arXiv subjects

Qifan Song

Publications and source records attributed to Qifan Song.

At least 37 records · Page 2Linked to original sources

arXiv2023

A New Paradigm for Generative Adversarial Networks based on Randomized Decision Rules

The Generative Adversarial Network (GAN) was recently introduced in the literature as a novel machine learning method for training generative models. It has many applications in statistics such as nonparametric clustering and nonparametric conditional independence tests. However, training the GAN is notoriously difficult due to the issue of mode collapse, which refers to the lack of diversity among generated data. In this paper, we identify the reasons why the GAN suffers from this issue, and to address it, we propose a new formulation for the GAN based on randomized decision rules. In the new formulation, the discriminator converges to a fixed point while the generator converges to a distribution at the Nash equilibrium. We propose to train the GAN by an empirical Bayes-like method by treating the discriminator as a hyper-parameter of the posterior distribution of the generator. Specifically, we simulate generators from its posterior distribution conditioned on the discriminator using a stochastic gradient Markov chain Monte Carlo (MCMC) algorithm, and update the discriminator using stochastic gradient descent along with simulations of the generators. We establish convergence of the proposed method to the Nash equilibrium. Apart from image generation, we apply the proposed method to nonparametric clustering and nonparametric conditional independence tests. A portion of the numerical results is presented in the supplementary material.

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arXiv2023

Matrix Completion from General Deterministic Sampling Patterns

Most of the existing works on provable guarantees for low-rank matrix completion algorithms rely on some unrealistic assumptions such that matrix entries are sampled randomly or the sampling pattern has a specific structure. In this work, we establish theoretical guarantee for the exact and approximate low-rank matrix completion problems which can be applied to any deterministic sampling schemes. For this, we introduce a graph having observed entries as its edge set, and investigate its graph properties involving the performance of the standard constrained nuclear norm minimization algorithm. We theoretically and experimentally show that the algorithm can be successful as the observation graph is well-connected and has similar node degrees. Our result can be viewed as an extension of the works by Bhojanapalli and Jain [2014] and Burnwal and Vidyasagar [2020], in which the node degrees of the observation graph were assumed to be the same. In particular, our theory significantly improves their results when the underlying matrix is symmetric.

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arXiv2023

Optimum-statistical Collaboration Towards General and Efficient Black-box Optimization

In this paper, we make the key delineation on the roles of resolution and statistical uncertainty in hierarchical bandits-based black-box optimization algorithms, guiding a more general analysis and a more efficient algorithm design. We introduce the \textit{optimum-statistical collaboration}, an algorithm framework of managing the interaction between optimization error flux and statistical error flux evolving in the optimization process. We provide a general analysis of this framework without specifying the forms of statistical error and uncertainty quantifier. Our framework and its analysis, due to their generality, can be applied to a large family of functions and partitions that satisfy different local smoothness assumptions and have different numbers of local optimums, which is much richer than the class of functions studied in prior works. Our framework also inspires us to propose a better measure of the statistical uncertainty and consequently a variance-adaptive algorithm \texttt{VHCT}. In theory, we prove the algorithm enjoys rate-optimal regret bounds under different local smoothness assumptions; in experiments, we show the algorithm outperforms prior efforts in different settings.

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arXiv2023

Federated X-Armed Bandit

This work establishes the first framework of federated $\mathcal{X}$-armed bandit, where different clients face heterogeneous local objective functions defined on the same domain and are required to collaboratively figure out the global optimum. We propose the first federated algorithm for such problems, named \texttt{Fed-PNE}. By utilizing the topological structure of the global objective inside the hierarchical partitioning and the weak smoothness property, our algorithm achieves sublinear cumulative regret with respect to both the number of clients and the evaluation budget. Meanwhile, it only requires logarithmic communications between the central server and clients, protecting the client privacy. Experimental results on synthetic functions and real datasets validate the advantages of \texttt{Fed-PNE} over various centralized and federated baseline algorithms.

