Functional Gradient Descent with Adaptive Representations
For practitioners of functional optimization, this provides the first implementable FGD method with strong convergence guarantees in a general setting, addressing a key bottleneck in applying FGD to real-world problems.
The paper proposes a new functional gradient descent algorithm that adaptively represents functional gradients during optimization, achieving convergence guarantees (to a stationary point or global minimizer) despite approximations. It outperforms fixed-approximation FGD and neural network baselines in efficiency and accuracy across regression, PDEs, and computer vision tasks.
Functional optimization problems are typically solved by optimizing the parameters of a fixed representation, such as a neural network, resulting in highly nonconvex losses that complicate both training and theoretical analysis. An interesting alternative is functional gradient descent (FGD), that is, gradient descent directly in function space, which benefits from strong convergence results and admits a clean theory. However, FGD is difficult to implement in practice because functional gradients are infinite-dimensional, and thus cannot be fully computed nor stored in memory. Existing implementations therefore rely on fixed approximations, which introduce approximation error. We propose a new, theoretically-grounded FGD algorithm that adapts the representation of the functional gradients over the course of optimization. By explicitly incorporating this approximation into the analysis, we establish convergence to a stationary point (for smooth losses) and to a global minimizer (under smoothness + a Polyak-Lojasiewicz-type condition) regardless of our approximations. To the best of our knowledge, this is the first implementable FGD method with such guarantees in a general setting. We demonstrate the effectiveness of our method on regression, numerical solution of PDEs, and modern computer vision. Across settings, our method consistently outperforms both FGD with fixed approximations and neural network baselines in efficiency and accuracy.