Diffusion models have made significant advancements in recent years. However, their performance often deteriorates when trained or fine-tuned on imbalanced datasets. This degradation is largely due to the disproportionate representation of majority and minority data in image-text pairs. In this paper, we propose a general fine-tuning approach, dubbed PoGDiff, to address this challenge. Rather than directly minimizing the KL divergence between the predicted and ground-truth distributions, PoGDiff replaces the ground-truth distribution with a Product of Gaussians (PoG), which is constructed by combining the original ground-truth targets with the predicted distribution conditioned on a neighboring text embedding. Experiments on real-world datasets demonstrate that our method effectively addresses the imbalance problem in diffusion models, improving both generation accuracy and quality.

Maximum Mean Discrepancy (MMD) is a probability metric that has found numerous applications in machine learning. In this work, we focus on its application in generative models, including the minimum MMD estimator, Generative Moment Matching Network (GMMN), and Generative Adversarial Network (GAN). In these cases, MMD is part of an objective function in a minimization or min-max optimization problem. Even if its empirical performance is competitive, the consistency and convergence rate analysis of the corresponding MMD-based estimators has yet to be carried out. We propose a uniform concentration inequality for a class of Maximum Mean Discrepancy (MMD)-based estimators, that is, a maximum deviation bound of empirical MMD values over a collection of generated distributions and adversarially learned kernels. Here, our inequality serves as an efficient tool in the theoretical analysis for MMD-based generative models. As elaborating examples, we applied our main result to provide the generalization error bounds for the MMD-based estimators in the context of the minimum MMD estimator and MMD GAN.

This paper studies the approximation of smooth functionals defined over a reproducing kernel Hilbert space (RKHS) using tanh neural networks. A functional maps from a space of functions that has infinite dimensions to R. In recent years, neural networks have been widely employed in operator learning tasks. We are interested in investigating their capability to approximate nonlinear functionals, a special type of operator. Neural networks have been known as universal approximators since [Cybenko, 1989], i.e., to approximate any continuous function, mapping a finite-dimensional input space into another finite-dimensional output space, to arbitrary accuracy. These days, many interesting tasks entail learning operators, i.e., mappings between an infinite-dimensional input Banach space and (possibly) an infinite-dimensional output space. A prototypical example in scientific computing is to map the initial datum into the (time series of) solution of a nonlinear time-dependent partial differential equation (PDE). A priori, it is unclear if neural networks can be successfully employed to learn such operators from data, given that their universality only pertains to finite-dimensional functions. One of the first successful uses of neural networks in the context of operator learning was provided by [Chen and Chen, 1995].

Adversarial training has been proposed to hedge against adversarial attacks in machine learning and statistical models. This paper focuses on adversarial training under $\ell_\infty$-perturbation, which has recently attracted much research attention. The asymptotic behavior of the adversarial training estimator is investigated in the generalized linear model. The results imply that the limiting distribution of the adversarial training estimator under $\ell_\infty$-perturbation could put a positive probability mass at $0$ when the true parameter is $0$, providing a theoretical guarantee of the associated sparsity-recovery ability. Alternatively, a two-step procedure is proposed -- adaptive adversarial training, which could further improve the performance of adversarial training under $\ell_\infty$-perturbation. Specifically, the proposed procedure could achieve asymptotic unbiasedness and variable-selection consistency. Numerical experiments are conducted to show the sparsity-recovery ability of adversarial training under $\ell_\infty$-perturbation and to compare the empirical performance between classic adversarial training and adaptive adversarial training.

In this paper, we first extend the result of FL93 and prove universal consistency for a classification rule based on wide and deep ReLU neural networks trained on the logistic loss. Unlike the approach in FL93 that decomposes the estimation and empirical error, we directly analyze the classification risk based on the observation that a realization of a neural network that is wide enough is capable of interpolating an arbitrary number of points. Secondly, we give sufficient conditions for a class of probability measures under which classifiers based on neural networks achieve minimax optimal rates of convergence. Our result is motivated from the practitioner's observation that neural networks are often trained to achieve 0 training error, which is the case for our proposed neural network classifiers. Our proofs hinge on recent developments in empirical risk minimization and on approximation rates of deep ReLU neural networks for various function classes of interest. Applications to classical function spaces of smoothness illustrate the usefulness of our result.

