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Wenlong Mou

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6 papers

2 author rows

JMLR Journal 2022 Journal Article

An Efficient Sampling Algorithm for Non-smooth Composite Potentials

Wenlong Mou
Nicolas Flammarion
Martin J. Wainwright
Peter L. Bartlett

We consider the problem of sampling from a density of the form $p(x) \propto \exp(-f(x)- g(x))$, where $f: \mathbb{R}^d \rightarrow \mathbb{R}$ is a smooth function and $g: \mathbb{R}^d \rightarrow \mathbb{R}$ is a convex and Lipschitz function. We propose a new algorithm based on the Metropolis--Hastings framework. Under certain isoperimetric inequalities on the target density, we prove that the algorithm mixes to within total variation (TV) distance $\varepsilon$ of the target density in at most $O(d \log (d/\varepsilon))$ iterations. This guarantee extends previous results on sampling from distributions with smooth log densities ($g = 0$) to the more general composite non-smooth case, with the same mixing time up to a multiple of the condition number. Our method is based on a novel proximal-based proposal distribution that can be efficiently computed for a large class of non-smooth functions $g$. Simulation results on posterior sampling problems that arise from the Bayesian Lasso show empirical advantage over previous proposal distributions. [abs] [ pdf ][ bib ] &copy JMLR 2022. ( edit, beta )

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JMLR Journal 2021 Journal Article

High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm

Wenlong Mou
Yi-An Ma
Martin J. Wainwright
Peter L. Bartlett
Michael I. Jordan

We propose a Markov chain Monte Carlo (MCMC) algorithm based on third-order Langevin dynamics for sampling from distributions with smooth, log-concave densities. The higher-order dynamics allow for more flexible discretization schemes, and we develop a specific method that combines splitting with more accurate integration. For a broad class of $d$-dimensional distributions arising from generalized linear models, we prove that the resulting third-order algorithm produces samples from a distribution that is at most $\varepsilon > 0$ in Wasserstein distance from the target distribution in $O\left(\frac{d^{1/4}}{ \varepsilon^{1/2}} \right)$ steps. This result requires only Lipschitz conditions on the gradient. For general strongly convex potentials with $\alpha$-th order smoothness, we prove that the mixing time scales as $O \left( \frac{d^{1/4}}{\varepsilon^{1/2}} + \frac{d^{1/2}}{ \varepsilon^{1/(\alpha - 1)}} \right)$. [abs] [ pdf ][ bib ] &copy JMLR 2021. ( edit, beta )

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ICML Conference 2018 Conference Paper

Dropout Training, Data-dependent Regularization, and Generalization Bounds

Wenlong Mou
Yuchen Zhou
Jun Gao 0004
Liwei Wang 0001

We study the problem of generalization guarantees for dropout training. A general framework is first proposed for learning procedures with random perturbation on model parameters. The generalization error is bounded by sum of two offset Rademacher complexities: the main term is Rademacher complexity of the hypothesis class with minus offset induced by the perturbation variance, which characterizes data-dependent regularization by the random perturbation; the auxiliary term is offset Rademacher complexity for the variance class, controlling the degree to which this regularization effect can be weakened. For neural networks, we estimate upper and lower bounds for the variance induced by truthful dropout, a variant of dropout that we propose to ensure unbiased output and fit into our framework, and the variance bounds exhibits connection to adaptive regularization methods. By applying our framework to ReLU networks with one hidden layer, a generalization upper bound is derived with no assumptions on the parameter norms or data distribution, with $O(1/n)$ fast rate and adaptivity to geometry of data points being achieved at the same time.

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ICML Conference 2017 Conference Paper

Collect at Once, Use Effectively: Making Non-interactive Locally Private Learning Possible

Kai Zheng 0007
Wenlong Mou
Liwei Wang 0001

Non-interactive Local Differential Privacy (LDP) requires data analysts to collect data from users through noisy channel at once. In this paper, we extend the frontiers of Non-interactive LDP learning and estimation from several aspects. For learning with smooth generalized linear losses, we propose an approximate stochastic gradient oracle estimated from non-interactive LDP channel using Chebyshev expansion, which is combined with inexact gradient methods to obtain an efficient algorithm with quasi-polynomial sample complexity bound. For the high-dimensional world, we discover that under $\ell_2$-norm assumption on data points, high-dimensional sparse linear regression and mean estimation can be achieved with logarithmic dependence on dimension, using random projection and approximate recovery. We also extend our methods to Kernel Ridge Regression. Our work is the first one that makes learning and estimation possible for a broad range of learning tasks under non-interactive LDP model.

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ICML Conference 2017 Conference Paper

Differentially Private Clustering in High-Dimensional Euclidean Spaces

Maria-Florina Balcan
Travis Dick
Yingyu Liang
Wenlong Mou
Hongyang Zhang 0001

We study the problem of clustering sensitive data while preserving the privacy of individuals represented in the dataset, which has broad applications in practical machine learning and data analysis tasks. Although the problem has been widely studied in the context of low-dimensional, discrete spaces, much remains unknown concerning private clustering in high-dimensional Euclidean spaces $\mathbb{R}^d$. In this work, we give differentially private and efficient algorithms achieving strong guarantees for $k$-means and $k$-median clustering when $d=\Omega(\mathsf{polylog}(n))$. Our algorithm achieves clustering loss at most $\log^3(n)\mathsf{OPT}+\mathsf{poly}(\log n, d, k)$, advancing the state-of-the-art result of $\sqrt{d}\mathsf{OPT}+\mathsf{poly}(\log n, d^d, k^d)$. We also study the case where the data points are $s$-sparse and show that the clustering loss can scale logarithmically with $d$, i. e. , $\log^3(n)\mathsf{OPT}+\mathsf{poly}(\log n, \log d, k, s)$. Experiments on both synthetic and real datasets verify the effectiveness of the proposed method.

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IJCAI Conference 2017 Conference Paper

Efficient Private ERM for Smooth Objectives

Jiaqi Zhang
Kai Zheng
Wenlong Mou
Liwei Wang

In this paper, we consider efficient differentially private empirical risk minimization from the viewpoint of optimization algorithms. For strongly convex and smooth objectives, we prove that gradient descent with output perturbation not only achieves nearly optimal utility, but also significantly improves the running time of previous state-of-the-art private optimization algorithms, for both $\epsilon$-DP and $(\epsilon, \delta)$-DP. For non-convex but smooth objectives, we propose an RRPSGD (Random Round Private Stochastic Gradient Descent) algorithm, which provably converges to a stationary point with privacy guarantee. Besides the expected utility bounds, we also provide guarantees in high probability form. Experiments demonstrate that our algorithm consistently outperforms existing method in both utility and running time.

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