Xin T. Tong

Matthias Morzfeld, Xin T. Tong, Youssef M. Marzouk

We investigate how ideas from covariance localization in numerical weather prediction can be used in Markov chain Monte Carlo (MCMC) sampling of high-dimensional posterior distributions arising in Bayesian inverse problems. To localize an inverse problem is to enforce an anticipated "local" structure by (i) neglecting small off-diagonal elements of the prior precision and covariance matrices; and (ii) restricting the influence of observations to their neighborhood. For linear problems we can specify the conditions under which posterior moments of the localized problem are close to those of the original problem. We explain physical interpretations of our assumptions about local structure and discuss the notion of high dimensionality in local problems, which is different from the usual notion of high dimensionality in function space MCMC. The Gibbs sampler is a natural choice of MCMC algorithm for localized inverse problems and we demonstrate that its convergence rate is independent of dimension for localized linear problems. Nonlinear problems can also be tackled efficiently by localization and, as a simple illustration of these ideas, we present a localized Metropolis-within-Gibbs sampler. Several linear and nonlinear numerical examples illustrate localization in the context of MCMC samplers for inverse problems.

Ruqi Zhang, Qiang Liu, Xin T. Tong

Sampling methods, as important inference and learning techniques, are typically designed for unconstrained domains. However, constraints are ubiquitous in machine learning problems, such as those on safety, fairness, robustness, and many other properties that must be satisfied to apply sampling results in real-life applications. Enforcing these constraints often leads to implicitly-defined manifolds, making efficient sampling with constraints very challenging. In this paper, we propose a new variational framework with a designed orthogonal-space gradient flow (O-Gradient) for sampling on a manifold $\mathcal{G}_0$ defined by general equality constraints. O-Gradient decomposes the gradient into two parts: one decreases the distance to $\mathcal{G}_0$ and the other decreases the KL divergence in the orthogonal space. While most existing manifold sampling methods require initialization on $\mathcal{G}_0$, O-Gradient does not require such prior knowledge. We prove that O-Gradient converges to the target constrained distribution with rate $\widetilde{O}(1/\text{the number of iterations})$ under mild conditions. Our proof relies on a new Stein characterization of conditional measure which could be of independent interest. We implement O-Gradient through both Langevin dynamics and Stein variational gradient descent and demonstrate its effectiveness in various experiments, including Bayesian deep neural networks.

Yi Chen, Jing Dong, Xin T. Tong

Many organizations have access to abundant data but lack the computational power to process the data. While they can outsource the computational task to other facilities, there are various constraints on the amount of data that can be shared. It is natural to ask what can data outsourcing accomplish under such constraints. We address this question from a machine learning perspective. When training a model with optimization algorithms, the quality of the results often relies heavily on the points where the algorithms are initialized. Random start is one of the most popular methods to tackle this issue, but it can be computationally expensive and not feasible for organizations lacking computing resources. Based on three different scenarios, we propose simulation-based algorithms that can utilize a small amount of outsourced data to find good initial points accordingly. Under suitable regularity conditions, we provide theoretical guarantees showing the algorithms can find good initial points with high probability. We also conduct numerical experiments to demonstrate that our algorithms perform significantly better than the random start approach.

Rui Xu, Jiepeng Wang, Hao Pan et al.

In this paper, we study an under-explored but important factor of diffusion generative models, i.e., the combinatorial complexity. Data samples are generally high-dimensional, and for various structured generation tasks, there are additional attributes which are combined to associate with data samples. We show that the space spanned by the combination of dimensions and attributes is insufficiently sampled by existing training scheme of diffusion generative models, causing degraded test time performance. We present a simple fix to this problem by constructing stochastic processes that fully exploit the combinatorial structures, hence the name ComboStoc. Using this simple strategy, we show that network training is significantly accelerated across diverse data modalities, including images and 3D structured shapes. Moreover, ComboStoc enables a new way of test time generation which uses insynchronized time steps for different dimensions and attributes, thus allowing for varying degrees of control over them.

Yang Lyu, Tan Minh Nguyen, Yuchun Qian et al.

Diffusion models are popular tools for generating new data samples, using a forward process that adds noise to data and a reverse process to denoise and produce samples. However, when the data distribution consists of n points, empirical diffusion models tend to reproduce existing data points, a phenomenon known as the memorization effect. Current literature often addresses this with complex machine learning techniques. This work shows that the memorization issue can be solved simply by applying an inertia update at the end of the empirical diffusion simulation. Our inertial diffusion model requires only the empirical score function and no additional training. We demonstrate that the distribution of samples from this model approximates the true data distribution on a $C^2$ manifold of dimension $d$, within a Wasserstein-1 distance of order $O(n^{-\frac{2}{d+4}})$. This bound significantly shrinks the Wasserstein distance between the population and empirical distributions, confirming that the inertial diffusion model produces new and diverse samples. Remarkably, this estimate is independent of the ambient space dimension, as no further training is needed. Our analysis shows that the inertial diffusion samples resemble Gaussian kernel density estimations on the manifold, revealing a novel connection between diffusion models and manifold learning.