The recent success of neural networks in pattern recognition and classification problems suggests that neural networks possess qualities distinct from other more classical classifiers such as SVMs or boosting classifiers. This paper studies the performance of plug-in classifiers based on neural networks in a binary classification setting as measured by their excess risks. Compared to the typical settings imposed in the literature, we consider a more general scenario that resembles actual practice in two respects: first, the function class to be approximated includes the Barron functions as a proper subset, and second, the neural network classifier constructed is the minimizer of a surrogate loss instead of the $0$-$1$ loss so that gradient descent-based numerical optimizations can be easily applied. While the class of functions we consider is quite large that optimal rates cannot be faster than $n^{-\frac{1}{3}}$, it is a regime in which dimension-free rates are possible and approximation power of neural networks can be taken advantage of. In particular, we analyze the estimation and approximation properties of neural networks to obtain a dimension-free, uniform rate of convergence for the excess risk. Finally, we show that the rate obtained is in fact minimax optimal up to a logarithmic factor, and the minimax lower bound shows the effect of the margin assumption in this regime.

It is frequently observed that overparameterized neural networks generalize well. Regarding such phenomena, existing theoretical work mainly devotes to linear settings or fully-connected neural networks. This paper studies the learning ability of an important family of deep neural networks, deep convolutional neural networks (DCNNs), under both underparameterized and overparameterized settings. We establish the first learning rates of underparameterized DCNNs without parameter or function variable structure restrictions presented in the literature. We also show that by adding well-defined layers to a non-interpolating DCNN, we can obtain some interpolating DCNNs that maintain the good learning rates of the non-interpolating DCNN. This result is achieved by a novel network deepening scheme designed for DCNNs. Our work provides theoretical verification of how overfitted DCNNs generalize well.

This paper studies the binary classification of unbounded data from ${\mathbb R}^d$ generated under Gaussian Mixture Models (GMMs) using deep ReLU neural networks. We obtain $\unicode{x2013}$ for the first time $\unicode{x2013}$ non-asymptotic upper bounds and convergence rates of the excess risk (excess misclassification error) for the classification without restrictions on model parameters. The convergence rates we derive do not depend on dimension $d$, demonstrating that deep ReLU networks can overcome the curse of dimensionality in classification. While the majority of existing generalization analysis of classification algorithms relies on a bounded domain, we consider an unbounded domain by leveraging the analyticity and fast decay of Gaussian distributions. To facilitate our analysis, we give a novel approximation error bound for general analytic functions using ReLU networks, which may be of independent interest. Gaussian distributions can be adopted nicely to model data arising in applications, e.g., speeches, images, and texts; our results provide a theoretical verification of the observed efficiency of deep neural networks in practical classification problems.

Given a sequence of observable variables $\{(x_1, y_1), \ldots, (x_n, y_n)\}$, the conformal prediction method estimates a confidence set for $y_{n+1}$ given $x_{n+1}$ that is valid for any finite sample size by merely assuming that the joint distribution of the data is permutation invariant. Although attractive, computing such a set is computationally infeasible in most regression problems. Indeed, in these cases, the unknown variable $y_{n+1}$ can take an infinite number of possible candidate values, and generating conformal sets requires retraining a predictive model for each candidate. In this paper, we focus on a sparse linear model with only a subset of variables for prediction and use numerical continuation techniques to approximate the solution path efficiently. The critical property we exploit is that the set of selected variables is invariant under a small perturbation of the input data. Therefore, it is sufficient to enumerate and refit the model only at the change points of the set of active features and smoothly interpolate the rest of the solution via a Predictor-Corrector mechanism. We show how our path-following algorithm accurately approximates conformal prediction sets and illustrate its performance using synthetic and real data examples.

In this paper, we derive a novel bound on the generalization error of Magnitude-Based pruning of overparameterized neural networks. Our work builds on the bounds in Arora et al. [2018] where the error depends on one, the approximation induced by pruning, and two, the number of parameters in the pruned model, and improves upon standard norm-based generalization bounds. The pruned estimates obtained using our new Magnitude-Based compression algorithm are close to the unpruned functions with high probability, which improves the first criteria. Using Sparse Matrix Sketching, the space of the pruned matrices can be efficiently represented in the space of dense matrices of much smaller dimensions, thereby lowering the second criterion. This leads to stronger generalization bound than many state-of-the-art methods, thereby breaking new ground in the algorithm development for pruning and bounding generalization error of overparameterized models. Beyond this, we extend our results to obtain generalization bound for Iterative Pruning [Frankle and Carbin, 2018]. We empirically verify the success of this new method on ReLU-activated Feed Forward Networks on the MNIST and CIFAR10 datasets.