Dimensionality Reduction for Stationary Time Series via Stochastic Nonconvex Optimization

Stochastic optimization naturally arises in machine learning. Efficient algorithms with provable guarantees, however, are still largely missing, when the objective function is nonconvex and the data points are dependent. This paper studies this fundamental challenge through a streaming PCA problem for stationary time series data. Specifically, our goal is to estimate the principle component of time series data with respect to the covariance matrix of the stationary distribution. Computationally, we propose a variant of Oja's algorithm combined with downsampling to control the bias of the stochastic gradient caused by the data dependency. Theoretically, we quantify the uncertainty of our proposed stochastic algorithm based on diffusion approximations. This allows us to prove the asymptotic rate of convergence and further implies near optimal asymptotic sample complexity. Numerical experiments are provided to support our analysis

Representation and statistical properties of deep neural networks on structured data

Significant success of deep learning has brought unprecedented challenges to conventional wisdom in statistics, optimization, and applied mathematics. In many high-dimensional applications, e.g., image data of hundreds of thousands of pixels, deep learning is remarkably scalable and mysteriously generalizes well. Although such appealing behavior stimulates wide applications, a fundamental theoretical challenge -- curse of data dimensionality -- naturally arises. Roughly put, the sample complexity in practical applications is significantly smaller than that predicted by theory. It is a common belief that deep neural networks are good at learning various geometric structures hidden in data sets. However, little theory has been established to explain such a power. This thesis aims to bridge the gap between theory and practice by studying function approximation and statistical theories of deep neural networks in exploitation of geometric structures in data. -- Function Approximation Theories on Low-dimensional Manifolds using Deep Neural Networks. We first develop an efficient universal approximation theory functions on a low-dimensional Riemannian manifold. A feedforward network architecture is constructed for function approximation, where the size of the network grows depending on the manifold dimension. Furthermore, we prove efficient approximation theory for convolutional residual networks in approximating Besov functions. Lastly, we demonstrate the benefit of overparameterized neural networks in function approximation. Specifically, we show that large neural networks are capable of accurately approximating a target function, and the network itself enjoys Lipschitz continuity. -- Statistical Theories on Low-dimensional Data using Deep Neural Networks. Efficient approximation theories of neural networks provide valuable guidelines to properly choose network architectures, when data exhibit geometric structures. In combination with statistical tools, we prove that neural networks can circumvent the curse of data dimensionality and enjoy fast statistical convergence in various learning problems, including nonparametric regression/classification, generative distribution estimation, and doubly-robust policy learning.Ph.D

Statistical Guarantees of Generative Adversarial Networks for Distribution Estimation

Generative Adversarial Networks (GANs) have achieved great success in unsupervised learning. Despite the remarkable empirical performance, there are limited theoretical understandings on the statistical properties of GANs. This paper provides statistical guarantees of GANs for the estimation of data distributions which have densities in a H\"{o}lder space. Our main result shows that, if the generator and discriminator network architectures are properly chosen (universally for all distributions with H\"{o}lder densities), GANs are consistent estimators of the data distributions under strong discrepancy metrics, such as the Wasserstein distance. To our best knowledge, this is the first statistical theory of GANs for H\"{o}lder densities. In comparison with existing works, our theory requires minimum assumptions on data distributions. Our generator and discriminator networks utilize general weight matrices and the non-invertible ReLU activation function, while many existing works only apply to invertible weight matrices and invertible activation functions. In our analysis, we decompose the error into a statistical error and an approximation error by a new oracle inequality, which may be of independent interest

Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data

Diffusion models achieve state-of-the-art performance in various generation tasks. However, their theoretical foundations fall far behind. This paper studies score approximation, estimation, and distribution recovery of diffusion models, when data are supported on an unknown low-dimensional linear subspace. Our result provides sample complexity bounds for distribution estimation using diffusion models. We show that with a properly chosen neural network architecture, the score function can be both accurately approximated and efficiently estimated. Furthermore, the generated distribution based on the estimated score function captures the data geometric structures and converges to a close vicinity of the data distribution. The convergence rate depends on the subspace dimension, indicating that diffusion models can circumvent the curse of data ambient dimensionality.Comment: 52 pages, 4 figure

Counterfactual Generative Models for Time-Varying Treatments

Estimating the counterfactual outcome of treatment is essential for decision-making in public health and clinical science, among others. Often, treatments are administered in a sequential, time-varying manner, leading to an exponentially increased number of possible counterfactual outcomes. Furthermore, in modern applications, the outcomes are high-dimensional and conventional average treatment effect estimation fails to capture disparities in individuals. To tackle these challenges, we propose a novel conditional generative framework capable of producing counterfactual samples under time-varying treatment, without the need for explicit density estimation. Our method carefully addresses the distribution mismatch between the observed and counterfactual distributions via a loss function based on inverse probability weighting. We present a thorough evaluation of our method using both synthetic and real-world data. Our results demonstrate that our method is capable of generating high-quality counterfactual samples and outperforms the state-of-the-art baselines

Efficient RL with Impaired Observability: Learning to Act with Delayed and Missing State Observations

In real-world reinforcement learning (RL) systems, various forms of impaired observability can complicate matters. These situations arise when an agent is unable to observe the most recent state of the system due to latency or lossy channels, yet the agent must still make real-time decisions. This paper introduces a theoretical investigation into efficient RL in control systems where agents must act with delayed and missing state observations. We establish near-optimal regret bounds, of the form $\tilde{\mathcal{O}}(\sqrt{{\rm poly}(H) SAK})$, for RL in both the delayed and missing observation settings. Despite impaired observability posing significant challenges to the policy class and planning, our results demonstrate that learning remains efficient, with the regret bound optimally depending on the state-action size of the original system. Additionally, we provide a characterization of the performance of the optimal policy under impaired observability, comparing it to the optimal value obtained with full observability