pwc-archive/papers-with-abstracts huggingface dataset:
Fast covariance parameter estimation of spatial Gaussian process models using neural networks
Authors: Florian Gerber, Douglas W. Nychka
Date: 2020-12-30 00:00:00
Tasks: Descriptive, Gaussian Processes, parameter estimation

Gaussian processes (GPs) are a popular model for spatially referenced data and allow descriptive statements, predictions at new locations, and simulation of new fields. Often a few parameters are sufficient to parameterize the covariance function, and maximum likelihood (ML) methods can be used to estimate these parameters from data. ML methods, however, are computationally demanding. For example, in the case of local likelihood estimation, even fitting covariance models on modest size windows can overwhelm typical computational resources for data analysis. This limitation motivates the idea of using neural network (NN) methods to approximate ML estimates. We train NNs to take moderate size spatial fields or variograms as input and return the range and noise-to-signal covariance parameters. Once trained, the NNs provide estimates with a similar accuracy compared to ML estimation and at a speedup by a factor of 100 or more. Although we focus on a specific covariance estimation problem motivated by a climate science application, this work can be easily extended to other, more complex, spatial problems and provides a proof-of-concept for this use of machine learning in computational statistics.
Fast Global Convergence for Low-rank Matrix Recovery via Riemannian Gradient Descent with Random Initialization
Authors: Thomas Y. Hou, Zhenzhen Li, Ziyun Zhang
Date: 2020-12-31 00:00:00
Tasks: Retrieval, Riemannian optimization

In this paper, we propose a new global analysis framework for a class of low-rank matrix recovery problems on the Riemannian manifold. We analyze the global behavior for the Riemannian optimization with random initialization. We use the Riemannian gradient descent algorithm to minimize a least squares loss function, and study the asymptotic behavior as well as the exact convergence rate. We reveal a previously unknown geometric property of the low-rank matrix manifold, which is the existence of spurious critical points for the simple least squares function on the manifold. We show that under some assumptions, the Riemannian gradient descent starting from a random initialization with high probability avoids these spurious critical points and only converges to the ground truth in nearly linear convergence rate, i.e. $\mathcal{O}(\text{log}(\frac{1}{\epsilon})+ \text{log}(n))$ iterations to reach an $\epsilon$-accurate solution. We use two applications as examples for our global analysis. The first one is a rank-1 matrix recovery problem. The second one is a generalization of the Gaussian phase retrieval problem. It only satisfies the weak isometry property, but has behavior similar to that of the first one except for an extra saddle set. Our convergence guarantee is nearly optimal and almost dimension-free, which fully explains the numerical observations. The global analysis can be potentially extended to other data problems with random measurement structures and empirical least squares loss functions.
Particle Dual Averaging: Optimization of Mean Field Neural Networks with Global Convergence Rate Analysis
Authors: Atsushi Nitanda, Denny Wu, Taiji Suzuki
Date: 2020-12-31 00:00:00
Proceeding: NeurIPS 2021 12

We propose the particle dual averaging (PDA) method, which generalizes the dual averaging method in convex optimization to the optimization over probability distributions with quantitative runtime guarantee. The algorithm consists of an inner loop and outer loop: the inner loop utilizes the Langevin algorithm to approximately solve for a stationary distribution, which is then optimized in the outer loop. The method can thus be interpreted as an extension of the Langevin algorithm to naturally handle nonlinear functional on the probability space. An important application of the proposed method is the optimization of neural network in the mean field regime, which is theoretically attractive due to the presence of nonlinear feature learning, but quantitative convergence rate can be challenging to obtain. By adapting finite-dimensional convex optimization theory into the space of measures, we analyze PDA in regularized empirical / expected risk minimization, and establish quantitative global convergence in learning two-layer mean field neural networks under more general settings. Our theoretical results are supported by numerical simulations on neural networks with reasonable size.
Straggler-Resilient Federated Learning: Leveraging the Interplay Between Statistical Accuracy and System Heterogeneity
Authors: Amirhossein Reisizadeh, Isidoros Tziotis, Hamed Hassani, Aryan Mokhtari, Ramtin Pedarsani
Date: 2020-12-28 00:00:00
Tasks: Federated Learning

