Xin T. Tong

Georg A. Gottwald, Shuigen Liu, Youssef Marzouk et al.

Diffusion models are state-of-the-art tools for various generative tasks. Yet training these models involves estimating high-dimensional score functions, which in principle suffers from the curse of dimensionality. It is therefore important to understand how low-dimensional structure in the target distribution can be exploited in these models. Here we consider locality structure, which describes certain sparse conditional dependencies among the target random variables. Given some locality structure, the score function is effectively low-dimensional, so that it can be estimated by a localized neural network with significantly reduced sample complexity. This observation motivates the localized diffusion model, where a localized score matching loss is used to train the score function within a localized hypothesis space. We prove that such localization enables diffusion models to circumvent the curse of dimensionality, at the price of additional localization error. Under realistic sample size scaling, we then show both theoretically and numerically that a moderate localization radius can balance the statistical and localization errors, yielding better overall performance. Localized structure also facilitates parallel training, making localized diffusion models potentially more efficient for large-scale applications.

Xi Chen, Qiang Liu, Xin T. Tong

One classical canon of statistics is that large models are prone to overfitting, and model selection procedures are necessary for high dimensional data. However, many overparameterized models, such as neural networks, perform very well in practice, although they are often trained with simple online methods and regularization. The empirical success of overparameterized models, which is often known as benign overfitting, motivates us to have a new look at the statistical generalization theory for online optimization. In particular, we present a general theory on the generalization error of stochastic gradient descent (SGD) solutions for both convex and locally convex loss functions. We further discuss data and model conditions that lead to a ``low effective dimension". Under these conditions, we show that the generalization error either does not depend on the ambient dimension $p$ or depends on $p$ via a poly-logarithmic factor. We also demonstrate that in several widely used statistical models, the ``low effective dimension'' arises naturally in overparameterized settings. The studied statistical applications include both convex models such as linear regression and logistic regression and non-convex models such as $M$-estimator and two-layer neural networks.

Jing Dong, Xin T. Tong

Gradient descent (GD) is known to converge quickly for convex objective functions, but it can be trapped at local minima. On the other hand, Langevin dynamics (LD) can explore the state space and find global minima, but in order to give accurate estimates, LD needs to run with a small discretization step size and weak stochastic force, which in general slow down its convergence. This paper shows that these two algorithms and their non-swapping variants. can ``collaborate" through a simple exchange mechanism, in which they swap their current positions if LD yields a lower objective function. This idea can be seen as the singular limit of the replica-exchange technique from the sampling literature. We show that this new algorithm converges to the global minimum linearly with high probability, assuming the objective function is strongly convex in a neighborhood of the unique global minimum. By replacing gradients with stochastic gradients, and adding a proper threshold to the exchange mechanism, our algorithm can also be used in online settings. We also study non-swapping variants of the algorithm, which achieve similar performance. We further verify our theoretical results through some numerical experiments and observe superior performance of the proposed algorithm over running GD or LD alone.

Xi Chen, Jason D. Lee, Xin T. Tong et al.

The stochastic gradient descent (SGD) algorithm has been widely used in statistical estimation for large-scale data due to its computational and memory efficiency. While most existing works focus on the convergence of the objective function or the error of the obtained solution, we investigate the problem of statistical inference of true model parameters based on SGD when the population loss function is strongly convex and satisfies certain smoothness conditions. Our main contributions are two-fold. First, in the fixed dimension setup, we propose two consistent estimators of the asymptotic covariance of the average iterate from SGD: (1) a plug-in estimator, and (2) a batch-means estimator, which is computationally more efficient and only uses the iterates from SGD. Both proposed estimators allow us to construct asymptotically exact confidence intervals and hypothesis tests. Second, for high-dimensional linear regression, using a variant of the SGD algorithm, we construct a debiased estimator of each regression coefficient that is asymptotically normal. This gives a one-pass algorithm for computing both the sparse regression coefficients and confidence intervals, which is computationally attractive and applicable to online data.