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We study a new two-time-scale stochastic gradient method for solving optimization problems, where the gradients are computed with the aid of an auxiliary variable under samples generated by time-varying Markov random processes parameterized by the underlying optimization variable. These time-varying samples make gradient directions in our update biased and dependent, which can potentially lead to the divergence of the iterates. In our two-time-scale approach, one scale is to estimate the true gradient from these samples, which is then used to update the estimate of the optimal solution. While these two iterates are implemented simultaneously, the former is updated "faster" (using bigger step sizes) than the latter (using smaller step sizes). Our first contribution is to characterize the finite-time complexity of the proposed two-time-scale stochastic gradient method. In particular, we provide explicit formulas for the convergence rates of this method under different structural assumptions, namely, strong convexity, convexity, the Polyak-Lojasiewicz condition, and general non-convexity. We apply our framework to two problems in control and reinforcement learning. First, we look at the standard online actor-critic algorithm over finite state and action spaces and derive a convergence rate of O(k^(-2/5)), which recovers the best known rate derived specifically for this problem. Second, we study an online actor-critic algorithm for the linear-quadratic regulator and show that a convergence rate of O(k^(-2/3)) is achieved. This is the first time such a result is known in the literature. Finally, we support our theoretical analysis with numerical simulations where the convergence rates are visualized.

Hyperspectral images often have hundreds of spectral bands of different wavelengths captured by aircraft or satellites that record land coverage. Identifying detailed classes of pixels becomes feasible due to the enhancement in spectral and spatial resolution of hyperspectral images. In this work, we propose a novel framework that utilizes both spatial and spectral information for classifying pixels in hyperspectral images. The method consists of three stages. In the first stage, the pre-processing stage, Nested Sliding Window algorithm is used to reconstruct the original data by {enhancing the consistency of neighboring pixels} and then Principal Component Analysis is used to reduce the dimension of data. In the second stage, Support Vector Machines are trained to estimate the pixel-wise probability map of each class using the spectral information from the images. Finally, a smoothed total variation model is applied to smooth the class probability vectors by {ensuring spatial connectivity} in the images. We demonstrate the superiority of our method against three state-of-the-art algorithms on six benchmark hyperspectral data sets with 10 to 50 training labels for each class. The results show that our method gives the overall best performance in accuracy. Especially, our gain in accuracy increases when the number of labeled pixels decreases and therefore our method is more advantageous to be applied to problems with small training set. Hence it is of great practical significance since expert annotations are often expensive and difficult to collect.

While algorithms for planar graphs have received a lot of attention, few papers have focused on the additional power that one gets from assuming an embedding of the graph is available. While in the classic sequential setting, this assumption gives no additional power (as a planar graph can be embedded in linear time), we show that this is far from being the case in other settings. We assume that the embedding is straight-line, but our methods also generalize to non-straight-line embeddings. Specifically, we focus on sublinear-time computation and massively parallel computation (MPC). Our main technical contribution is a sublinear-time algorithm for computing a relaxed version of an $r$-division. We then show how this can be used to estimate Lipschitz additive graph parameters. This includes, for example, the maximum matching, maximum independent set, or the minimum dominating set. We also show how this can be used to solve some property testing problems with respect to the vertex edit distance. In the second part of our paper, we show an MPC algorithm that computes an $r$-division of the input graph. We show how this can be used to solve various classical graph problems with space per machine of $O(n^{2/3+\epsilon})$ for some $\epsilon>0$, and while performing $O(1)$ rounds. This includes for example approximate shortest paths or the minimum spanning tree. Our results also imply an improved MPC algorithm for Euclidean minimum spanning tree.

This paper considers the problem of inference in cluster randomized experiments when cluster sizes are non-ignorable. Here, by a cluster randomized experiment, we mean one in which treatment is assigned at the level of the cluster; by non-ignorable cluster sizes we mean that "large" clusters and "small" clusters may be heterogeneous, and, in particular, the effects of the treatment may vary across clusters of differing sizes. In order to permit this sort of flexibility, we consider a sampling framework in which cluster sizes themselves are random. In this way, our analysis departs from earlier analyses of cluster randomized experiments in which cluster sizes are treated as non-random. We distinguish between two different parameters of interest: the equally-weighted cluster-level average treatment effect, and the size-weighted cluster-level average treatment effect. For each parameter, we provide methods for inference in an asymptotic framework where the number of clusters tends to infinity and treatment is assigned using simple random sampling. We additionally permit the experimenter to sample only a subset of the units within each cluster rather than the entire cluster and demonstrate the implications of such sampling for some commonly used estimators. A small simulation study shows the practical relevance of our theoretical results.