Federated Learning is a novel paradigm that involves learning from data samples distributed across a large network of clients while the data remains local. It is, however, known that federated learning is prone to multiple system challenges including system heterogeneity where clients have different computation and communication capabilities. Such heterogeneity in clients' computation speeds has a negative effect on the scalability of federated learning algorithms and causes significant slow-down in their runtime due to the existence of stragglers. In this paper, we propose a novel straggler-resilient federated learning method that incorporates statistical characteristics of the clients' data to adaptively select the clients in order to speed up the learning procedure. The key idea of our algorithm is to start the training procedure with faster nodes and gradually involve the slower nodes in the model training once the statistical accuracy of the data corresponding to the current participating nodes is reached. The proposed approach reduces the overall runtime required to achieve the statistical accuracy of data of all nodes, as the solution for each stage is close to the solution of the subsequent stage with more samples and can be used as a warm-start. Our theoretical results characterize the speedup gain in comparison to standard federated benchmarks for strongly convex objectives, and our numerical experiments also demonstrate significant speedups in wall-clock time of our straggler-resilient method compared to federated learning benchmarks.
Multivariate Smoothing via the Fourier Integral Theorem and Fourier Kernel
Authors: Nhat Ho, Stephen G. Walker
Date: 2020-12-28 00:00:00
Tasks: regression

Starting with the Fourier integral theorem, we present natural Monte Carlo estimators of multivariate functions including densities, mixing densities, transition densities, regression functions, and the search for modes of multivariate density functions (modal regression). Rates of convergence are established and, in many cases, provide superior rates to current standard estimators such as those based on kernels, including kernel density estimators and kernel regression functions. Numerical illustrations are presented.
Source Identification for Mixtures of Product Distributions
Authors: Spencer L. Gordon, Bijan Mazaheri, Yuval Rabani, Leonard J. Schulman
Date: 2020-12-29 00:00:00

We give an algorithm for source identification of a mixture of $k$ product distributions on $n$ bits. This is a fundamental problem in machine learning with many applications. Our algorithm identifies the source parameters of an identifiable mixture, given, as input, approximate values of multilinear moments (derived, for instance, from a sufficiently large sample), using $2^{O(k^2)} n^{O(k)}$ arithmetic operations. Our result is the first explicit bound on the computational complexity of source identification of such mixtures. The running time improves previous results by Feldman, O'Donnell, and Servedio (FOCS 2005) and Chen and Moitra (STOC 2019) that guaranteed only learning the mixture (without parametric identification of the source). Our analysis gives a quantitative version of a qualitative characterization of identifiable sources that is due to Tahmasebi, Motahari, and Maddah-Ali (ISIT 2018).
Sparse PCA via $l_{2,p}$-Norm Regularization for Unsupervised Feature Selection
Authors: Zhengxin Li, Feiping Nie, Jintang Bian, Xuelong Li
Date: 2020-12-29 00:00:00
Tasks: feature selection

In the field of data mining, how to deal with high-dimensional data is an inevitable problem. Unsupervised feature selection has attracted more and more attention because it does not rely on labels. The performance of spectral-based unsupervised methods depends on the quality of constructed similarity matrix, which is used to depict the intrinsic structure of data. However, real-world data contain a large number of noise samples and features, making the similarity matrix constructed by original data cannot be completely reliable. Worse still, the size of similarity matrix expands rapidly as the number of samples increases, making the computational cost increase significantly. Inspired by principal component analysis, we propose a simple and efficient unsupervised feature selection method, by combining reconstruction error with $l_{2,p}$-norm regularization. The projection matrix, which is used for feature selection, is learned by minimizing the reconstruction error under the sparse constraint. Then, we present an efficient optimization algorithm to solve the proposed unsupervised model, and analyse the convergence and computational complexity of the algorithm theoretically. Finally, extensive experiments on real-world data sets demonstrate the effectiveness of our proposed method.
Improved Sample Complexity for Incremental Autonomous Exploration in MDPs
Authors: Jean Tarbouriech, Matteo Pirotta, Michal Valko, Alessandro Lazaric
Date: 2020-12-29 00:00:00
Proceeding: NeurIPS 2020 12