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arXiv2023

PyXAB -- A Python Library for $\mathcal{X}$-Armed Bandit and Online Blackbox Optimization Algorithms

We introduce a Python open-source library for $\mathcal{X}$-armed bandit and online blackbox optimization named PyXAB. PyXAB contains the implementations for more than 10 $\mathcal{X}$-armed bandit algorithms, such as HOO, StoSOO, HCT, and the most recent works GPO and VHCT. PyXAB also provides the most commonly-used synthetic objectives to evaluate the performance of different algorithms and the various choices of the hierarchical partitions on the parameter space. The online documentation for PyXAB includes clear instructions for installation, straight-forward examples, detailed feature descriptions, and a complete reference of the API. PyXAB is released under the MIT license in order to encourage both academic and industrial usage. The library can be directly installed from PyPI with its source code available at https://github.com/WilliamLwj/PyXAB

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arXiv2023

Support Recovery in Sparse PCA with Non-Random Missing Data

We analyze a practical algorithm for sparse PCA on incomplete and noisy data under a general non-random sampling scheme. The algorithm is based on a semidefinite relaxation of the $\ell_1$-regularized PCA problem. We provide theoretical justification that under certain conditions, we can recover the support of the sparse leading eigenvector with high probability by obtaining a unique solution. The conditions involve the spectral gap between the largest and second-largest eigenvalues of the true data matrix, the magnitude of the noise, and the structural properties of the observed entries. The concepts of algebraic connectivity and irregularity are used to describe the structural properties of the observed entries. We empirically justify our theorem with synthetic and real data analysis. We also show that our algorithm outperforms several other sparse PCA approaches especially when the observed entries have good structural properties. As a by-product of our analysis, we provide two theorems to handle a deterministic sampling scheme, which can be applied to other matrix-related problems.

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arXiv2022

Nearly optimal Bayesian Shrinkage for High Dimensional Regression

During the past decade, shrinkage priors have received much attention in Bayesian analysis of high-dimensional data. This paper establishes the posterior consistency for high-dimensional linear regression with a class of shrinkage priors, which has a heavy and flat tail and allocates a sufficiently large probability mass in a very small neighborhood of zero. While enjoying its efficiency in posterior simulations, the shrinkage prior can lead to a nearly optimal posterior contraction rate and variable selection consistency as the spike-and-slab prior. Our numerical results show that under the posterior consistency, Bayesian methods can yield much better results in variable selection than the regularization methods such as Lasso and SCAD. This paper also establishes a Bernstein von-Mises type result, which leads to a convenient way of uncertainty quantification for regression coefficient estimates.

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arXiv2022

Optimal False Discovery Control of Minimax Estimator

Two major research tasks lie at the heart of high dimensional data analysis: accurate parameter estimation and correct support recovery. The existing literature mostly aims for either the best parameter estimation or the best model selection result, however little has been done to understand the potential interaction between the estimation precision and the selection behavior. In this work, our minimax result shows that an estimator's performance of type I error control directly links with its $L_2$ estimation error rate, and reveals a trade-off phenomenon between the rate of convergence and the false discovery control: to achieve better accuracy, one risks yielding more false discoveries. In particular, we characterize the false discovery control behavior of rate optimal and rate suboptimal estimators under different sparsity regimes, and discover a rigid dichotomy between these two estimators under near-linear and linear sparsity settings. In addition, this work provides a rigorous explanation to the incompatibility phenomenon between selection consistency and rate minimaxity which has been frequently observed in the high dimensional literature.

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arXiv2022

Support Recovery in Sparse PCA with Incomplete Data

We study a practical algorithm for sparse principal component analysis (PCA) of incomplete and noisy data. Our algorithm is based on the semidefinite program (SDP) relaxation of the non-convex $l_1$-regularized PCA problem. We provide theoretical and experimental evidence that SDP enables us to exactly recover the true support of the sparse leading eigenvector of the unknown true matrix, despite only observing an incomplete (missing uniformly at random) and noisy version of it. We derive sufficient conditions for exact recovery, which involve matrix incoherence, the spectral gap between the largest and second-largest eigenvalues, the observation probability and the noise variance. We validate our theoretical results with incomplete synthetic data, and show encouraging and meaningful results on a gene expression dataset.

