arXiv subjects
Publications and source records attributed to Chao Gao.
We point out that the computation of true \emph{proof} and \emph{disproof} numbers for proof number search in arbitrary directed acyclic graphs is NP-hard, an important theoretical result for proof number search. The proof requires a reduction from SAT, which demonstrates that finding true proof/disproof number for arbitrary DAG is at least as hard as deciding if arbitrary SAT instance is satisfiable, thus NP-hard.
We consider the problem of ranking $n$ players from partial pairwise comparison data under the Bradley-Terry-Luce model. For the first time in the literature, the minimax rate of this ranking problem is derived with respect to the Kendall's tau distance that measures the difference between two rank vectors by counting the number of inversions. The minimax rate of ranking exhibits a transition between an exponential rate and a polynomial rate depending on the magnitude of the signal-to-noise ratio of the problem. To the best of our knowledge, this phenomenon is unique to full ranking and has not been seen in any other statistical estimation problem. To achieve the minimax rate, we propose a divide-and-conquer ranking algorithm that first divides the $n$ players into groups of similar skills and then computes local MLE within each group. The optimality of the proposed algorithm is established by a careful approximate independence argument between the two steps.
We study the phase synchronization problem with measurements $Y=z^*z^{*H}+σW\in\mathbb{C}^{n\times n}$, where $z^*$ is an $n$-dimensional complex unit-modulus vector and $W$ is a complex-valued Gaussian random matrix. It is assumed that each entry $Y_{jk}$ is observed with probability $p$. We prove that the minimax lower bound of estimating $z^*$ under the squared $\ell_2$ loss is $(1-o(1))\frac{σ^2}{2p}$. We also show that both generalized power method and maximum likelihood estimator achieve the error bound $(1+o(1))\frac{σ^2}{2p}$. Thus, $\frac{σ^2}{2p}$ is the exact asymptotic minimax error of the problem. Our upper bound analysis involves a precise characterization of the statistical property of the power iteration. The lower bound is derived through an application of van Trees' inequality.
We study the detection of a sparse change in a high-dimensional mean vector as a minimax testing problem. Our first main contribution is to derive the exact minimax testing rate across all parameter regimes for $n$ independent, $p$-variate Gaussian observations. This rate exhibits a phase transition when the sparsity level is of order $\sqrt{p \log \log (8n)}$ and has a very delicate dependence on the sample size: in a certain sparsity regime it involves a triple iterated logarithmic factor in~$n$. Further, in a dense asymptotic regime, we identify the sharp leading constant, while in the corresponding sparse asymptotic regime, this constant is determined to within a factor of $\sqrt{2}$. Extensions that cover spatial and temporal dependence, primarily in the dense case, are also provided.
We investigate the relationship between the cosmic curvature and the model of dark energy (hereafter DE) with the recent Type Ia supernovae (hereafter SNe Ia) data, i.e., the Pantheon sample including 1048 SNe Ia with $0.01 < z < 2.3$. We obtain the measurements of the dimensionless spatial curvature density today, i.e., $Ω_{k0} = -0.062^{+0.189}_{-0.169}, -0.004^{+0.228}_{-0.134}, 0.127^{+0.280}_{-0.276}$ and $0.422^{+0.213}_{-0.338}$ at 68\% confidence level (CL), respectively, in the scenarios of $Λ$CDM, $ϕ$CDM (i.e., scalar field dark energy), $ω$CDM and $ω_0ω_a$CDM models. In the scenario of $Λ$CDM model, a closed universe is preferred by the Pantheon sample, which is consistent with that from the Planck CMB spectra. However, the uncertainty of $Ω_{k0}$ from the Pantheon SNe sample is about 8 times larger than that from the Planck data, so the former one supports a closed universe at a much lower CL than that from the latter one. An open unverse is supported by the Pantheon sample at $\sim$32\% and $\sim$78\% CLs, respectively, in the $ω$CDM and $ω_0ω_a$CDM models. Among these models, the $ϕ$CDM model is the one which supports the flat universe most strongly. It shows that $Ω_{k0}$ is significantly dependent on the adopted model of dark energy, and there is a negative correlation between $Ω_{k0}$ and the equation of state of DE.
We propose a general modeling and algorithmic framework for discrete structure recovery that can be applied to a wide range of problems. Under this framework, we are able to study the recovery of clustering labels, ranks of players, signs of regression coefficients, cyclic shifts, and even group elements from a unified perspective. A simple iterative algorithm is proposed for discrete structure recovery, which generalizes methods including Lloyd's algorithm and the power method. A linear convergence result for the proposed algorithm is established in this paper under appropriate abstract conditions on stochastic errors and initialization. We illustrate our general theory by applying it on several representative problems: (1) clustering in Gaussian mixture model, (2) approximate ranking, (3) sign recovery in compressed sensing, (4) multireference alignment, and (5) group synchronization, and show that minimax rate is achieved in each case.