We investigate the exploration of an unknown environment when no reward function is provided. Building on the incremental exploration setting introduced by Lim and Auer [1], we define the objective of learning the set of $\epsilon$-optimal goal-conditioned policies attaining all states that are incrementally reachable within $L$ steps (in expectation) from a reference state $s_0$. In this paper, we introduce a novel model-based approach that interleaves discovering new states from $s_0$ and improving the accuracy of a model estimate that is used to compute goal-conditioned policies to reach newly discovered states. The resulting algorithm, DisCo, achieves a sample complexity scaling as $\tilde{O}(L^5 S_{L+\epsilon} \Gamma_{L+\epsilon} A \epsilon^{-2})$, where $A$ is the number of actions, $S_{L+\epsilon}$ is the number of states that are incrementally reachable from $s_0$ in $L+\epsilon$ steps, and $\Gamma_{L+\epsilon}$ is the branching factor of the dynamics over such states. This improves over the algorithm proposed in [1] in both $\epsilon$ and $L$ at the cost of an extra $\Gamma_{L+\epsilon}$ factor, which is small in most environments of interest. Furthermore, DisCo is the first algorithm that can return an $\epsilon/c_{\min}$-optimal policy for any cost-sensitive shortest-path problem defined on the $L$-reachable states with minimum cost $c_{\min}$. Finally, we report preliminary empirical results confirming our theoretical findings.
Inference for Low-rank Tensors -- No Need to Debias
Authors: Dong Xia, Anru R. Zhang, Yuchen Zhou
Date: 2020-12-29 00:00:00
Tasks: regression

In this paper, we consider the statistical inference for several low-rank tensor models. Specifically, in the Tucker low-rank tensor PCA or regression model, provided with any estimates achieving some attainable error rate, we develop the data-driven confidence regions for the singular subspace of the parameter tensor based on the asymptotic distribution of an updated estimate by two-iteration alternating minimization. The asymptotic distributions are established under some essential conditions on the signal-to-noise ratio (in PCA model) or sample size (in regression model). If the parameter tensor is further orthogonally decomposable, we develop the methods and non-asymptotic theory for inference on each individual singular vector. For the rank-one tensor PCA model, we establish the asymptotic distribution for general linear forms of principal components and confidence interval for each entry of the parameter tensor. Finally, numerical simulations are presented to corroborate our theoretical discoveries. In all these models, we observe that different from many matrix/vector settings in existing work, debiasing is not required to establish the asymptotic distribution of estimates or to make statistical inference on low-rank tensors. In fact, due to the widely observed statistical-computational-gap for low-rank tensor estimation, one usually requires stronger conditions than the statistical (or information-theoretic) limit to ensure the computationally feasible estimation is achievable. Surprisingly, such conditions ``incidentally" render a feasible low-rank tensor inference without debiasing.
Twin Neural Network Regression
Authors: Sebastian J. Wetzel, Kevin Ryczko, Roger G. Melko, Isaac Tamblyn
Date: 2020-12-29 00:00:00
Tasks: regression

We introduce twin neural network (TNN) regression. This method predicts differences between the target values of two different data points rather than the targets themselves. The solution of a traditional regression problem is then obtained by averaging over an ensemble of all predicted differences between the targets of an unseen data point and all training data points. Whereas ensembles are normally costly to produce, TNN regression intrinsically creates an ensemble of predictions of twice the size of the training set while only training a single neural network. Since ensembles have been shown to be more accurate than single models this property naturally transfers to TNN regression. We show that TNNs are able to compete or yield more accurate predictions for different data sets, compared to other state-of-the-art methods. Furthermore, TNN regression is constrained by self-consistency conditions. We find that the violation of these conditions provides an estimate for the prediction uncertainty.
Growing Deep Forests Efficiently with Soft Routing and Learned Connectivity
Authors: Jianghao Shen, Sicheng Wang, Zhangyang Wang
Date: 2020-12-29 00:00:00