Bihan Xu, Shiwei Zhao, Runze Wu et al.

Within the domain of Massively Multiplayer Online (MMO) economy research, Agent-Based Modeling (ABM) has emerged as a robust tool for analyzing game economics, evolving from rule-based agents to decision-making agents enhanced by reinforcement learning. Nevertheless, existing works encounter significant challenges when attempting to emulate human-like economic activities among agents, particularly regarding agent reliability, sociability, and interpretability. In this study, we take a preliminary step in introducing a novel approach using Large Language Models (LLMs) in MMO economy simulation. Leveraging LLMs' role-playing proficiency, generative capacity, and reasoning aptitude, we design LLM-driven agents with human-like decision-making and adaptability. These agents are equipped with the abilities of role-playing, perception, memory, and reasoning, addressing the aforementioned challenges effectively. Simulation experiments focusing on in-game economic activities demonstrate that LLM-empowered agents can promote emergent phenomena like role specialization and price fluctuations in line with market rules.

Jiangze Han, Christopher Thomas Ryan, Xin T. Tong

We examine the infinite-dimensional optimization problem of finding a decomposition of a probability measure into K probability sub-measures to minimize specific loss functions inspired by applications in clustering and user grouping. We analytically explore the structures of the support of optimal sub-measures and introduce algorithms based on Wasserstein gradient flow, demonstrating their convergence. Numerical results illustrate the implementability of our algorithms and provide further insights.

Jing Dong, Xin T. Tong

In reinforcement learning (RL), offline learning decoupled learning from data collection and is useful in dealing with exploration-exploitation tradeoff and enables data reuse in many applications. In this work, we study two offline learning tasks: policy evaluation and policy learning. For policy evaluation, we formulate it as a stochastic optimization problem and show that it can be solved using approximate stochastic gradient descent (aSGD) with time-dependent data. We show aSGD achieves $\tilde O(1/t)$ convergence when the loss function is strongly convex and the rate is independent of the discount factor $γ$. This result can be extended to include algorithms making approximately contractive iterations such as TD(0). The policy evaluation algorithm is then combined with the policy iteration algorithm to learn the optimal policy. To achieve an $ε$ accuracy, the complexity of the algorithm is $\tilde O(ε^{-2}(1-γ)^{-5})$, which matches the complexity bound for classic online RL algorithms such as Q-learning.

Neil K. Chada, Claudia Schillings, Xin T. Tong et al.

One fundamental problem when solving inverse problems is how to find regularization parameters. This article considers solving this problem using data-driven bilevel optimization, i.e. we consider the adaptive learning of the regularization parameter from data by means of optimization. This approach can be interpreted as solving an empirical risk minimization problem, and we analyze its performance in the large data sample size limit for general nonlinear problems. We demonstrate how to implement our framework on linear inverse problems, where we can further show the inverse accuracy does not depend on the ambient space dimension. To reduce the associated computational cost, online numerical schemes are derived using the stochastic gradient descent method. We prove convergence of these numerical schemes under suitable assumptions on the forward problem. Numerical experiments are presented illustrating the theoretical results and demonstrating the applicability and efficiency of the proposed approaches for various linear and nonlinear inverse problems, including Darcy flow, the eikonal equation, and an image denoising example.

Xi Chen, Simon S. Du, Xin T. Tong

Stochastic gradient Langevin dynamics (SGLD) is a fundamental algorithm in stochastic optimization. Recent work by Zhang et al. [2017] presents an analysis for the hitting time of SGLD for the first and second order stationary points. The proof in Zhang et al. [2017] is a two-stage procedure through bounding the Cheeger's constant, which is rather complicated and leads to loose bounds. In this paper, using intuitions from stochastic differential equations, we provide a direct analysis for the hitting times of SGLD to the first and second order stationary points. Our analysis is straightforward. It only relies on basic linear algebra and probability theory tools. Our direct analysis also leads to tighter bounds comparing to Zhang et al. [2017] and shows the explicit dependence of the hitting time on different factors, including dimensionality, smoothness, noise strength, and step size effects. Under suitable conditions, we show that the hitting time of SGLD to first-order stationary points can be dimension-independent. Moreover, we apply our analysis to study several important online estimation problems in machine learning, including linear regression, matrix factorization, and online PCA.