We study the convergence rates of empirical Bayes posterior distributions for nonparametric and high-dimensional inference. We show that as long as the hyperparameter set is discrete, the empirical Bayes posterior distribution induced by the maximum marginal likelihood estimator can be regarded as a variational approximation to a hierarchical Bayes posterior distribution. This connection between empirical Bayes and variational Bayes allows us to leverage the recent results in the variational Bayes literature, and directly obtains the convergence rates of empirical Bayes posterior distributions from a variational perspective. For a more general hyperparameter set that is not necessarily discrete, we introduce a new technique called "prior decomposition" to deal with prior distributions that can be written as convex combinations of probability measures whose supports are low-dimensional subspaces. This leads to generalized versions of the classical "prior mass and testing" conditions for the convergence rates of empirical Bayes. Our theory is applied to a number of statistical estimation problems including nonparametric density estimation and sparse linear regression.
We propose a general algorithmic framework for Bayesian model selection. A spike-and-slab Laplacian prior is introduced to model the underlying structural assumption. Using the notion of effective resistance, we derive an EM-type algorithm with closed-form iterations to efficiently explore possible candidates for Bayesian model selection. The deterministic nature of the proposed algorithm makes it more scalable to large-scale and high-dimensional data sets compared with existing stochastic search algorithms. When applied to sparse linear regression, our framework recovers the EMVS algorithm [Rockova and George, 2014] as a special case. We also discuss extensions of our framework using tools from graph algebra to incorporate complex Bayesian models such as biclustering and submatrix localization. Extensive simulation studies and real data applications are conducted to demonstrate the superior performance of our methods over its frequentist competitors such as $\ell_0$ or $\ell_1$ penalization.
A new type of robust estimation problem is introduced where the goal is to recover a statistical model that has been corrupted after it has been estimated from data. Methods are proposed for "repairing" the model using only the design and not the response values used to fit the model in a supervised learning setting. Theory is developed which reveals that two important ingredients are necessary for model repair---the statistical model must be over-parameterized, and the estimator must incorporate redundancy. In particular, estimators based on stochastic gradient descent are seen to be well suited to model repair, but sparse estimators are not in general repairable. After formulating the problem and establishing a key technical lemma related to robust estimation, a series of results are presented for repair of over-parameterized linear models, random feature models, and artificial neural networks. Simulation studies are presented that corroborate and illustrate the theoretical findings.
Consider the heteroscedastic nonparametric regression model with random design \begin{align*} Y_i = f(X_i) + V^{1/2}(X_i)\varepsilon_i, \quad i=1,2,\ldots,n, \end{align*} with $f(\cdot)$ and $V(\cdot)$ $α$- and $β$-Hölder smooth, respectively. We show that the minimax rate of estimating $V(\cdot)$ under both local and global squared risks is of the order \begin{align*} n^{-\frac{8αβ}{4αβ+ 2α+ β}} \vee n^{-\frac{2β}{2β+1}}, \end{align*} where $a\vee b := \max\{a,b\}$ for any two real numbers $a,b$. This result extends the fixed design rate $n^{-4α} \vee n^{-2β/(2β+1)}$ derived in Wang et al. [2008] in a non-trivial manner, as indicated by the appearances of both $α$ and $β$ in the first term. In the special case of constant variance, we show that the minimax rate is $n^{-8α/(4α+1)}\vee n^{-1}$ for variance estimation, which further implies the same rate for quadratic functional estimation and thus unifies the minimax rate under the nonparametric regression model with those under the density model and the white noise model. To achieve the minimax rate, we develop a U-statistic-based local polynomial estimator and a lower bound that is constructed over a specified distribution family of randomness designed for both $\varepsilon_i$ and $X_i$.
Motivated by an unexpected experimental observation from the Cambridge group, [Eigen {\it et al.,} Nature {\bf563}, 221 (2018)], we study the evolution of the momentum distribution of a degenerate Bose gas quenched from the weakly interacting to the unitarity regime. For the two-body problem, we establish a relation that connects the momentum distribution at a long time to a sub-leading term in the initial wave function. For the many-body problem, we employ the time-dependent Bogoliubov variational wave function and find that, in certain momentum regimes, the momentum distribution at long times displays the same exponential behavior found by the experiment. Moreover, we find that this behavior is universal and independent of the short-range details of the interaction potential. Consistent with the relation found in the two-body problem, we also numerically show that this exponential form is hidden in the same sub-leading term of the Bogoliubov wave function in the initial stages. Our results establish a consistent picture to understand the universal dynamics observed in the Cambridge experiment.
We study the sample complexity of canonical correlation analysis (CCA), \ie, the number of samples needed to estimate the population canonical correlation and directions up to arbitrarily small error. With mild assumptions on the data distribution, we show that in order to achieve $ε$-suboptimality in a properly defined measure of alignment between the estimated canonical directions and the population solution, we can solve the empirical objective exactly with $N(ε, Δ, γ)$ samples, where $Δ$ is the singular value gap of the whitened cross-covariance matrix and $1/γ$ is an upper bound of the condition number of auto-covariance matrices. Moreover, we can achieve the same learning accuracy by drawing the same level of samples and solving the empirical objective approximately with a stochastic optimization algorithm; this algorithm is based on the shift-and-invert power iterations and only needs to process the dataset for $\mathcal{O}\left(\log \frac{1}ε \right)$ passes. Finally, we show that, given an estimate of the canonical correlation, the streaming version of the shift-and-invert power iterations achieves the same learning accuracy with the same level of sample complexity, by processing the data only once.
