We prove non-asymptotic error bounds for particle gradient descent (PGD, Kuntz et al., 2023), a recently introduced algorithm for maximum likelihood estimation of large latent variable models obtained by discretizing a gradient flow of the free energy. We begin by showing that the flow converges exponentially fast to the free energy's minimizers for models satisfying a condition that generalizes both the log-Sobolev and the Polyak--Łojasiewicz inequalities (LSI and PŁI, respectively). We achieve this by extending a result well-known in the optimal transport literature (that the LSI implies the Talagrand inequality) and its counterpart in the optimization literature (that the PŁI implies the so-called quadratic growth condition), and applying the extension to our new setting. We also generalize the Bakry--Émery Theorem and show that the LSI/PŁI extension holds for models with strongly concave log-likelihoods. For such models, we further control PGD's discretization error and obtain the non-asymptotic error bounds. While we are motivated by the study of PGD, we believe that the inequalities and results we extend may be of independent interest.
Microsoft Research（微软研究院）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
Stein Variational Gradient Descent (SVGD) is an algorithm for sampling from a target density which is known up to a multiplicative constant. Although SVGD is a popular algorithm in practice, its theoretical study is limited to a few recent works. We study the convergence of SVGD in the population limit, (i.e., with an infinite number of particles) to sample from a non-logconcave target distribution satisfying Talagrand's inequality T1. We first establish the convergence of the algorithm. Then, we establish a dimension-dependent complexity bound in terms of the Kernelized Stein Discrepancy (KSD…
IBM Research（IBM研究院）；MIT-IBM Watson Lab（麻省理工学院-IBM沃森实验室）；MPI（马克斯·普朗克研究所）
We study a simplification of GAN training: the problem of transporting particles from a source to a target distribution. Starting from the Sobolev GAN critic, part of the gradient regularized GAN family, we show a strong relation with Optimal Transport (OT). Specifically with the less popular dynamic formulation of OT that finds a path of distributions from source to target minimizing a ``kinetic energy''. We introduce Sobolev descent that constructs similar paths by following gradient flows of a critic function in a kernel space or parametrized by a neural network. In the kernel version, we s…
In this paper, we study the statistical limits in terms of Sobolev norms of gradient descent for solving inverse problem from randomly sampled noisy observations using a general class of objective functions. Our class of objective functions includes Sobolev training for kernel regression, Deep Ritz Methods (DRM), and Physics Informed Neural Networks (PINN) for solving elliptic partial differential equations (PDEs) as special cases. We consider a potentially infinite-dimensional parameterization of our model using a suitable Reproducing Kernel Hilbert Space and a continuous parameterization of …
Random Smoothing Regularization in Kernel Gradient Descent Learning
Fudan University（复旦大学）；Purdue University（普渡大学）；The Hong Kong University of Science and Technology（香港科技大学）；The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））
Sobolev Gradient Ascent for Optimal Transport: Barycenter Optimization and Convergence Analysis
Department of ACMS University of Notre Dame（ACMS系，诺特大学）；Department of Mathematics University of California（数学系，加州大学）；Department of Mathematics University of Southern California（数学系，南加州大学）
Technical University Berlin（柏林工业大学）；University of Potsdam（波茨坦大学）
Stochastic Gradient Descent (SGD) has become the method of choice for solving a broad range of machine learning problems. However, some of its learning properties are still not fully understood. We consider least squares learning in reproducing kernel Hilbert spaces (RKHSs) and extend the classical SGD analysis to a learning setting in Hilbert scales, including Sobolev spaces and Diffusion spaces on compact Riemannian manifolds. We show that even for well-specified models, violation of a traditional benchmark smoothness assumption has a tremendous effect on the learning rate. In addition, we s…
We study the behavior of stochastic gradient descent applied to $\|Ax -b \|_2^2 \rightarrow \min$ for invertible $A \in \mathbb{R}^{n \times n}$. We show that there is an explicit constant $c_{A}$ depending (mildly) on $A$ such that $$ \mathbb{E} ~\left\| Ax_{k+1}-b\right\|^2_{2} \leq \left(1 + \frac{c_{A}}{\|A\|_F^2}\right) \left\|A x_k -b \right\|^2_{2} - \frac{2}{\|A\|_F^2} \left\|A^T A (x_k - x)\right\|^2_{2}.$$ This is a curious inequality: the last term has one more matrix applied to the residual $u_k - u$ than the remaining terms: if $x_k - x$ is mainly comprised of large singular vecto…
