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Given its status as a classic problem and its importance to both theoreticians and practitioners, edit distance provides an excellent lens through which to understand how the theoretical analysis of algorithms impacts practical implementations. From an applied perspective, the goals of theoretical analysis are to predict the empirical performance of an algorithm and to serve as a yardstick to design novel algorithms that perform well in practice. In this paper, we systematically survey the types of theoretical analysis techniques that have been applied to edit distance and evaluate the extent to which each one has achieved these two goals. These techniques include traditional worst-case analysis, worst-case analysis parametrized by edit distance or entropy or compressibility, average-case analysis, semi-random models, and advice-based models. We find that the track record is mixed. On one hand, two algorithms widely used in practice have been born out of theoretical analysis and their empirical performance is captured well by theoretical predictions. On the other hand, all the algorithms developed using theoretical analysis as a yardstick since then have not had any practical relevance. We conclude by discussing the remaining open problems and how they can be tackled.

We study online convex optimization with switching costs, a practically important but also extremely challenging problem due to the lack of complete offline information. By tapping into the power of machine learning (ML) based optimizers, ML-augmented online algorithms (also referred to as expert calibration in this paper) have been emerging as state of the art, with provable worst-case performance guarantees. Nonetheless, by using the standard practice of training an ML model as a standalone optimizer and plugging it into an ML-augmented algorithm, the average cost performance can be even worse than purely using ML predictions. In order to address the "how to learn" challenge, we propose EC-L2O (expert-calibrated learning to optimize), which trains an ML-based optimizer by explicitly taking into account the downstream expert calibrator. To accomplish this, we propose a new differentiable expert calibrator that generalizes regularized online balanced descent and offers a provably better competitive ratio than pure ML predictions when the prediction error is large. For training, our loss function is a weighted sum of two different losses -- one minimizing the average ML prediction error for better robustness, and the other one minimizing the post-calibration average cost. We also provide theoretical analysis for EC-L2O, highlighting that expert calibration can be even beneficial for the average cost performance and that the high-percentile tail ratio of the cost achieved by EC-L2O to that of the offline optimal oracle (i.e., tail cost ratio) can be bounded. Finally, we test EC-L2O by running simulations for sustainable datacenter demand response. Our results demonstrate that EC-L2O can empirically achieve a lower average cost as well as a lower competitive ratio than the existing baseline algorithms.

Covariance estimation for matrix-valued data has received an increasing interest in applications. Unlike previous works that rely heavily on matrix normal distribution assumption and the requirement of fixed matrix size, we propose a class of distribution-free regularized covariance estimation methods for high-dimensional matrix data under a separability condition and a bandable covariance structure. Under these conditions, the original covariance matrix is decomposed into a Kronecker product of two bandable small covariance matrices representing the variability over row and column directions. We formulate a unified framework for estimating bandable covariance, and introduce an efficient algorithm based on rank one unconstrained Kronecker product approximation. The convergence rates of the proposed estimators are established, and the derived minimax lower bound shows our proposed estimator is rate-optimal under certain divergence regimes of matrix size. We further introduce a class of robust covariance estimators and provide theoretical guarantees to deal with heavy-tailed data. We demonstrate the superior finite-sample performance of our methods using simulations and real applications from a gridded temperature anomalies dataset and a S&P 500 stock data analysis.

We provide a decision theoretic analysis of bandit experiments. The setting corresponds to a dynamic programming problem, but solving this directly is typically infeasible. Working within the framework of diffusion asymptotics, we define suitable notions of asymptotic Bayes and minimax risk for bandit experiments. For normally distributed rewards, the minimal Bayes risk can be characterized as the solution to a nonlinear second-order partial differential equation (PDE). Using a limit of experiments approach, we show that this PDE characterization also holds asymptotically under both parametric and non-parametric distribution of the rewards. The approach further describes the state variables it is asymptotically sufficient to restrict attention to, and therefore suggests a practical strategy for dimension reduction. The upshot is that we can approximate the dynamic programming problem defining the bandit experiment with a PDE which can be efficiently solved using sparse matrix routines. We derive the optimal Bayes and minimax policies from the numerical solutions to these equations. The proposed policies substantially dominate existing methods such as Thompson sampling. The framework also allows for substantial generalizations to the bandit problem such as time discounting and pure exploration motives.

Let ${\mathcal M}\subset {\mathbb R}^n$ be a $C^2$-smooth compact submanifold of dimension $d$. Assume that the volume of ${\mathcal M}$ is at most $V$ and the reach (i.e. the normal injectivity radius) of ${\mathcal M}$ is greater than $\tau$. Moreover, let $\mu$ be a probability measure on ${\mathcal M}$ whose density on ${\mathcal M}$ is a strictly positive Lipschitz-smooth function. Let $x_j\in {\mathcal M}$, $j=1,2,\dots,N$ be $N$ independent random samples from distribution $\mu$. Also, let $\xi_j$, $j=1,2,\dots, N$ be independent random samples from a Gaussian random variable in ${\mathbb R}^n$ having covariance $\sigma^2I$, where $\sigma$ is less than a certain specified function of $d, V$ and $\tau$. We assume that we are given the data points $y_j=x_j+\xi_j,$ $j=1,2,\dots,N$, modelling random points of ${\mathcal M}$ with measurement noise. We develop an algorithm which produces from these data, with high probability, a $d$ dimensional submanifold ${\mathcal M}_o\subset {\mathbb R}^n$ whose Hausdorff distance to ${\mathcal M}$ is less than $Cd\sigma^2/\tau$ and whose reach is greater than $c{\tau}/d^6$ with universal constants $C,c > 0$. The number $N$ of random samples required depends almost linearly on $n$, polynomially on $\sigma^{-1}$ and exponentially on $d$.

