Alexandre Galashov, Nathaël Da Costa, Liyuan Xu, Philipp Hennig, Arthur Gretton
原文摘要
Neural networks are typically optimized with variants of stochastic gradient descent. Under a squared loss, however, the optimal solution to the linear last layer weights is known in closed-form. We propose to leverage this during optimization, treating the last layer as a function of the backbone parameters, and optimizing solely for these parameters. We show this is equivalent to alternating between gradient descent steps on the backbone and closed-form updates on the last layer. We adapt the method for the setting of stochastic gradient descent, by trading off the loss on the current batch against the accumulated information from previous batches. We provide theoretical analyses showing convergence of the method to an optimal solution in the neural tangent kernel regime, as well as quantifying the gains compared to standard SGD in a one-step analysis. Finally, we demonstrate the effectiveness of our approach compared with SGD and Adam on a squared loss in several regression tasks, including neural operators and causal inference.
Many causal inference problems require sampling from interventional outcome distributions rather than estimating low-dimensional summaries. Existing approaches often rely on fitting generative models of the target distribution together with additional nuisance components, which can be computationally demanding for high-dimensional outcomes. We propose Interventional Maximum Mean Discrepancy Flow (IMMD-Flow), a particle-based method that instead represents the target interventional distribution through its kernel mean embedding. This kernel embedding admits a doubly robust, Neyman-orthogonal estimator with finite-sample guarantees and can be efficiently reused to sample from the interventional distribution via MMD gradient flow. IMMD-Flow accommodates both hard and soft interventions, admits theoretical guarantees for estimation and particle approximation, and extends to high-dimensional image outcomes through frozen encoders. Across image-based benchmarks, IMMD-Flow matches or improves upon state-of-the-art generative approaches while showing significant speed advantages.