Despite the latest prevailing success of deep neural networks (DNNs), several concerns have been raised against their usage, including the lack of intepretability the gap between DNNs and other well-established machine learning models, and the growingly expensive computational costs. A number of recent works [1], [2], [3] explored the alternative to sequentially stacking decision tree/random forest building blocks in a purely feed-forward way, with no need of back propagation. Since decision trees enjoy inherent reasoning transparency, such deep forest models can also facilitate the understanding of the internaldecision making process. This paper further extends the deep forest idea in several important aspects. Firstly, we employ a probabilistic tree whose nodes make probabilistic routing decisions, a.k.a., soft routing, rather than hard binary decisions.Besides enhancing the flexibility, it also enables non-greedy optimization for each tree. Second, we propose an innovative topology learning strategy: every node in the ree now maintains a new learnable hyperparameter indicating the probability that it will be a leaf node. In that way, the tree will jointly optimize both its parameters and the tree topology during training. Experiments on the MNIST dataset demonstrate that our empowered deep forests can achieve better or comparable performance than [1],[3] , with dramatically reduced model complexity. For example,our model with only 1 layer of 15 trees can perform comparably with the model in [3] with 2 layers of 2000 trees each.
DeepSphere: a graph-based spherical CNN
Authors: Michaël Defferrard, Martino Milani, Frédérick Gusset, Nathanaël Perraudin
Date: 2020-12-30 00:00:00
Proceeding: ICLR 2020 1

Designing a convolution for a spherical neural network requires a delicate tradeoff between efficiency and rotation equivariance. DeepSphere, a method based on a graph representation of the sampled sphere, strikes a controllable balance between these two desiderata. This contribution is twofold. First, we study both theoretically and empirically how equivariance is affected by the underlying graph with respect to the number of vertices and neighbors. Second, we evaluate DeepSphere on relevant problems. Experiments show state-of-the-art performance and demonstrates the efficiency and flexibility of this formulation. Perhaps surprisingly, comparison with previous work suggests that anisotropic filters might be an unnecessary price to pay. Our code is available at https://github.com/deepsphere
Infer-AVAE: An Attribute Inference Model Based on Adversarial Variational Autoencoder
Authors: Yadong Zhou, Zhihao Ding, Xiaoming Liu, Chao Shen, Lingling Tong, Xiaohong Guan
Date: 2020-12-30 00:00:00
Tasks: Attribute

User attributes, such as gender and education, face severe incompleteness in social networks. In order to make this kind of valuable data usable for downstream tasks like user profiling and personalized recommendation, attribute inference aims to infer users' missing attribute labels based on observed data. Recently, variational autoencoder (VAE), an end-to-end deep generative model, has shown promising performance by handling the problem in a semi-supervised way. However, VAEs can easily suffer from over-fitting and over-smoothing when applied to attribute inference. To be specific, VAE implemented with multi-layer perceptron (MLP) can only reconstruct input data but fail in inferring missing parts. While using the trending graph neural networks (GNNs) as encoder has the problem that GNNs aggregate redundant information from neighborhood and generate indistinguishable user representations, which is known as over-smoothing. In this paper, we propose an attribute \textbf{Infer}ence model based on \textbf{A}dversarial \textbf{VAE} (Infer-AVAE) to cope with these issues. Specifically, to overcome over-smoothing, Infer-AVAE unifies MLP and GNNs in encoder to learn positive and negative latent representations respectively. Meanwhile, an adversarial network is trained to distinguish the two representations and GNNs are trained to aggregate less noise for more robust representations through adversarial training. Finally, to relieve over-fitting, mutual information constraint is introduced as a regularizer for decoder, so that it can make better use of auxiliary information in representations and generate outputs not limited by observations. We evaluate our model on 4 real-world social network datasets, experimental results demonstrate that our model averagely outperforms baselines by 7.0$\%$ in accuracy.
SGD Distributional Dynamics of Three Layer Neural Networks
Authors: Victor Luo, Yazhen Wang, Glenn Fung
Date: 2020-12-30 00:00:00

