Zhao Ding

Li Jiang, Sijie Cheng, Jielin Qiu et al. · tsinghua

The prevalent use of benchmarks in current offline reinforcement learning (RL) research has led to a neglect of the imbalance of real-world dataset distributions in the development of models. The real-world offline RL dataset is often imbalanced over the state space due to the challenge of exploration or safety considerations. In this paper, we specify properties of imbalanced datasets in offline RL, where the state coverage follows a power law distribution characterized by skewed policies. Theoretically and empirically, we show that typically offline RL methods based on distributional constraints, such as conservative Q-learning (CQL), are ineffective in extracting policies under the imbalanced dataset. Inspired by natural intelligence, we propose a novel offline RL method that utilizes the augmentation of CQL with a retrieval process to recall past related experiences, effectively alleviating the challenges posed by imbalanced datasets. We evaluate our method on several tasks in the context of imbalanced datasets with varying levels of imbalance, utilizing the variant of D4RL. Empirical results demonstrate the superiority of our method over other baselines.

Zhao Ding, Chenguang Duan, Yuling Jiao et al.

This paper introduces score-based sequential Langevin sampling (SSLS), a novel approach to nonlinear data assimilation within a recursive Bayesian filtering framework. The proposed method decomposes the assimilation process into alternating prediction and update steps, using dynamic models for state prediction and incorporating observational data via score-based Langevin Monte Carlo during the updates. To overcome inherent challenges in highly non-log-concave posterior sampling, we integrate an annealing strategy into the update mechanism. Theoretically, we establish convergence guarantees for SSLS in total variation (TV) distance, yielding concrete insights into the algorithm's error behavior with respect to key hyperparameters. Crucially, our derived error bounds demonstrate the asymptotic stability of SSLS, guaranteeing that local posterior sampling errors do not accumulate indefinitely over time. Extensive numerical experiments across challenging scenarios, including high-dimensional systems, strong nonlinearity, and sparse observations, highlight the robust performance of the proposed method. Furthermore, SSLS effectively quantifies the uncertainty associated with state estimates, rendering it particularly valuable for reliable error calibration.

Zhao Ding, Chenguang Duan, Yuling Jiao et al.

We propose SDORE, a Semi-supervised Deep Sobolev Regressor, for the nonparametric estimation of the underlying regression function and its gradient. SDORE employs deep ReQU neural networks to minimize the empirical risk with gradient norm regularization, allowing the approximation of the regularization term by unlabeled data. Our study includes a thorough analysis of the convergence rates of SDORE in $L^{2}$-norm, achieving the minimax optimality. Further, we establish a convergence rate for the associated plug-in gradient estimator, even in the presence of significant domain shift. These theoretical findings offer valuable insights for selecting regularization parameters and determining the size of the neural network, while showcasing the provable advantage of leveraging unlabeled data in semi-supervised learning. To the best of our knowledge, SDORE is the first provable neural network-based approach that simultaneously estimates the regression function and its gradient, with diverse applications such as nonparametric variable selection. The effectiveness of SDORE is validated through an extensive range of numerical simulations.

Jinyuan Chang, Zhao Ding, Yuling Jiao et al.

We introduce an ordinary differential equation (ODE) based deep generative method for learning conditional distributions, named Conditional Föllmer Flow. Starting from a standard Gaussian distribution, the proposed flow could approximate the target conditional distribution very well when the time is close to 1. For effective implementation, we discretize the flow with Euler's method where we estimate the velocity field nonparametrically using a deep neural network. Furthermore, we also establish the convergence result for the Wasserstein-2 distance between the distribution of the learned samples and the target conditional distribution, providing the first comprehensive end-to-end error analysis for conditional distribution learning via ODE flow. Our numerical experiments showcase its effectiveness across a range of scenarios, from standard nonparametric conditional density estimation problems to more intricate challenges involving image data, illustrating its superiority over various existing conditional density estimation methods.

Zhao Ding, Chenguang Duan, Yuling Jiao et al.

We propose the characteristic generator, a novel one-step generative model that combines the efficiency of sampling in Generative Adversarial Networks (GANs) with the stable performance of flow-based models. Our model is driven by characteristics, along which the probability density transport can be described by ordinary differential equations (ODEs). Specifically, we first estimate the underlying velocity field and use the Euler method to solve the probability flow ODE, generating discrete approximations of the characteristics. A deep neural network is then trained to fit these characteristics, creating a one-step map that pushes a simple Gaussian distribution to the target distribution. In the theoretical aspect, we provide a comprehensive analysis of the errors arising from velocity matching, Euler discretization, and characteristic fitting to establish a non-asymptotic convergence rate in the 2-Wasserstein distance under mild data assumptions. Crucially, we demonstrate that under a standard manifold assumption, this convergence rate depends only on the intrinsic dimension of data rather than the much larger ambient dimension, proving our model's ability to mitigate the curse of dimensionality. To our knowledge, this is the first rigorous convergence analysis for a flow-based one-step generative model. Experiments on both synthetic and real-world datasets demonstrate that the characteristic generator achieves high-quality and high-resolution sample generation with the efficiency of just a single neural network evaluation.