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We consider the sparse moment problem of learning a $k$-spike mixture in high dimensional space from its noisy moment information in any dimension. We measure the accuracy of the learned mixtures using transportation distance. Previous algorithms either assume certain separation assumptions, use more recovery moments, or run in (super) exponential time. Our algorithm for the 1-dimension problem (also called the sparse Hausdorff moment problem) is a robust version of the classic Prony's method, and our contribution mainly lies in the analysis. We adopt a global and much tighter analysis than previous work (which analyzes the perturbation of the intermediate results of Prony's method). A useful technical ingredient is a connection between the linear system defined by the Vandermonde matrix and the Schur polynomial, which allows us to provide tight perturbation bound independent of the separation and may be useful in other contexts. To tackle the high dimensional problem, we first solve the 2-dimensional problem by extending the 1-dimension algorithm and analysis to complex numbers. Our algorithm for the high dimensional case determines the coordinates of each spike by aligning a 1-d projection of the mixture to a random vector and a set of 2d-projections of the mixture. Our results have applications to learning topic models and Gaussian mixtures, implying improved sample complexity results or running time over prior work.

In this work we are interested in general linear inverse problems where the corresponding forward problem is solved iteratively using fixed point methods. Then one-shot methods, which iterate at the same time on the forward problem solution and on the inverse problem unknown, can be applied. We analyze two variants of the so-called multi-step one-shot methods and establish sufficient conditions on the descent step for their convergence, by studying the eigenvalues of the block matrix of the coupled iterations. Several numerical experiments are provided to illustrate the convergence of these methods in comparison with the classical usual and shifted gradient descent. In particular, we observe that very few inner iterations on the forward problem are enough to guarantee good convergence of the inversion algorithm.

The generalized Lanczos trust-region (GLTR) method is one of the most popular approaches for solving large-scale trust-region subproblem (TRS). Recently, Jia and Wang [Z. Jia and F. Wang, \emph{SIAM J. Optim., 31 (2021), pp. 887--914}] considered the convergence of this method and established some {\it a prior} error bounds on the residual, the solution and the Largrange multiplier. In this paper, we revisit the convergence of the GLTR method and try to improve these bounds. First, we establish a sharper upper bound on the residual. Second, we give a new bound on the distance between the approximation and the exact solution, and show that the convergence of the approximation has nothing to do with the associated spectral separation. Third, we present some non-asymptotic bounds for the convergence of the Largrange multiplier, and define a factor that plays an important role on the convergence of the Largrange multiplier. Numerical experiments demonstrate the effectiveness of our theoretical results.

We present a novel hybrid algorithm for training Deep Neural Networks that combines the state-of-the-art Gradient Descent (GD) method with a Mixed Integer Linear Programming (MILP) solver, outperforming GD and variants in terms of accuracy, as well as resource and data efficiency for both regression and classification tasks. Our GD+Solver hybrid algorithm, called GDSolver, works as follows: given a DNN $D$ as input, GDSolver invokes GD to partially train $D$ until it gets stuck in a local minima, at which point GDSolver invokes an MILP solver to exhaustively search a region of the loss landscape around the weight assignments of $D$'s final layer parameters with the goal of tunnelling through and escaping the local minima. The process is repeated until desired accuracy is achieved. In our experiments, we find that GDSolver not only scales well to additional data and very large model sizes, but also outperforms all other competing methods in terms of rates of convergence and data efficiency. For regression tasks, GDSolver produced models that, on average, had 31.5% lower MSE in 48% less time, and for classification tasks on MNIST and CIFAR10, GDSolver was able to achieve the highest accuracy over all competing methods, using only 50% of the training data that GD baselines required.

General purpose optimization techniques can be used to solve many problems in engineering computations, although their cost is often prohibitive when the number of degrees of freedom is very large. We describe a multilevel approach to speed up the computation of the solution of a large-scale optimization problem by a given optimization technique. By embedding the problem within Harten's Multiresolution Framework (MRF), we set up a procedure that leads to the desired solution, after the computation of a finite sequence of sub-optimal solutions, which solve auxiliary optimization problems involving a smaller number of variables. For convex optimization problems having smooth solutions, we prove that the distance between the optimal solution and each sub-optimal approximation is related to the accuracy of the interpolation technique used within the MRF and analyze its relation with the performance of the proposed algorithm. Several numerical experiments confirm that our technique provides a computationally efficient strategy that allows the end user to treat both the optimizer and the objective function as black boxes throughout the optimization process.