In this paper, we investigate the effect of different hyperparameters as well as different combinations of hyperparameters settings on the performance of the Attention-Gated Convolutional Neural Networks (AGCNNs), e.g., the kernel window size, the number of feature maps, the keep rate of the dropout layer, and the activation function. We draw practical advice from a wide range of empirical results. Through the sensitivity analysis, we further improve the hyperparameters settings of AGCNNs. Experiments show that our proposals could achieve an average of 0.81% and 0.67% improvements on AGCNN-NLReLU-rand and AGCNN-SELU-rand, respectively; and an average of 0.47% and 0.45% improvements on AGCNN-NLReLU-static and AGCNN-SELU-static, respectively.
Activation functions play a key role in providing remarkable performance in deep neural networks, and the rectified linear unit (ReLU) is one of the most widely used activation functions. Various new activation functions and improvements on ReLU have been proposed, but each carry performance drawbacks. In this paper, we propose an improved activation function, which we name the natural-logarithm-rectified linear unit (NLReLU). This activation function uses the parametric natural logarithmic transform to improve ReLU and is simply defined as. NLReLU not only retains the sparse activation characteristic of ReLU, but it also alleviates the "dying ReLU" and vanishing gradient problems to some extent. It also reduces the bias shift effect and heteroscedasticity of neuron data distributions among network layers in order to accelerate the learning process. The proposed method was verified across ten convolutional neural networks with different depths for two essential datasets. Experiments illustrate that convolutional neural networks with NLReLU exhibit higher accuracy than those with ReLU, and that NLReLU is comparable to other well-known activation functions. NLReLU provides 0.16% and 2.04% higher classification accuracy on average compared to ReLU when used in shallow convolutional neural networks with the MNIST and CIFAR-10 datasets, respectively. The average accuracy of deep convolutional neural networks with NLReLU is 1.35% higher on average with the CIFAR-10 dataset.
Consider a sequence of real data points $X_1,\ldots, X_n$ with underlying means $θ^*_1,\dots,θ^*_n$. This paper starts from studying the setting that $θ^*_i$ is both piecewise constant and monotone as a function of the index $i$. For this, we establish the exact minimax rate of estimating such monotone functions, and thus give a non-trivial answer to an open problem in the shape-constrained analysis literature. The minimax rate involves an interesting iterated logarithmic dependence on the dimension, a phenomenon that is revealed through characterizing the interplay between the isotonic shape constraint and model selection complexity. We then develop a penalized least-squares procedure for estimating the vector $θ^*=(θ^*_1,\dots,θ^*_n)^T$. This estimator is shown to achieve the derived minimax rate adaptively. For the proposed estimator, we further allow the model to be misspecified and derive oracle inequalities with the optimal rates, and show there exists a computationally efficient algorithm to compute the exact solution.
How to best explore in domains with sparse, delayed, and deceptive rewards is an important open problem for reinforcement learning (RL). This paper considers one such domain, the recently-proposed multi-agent benchmark of Pommerman. This domain is very challenging for RL --- past work has shown that model-free RL algorithms fail to achieve significant learning without artificially reducing the environment's complexity. In this paper, we illuminate reasons behind this failure by providing a thorough analysis on the hardness of random exploration in Pommerman. While model-free random exploration is typically futile, we develop a model-based automatic reasoning module that can be used for safer exploration by pruning actions that will surely lead the agent to death. We empirically demonstrate that this module can significantly improve learning.
We study the problem of approximate ranking from observations of pairwise interactions. The goal is to estimate the underlying ranks of $n$ objects from data through interactions of comparison or collaboration. Under a general framework of approximate ranking models, we characterize the exact optimal statistical error rates of estimating the underlying ranks. We discover important phase transition boundaries of the optimal error rates. Depending on the value of the signal-to-noise ratio (SNR) parameter, the optimal rate, as a function of SNR, is either trivial, polynomial, exponential or zero. The four corresponding regimes thus have completely different error behaviors. To the best of our knowledge, this phenomenon, especially the phase transition between the polynomial and the exponential rates, has not been discovered before.
We study convergence rates of variational posterior distributions for nonparametric and high-dimensional inference. We formulate general conditions on prior, likelihood, and variational class that characterize the convergence rates. Under similar "prior mass and testing" conditions considered in the literature, the rate is found to be the sum of two terms. The first term stands for the convergence rate of the true posterior distribution, and the second term is contributed by the variational approximation error. For a class of priors that admit the structure of a mixture of product measures, we propose a novel prior mass condition, under which the variational approximation error of the mean-field class is dominated by convergence rate of the true posterior. We demonstrate the applicability of our general results for various models, prior distributions and variational classes by deriving convergence rates of the corresponding variational posteriors.