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arXiv2022

Benefit of Interpolation in Nearest Neighbor Algorithms

In some studies \citep[e.g.,][]{zhang2016understanding} of deep learning, it is observed that over-parametrized deep neural networks achieve a small testing error even when the training error is almost zero. Despite numerous works towards understanding this so-called "double descent" phenomenon \citep[e.g.,][]{belkin2018reconciling,belkin2019two}, in this paper, we turn into another way to enforce zero training error (without over-parametrization) through a data interpolation mechanism. Specifically, we consider a class of interpolated weighting schemes in the nearest neighbors (NN) algorithms. By carefully characterizing the multiplicative constant in the statistical risk, we reveal a U-shaped performance curve for the level of data interpolation in both classification and regression setups. This sharpens the existing result \citep{belkin2018does} that zero training error does not necessarily jeopardize predictive performances and claims a counter-intuitive result that a mild degree of data interpolation actually {\em strictly} improve the prediction performance and statistical stability over those of the (un-interpolated) $k$-NN algorithm. In the end, the universality of our results, such as change of distance measure and corrupted testing data, will also be discussed.

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arXiv2022

Data-guided Treatment Recommendation with Feature Scores

Despite the availability of large amounts of genomics data, medical treatment recommendations have not successfully used them. In this paper, we consider the utility of high dimensional genomic-clinical data and nonparametric methods for making cancer treatment recommendations. This builds upon the framework of the individualized treatment rule [Qian and Murphy 2011] but we aim to overcome their method's limitations, specifically in the instances when the method encounters a large number of covariates and an issue of model misspecification. We tackle this problem using a dimension reduction method, namely Sliced Inverse Regression (SIR, [Li 1991]), with a rich class of models for the treatment response. Notably, SIR defines a feature space for high-dimensional data, offering an advantage similar to those found in the popular neural network models. With the features obtained from SIR, a simple visualization is used to compare different treatment options and present the recommended treatment. Additionally, we derive the consistency and the convergence rate of the proposed recommendation approach through a value function. The effectiveness of the proposed approach is demonstrated through simulation studies and the promising results from a real-data example of the treatment of multiple myeloma.

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arXiv2022

Unlabeled Data Help: Minimax Analysis and Adversarial Robustness

The recent proposed self-supervised learning (SSL) approaches successfully demonstrate the great potential of supplementing learning algorithms with additional unlabeled data. However, it is still unclear whether the existing SSL algorithms can fully utilize the information of both labelled and unlabeled data. This paper gives an affirmative answer for the reconstruction-based SSL algorithm \citep{lee2020predicting} under several statistical models. While existing literature only focuses on establishing the upper bound of the convergence rate, we provide a rigorous minimax analysis, and successfully justify the rate-optimality of the reconstruction-based SSL algorithm under different data generation models. Furthermore, we incorporate the reconstruction-based SSL into the existing adversarial training algorithms and show that learning from unlabeled data helps improve the robustness.

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arXiv2021

A Langevinized Ensemble Kalman Filter for Large-Scale Static and Dynamic Learning

The Ensemble Kalman Filter (EnKF) has achieved great successes in data assimilation in atmospheric and oceanic sciences, but its failure in convergence to the right filtering distribution precludes its use for uncertainty quantification. We reformulate the EnKF under the framework of Langevin dynamics, which leads to a new particle filtering algorithm, the so-called Langevinized EnKF. The Langevinized EnKF inherits the forecast-analysis procedure from the EnKF and the use of mini-batch data from the stochastic gradient Langevin-type algorithms, which make it scalable with respect to both the dimension and sample size. We prove that the Langevinized EnKF converges to the right filtering distribution in Wasserstein distance under the big data scenario that the dynamic system consists of a large number of stages and has a large number of samples observed at each stage. We reformulate the Bayesian inverse problem as a dynamic state estimation problem based on the techniques of subsampling and Langevin diffusion process. We illustrate the performance of the Langevinized EnKF using a variety of examples, including the Lorenz-96 model, high-dimensional variable selection, Bayesian deep learning, and Long Short Term Memory (LSTM) network learning with dynamic data.

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arXiv2021

On the Generalization Properties of Adversarial Training

Modern machine learning and deep learning models are shown to be vulnerable when testing data are slightly perturbed. Existing theoretical studies of adversarial training algorithms mostly focus on either adversarial training losses or local convergence properties. In contrast, this paper studies the generalization performance of a generic adversarial training algorithm. Specifically, we consider linear regression models and two-layer neural networks (with lazy training) using squared loss under low-dimensional and high-dimensional regimes. In the former regime, after overcoming the non-smoothness of adversarial training, the adversarial risk of the trained models can converge to the minimal adversarial risk. In the latter regime, we discover that data interpolation prevents the adversarially robust estimator from being consistent. Therefore, inspired by successes of the least absolute shrinkage and selection operator (LASSO), we incorporate the L1 penalty in the high dimensional adversarial learning and show that it leads to consistent adversarially robust estimation. A series of numerical studies are conducted to demonstrate how the smoothness and L1 penalization help improve the adversarial robustness of DNN models.