There are various cluster validity indices used for evaluating clustering results. One of the main objectives of using these indices is to seek the optimal unknown number of clusters. Some indices work well for clusters with different densities, sizes, and shapes. Yet, one shared weakness of those validity indices is that they often provide only one optimal number of clusters. That number is unknown in real-world problems, and there might be more than one possible option. We develop a new cluster validity index based on a correlation between an actual distance between a pair of data points and a centroid distance of clusters that the two points occupy. Our proposed index constantly yields several local peaks and overcomes the previously stated weakness. Several experiments in different scenarios, including UCI real-world data sets, have been conducted to compare the proposed validity index with several well-known ones. An R package related to this new index called NCvalid is available at //github.com/nwiroonsri/NCvalid.

Our world is full of asymmetries. Gravity and wind can make reaching a place easier than coming back. Social artifacts such as genealogy charts and citation graphs are inherently directed. In reinforcement learning and control, optimal goal-reaching strategies are rarely reversible (symmetrical). Distance functions supported on these asymmetrical structures are called quasimetrics. Despite their common appearance, little research has been done on the learning of quasimetrics. Our theoretical analysis reveals that a common class of learning algorithms, including unconstrained multilayer perceptrons (MLPs), provably fails to learn a quasimetric consistent with training data. In contrast, our proposed Poisson Quasimetric Embedding (PQE) is the first quasimetric learning formulation that both is learnable with gradient-based optimization and enjoys strong performance guarantees. Experiments on random graphs, social graphs, and offline Q-learning demonstrate its effectiveness over many common baselines.

LU and Cholesky matrix factorization algorithms are core subroutines used to solve systems of linear equations (SLEs) encountered while solving an optimization problem. Standard factorization algorithms are highly efficient but remain susceptible to the accumulation of roundoff errors, which can lead solvers to return feasibility and optimality claims that are actually invalid. This paper introduces a novel approach for solving sequences of closely related SLEs encountered in nonlinear programming efficiently and without roundoff errors. Specifically, it introduces rank-one update algorithms for the roundoff-error-free (REF) factorization framework, a toolset built on integer-preserving arithmetic that has led to the development and implementation of fail-proof SLE solution subroutines for linear programming. The formal guarantees of the proposed algorithms are established through the derivation of theoretical insights. Their advantages are supported with computational experiments, which demonstrate upwards of 75x-improvements over exact factorization run-times on fully dense matrices with over one million entries. A significant advantage of the methodology is that the length of any coefficient calculated via the proposed algorithms is bounded polynomially in the size of the inputs without having to resort to greatest common divisor operations, which are required by and thereby hinder an efficient implementation of exact rational arithmetic approaches.

Deep Neural Network (DNN) models are usually trained sequentially from one layer to another, which causes forward, backward and update locking's problems, leading to poor performance in terms of training time. The existing parallel strategies to mitigate these problems provide suboptimal runtime performance. In this work, we have proposed a novel layer-wise partitioning and merging, forward and backward pass parallel framework to provide better training performance. The novelty of the proposed work consists of 1) a layer-wise partition and merging model which can minimise communication overhead between devices without the memory cost of existing strategies during the training process; 2) a forward pass and backward pass parallelisation and optimisation to address the update locking problem and minimise the total training cost. The experimental evaluation on real use cases shows that the proposed method outperforms the state-of-the-art approaches in terms of training speed; and achieves almost linear speedup without compromising the accuracy performance of the non-parallel approach.

We study the performance -- and specifically the rate at which the error probability converges to zero -- of Machine Learning (ML) classification techniques. Leveraging the theory of large deviations, we provide the mathematical conditions for a ML classifier to exhibit error probabilities that vanish exponentially, say $\sim \exp\left(-n\,I + o(n) \right)$, where $n$ is the number of informative observations available for testing (or another relevant parameter, such as the size of the target in an image) and $I$ is the error rate. Such conditions depend on the Fenchel-Legendre transform of the cumulant-generating function of the Data-Driven Decision Function (D3F, i.e., what is thresholded before the final binary decision is made) learned in the training phase. As such, the D3F and, consequently, the related error rate $I$, depend on the given training set, which is assumed of finite size. Interestingly, these conditions can be verified and tested numerically exploiting the available dataset, or a synthetic dataset, generated according to the available information on the underlying statistical model. In other words, the classification error probability convergence to zero and its rate can be computed on a portion of the dataset available for training. Coherently with the large deviations theory, we can also establish the convergence, for $n$ large enough, of the normalized D3F statistic to a Gaussian distribution. This property is exploited to set a desired asymptotic false alarm probability, which empirically turns out to be accurate even for quite realistic values of $n$. Furthermore, approximate error probability curves $\sim \zeta_n \exp\left(-n\,I \right)$ are provided, thanks to the refined asymptotic derivation (often referred to as exact asymptotics), where $\zeta_n$ represents the most representative sub-exponential terms of the error probabilities.