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Yuchen Zhang

Publications and source records attributed to Yuchen Zhang.

208 records · Page 12Linked to original sources
arXiv2015

Distributed Estimation of Generalized Matrix Rank: Efficient Algorithms and Lower Bounds

We study the following generalized matrix rank estimation problem: given an $n \times n$ matrix and a constant $c \geq 0$, estimate the number of eigenvalues that are greater than $c$. In the distributed setting, the matrix of interest is the sum of $m$ matrices held by separate machines. We show that any deterministic algorithm solving this problem must communicate $Ω(n^2)$ bits, which is order-equivalent to transmitting the whole matrix. In contrast, we propose a randomized algorithm that communicates only $\widetilde O(n)$ bits. The upper bound is matched by an $Ω(n)$ lower bound on the randomized communication complexity. We demonstrate the practical effectiveness of the proposed algorithm with some numerical experiments.

cs.DS↗
arXiv2015

Communication-Efficient Distributed Optimization of Self-Concordant Empirical Loss

We consider distributed convex optimization problems originated from sample average approximation of stochastic optimization, or empirical risk minimization in machine learning. We assume that each machine in the distributed computing system has access to a local empirical loss function, constructed with i.i.d. data sampled from a common distribution. We propose a communication-efficient distributed algorithm to minimize the overall empirical loss, which is the average of the local empirical losses. The algorithm is based on an inexact damped Newton method, where the inexact Newton steps are computed by a distributed preconditioned conjugate gradient method. We analyze its iteration complexity and communication efficiency for minimizing self-concordant empirical loss functions, and discuss the results for distributed ridge regression, logistic regression and binary classification with a smoothed hinge loss. In a standard setting for supervised learning, the required number of communication rounds of the algorithm does not increase with the sample size, and only grows slowly with the number of machines.

math.OC↗
arXiv2014

Spectral Methods meet EM: A Provably Optimal Algorithm for Crowdsourcing

Crowdsourcing is a popular paradigm for effectively collecting labels at low cost. The Dawid-Skene estimator has been widely used for inferring the true labels from the noisy labels provided by non-expert crowdsourcing workers. However, since the estimator maximizes a non-convex log-likelihood function, it is hard to theoretically justify its performance. In this paper, we propose a two-stage efficient algorithm for multi-class crowd labeling problems. The first stage uses the spectral method to obtain an initial estimate of parameters. Then the second stage refines the estimation by optimizing the objective function of the Dawid-Skene estimator via the EM algorithm. We show that our algorithm achieves the optimal convergence rate up to a logarithmic factor. We conduct extensive experiments on synthetic and real datasets. Experimental results demonstrate that the proposed algorithm is comparable to the most accurate empirical approach, while outperforming several other recently proposed methods.

stat.ML↗
arXiv2014

Optimality guarantees for distributed statistical estimation

Large data sets often require performing distributed statistical estimation, with a full data set split across multiple machines and limited communication between machines. To study such scenarios, we define and study some refinements of the classical minimax risk that apply to distributed settings, comparing to the performance of estimators with access to the entire data. Lower bounds on these quantities provide a precise characterization of the minimum amount of communication required to achieve the centralized minimax risk. We study two classes of distributed protocols: one in which machines send messages independently over channels without feedback, and a second allowing for interactive communication, in which a central server broadcasts the messages from a given machine to all other machines. We establish lower bounds for a variety of problems, including location estimation in several families and parameter estimation in different types of regression models. Our results include a novel class of quantitative data-processing inequalities used to characterize the effects of limited communication.

cs.IT↗
arXiv2014

Lower bounds on the performance of polynomial-time algorithms for sparse linear regression

Under a standard assumption in complexity theory (NP not in P/poly), we demonstrate a gap between the minimax prediction risk for sparse linear regression that can be achieved by polynomial-time algorithms, and that achieved by optimal algorithms. In particular, when the design matrix is ill-conditioned, the minimax prediction loss achievable by polynomial-time algorithms can be substantially greater than that of an optimal algorithm. This result is the first known gap between polynomial and optimal algorithms for sparse linear regression, and does not depend on conjectures in average-case complexity.

math.ST↗
arXiv2014

Divide and Conquer Kernel Ridge Regression: A Distributed Algorithm with Minimax Optimal Rates

We establish optimal convergence rates for a decomposition-based scalable approach to kernel ridge regression. The method is simple to describe: it randomly partitions a dataset of size N into m subsets of equal size, computes an independent kernel ridge regression estimator for each subset, then averages the local solutions into a global predictor. This partitioning leads to a substantial reduction in computation time versus the standard approach of performing kernel ridge regression on all N samples. Our two main theorems establish that despite the computational speed-up, statistical optimality is retained: as long as m is not too large, the partition-based estimator achieves the statistical minimax rate over all estimators using the set of N samples. As concrete examples, our theory guarantees that the number of processors m may grow nearly linearly for finite-rank kernels and Gaussian kernels and polynomially in N for Sobolev spaces, which in turn allows for substantial reductions in computational cost. We conclude with experiments on both simulated data and a music-prediction task that complement our theoretical results, exhibiting the computational and statistical benefits of our approach.

math.ST↗
arXiv2014

On the generic vanishing theorem of Cartier modules

We generalize the Generic Vanishing theorem by Hacon and Patakfalvi in the spirit of Pareschi and Popa. We give several examples illustrating the pathologies appearing in the positive characteristic setting.

math.AG↗
arXiv2013

On the Volume of Isolated Singularities

We give an equivalent definition of the local volume of an isolated singularity Vol_{BdFF}(X,0) given in [BdFF12] in the Q-Gorenstein case and we generalize it to the non-Q-Gorenstein case. We prove that there is a positive lower bound depending only on the dimension for the non-zero local volume of an isolated singularity if X is Gorenstein. We also give a non-Q-Gorenstein example with Vol_{BdFF}(X,0)=0, which does not allow a boundary Δsuch that the pair (X,Δ) is log canonical.

math.AG↗
arXiv2013

Comunication-Efficient Algorithms for Statistical Optimization

We analyze two communication-efficient algorithms for distributed statistical optimization on large-scale data sets. The first algorithm is a standard averaging method that distributes the $N$ data samples evenly to $\nummac$ machines, performs separate minimization on each subset, and then averages the estimates. We provide a sharp analysis of this average mixture algorithm, showing that under a reasonable set of conditions, the combined parameter achieves mean-squared error that decays as $\order(N^{-1}+(N/m)^{-2})$. Whenever $m \le \sqrt{N}$, this guarantee matches the best possible rate achievable by a centralized algorithm having access to all $\totalnumobs$ samples. The second algorithm is a novel method, based on an appropriate form of bootstrap subsampling. Requiring only a single round of communication, it has mean-squared error that decays as $\order(N^{-1} + (N/m)^{-3})$, and so is more robust to the amount of parallelization. In addition, we show that a stochastic gradient-based method attains mean-squared error decaying as $O(N^{-1} + (N/ m)^{-3/2})$, easing computation at the expense of penalties in the rate of convergence. We also provide experimental evaluation of our methods, investigating their performance both on simulated data and on a large-scale regression problem from the internet search domain. In particular, we show that our methods can be used to efficiently solve an advertisement prediction problem from the Chinese SoSo Search Engine, which involves logistic regression with $N \approx 2.4 \times 10^8$ samples and $d \approx 740,000$ covariates.

stat.ML↗