Multigrid is a powerful solver for large-scale linear systems arising from discretized partial differential equations. The convergence theory of multigrid methods for symmetric positive definite problems has been well developed over the past decades, while, for nonsymmetric problems, such theory is still not mature. As a foundation for multigrid analysis, two-grid convergence theory plays an important role in motivating multigrid algorithms. Regarding two-grid methods for nonsymmetric problems, most previous works focus on the spectral radius of iteration matrix or rely on convergence measures that are typically difficult to compute in practice. Moreover, the existing results are confined to two-grid methods with exact solution of the coarse-grid system. In this paper, we analyze the convergence of a two-grid method for nonsymmetric positive definite problems (e.g., linear systems arising from the discretizations of convection-diffusion equations). In the case of exact coarse solver, we establish an elegant identity for characterizing two-grid convergence factor, which is measured by a smoother-induced norm. The identity can be conveniently used to derive a class of optimal restriction operators and analyze how the convergence factor is influenced by restriction. More generally, we present some convergence estimates for an inexact variant of the two-grid method, in which both linear and nonlinear coarse solvers are considered.

We study the acceleration of the Local Polynomial Interpolation-based Gradient Descent method (LPI-GD) recently proposed for the approximate solution of empirical risk minimization problems (ERM). We focus on loss functions that are strongly convex and smooth with condition number $\sigma$. We additionally assume the loss function is $\eta$-H\"older continuous with respect to the data. The oracle complexity of LPI-GD is $\tilde{O}\left(\sigma m^d \log(1/\varepsilon)\right)$ for a desired accuracy $\varepsilon$, where $d$ is the dimension of the parameter space, and $m$ is the cardinality of an approximation grid. The factor $m^d$ can be shown to scale as $O((1/\varepsilon)^{d/2\eta})$. LPI-GD has been shown to have better oracle complexity than gradient descent (GD) and stochastic gradient descent (SGD) for certain parameter regimes. We propose two accelerated methods for the ERM problem based on LPI-GD and show an oracle complexity of $\tilde{O}\left(\sqrt{\sigma} m^d \log(1/\varepsilon)\right)$. Moreover, we provide the first empirical study on local polynomial interpolation-based gradient methods and corroborate that LPI-GD has better performance than GD and SGD in some scenarios, and the proposed methods achieve acceleration.

Tensor PCA is a stylized statistical inference problem introduced by Montanari and Richard to study the computational difficulty of estimating an unknown parameter from higher-order moment tensors. Unlike its matrix counterpart, Tensor PCA exhibits a statistical-computational gap, i.e., a sample size regime where the problem is information-theoretically solvable but conjectured to be computationally hard. This paper derives computational lower bounds on the run-time of memory bounded algorithms for Tensor PCA using communication complexity. These lower bounds specify a trade-off among the number of passes through the data sample, the sample size, and the memory required by any algorithm that successfully solves Tensor PCA. While the lower bounds do not rule out polynomial-time algorithms, they do imply that many commonly-used algorithms, such as gradient descent and power method, must have a higher iteration count when the sample size is not large enough. Similar lower bounds are obtained for Non-Gaussian Component Analysis, a family of statistical estimation problems in which low-order moment tensors carry no information about the unknown parameter. Finally, stronger lower bounds are obtained for an asymmetric variant of Tensor PCA and related statistical estimation problems. These results explain why many estimators for these problems use a memory state that is significantly larger than the effective dimensionality of the parameter of interest.

The minimum energy path (MEP) describes the mechanism of reaction, and the energy barrier along the path can be used to calculate the reaction rate in thermal systems. The nudged elastic band (NEB) method is one of the most commonly used schemes to compute MEPs numerically. It approximates an MEP by a discrete set of configuration images, where the discretization size determines both computational cost and accuracy of the simulations. In this paper, we consider a discrete MEP to be a stationary state of the NEB method and prove an optimal convergence rate of the discrete MEP with respect to the number of images. Numerical simulations for the transitions of some several proto-typical model systems are performed to support the theory.

There are many important high dimensional function classes that have fast agnostic learning algorithms when strong assumptions on the distribution of examples can be made, such as Gaussianity or uniformity over the domain. But how can one be sufficiently confident that the data indeed satisfies the distributional assumption, so that one can trust in the output quality of the agnostic learning algorithm? We propose a model by which to systematically study the design of tester-learner pairs $(\mathcal{A},\mathcal{T})$, such that if the distribution on examples in the data passes the tester $\mathcal{T}$ then one can safely trust the output of the agnostic learner $\mathcal{A}$ on the data. To demonstrate the power of the model, we apply it to the classical problem of agnostically learning halfspaces under the standard Gaussian distribution and present a tester-learner pair with a combined run-time of $n^{\tilde{O}(1/\epsilon^4)}$. This qualitatively matches that of the best known ordinary agnostic learning algorithms for this task. In contrast, finite sample Gaussian distribution testers do not exist for the $L_1$ and EMD distance measures. A key step in the analysis is a novel characterization of concentration and anti-concentration properties of a distribution whose low-degree moments approximately match those of a Gaussian. We also use tools from polynomial approximation theory. In contrast, we show strong lower bounds on the combined run-times of tester-learner pairs for the problems of agnostically learning convex sets under the Gaussian distribution and for monotone Boolean functions under the uniform distribution over $\{0,1\}^n$. Through these lower bounds we exhibit natural problems where there is a dramatic gap between standard agnostic learning run-time and the run-time of the best tester-learner pair.