INRIA（法国国家信息与自动化研究所）；Ecole Normale Supérieure（高等师范学校）；PSL Research University（巴黎文理研究大学）
In the context of statistical supervised learning, the noiseless linear model assumes that there exists a deterministic linear relation $Y = \langle θ_*, X \rangle$ between the random output $Y$ and the random feature vector $Φ(U)$, a potentially non-linear transformation of the inputs $U$. We analyze the convergence of single-pass, fixed step-size stochastic gradient descent on the least-square risk under this model. The convergence of the iterates to the optimum $θ_*$ and the decay of the generalization error follow polynomial convergence rates with exponents that both depend on the regulari…
CASUS - Center for Advanced System Understanding, Helmholtz-Zentrum Dresden-Rossendorf e.V. (HZDR)（德累斯顿-罗森多夫亥姆霍兹中心（HZDR）高级系统理解中心（CASUS））
We introduce a novel spectral, finite-dimensional approximation of general Sobolev spaces in terms of Chebyshev polynomials. Based on this polynomial surrogate model (PSM), we realise a variational formulation, solving a vast class of linear and non-linear partial differential equations (PDEs). The PSMs are as flexible as the physics-informed neural nets (PINNs) and provide an alternative for addressing inverse PDE problems, such as PDE-parameter inference. In contrast to PINNs, the PSMs result in a convex optimisation problem for a vast class of PDEs, including all linear ones, in which case …
A generalized linear point process is specified in terms of an intensity that depends upon a linear predictor process through a fixed non-linear function. We present a framework where the linear predictor is parametrized by a Banach space and give results on Gateaux differentiability of the log-likelihood. Of particular interest is when the intensity is expressed in terms of a linear filter parametrized by a Sobolev space. Using that the Sobolev spaces are reproducing kernel Hilbert spaces we derive results on the representation of the penalized maximum likelihood estimator in a special case a…
We introduce Unbalanced Sobolev Descent (USD), a particle descent algorithm for transporting a high dimensional source distribution to a target distribution that does not necessarily have the same mass. We define the Sobolev-Fisher discrepancy between distributions and show that it relates to advection-reaction transport equations and the Wasserstein-Fisher-Rao metric between distributions. USD transports particles along gradient flows of the witness function of the Sobolev-Fisher discrepancy (advection step) and reweighs the mass of particles with respect to this witness function (reaction st…
The multiplicative structure of parameters and input data in the first layer of neural networks is explored to build connection between the landscape of the loss function with respect to parameters and the landscape of the model function with respect to input data. By this connection, it is shown that flat minima regularize the gradient of the model function, which explains the good generalization performance of flat minima. Then, we go beyond the flatness and consider high-order moments of the gradient noise, and show that Stochastic Gradient Descent (SGD) tends to impose constraints on these…
In this article, we describe a function fitting method that has potential applications in machine learning and also prove relevant theorems. The described function fitting method is a convex minimization problem and can be solved using a gradient descent algorithm. We also provide qualitative analysis on fitness to data of this function fitting method. The function fitting problem is also shown to be a solution of a linear, weak partial differential equation(PDE). We describe a way to fit a Sobolev function by giving a method to choose the optimal $λ$ parameter. We describe a closed-form solut…
Propagation of Chaos for Mean-Field Langevin Dynamics and its Application to Model Ensemble
Nanyang Technological University（南洋理工大学）；National University of Singapore（新加坡国立大学）；Kyushu Institute of Technology（九州理工学院）；The University of Tokyo（东京大学）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进人工智能项目中心）
Linear convergence of proximal descent schemes on the Wasserstein space
Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan（日本东京RIKEN先进智能研究中心）；School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, UK（赫瑞-瓦特大学数学与计算机科学学院）；Maxwell Institute for Mathematical Sciences, Edinburgh, UK（爱丁堡数学科学麦克斯韦研究所）；School of Mathematics, University of Edinburgh, UK（爱丁堡大学数学学院）；School of Mathematics, University of Edinburgh, UK, The Alan Turing Institute, UK（爱丁堡大学数学学院、英国艾伦·图灵研究所）
Uniform-in-Time Weak Propagation-of-Chaos in Shallow Neural Networks
Massachusetts Institute of Technology（麻省理工学院）；Courant Institute School of Mathematics, Computing and Data Science（Courant研究所数学、计算与数据科学学院）；New York University（纽约大学）
