arXiv subjects

Shengyu Zhu

Publications and source records attributed to Shengyu Zhu.

29 records · Page 2Linked to original sources

arXiv2021

Asymptotically Optimal One- and Two-Sample Testing with Kernels

We characterize the asymptotic performance of nonparametric one- and two-sample testing. The exponential decay rate or error exponent of the type-II error probability is used as the asymptotic performance metric, and an optimal test achieves the maximum rate subject to a constant level constraint on the type-I error probability. With Sanov's theorem, we derive a sufficient condition for one-sample tests to achieve the optimal error exponent in the universal setting, i.e., for any distribution defining the alternative hypothesis. We then show that two classes of Maximum Mean Discrepancy (MMD) based tests attain the optimal type-II error exponent on $\mathbb R^d$, while the quadratic-time Kernel Stein Discrepancy (KSD) based tests achieve this optimality with an asymptotic level constraint. For general two-sample testing, however, Sanov's theorem is insufficient to obtain a similar sufficient condition. We proceed to establish an extended version of Sanov's theorem and derive an exact error exponent for the quadratic-time MMD based two-sample tests. The obtained error exponent is further shown to be optimal among all two-sample tests satisfying a given level constraint. Our work hence provides an achievability result for optimal nonparametric one- and two-sample testing in the universal setting. Application to off-line change detection and related issues are also discussed.

cs.IT↗

arXiv2020

Causal Discovery with Reinforcement Learning

Discovering causal structure among a set of variables is a fundamental problem in many empirical sciences. Traditional score-based casual discovery methods rely on various local heuristics to search for a Directed Acyclic Graph (DAG) according to a predefined score function. While these methods, e.g., greedy equivalence search, may have attractive results with infinite samples and certain model assumptions, they are usually less satisfactory in practice due to finite data and possible violation of assumptions. Motivated by recent advances in neural combinatorial optimization, we propose to use Reinforcement Learning (RL) to search for the DAG with the best scoring. Our encoder-decoder model takes observable data as input and generates graph adjacency matrices that are used to compute rewards. The reward incorporates both the predefined score function and two penalty terms for enforcing acyclicity. In contrast with typical RL applications where the goal is to learn a policy, we use RL as a search strategy and our final output would be the graph, among all graphs generated during training, that achieves the best reward. We conduct experiments on both synthetic and real datasets, and show that the proposed approach not only has an improved search ability but also allows a flexible score function under the acyclicity constraint.

cs.LG↗

arXiv2019

A Graph Autoencoder Approach to Causal Structure Learning

Causal structure learning has been a challenging task in the past decades and several mainstream approaches such as constraint- and score-based methods have been studied with theoretical guarantees. Recently, a new approach has transformed the combinatorial structure learning problem into a continuous one and then solved it using gradient-based optimization methods. Following the recent state-of-the-arts, we propose a new gradient-based method to learn causal structures from observational data. The proposed method generalizes the recent gradient-based methods to a graph autoencoder framework that allows nonlinear structural equation models and is easily applicable to vector-valued variables. We demonstrate that on synthetic datasets, our proposed method outperforms other gradient-based methods significantly, especially on large causal graphs. We further investigate the scalability and efficiency of our method, and observe a near linear training time when scaling up the graph size.

cs.LG↗

arXiv2019

Causal Discovery by Kernel Intrinsic Invariance Measure

Reasoning based on causality, instead of association has been considered as a key ingredient towards real machine intelligence. However, it is a challenging task to infer causal relationship/structure among variables. In recent years, an Independent Mechanism (IM) principle was proposed, stating that the mechanism generating the cause and the one mapping the cause to the effect are independent. As the conjecture, it is argued that in the causal direction, the conditional distributions instantiated at different value of the conditioning variable have less variation than the anti-causal direction. Existing state-of-the-arts simply compare the variance of the RKHS mean embedding norms of these conditional distributions. In this paper, we prove that this norm-based approach sacrifices important information of the original conditional distributions. We propose a Kernel Intrinsic Invariance Measure (KIIM) to capture higher order statistics corresponding to the shapes of the density functions. We show our algorithm can be reduced to an eigen-decomposition task on a kernel matrix measuring intrinsic deviance/invariance. Causal directions can then be inferred by comparing the KIIM scores of two hypothetic directions. Experiments on synthetic and real data are conducted to show the advantages of our methods over existing solutions.

cs.LG↗

arXiv2019

Universal Hypothesis Testing with Kernels: Asymptotically Optimal Tests for Goodness of Fit

We characterize the asymptotic performance of nonparametric goodness of fit testing. The exponential decay rate of the type-II error probability is used as the asymptotic performance metric, and a test is optimal if it achieves the maximum rate subject to a constant level constraint on the type-I error probability. We show that two classes of Maximum Mean Discrepancy (MMD) based tests attain this optimality on $\mathbb R^d$, while the quadratic-time Kernel Stein Discrepancy (KSD) based tests achieve the maximum exponential decay rate under a relaxed level constraint. Under the same performance metric, we proceed to show that the quadratic-time MMD based two-sample tests are also optimal for general two-sample problems, provided that kernels are bounded continuous and characteristic. Key to our approach are Sanov's theorem from large deviation theory and the weak metrizable properties of the MMD and KSD.

