Estimation of Markov Chain via Rank-Constrained Likelihood

This paper studies the estimation of low-rank Markov chains from empirical trajectories. We propose a non-convex estimator based on rank-constrained likelihood maximization. Statistical upper bounds are provided for the Kullback-Leiber divergence and the $\ell_2$ risk between the estimator and the true transition matrix. The estimator reveals a compressed state space of the Markov chain. We also develop a novel DC (difference of convex function) programming algorithm to tackle the rank-constrained non-smooth optimization problem. Convergence results are established. Experiments show that the proposed estimator achieves better empirical performance than other popular approaches.Comment: Accepted at ICML 201

Diffusion Approximations for Online Principal Component Estimation and Global Convergence

In this paper, we propose to adopt the diffusion approximation tools to study the dynamics of Oja's iteration which is an online stochastic gradient descent method for the principal component analysis. Oja's iteration maintains a running estimate of the true principal component from streaming data and enjoys less temporal and spatial complexities. We show that the Oja's iteration for the top eigenvector generates a continuous-state discrete-time Markov chain over the unit sphere. We characterize the Oja's iteration in three phases using diffusion approximation and weak convergence tools. Our three-phase analysis further provides a finite-sample error bound for the running estimate, which matches the minimax information lower bound for principal component analysis under the additional assumption of bounded samples.Comment: Appeared in NIPS 201

Federated Multi-Level Optimization over Decentralized Networks

Multi-level optimization has gained increasing attention in recent years, as it provides a powerful framework for solving complex optimization problems that arise in many fields, such as meta-learning, multi-player games, reinforcement learning, and nested composition optimization. In this paper, we study the problem of distributed multi-level optimization over a network, where agents can only communicate with their immediate neighbors. This setting is motivated by the need for distributed optimization in large-scale systems, where centralized optimization may not be practical or feasible. To address this problem, we propose a novel gossip-based distributed multi-level optimization algorithm that enables networked agents to solve optimization problems at different levels in a single timescale and share information through network propagation. Our algorithm achieves optimal sample complexity, scaling linearly with the network size, and demonstrates state-of-the-art performance on various applications, including hyper-parameter tuning, decentralized reinforcement learning, and risk-averse optimization.Comment: arXiv admin note: substantial text overlap with arXiv:2206.1087

A sensitive and rapid HPLC-DAD method for the determination of 3-hydroxy-1,2-dimethyl-4-pyridone and its distribution in rats

Purpose: To establish a sensitive and rapid method for the determination of the tissue distribution of 3-hydroxy-1,2-dimethyl-4-pyridone (L1) in vivo, and its plasma protein binding capacity.Methods: This study optimized a reverse-phase HPLC method for specific and sensitive determination of L1 as well as its plasma and tissue  distributions. The optimized method was used to determine the plasma protein-binding capacity of L1 in Wistar rats.Results: A rapid, sensitive and simple HPLC-DAD method was established for studying the plasma and tissue distribution of L1. Following TI  administration, its liver concentrations peaked at 60 min and 360min, followed 360 min later with peak level in the kidney (second highest). The L1 concentration was significantly lower after 360 min than after 60 min, and values of its mean binding to plasma proteins was 5.2 % at different L1 concentrations.Conclusion: These results indicate that L1 is a drug with rapid-absorption and rapid-elimination thath is distributed widely in vivo in rats. Moreover, the drug has a weak plasma protein-binding capacity. Keywords: 3-Hydroxy-1,2-dimethyl-4-pyridone, Distribution, Alzheimer’s disease, Therap