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This paper provides a finite-time analysis of linear stochastic approximation (LSA) algorithms with fixed step size, a core method in statistics and machine learning. LSA is used to compute approximate solutions of a $d$-dimensional linear system $\bar{\mathbf{A}} \theta = \bar{\mathbf{b}}$ for which $(\bar{\mathbf{A}}, \bar{\mathbf{b}})$ can only be estimated by (asymptotically) unbiased observations $\{(\mathbf{A}(Z_n),\mathbf{b}(Z_n))\}_{n \in \mathbb{N}}$. We consider here the case where $\{Z_n\}_{n \in \mathbb{N}}$ is an i.i.d. sequence or a uniformly geometrically ergodic Markov chain. We derive $p$-th moment and high-probability deviation bounds for the iterates defined by LSA and its Polyak-Ruppert-averaged version. Our finite-time instance-dependent bounds for the averaged LSA iterates are sharp in the sense that the leading term we obtain coincides with the local asymptotic minimax limit. Moreover, the remainder terms of our bounds admit a tight dependence on the mixing time $t_{\operatorname{mix}}$ of the underlying chain and the norm of the noise variables. We emphasize that our result requires the SA step size to scale only with logarithm of the problem dimension $d$.

Many streaming algorithms provide only a high-probability relative approximation. These two relaxations, of allowing approximation and randomization, seem necessary -- for many streaming problems, both relaxations must be employed simultaneously, to avoid an exponentially larger (and often trivial) space complexity. A common drawback of these randomized approximate algorithms is that independent executions on the same input have different outputs, that depend on their random coins. Pseudo-deterministic algorithms combat this issue, and for every input, they output with high probability the same ``canonical'' solution. We consider perhaps the most basic problem in data streams, of counting the number of items in a stream of length at most $n$. Morris's counter [CACM, 1978] is a randomized approximation algorithm for this problem that uses $O(\log\log n)$ bits of space, for every fixed approximation factor (greater than $1$). Goldwasser, Grossman, Mohanty and Woodruff [ITCS 2020] asked whether pseudo-deterministic approximation algorithms can match this space complexity. Our main result answers their question negatively, and shows that such algorithms must use $\Omega(\sqrt{\log n / \log\log n})$ bits of space. Our approach is based on a problem that we call Shift Finding, and may be of independent interest. In this problem, one has query access to a shifted version of a known string $F\in\{0,1\}^{3n}$, which is guaranteed to start with $n$ zeros and end with $n$ ones, and the goal is to find the unknown shift using a small number of queries. We provide for this problem an algorithm that uses $O(\sqrt{n})$ queries. It remains open whether $poly(\log n)$ queries suffice; if true, then our techniques immediately imply a nearly-tight $\Omega(\log n/\log\log n)$ space bound for pseudo-deterministic approximate counting.

The aim of this paper is twofold. First, we prove $L^p$ estimates for a regularized Green's function in three dimensions. We then establish new estimates for the discrete Green's function and obtain some positivity results. In particular, we prove that the discrete Green's functions with singularity in the interior of the domain cannot be bounded uniformly with respect of the mesh parameter $h$. Actually, we show that at the singularity the discrete Green's function is of order $h^{-1}$, which is consistent with the behavior of the continuous Green's function. In addition, we also show that the discrete Green's function is positive and decays exponentially away from the singularity. We also provide numerically persistent negative values of the discrete Green's function on Delaunay meshes which then implies a discrete Harnack inequality cannot be established for unstructured finite element discretizations.

Causal Discovery (CD) is the process of identifying the cause-effect relationships among the variables from data. Over the years, several methods have been developed primarily based on the statistical properties of data to uncover the underlying causal mechanism. In this study we introduce the common terminologies in causal discovery, and provide a comprehensive discussion of the approaches designed to identify the causal edges in different settings. We further discuss some of the benchmark datasets available for evaluating the performance of the causal discovery algorithms, available tools to perform causal discovery readily, and the common metrics used to evaluate these methods. Finally, we conclude by presenting the common challenges involved in CD and also, discuss the applications of CD in multiple areas of interest.

We study the convergence of a family of numerical integration methods where the numerical integral is formulated as a finite matrix approximation to a multiplication operator. For bounded functions, the convergence has already been established using the theory of strong operator convergence. In this article, we consider unbounded functions and domains which pose several difficulties compared to the bounded case. A natural choice of method for this study is the theory of strong resolvent convergence which has previously been mostly applied to study the convergence of approximations of differential operators. The existing theory already includes convergence theorems that can be used as proofs as such for a limited class of functions and extended for wider class of functions in terms of function growth or discontinuity. The extended results apply to all self-adjoint operators, not just multiplication operators. We also show how Jensen's operator inequality can be used to analyse the convergence of an improper numerical integral of a function bounded by an operator convex function.

