Xiaoyu Wang

Ruinan Jin, Xiaoyu Wang, Baoxiang Wang

Adaptive gradient optimizers (AdaGrad), which dynamically adjust the learning rate based on iterative gradients, have emerged as powerful tools in deep learning. These adaptive methods have significantly succeeded in various deep learning tasks, outperforming stochastic gradient descent. However, despite AdaGrad's status as a cornerstone of adaptive optimization, its theoretical analysis has not adequately addressed key aspects such as asymptotic convergence and non-asymptotic convergence rates in non-convex optimization scenarios. This study aims to provide a comprehensive analysis of AdaGrad and bridge the existing gaps in the literature. We introduce a new stopping time technique from probability theory, which allows us to establish the stability of AdaGrad under mild conditions. We further derive the asymptotically almost sure and mean-square convergence for AdaGrad. In addition, we demonstrate the near-optimal non-asymptotic convergence rate measured by the average-squared gradients in expectation, which is stronger than the existing high-probability results. The techniques developed in this work are potentially of independent interest for future research on other adaptive stochastic algorithms.

Chengchun Shi, Xiaoyu Wang, Shikai Luo et al.

A/B testing, or online experiment is a standard business strategy to compare a new product with an old one in pharmaceutical, technological, and traditional industries. Major challenges arise in online experiments of two-sided marketplace platforms (e.g., Uber) where there is only one unit that receives a sequence of treatments over time. In those experiments, the treatment at a given time impacts current outcome as well as future outcomes. The aim of this paper is to introduce a reinforcement learning framework for carrying A/B testing in these experiments, while characterizing the long-term treatment effects. Our proposed testing procedure allows for sequential monitoring and online updating. It is generally applicable to a variety of treatment designs in different industries. In addition, we systematically investigate the theoretical properties (e.g., size and power) of our testing procedure. Finally, we apply our framework to both simulated data and a real-world data example obtained from a technological company to illustrate its advantage over the current practice. A Python implementation of our test is available at https://github.com/callmespring/CausalRL.

Xiaoyu Wang, Mikael Johansson

A theoretical, and potentially also practical, problem with stochastic gradient descent is that trajectories may escape to infinity. In this note, we investigate uniform boundedness properties of iterates and function values along the trajectories of the stochastic gradient descent algorithm and its important momentum variant. Under smoothness and $R$-dissipativity of the loss function, we show that broad families of step-sizes, including the widely used step-decay and cosine with (or without) restart step-sizes, result in uniformly bounded iterates and function values. Several important applications that satisfy these assumptions, including phase retrieval problems, Gaussian mixture models, and some neural network classifiers, are discussed in detail. We further extend the uniform boundedness of SGD and its momentum variant under the generalized dissipativity for the functions whose tails grow slower than quadratic functions. This includes some interesting applications, for example, Bayesian logistic regression and logistic regression with $\ell_1$ regularization.

Yuanhan Hu, Xiaoyu Wang, Xuefeng Gao et al.

Stochastic Gradient Langevin Dynamics (SGLD) is a powerful algorithm for optimizing a non-convex objective, where a controlled and properly scaled Gaussian noise is added to the stochastic gradients to steer the iterates towards a global minimum. SGLD is based on the overdamped Langevin diffusion which is reversible in time. By adding an anti-symmetric matrix to the drift term of the overdamped Langevin diffusion, one gets a non-reversible diffusion that converges to the same stationary distribution with a faster convergence rate. In this paper, we study the non reversible Stochastic Gradient Langevin Dynamics (NSGLD) which is based on discretization of the non-reversible Langevin diffusion. We provide finite-time performance bounds for the global convergence of NSGLD for solving stochastic non-convex optimization problems. Our results lead to non-asymptotic guarantees for both population and empirical risk minimization problems. Numerical experiments for Bayesian independent component analysis and neural network models show that NSGLD can outperform SGLD with proper choices of the anti-symmetric matrix.