Active Learning of Conditional Generative Models via the Transport Neural Tangent Kernel

Conditional generative models have emerged as powerful tools for sampling from target conditional distributions, driving substantial advances across a wide range of scientific and applied domains. As these models proliferate, practitioners often face multiple plausible generators whose performance can vary with the task, data, or input condition. We propose an optimal model averaging framework for conditional generative models, allowing candidate generators to be combined even when they are accessible only through conditional samples without tractable densities. Specifically, we use a sample-based maximum mean discrepancy between conditional distributions, which first leads to a static model averaging method, StaticMA, assigning fixed weights to different candidates. In addition, we develop MoEMA (mixture-of-experts model averaging), an input-adaptive method that parameterizes covariate-dependent weights through a softmax neural-network gate. We establish in-sample and out-of-sample asymptotic optimality for the proposed methods, together with consistency of the estimated adaptive weight function under regularity conditions. The framework applies directly to Euclidean responses and extends to unstructured data by combining our formulation with fixed representation maps. Across a broad set of simulations and real-data studies spanning tabular, image, and text modalities, MoEMA generally improves over competing baselines, demonstrating the effectiveness of our proposed methods.

One-step generative models construct a static generator through iterative training-time transport. Existing transport objectives primarily assess distributional motion, although a neural generator needs to realize the requested sample displacements jointly through shared parameter updates. The training-time construction raises the question: once training becomes the iterative process that constructs the final one-step map, what to optimize: the next distributional move, or the route by which the finite generator learns the final map? To address the question, we introduce Training Dynamics Action (TDAction), which selects transport targets according to local shared-parameter realization cost while retaining a prescribed level of distributional progress. We formulate the cost as a soft-terminal control problem and derive a closed-form Batch Tangent Action-to-Go value that accounts for parameter effort and terminal mismatch. The criterion captures cross-sample interactions omitted by independent pairwise costs; under isotropic mobility, the criterion agrees with quadratic Euclidean assignment for deterministic balanced couplings. Randomized tangent probes provide a low-rank implementation that constructs shared detached targets without adding an inference-time trajectory. Controlled studies examine the relationship between generator geometry, transport selection, and realized local action. On ImageNet $256\times256$, TDAction attains an FID below $1.1$ without distillation.

Conditional generative models parameterized by triangular transport maps are widely used for conditional sampling, yet theoretical guarantees on the quality of the conditional measures they produce remain scarce. We address this question for minimum divergence estimators of triangular maps. We use MMD-based objectives as a tractable setting in which the gap between joint and conditional errors can be analyzed precisely, and we measure performance through the average conditional estimation error, defined as the squared maximum mean discrepancy (MMD) between the estimated and the true conditionals, averaged over the conditioning variable. We first show that the common practice of minimizing a joint MMD may fail to recover the conditionals: writing the joint and the conditional criteria as quadratic energies of the same conditional mean embedding residual, we prove that, for parametric classes, joint minimizers are generically inconsistent in the misspecified setting. We then analyze an estimator that directly minimizes the conditional criterion through a surrogate loss computable from joint target samples and reference samples. We prove a non-asymptotic error bound that decomposes the average conditional estimation error into approximation, estimation, and reference sampling errors, with fast rates independent of the ambient dimension in the parametric setting. Finally, for joint MMDs built from concentrating Gaussian kernels on the conditioning variable, we show that the joint criterion approximates a density-reweighted version of the conditional one, and we derive an error bound for the joint minimizer whose rate suffers from a curse of dimensionality in the conditioning variable. Our results suggest that training criteria for conditional generative models should target the conditional rather than the joint error, a principle that extends to other divergences and to dynamic transport.