Grammatical Error Correction (GEC) is the task of automatically detecting and correcting errors in text. The task not only includes the correction of grammatical errors, such as missing prepositions and mismatched subject-verb agreement, but also orthographic and semantic errors, such as misspellings and word choice errors respectively. The field has seen significant progress in the last decade, motivated in part by a series of five shared tasks, which drove the development of rule-based methods, statistical classifiers, statistical machine translation, and finally neural machine translation systems which represent the current dominant state of the art. In this survey paper, we condense the field into a single article and first outline some of the linguistic challenges of the task, introduce the most popular datasets that are available to researchers (for both English and other languages), and summarise the various methods and techniques that have been developed with a particular focus on artificial error generation. We next describe the many different approaches to evaluation as well as concerns surrounding metric reliability, especially in relation to subjective human judgements, before concluding with an overview of recent progress and suggestions for future work and remaining challenges. We hope that this survey will serve as comprehensive resource for researchers who are new to the field or who want to be kept apprised of recent developments.

In real-world applications of reinforcement learning, it is often challenging to obtain a state representation that is parsimonious and satisfies the Markov property without prior knowledge. Consequently, it is common practice to construct a state which is larger than necessary, e.g., by concatenating measurements over contiguous time points. However, needlessly increasing the dimension of the state can slow learning and obfuscate the learned policy. We introduce the notion of a minimal sufficient state in a Markov decision process (MDP) as the smallest subvector of the original state under which the process remains an MDP and shares the same optimal policy as the original process. We propose a novel sequential knockoffs (SEEK) algorithm that estimates the minimal sufficient state in a system with high-dimensional complex nonlinear dynamics. In large samples, the proposed method controls the false discovery rate, and selects all sufficient variables with probability approaching one. As the method is agnostic to the reinforcement learning algorithm being applied, it benefits downstream tasks such as policy optimization. Empirical experiments verify theoretical results and show the proposed approach outperforms several competing methods in terms of variable selection accuracy and regret.

This paper focuses on parameter estimation and introduces a new method for lower bounding the Bayesian risk. The method allows for the use of virtually \emph{any} information measure, including R\'enyi's $\alpha$, $\varphi$-Divergences, and Sibson's $\alpha$-Mutual Information. The approach considers divergences as functionals of measures and exploits the duality between spaces of measures and spaces of functions. In particular, we show that one can lower bound the risk with any information measure by upper bounding its dual via Markov's inequality. We are thus able to provide estimator-independent impossibility results thanks to the Data-Processing Inequalities that divergences satisfy. The results are then applied to settings of interest involving both discrete and continuous parameters, including the ``Hide-and-Seek'' problem, and compared to the state-of-the-art techniques. An important observation is that the behaviour of the lower bound in the number of samples is influenced by the choice of the information measure. We leverage this by introducing a new divergence inspired by the ``Hockey-Stick'' Divergence, which is demonstrated empirically to provide the largest lower-bound across all considered settings. If the observations are subject to privatisation, stronger impossibility results can be obtained via Strong Data-Processing Inequalities. The paper also discusses some generalisations and alternative directions.

Generalization error predictors (GEPs) aim to predict model performance on unseen distributions by deriving dataset-level error estimates from sample-level scores. However, GEPs often utilize disparate mechanisms (e.g., regressors, thresholding functions, calibration datasets, etc), to derive such error estimates, which can obfuscate the benefits of a particular scoring function. Therefore, in this work, we rigorously study the effectiveness of popular scoring functions (confidence, local manifold smoothness, model agreement), independent of mechanism choice. We find, absent complex mechanisms, that state-of-the-art confidence- and smoothness- based scores fail to outperform simple model-agreement scores when estimating error under distribution shifts and corruptions. Furthermore, on realistic settings where the training data has been compromised (e.g., label noise, measurement noise, undersampling), we find that model-agreement scores continue to perform well and that ensemble diversity is important for improving its performance. Finally, to better understand the limitations of scoring functions, we demonstrate that simplicity bias, or the propensity of deep neural networks to rely upon simple but brittle features, can adversely affect GEP performance. Overall, our work carefully studies the effectiveness of popular scoring functions in realistic settings and helps to better understand their limitations.

The solution of the governing equation representing the drawdown in a horizontal confined aquifer, where groundwater flow is unsteady, is provided in terms of the exponential integral, which is famously known as the Well function. For the computation of this function in practical applications, it is important to develop not only accurate but also a simple approximation that requires evaluation of the fewest possible terms. To that end, introducing Ramanujan's series expression, this work proposes a full-range approximation to the exponential integral using Ramanujan's series for the small argument (u \leq 1) and an approximation based on the bound of the integral for the other range (u \in (1,100]). The evaluation of the proposed approximation results in the most accurate formulae compared to the existing studies, which possess the maximum percentage error of 0.05\%. Further, the proposed formula is much simpler to apply as it contains just the product of exponential and logarithm functions. To further check the efficiency of the proposed approximation, we consider a practical example for evaluating the discrete pumping kernel, which shows the superiority of this approximation over the others. Finally, the authors hope that the proposed efficient approximation can be useful for groundwater and hydrogeological applications.