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Most existing theoretical investigations of the accuracy of diffusion models, albeit significant, assume the score function has been approximated to a certain accuracy, and then use this a priori bound to control the error of generation. This article instead provides a first quantitative understanding of the whole generation process, i.e., both training and sampling. More precisely, it conducts a non-asymptotic convergence analysis of denoising score matching under gradient descent. In addition, a refined sampling error analysis for variance exploding models is also provided. The combination of these two results yields a full error analysis, which elucidates (again, but this time theoretically) how to design the training and sampling processes for effective generation. For instance, our theory implies a preference toward noise distribution and loss weighting in training that qualitatively agree with the ones used in [Karras et al., 2022]. It also provides perspectives on the choices of time and variance schedules in sampling: when the score is well trained, the design in [Song et al., 2021] is more preferable, but when it is less trained, the design in [Karras et al., 2022] becomes more preferable., Comment: Comments are welcome. Out of admiration we titled our paper after EDM, and hoped theorists' humor is not too corny

We study the complexity of heavy-tailed sampling and present a separation result in terms of obtaining high-accuracy versus low-accuracy guarantees i.e., samplers that require only $O(\log(1/\varepsilon))$ versus $\Omega(\text{poly}(1/\varepsilon))$ iterations to output a sample which is $\varepsilon$-close to the target in $\chi^2$-divergence. Our results are presented for proximal samplers that are based on Gaussian versus stable oracles. We show that proximal samplers based on the Gaussian oracle have a fundamental barrier in that they necessarily achieve only low-accuracy guarantees when sampling from a class of heavy-tailed targets. In contrast, proximal samplers based on the stable oracle exhibit high-accuracy guarantees, thereby overcoming the aforementioned limitation. We also prove lower bounds for samplers under the stable oracle and show that our upper bounds cannot be fundamentally improved.

A high precision beam monitor system based on silicon carbide PIN sensor is designed for China Spallation Neutron Source 1.6 GeV proton beam to monitor the proton beam fluence.The concept design of the beam monitor system is finished together with front-end electronics with silicon carbide PIN sensors, readout system and mechanical system.Several tests are performed to study the performance of each component of the system.The charge collection of the SiC PIN sensors after proton radiation is studied with 80 MeV proton beam for continuous running. Research on the performance of the front-end electronics and readout system is finished for better data acquisition.The uncertainty of proton beam fluence is below 1% in the beam monitor system.

This paper considers the problem of sampling from non-logconcave distribution, based on queries of its unnormalized density. It first describes a framework, Denoising Diffusion Monte Carlo (DDMC), based on the simulation of a denoising diffusion process with its score function approximated by a generic Monte Carlo estimator. DDMC is an oracle-based meta-algorithm, where its oracle is the assumed access to samples that generate a Monte Carlo score estimator. Then we provide an implementation of this oracle, based on rejection sampling, and this turns DDMC into a true algorithm, termed Zeroth-Order Diffusion Monte Carlo (ZOD-MC). We provide convergence analyses by first constructing a general framework, i.e. a performance guarantee for DDMC, without assuming the target distribution to be log-concave or satisfying any isoperimetric inequality. Then we prove that ZOD-MC admits an inverse polynomial dependence on the desired sampling accuracy, albeit still suffering from the curse of dimensionality. Consequently, for low dimensional distributions, ZOD-MC is a very efficient sampler, with performance exceeding latest samplers, including also-denoising-diffusion-based RDMC and RSDMC. Last, we experimentally demonstrate the insensitivity of ZOD-MC to increasingly higher barriers between modes or discontinuity in non-convex potential., Comment: updated related works and experiments

We derive high-dimensional scaling limits and fluctuations for the online least-squares Stochastic Gradient Descent (SGD) algorithm by taking the properties of the data generating model explicitly into consideration. Our approach treats the SGD iterates as an interacting particle system, where the expected interaction is characterized by the covariance structure of the input. Assuming smoothness conditions on moments of order up to eight orders, and without explicitly assuming Gaussianity, we establish the high-dimensional scaling limits and fluctuations in the form of infinite-dimensional Ordinary Differential Equations (ODEs) or Stochastic Differential Equations (SDEs). Our results reveal a precise three-step phase transition of the iterates; it goes from being ballistic, to diffusive, and finally to purely random behavior, as the noise variance goes from low, to moderate and finally to very-high noise setting. In the low-noise setting, we further characterize the precise fluctuations of the (scaled) iterates as infinite-dimensional SDEs. We also show the existence and uniqueness of solutions to the derived limiting ODEs and SDEs. Our results have several applications, including characterization of the limiting mean-square estimation or prediction errors and their fluctuations, which can be obtained by analytically or numerically solving the limiting equations.

Langevin diffusions are rapidly convergent under appropriate functional inequality assumptions. Hence, it is natural to expect that with additional smoothness conditions to handle the discretization errors, their discretizations like the Langevin Monte Carlo (LMC) converge in a similar fashion. This research program was initiated by Vempala and Wibisono (2019), who established results under log-Sobolev inequalities. Chewi et al. (2022) extended the results to handle the case of Poincar\'e inequalities. In this paper, we go beyond Poincar\'e inequalities, and push this research program to its limit. We do so by establishing upper and lower bounds for Langevin diffusions and LMC under weak Poincar\'e inequalities that are satisfied by a large class of densities including polynomially-decaying heavy-tailed densities (i.e., Cauchy-type). Our results explicitly quantify the effect of the initializer on the performance of the LMC algorithm. In particular, we show that as the tail goes from sub-Gaussian, to sub-exponential, and finally to Cauchy-like, the dependency on the initial error goes from being logarithmic, to polynomial, and then finally to being exponential. This three-step phase transition is in particular unavoidable as demonstrated by our lower bounds, clearly defining the boundaries of LMC., Comment: 28 pages. To appear in the Conference on Learning Theory (COLT) 2023

We analyze the complexity of sampling from a class of heavy-tailed distributions by discretizing a natural class of It\^o diffusions associated with weighted Poincar\'e inequalities. Based on a mean-square analysis, we establish the iteration complexity for obtaining a sample whose distribution is $\epsilon$ close to the target distribution in the Wasserstein-2 metric. In this paper, our results take the mean-square analysis to its limits, i.e., we invariably only require that the target density has finite variance, the minimal requirement for a mean-square analysis. To obtain explicit estimates, we compute upper bounds on certain moments associated with heavy-tailed targets under various assumptions. We also provide similar iteration complexity results for the case where only function evaluations of the unnormalized target density are available by estimating the gradients using a Gaussian smoothing technique. We provide illustrative examples based on the multivariate $t$-distribution.

The Stein Variational Gradient Descent (SVGD) algorithm is a deterministic particle method for sampling. However, a mean-field analysis reveals that the gradient flow corresponding to the SVGD algorithm (i.e., the Stein Variational Gradient Flow) only provides a constant-order approximation to the Wasserstein Gradient Flow corresponding to the KL-divergence minimization. In this work, we propose the Regularized Stein Variational Gradient Flow, which interpolates between the Stein Variational Gradient Flow and the Wasserstein Gradient Flow. We establish various theoretical properties of the Regularized Stein Variational Gradient Flow (and its time-discretization) including convergence to equilibrium, existence and uniqueness of weak solutions, and stability of the solutions. We provide preliminary numerical evidence of the improved performance offered by the regularization.