arXiv subjects

Yuhua Zhu

Publications and source records attributed to Yuhua Zhu.

33 records · Page 2Linked to original sources

arXiv2023

FedCBO: Reaching Group Consensus in Clustered Federated Learning through Consensus-based Optimization

Federated learning is an important framework in modern machine learning that seeks to integrate the training of learning models from multiple users, each user having their own local data set, in a way that is sensitive to data privacy and to communication loss constraints. In clustered federated learning, one assumes an additional unknown group structure among users, and the goal is to train models that are useful for each group, rather than simply training a single global model for all users. In this paper, we propose a novel solution to the problem of clustered federated learning that is inspired by ideas in consensus-based optimization (CBO). Our new CBO-type method is based on a system of interacting particles that is oblivious to group memberships. Our model is motivated by rigorous mathematical reasoning, including a mean field analysis describing the large number of particles limit of our particle system, as well as convergence guarantees for the simultaneous global optimization of general non-convex objective functions (corresponding to the loss functions of each cluster of users) in the mean-field regime. Experimental results demonstrate the efficacy of our FedCBO algorithm compared to other state-of-the-art methods and help validate our methodological and theoretical work.

cs.LG↗

arXiv2023

Operator Shifting for Model-based Policy Evaluation

In model-based reinforcement learning, the transition matrix and reward vector are often estimated from random samples subject to noise. Even if the estimated model is an unbiased estimate of the true underlying model, the value function computed from the estimated model is biased. We introduce an operator shifting method for reducing the error introduced by the estimated model. When the error is in the residual norm, we prove that the shifting factor is always positive and upper bounded by $1+O\left(1/n\right)$, where $n$ is the number of samples used in learning each row of the transition matrix. We also propose a practical numerical algorithm for implementing the operator shifting.

cs.LG↗

arXiv2023

Variational Actor-Critic Algorithms

We introduce a class of variational actor-critic algorithms based on a variational formulation over both the value function and the policy. The objective function of the variational formulation consists of two parts: one for maximizing the value function and the other for minimizing the Bellman residual. Besides the vanilla gradient descent with both the value function and the policy updates, we propose two variants, the clipping method and the flipping method, in order to speed up the convergence. We also prove that, when the prefactor of the Bellman residual is sufficiently large, the fixed point of the algorithm is close to the optimal policy.

cs.LG↗

arXiv2021

On Large Batch Training and Sharp Minima: A Fokker-Planck Perspective

We study the statistical properties of the dynamic trajectory of stochastic gradient descent (SGD). We approximate the mini-batch SGD and the momentum SGD as stochastic differential equations (SDEs). We exploit the continuous formulation of SDE and the theory of Fokker-Planck equations to develop new results on the escaping phenomenon and the relationship with large batch and sharp minima. In particular, we find that the stochastic process solution tends to converge to flatter minima regardless of the batch size in the asymptotic regime. However, the convergence rate is rigorously proven to depend on the batch size. These results are validated empirically with various datasets and models.

cs.LG↗

arXiv2021

Why resampling outperforms reweighting for correcting sampling bias with stochastic gradients

A data set sampled from a certain population is biased if the subgroups of the population are sampled at proportions that are significantly different from their underlying proportions. Training machine learning models on biased data sets requires correction techniques to compensate for the bias. We consider two commonly-used techniques, resampling and reweighting, that rebalance the proportions of the subgroups to maintain the desired objective function. Though statistically equivalent, it has been observed that resampling outperforms reweighting when combined with stochastic gradient algorithms. By analyzing illustrative examples, we explain the reason behind this phenomenon using tools from dynamical stability and stochastic asymptotics. We also present experiments from regression, classification, and off-policy prediction to demonstrate that this is a general phenomenon. We argue that it is imperative to consider the objective function design and the optimization algorithm together while addressing the sampling bias.

cs.LG↗

arXiv2020

A Note on Optimization Formulations of Markov Decision Processes

This note summarizes the optimization formulations used in the study of Markov decision processes. We consider both the discounted and undiscounted processes under the standard and the entropy-regularized settings. For each setting, we first summarize the primal, dual, and primal-dual problems of the linear programming formulation. We then detail the connections between these problems and other formulations for Markov decision processes such as the Bellman equation and the policy gradient method.

math.OC↗

arXiv2020

Borrowing From the Future: An Attempt to Address Double Sampling

For model-free reinforcement learning, one of the main difficulty of stochastic Bellman residual minimization is the double sampling problem, i.e., while only one single sample for the next state is available in the model-free setting, two independent samples for the next state are required in order to perform unbiased stochastic gradient descent. We propose new algorithms for addressing this problem based on the idea of borrowing extra randomness from the future. When the transition kernel varies slowly with respect to the state, it is shown that the training trajectory of new algorithms is close to the one of unbiased stochastic gradient descent. Numerical results for policy evaluation in both tabular and neural network settings are provided to confirm the theoretical findings.

math.OC↗

arXiv2020

Borrowing From the Future: Addressing Double Sampling in Model-free Control

In model-free reinforcement learning, the temporal difference method and its variants become unstable when combined with nonlinear function approximations. Bellman residual minimization with stochastic gradient descent (SGD) is more stable, but it suffers from the double sampling problem: given the current state, two independent samples for the next state are required, but often only one sample is available. Recently, the authors of [Zhu et al, 2020] introduced the borrowing from the future (BFF) algorithm to address this issue for the prediction problem. The main idea is to borrow extra randomness from the future to approximately re-sample the next state when the underlying dynamics of the problem are sufficiently smooth. This paper extends the BFF algorithm to action-value function based model-free control. We prove that BFF is close to unbiased SGD when the underlying dynamics vary slowly with respect to actions. We confirm our theoretical findings with numerical simulations.