stat.ML↗

arXiv2018

Distributed Detection in Ad Hoc Networks Through Quantized Consensus

We study asymptotic performance of distributed detection in large scale connected sensor networks. Contrasting to the canonical parallel network where a single node has access to local decisions from all other nodes, each node can only exchange information with its direct neighbors in the present setting. We establish that, with each node employing an identical one-bit quantizer for local information exchange, a novel consensus reaching approach can achieve the optimal asymptotic performance of centralized detection as the network size scales. The statement is true under three different detection frameworks: the Bayesian criterion where the maximum a posteriori detector is optimal, the Neyman-Pearson criterion with a constant type-I error probability constraint, and the Neyman-Pearson criterion with an exponential type-I error probability constraint. Leveraging recent development in distributed consensus reaching using bounded quantizers with possibly unbounded data (which are log-likelihood ratios of local observations in the context of distributed detection), we design a one-bit deterministic quantizer with controllable threshold that leads to desirable consensus error bounds. The obtained bounds are key to establishing the optimal asymptotic detection performance. In addition, we examine non-asymptotic performance of the proposed approach, and show that the type-I and type-II error probabilities at each node can be made arbitrarily close to the centralized ones simultaneously when a continuity condition is satisfied.

cs.IT↗

arXiv2018

Exponentially Consistent Kernel Two-Sample Tests

Given two sets of independent samples from unknown distributions $P$ and $Q$, a two-sample test decides whether to reject the null hypothesis that $P=Q$. Recent attention has focused on kernel two-sample tests as the test statistics are easy to compute, converge fast, and have low bias with their finite sample estimates. However, there still lacks an exact characterization on the asymptotic performance of such tests, and in particular, the rate at which the type-II error probability decays to zero in the large sample limit. In this work, we establish that a class of kernel two-sample tests are exponentially consistent with Polish, locally compact Hausdorff sample space, e.g., $\mathbb R^d$. The obtained exponential decay rate is further shown to be optimal among all two-sample tests satisfying the level constraint, and is independent of particular kernels provided that they are bounded continuous and characteristic. Our results gain new insights into related issues such as fair alternative for testing and kernel selection strategy. Finally, as an application, we show that a kernel based test achieves the optimal detection for off-line change detection in the nonparametric setting.

stat.ML↗

arXiv2017

Distributed Average Consensus with Bounded Quantizer and Unbounded Input

This paper considers distributed average consensus using finite-bit bounded quantizer with possibly unbounded data. Under the framework of the alternating direction method of multipliers (ADMM), we develop distributed averaging algorithms where each node iteratively updates using only the local information and finitely quantized outputs from its neighbors. It is shown that all the agent variables either converge to the same quantization level or cycle around the data average after finite iterations. An error bound for the consensus value is established, which turns out to be the same as that of using the unbounded rounding quantizer provided that an algorithm parameter (i.e., ADMM step size) is small enough. We also analyze the effect of the algorithm parameter and propose an adaptive parameter selection strategy that only requires knowledge of the number of agents in order to accelerate the algorithm with certain consensus accuracy guarantee. Finally, simulations are performed to illustrate the effectiveness of the proposed algorithms.

math.OC↗

arXiv2016

Quantized Consensus by the ADMM: Probabilistic versus Deterministic Quantizers

This paper develops efficient algorithms for distributed average consensus with quantized communication using the alternating direction method of multipliers (ADMM). We first study the effects of probabilistic and deterministic quantizations on a distributed ADMM algorithm. With probabilistic quantization, this algorithm yields linear convergence to the desired average in the mean sense with a bounded variance. When deterministic quantization is employed, the distributed ADMM either converges to a consensus or cycles with a finite period after a finite-time iteration. In the cyclic case, local quantized variables have the same mean over one period and hence each node can also reach a consensus. We then obtain an upper bound on the consensus error which depends only on the quantization resolution and the average degree of the network. Finally, we propose a two-stage algorithm which combines both probabilistic and deterministic quantizations. Simulations show that the two-stage algorithm, without picking small algorithm parameter, has consensus errors that are typically less than one quantization resolution for all connected networks where agents' data can be of arbitrary magnitudes.

eess.SY↗

arXiv2015

Quantized Consensus ADMM for Multi-Agent Distributed Optimization

Multi-agent distributed optimization over a network minimizes a global objective formed by a sum of local convex functions using only local computation and communication. We develop and analyze a quantized distributed algorithm based on the alternating direction method of multipliers (ADMM) when inter-agent communications are subject to finite capacity and other practical constraints. While existing quantized ADMM approaches only work for quadratic local objectives, the proposed algorithm can deal with more general objective functions (possibly non-smooth) including the LASSO. Under certain convexity assumptions, our algorithm converges to a consensus within $\log_{1+η}Ω$ iterations, where $η>0$ depends on the local objectives and the network topology, and $Ω$ is a polynomial determined by the quantization resolution, the distance between initial and optimal variable values, the local objective functions and the network topology. A tight upper bound on the consensus error is also obtained which does not depend on the size of the network.

math.OC↗

arXiv2013

Decentralized Data Reduction with Quantization Constraints

A guiding principle for data reduction in statistical inference is the sufficiency principle. This paper extends the classical sufficiency principle to decentralized inference, i.e., data reduction needs to be achieved in a decentralized manner. We examine the notions of local and global sufficient statistics and the relationship between the two for decentralized inference under different observation models. We then consider the impacts of quantization on decentralized data reduction which is often needed when communications among sensors are subject to finite capacity constraints. The central question we intend to ask is: if each node in a decentralized inference system has to summarize its data using a finite number of bits, is it still optimal to implement data reduction using global sufficient statistics prior to quantization? We show that the answer is negative using a simple example and proceed to identify conditions under which sufficiency based data reduction followed by quantization is indeed optimal. They include the well known case when the data at decentralized nodes are conditionally independent as well as a class of problems with conditionally dependent observations that admit conditional independence structure through the introduction of an appropriately chosen hidden variable.

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