We present a new sublinear time algorithm for approximating the spectral density (eigenvalue distribution) of an $n\times n$ normalized graph adjacency or Laplacian matrix. The algorithm recovers the spectrum up to $\epsilon$ accuracy in the Wasserstein-1 distance in $O(n\cdot \text{poly}(1/\epsilon))$ time given sample access to the graph. This result compliments recent work by David Cohen-Steiner, Weihao Kong, Christian Sohler, and Gregory Valiant (2018), which obtains a solution with runtime independent of $n$, but exponential in $1/\epsilon$. We conjecture that the trade-off between dimension dependence and accuracy is inherent. Our method is simple and works well experimentally. It is based on a Chebyshev polynomial moment matching method that employees randomized estimators for the matrix trace. We prove that, for any Hermitian $A$, this moment matching method returns an $\epsilon$ approximation to the spectral density using just $O({1}/{\epsilon})$ matrix-vector products with $A$. By leveraging stability properties of the Chebyshev polynomial three-term recurrence, we then prove that the method is amenable to the use of coarse approximate matrix-vector products. Our sublinear time algorithm follows from combining this result with a novel sampling algorithm for approximating matrix-vector products with a normalized graph adjacency matrix. Of independent interest, we show a similar result for the widely used \emph{kernel polynomial method} (KPM), proving that this practical algorithm nearly matches the theoretical guarantees of our moment matching method. Our analysis uses tools from Jackson's seminal work on approximation with positive polynomial kernels.

Spectral clustering (SC) is a popular clustering technique to find strongly connected communities on a graph. SC can be used in Graph Neural Networks (GNNs) to implement pooling operations that aggregate nodes belonging to the same cluster. However, the eigendecomposition of the Laplacian is expensive and, since clustering results are graph-specific, pooling methods based on SC must perform a new optimization for each new sample. In this paper, we propose a graph clustering approach that addresses these limitations of SC. We formulate a continuous relaxation of the normalized minCUT problem and train a GNN to compute cluster assignments that minimize this objective. Our GNN-based implementation is differentiable, does not require to compute the spectral decomposition, and learns a clustering function that can be quickly evaluated on out-of-sample graphs. From the proposed clustering method, we design a graph pooling operator that overcomes some important limitations of state-of-the-art graph pooling techniques and achieves the best performance in several supervised and unsupervised tasks.

When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.

The area of Data Analytics on graphs promises a paradigm shift as we approach information processing of classes of data, which are typically acquired on irregular but structured domains (social networks, various ad-hoc sensor networks). Yet, despite its long history, current approaches mostly focus on the optimization of graphs themselves, rather than on directly inferring learning strategies, such as detection, estimation, statistical and probabilistic inference, clustering and separation from signals and data acquired on graphs. To fill this void, we first revisit graph topologies from a Data Analytics point of view, and establish a taxonomy of graph networks through a linear algebraic formalism of graph topology (vertices, connections, directivity). This serves as a basis for spectral analysis of graphs, whereby the eigenvalues and eigenvectors of graph Laplacian and adjacency matrices are shown to convey physical meaning related to both graph topology and higher-order graph properties, such as cuts, walks, paths, and neighborhoods. Next, to illustrate estimation strategies performed on graph signals, spectral analysis of graphs is introduced through eigenanalysis of mathematical descriptors of graphs and in a generic way. Finally, a framework for vertex clustering and graph segmentation is established based on graph spectral representation (eigenanalysis) which illustrates the power of graphs in various data association tasks. The supporting examples demonstrate the promise of Graph Data Analytics in modeling structural and functional/semantic inferences. At the same time, Part I serves as a basis for Part II and Part III which deal with theory, methods and applications of processing Data on Graphs and Graph Topology Learning from data.

High spectral dimensionality and the shortage of annotations make hyperspectral image (HSI) classification a challenging problem. Recent studies suggest that convolutional neural networks can learn discriminative spatial features, which play a paramount role in HSI interpretation. However, most of these methods ignore the distinctive spectral-spatial characteristic of hyperspectral data. In addition, a large amount of unlabeled data remains an unexploited gold mine for efficient data use. Therefore, we proposed an integration of generative adversarial networks (GANs) and probabilistic graphical models for HSI classification. Specifically, we used a spectral-spatial generator and a discriminator to identify land cover categories of hyperspectral cubes. Moreover, to take advantage of a large amount of unlabeled data, we adopted a conditional random field to refine the preliminary classification results generated by GANs. Experimental results obtained using two commonly studied datasets demonstrate that the proposed framework achieved encouraging classification accuracy using a small number of data for training.

Image segmentation is considered to be one of the critical tasks in hyperspectral remote sensing image processing. Recently, convolutional neural network (CNN) has established itself as a powerful model in segmentation and classification by demonstrating excellent performances. The use of a graphical model such as a conditional random field (CRF) contributes further in capturing contextual information and thus improving the segmentation performance. In this paper, we propose a method to segment hyperspectral images by considering both spectral and spatial information via a combined framework consisting of CNN and CRF. We use multiple spectral cubes to learn deep features using CNN, and then formulate deep CRF with CNN-based unary and pairwise potential functions to effectively extract the semantic correlations between patches consisting of three-dimensional data cubes. Effective piecewise training is applied in order to avoid the computationally expensive iterative CRF inference. Furthermore, we introduce a deep deconvolution network that improves the segmentation masks. We also introduce a new dataset and experimented our proposed method on it along with several widely adopted benchmark datasets to evaluate the effectiveness of our method. By comparing our results with those from several state-of-the-art models, we show the promising potential of our method.