With the rise of big data analytics, multi-layer neural networks have surfaced as one of the most powerful machine learning methods. However, their theoretical mathematical properties are still not fully understood. Training a neural network requires optimizing a non-convex objective function, typically done using stochastic gradient descent (SGD). In this paper, we seek to extend the mean field results of Mei et al. (2018) from two-layer neural networks with one hidden layer to three-layer neural networks with two hidden layers. We will show that the SGD dynamics is captured by a set of non-linear partial differential equations, and prove that the distributions of weights in the two hidden layers are independent. We will also detail exploratory work done based on simulation and real-world data.
Risk Guarantees for End-to-End Prediction and Optimization Processes
Authors: Nam Ho-Nguyen, Fatma Kılınç-Karzan
Date: 2020-12-30 00:00:00
Tasks: Learning Theory, Portfolio Optimization, Prediction

Prediction models are often employed in estimating parameters of optimization models. Despite the fact that in an end-to-end view, the real goal is to achieve good optimization performance, the prediction performance is measured on its own. While it is usually believed that good prediction performance in estimating the parameters will result in good subsequent optimization performance, formal theoretical guarantees on this are notably lacking. In this paper, we explore conditions that allow us to explicitly describe how the prediction performance governs the optimization performance. Our weaker condition allows for an asymptotic convergence result, while our stronger condition allows for exact quantification of the optimization performance in terms of the prediction performance. In general, verification of these conditions is a non-trivial task. Nevertheless, we show that our weaker condition is equivalent to the well-known Fisher consistency concept from the learning theory literature. This then allows us to easily check our weaker condition for several loss functions. We also establish that the squared error loss function satisfies our stronger condition. Consequently, we derive the exact theoretical relationship between prediction performance measured with the squared loss, as well as a class of symmetric loss functions, and the subsequent optimization performance. In a computational study on portfolio optimization, fractional knapsack and multiclass classification problems, we compare the optimization performance of using of several prediction loss functions (some that are Fisher consistent and some that are not) and demonstrate that lack of consistency of the loss function can indeed have a detrimental effect on performance.
Adjusted chi-square test for degree-corrected block models
Authors: Linfan Zhang, Arash A. Amini
Date: 2020-12-30 00:00:00
Tasks: Computational Efficiency

We propose a goodness-of-fit test for degree-corrected stochastic block models (DCSBM). The test is based on an adjusted chi-square statistic for measuring equality of means among groups of $n$ multinomial distributions with $d_1,\dots,d_n$ observations. In the context of network models, the number of multinomials, $n$, grows much faster than the number of observations, $d_i$, corresponding to the degree of node $i$, hence the setting deviates from classical asymptotics. We show that a simple adjustment allows the statistic to converge in distribution, under null, as long as the harmonic mean of $\{d_i\}$ grows to infinity. When applied sequentially, the test can also be used to determine the number of communities. The test operates on a compressed version of the adjacency matrix, conditional on the degrees, and as a result is highly scalable to large sparse networks. We incorporate a novel idea of compressing the rows based on a $(K+1)$-community assignment when testing for $K$ communities. This approach increases the power in sequential applications without sacrificing computational efficiency, and we prove its consistency in recovering the number of communities. Since the test statistic does not rely on a specific alternative, its utility goes beyond sequential testing and can be used to simultaneously test against a wide range of alternatives outside the DCSBM family. In particular, we prove that the test is consistent against a general family of latent-variable network models with community structure.