arXiv Statistics ML
A functional central limit theorem for kernel gradient flow and infinitesimal gradient boosting
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Stochastic Gradient Descent over P2
The paper develops a diffusion approximation for stochastic gradient descent (SGD) when the optimization target is a functional on the Wasserstein space ℝ2. By lifting the problem to a Hilbert space via Lions differentiability, the authors construct a Gaussian random-field approximation whose velocity field matches the mean and covariance of the original stochastic gradient. They prove that this Gaussian approximation achieves second‑order weak accuracy, providing a rigorous basis for replacing sample‑driven randomness with analytically tractable Gaussian fluctuations in stochastic optimization over probability measures.
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Weak Correlations as the Underlying Principle for Linearization of Gradient-Based Learning Systems
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Constant-Stepsize Stochastic Approximation: Finite-Time Convergence, Gaussian Approximation, and Tail Bounds
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Generalization Error Curves for Analytic Spectral Algorithms under Power-law Decay
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Stochastic Gradient Descent for Operator Learning in Hilbert Spaces: Convergence Rates and Minimax Lower Bounds
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Gaussian Processes and Reproducing Kernel Hilbert Spaces: Connections and Equivalences
The monograph explores the relationships between Gaussian processes and reproducing kernel Hilbert spaces (RKHS), two widely used approaches that rely on positive definite kernels. It examines how these frameworks connect and are equivalent across key topics such as regression, interpolation, numerical integration, distributional discrepancies, statistical dependence, and Gaussian process sample path properties. By establishing a unifying perspective based on the equivalence between the Gaussian Hilbert space and the RKHS, the work aims to bridge methods developed independently by the machine learning, statistics, and numerical analysis communities.