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arXiv2021

Consistent Sparse Deep Learning: Theory and Computation

Deep learning has been the engine powering many successes of data science. However, the deep neural network (DNN), as the basic model of deep learning, is often excessively over-parameterized, causing many difficulties in training, prediction and interpretation. We propose a frequentist-like method for learning sparse DNNs and justify its consistency under the Bayesian framework: the proposed method could learn a sparse DNN with at most $O(n/\log(n))$ connections and nice theoretical guarantees such as posterior consistency, variable selection consistency and asymptotically optimal generalization bounds. In particular, we establish posterior consistency for the sparse DNN with a mixture Gaussian prior, show that the structure of the sparse DNN can be consistently determined using a Laplace approximation-based marginal posterior inclusion probability approach, and use Bayesian evidence to elicit sparse DNNs learned by an optimization method such as stochastic gradient descent in multiple runs with different initializations. The proposed method is computationally more efficient than standard Bayesian methods for large-scale sparse DNNs. The numerical results indicate that the proposed method can perform very well for large-scale network compression and high-dimensional nonlinear variable selection, both advancing interpretable machine learning.

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arXiv2020

Efficient Variational Inference for Sparse Deep Learning with Theoretical Guarantee

Sparse deep learning aims to address the challenge of huge storage consumption by deep neural networks, and to recover the sparse structure of target functions. Although tremendous empirical successes have been achieved, most sparse deep learning algorithms are lacking of theoretical support. On the other hand, another line of works have proposed theoretical frameworks that are computationally infeasible. In this paper, we train sparse deep neural networks with a fully Bayesian treatment under spike-and-slab priors, and develop a set of computationally efficient variational inferences via continuous relaxation of Bernoulli distribution. The variational posterior contraction rate is provided, which justifies the consistency of the proposed variational Bayes method. Notably, our empirical results demonstrate that this variational procedure provides uncertainty quantification in terms of Bayesian predictive distribution and is also capable to accomplish consistent variable selection by training a sparse multi-layer neural network.

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arXiv2020

Nearly Optimal Variational Inference for High Dimensional Regression with Shrinkage Priors

We propose a variational Bayesian (VB) procedure for high-dimensional linear model inferences with heavy tail shrinkage priors, such as student-t prior. Theoretically, we establish the consistency of the proposed VB method and prove that under the proper choice of prior specifications, the contraction rate of the VB posterior is nearly optimal. It justifies the validity of VB inference as an alternative of Markov Chain Monte Carlo (MCMC) sampling. Meanwhile, comparing to conventional MCMC methods, the VB procedure achieves much higher computational efficiency, which greatly alleviates the computing burden for modern machine learning applications such as massive data analysis. Through numerical studies, we demonstrate that the proposed VB method leads to shorter computing time, higher estimation accuracy, and lower variable selection error than competitive sparse Bayesian methods.

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arXiv2020

Stochastic Gradient Langevin Dynamics Algorithms with Adaptive Drifts

Bayesian deep learning offers a principled way to address many issues concerning safety of artificial intelligence (AI), such as model uncertainty,model interpretability, and prediction bias. However, due to the lack of efficient Monte Carlo algorithms for sampling from the posterior of deep neural networks (DNNs), Bayesian deep learning has not yet powered our AI system. We propose a class of adaptive stochastic gradient Markov chain Monte Carlo (SGMCMC) algorithms, where the drift function is biased to enhance escape from saddle points and the bias is adaptively adjusted according to the gradient of past samples. We establish the convergence of the proposed algorithms under mild conditions, and demonstrate via numerical examples that the proposed algorithms can significantly outperform the existing SGMCMC algorithms, such as stochastic gradient Langevin dynamics (SGLD), stochastic gradient Hamiltonian Monte Carlo (SGHMC) and preconditioned SGLD, in both simulation and optimization tasks.

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