HTI), to learn, from observed data, how a NN's conditional output distribution changes as a function of its hyperparameters, such that a surrogate model can approximate the NN at unobserved hyperparameter settings. HTI requires extending existing trajectory inference approaches to incorporate conditions, posing key challenges to ensure meaningful inferred conditional probability paths. We propose an approach grounded in conditional Lagrangian optimal transport theory, jointly learning the Lagrangian function governing hyperparameter-induced dynamics along with the associated optimal transport maps and geodesics, which form the surrogate model. We incorporate inductive biases based on the manifold hypothesis and least-action principles into the learned Lagrangian, improving surrogate model feasibility. We empirically demonstrate that our approach reconstructs NN behaviour across hyperparameter spectrums better than other alternatives, enabling effective inference-time adaptation of NNs.