math.OC↗

arXiv2020

A consensus-based global optimization method for high dimensional machine learning problems

We improve recently introduced consensus-based optimization method, proposed in [R. Pinnau, C. Totzeck, O. Tse and S. Martin, Math. Models Methods Appl. Sci., 27(01):183--204, 2017], which is a gradient-free optimization method for general non-convex functions. We first replace the isotropic geometric Brownian motion by the component-wise one, thus removing the dimensionality dependence of the drift rate, making the method more competitive for high dimensional optimization problems. Secondly, we utilize the random mini-batch ideas to reduce the computational cost of calculating the weighted average which the individual particles tend to relax toward. For its mean-field limit--a nonlinear Fokker-Planck equation--we prove, in both time continuous and semi-discrete settings, that the convergence of the method, which is exponential in time, is guaranteed with parameter constraints {\it independent} of the dimensionality. We also conduct numerical tests to high dimensional problems to check the success rate of the method.

math.OC↗

arXiv2020

Boundary Control of Vlasov--Fokker--Planck Equations

We introduce a novel Lyapunov function for stabilization of linear Vlasov--Fokker--Planck type equations with stiff source term. Contrary to existing results relying on transport properties to obtain stabilization, we present results based on hypocoercivity analysis for the Fokker--Planck operator. The existing estimates are extended to derive suitable feedback boundary control to guarantee the exponential stabilization. Further, we study the associated macroscopic limit and derive conditions on the feedback boundary control such that in the formal limit no boundary layer exists.

math.OC↗

arXiv2020

A Sharp Convergence Rate for the Asynchronous Stochastic Gradient Descent

We give a sharp convergence rate for the asynchronous stochastic gradient descent (ASGD) algorithms when the loss function is a perturbed quadratic function based on the stochastic modified equations introduced in [An et al. Stochastic modified equations for the asynchronous stochastic gradient descent, arXiv:1805.08244]. We prove that when the number of local workers is larger than the expected staleness, then ASGD is more efficient than stochastic gradient descent. Our theoretical result also suggests that longer delays result in slower convergence rate. Besides, the learning rate cannot be smaller than a threshold inversely proportional to the expected staleness.

math.NA↗

arXiv2019

The Vlasov-Fokker-Planck Equation with High Dimensional Parametric Forcing Term

We consider the Vlasov-Fokker-Planck equation with random electric field where the random field is parametrized by countably many infinite random variables due to uncertainty. At the theoretical level, with suitable assumption on the anisotropy of the randomness, adopting the technique employed in elliptic PDEs [Cohen, DeVore, 2015], we prove the best N approximation in the random space breaks the dimension curse and the convergence rate is faster than the Monte Carlo method. For the numerical method, based on the adaptive sparse polynomial interpolation (ASPI) method introduced in [Chkifa, Cohen, Schwab, 2014], we develop a residual-based adaptive sparse polynomial interpolation (RASPI) method which is more efficient for multi-scale linear kinetic equation, when using numerical schemes that are time-dependent and implicit. Numerical experiments show that the numerical error of the RASPI decays faster than the Monte-Carlo method and is also dimension independent.

math.NA↗

arXiv2019

An Uncertainty Quantification Approach to the Study of Gene Expression Robustness

We study a chemical kinetic system with uncertainty modeling a gene regulatory network in biology. Specifically, we consider a system of two equations for the messenger RNA and micro RNA content of a cell. Our target is to provide a simple framework for noise buffering in gene expression through micro RNA production. Here the uncertainty, modeled by random variables, enters the system through the initial data and the source term. We obtain a sharp decay rate of the solution to the steady state, which reveals that the biology system is not sensitive to the initial perturbation around the steady state. The sharp regularity estimate leads to the stability of the generalized Polynomial Chaos stochastic Galerkin (gPC-SG) method. Based on the smoothness of the solution in the random space and the stability of the numerical method, we conclude the gPC-SG method has spectral accuracy. Numerical experiments are conducted to verify the theoretical findings.

math.NA↗

arXiv2018

Towards Theoretical Understanding of Large Batch Training in Stochastic Gradient Descent

Stochastic gradient descent (SGD) is almost ubiquitously used for training non-convex optimization tasks. Recently, a hypothesis proposed by Keskar et al. [2017] that large batch methods tend to converge to sharp minimizers has received increasing attention. We theoretically justify this hypothesis by providing new properties of SGD in both finite-time and asymptotic regimes. In particular, we give an explicit escaping time of SGD from a local minimum in the finite-time regime and prove that SGD tends to converge to flatter minima in the asymptotic regime (although may take exponential time to converge) regardless of the batch size. We also find that SGD with a larger ratio of learning rate to batch size tends to converge to a flat minimum faster, however, its generalization performance could be worse than the SGD with a smaller ratio of learning rate to batch size. We include numerical experiments to corroborate these theoretical findings.

stat.ML↗

arXiv2017

Hypocoercivity and Uniform Regularity for the Vlasov-Poisson-Fokker-Planck System with Uncertainty and Multiple Scales

We study the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple scales. Here the uncertainty, modeled by random variables, enters the solution through initial data, while the multiple scales lead the system to its high-field or parabolic regimes. With the help of proper Lyapunov-type inequalities, under some mild conditions on the initial data, the regularity of the solution in the random space, as well as exponential decay of the solution to the global Maxwellian, are established under Sobolev norms, which are ${\it uniform}$ in terms of the scaling parameters. These are the first hypocoercivity results for a nonlinear kinetic system with random input, which are important for the understanding of the sensitivity of the system under random perturbations, and for the establishment of spectral convergence of popular numerical methods for uncertainty quantification based on (spectrally accurate) polynomial chaos expansions.

math.AP↗