We consider the problem of estimating the factors of a rank-$1$ matrix with i.i.d. Gaussian, rank-$1$ measurements that are nonlinearly transformed and corrupted by noise. Considering two prototypical choices for the nonlinearity, we study the convergence properties of a natural alternating update rule for this nonconvex optimization problem starting from a random initialization. We show sharp convergence guarantees for a sample-split version of the algorithm by deriving a deterministic recursion that is accurate even in high-dimensional problems. Notably, while the infinite-sample population update is uninformative and suggests exact recovery in a single step, the algorithm -- and our deterministic prediction -- converges geometrically fast from a random initialization. Our sharp, non-asymptotic analysis also exposes several other fine-grained properties of this problem, including how the nonlinearity and noise level affect convergence behavior. On a technical level, our results are enabled by showing that the empirical error recursion can be predicted by our deterministic sequence within fluctuations of the order $n^{-1/2}$ when each iteration is run with $n$ observations. Our technique leverages leave-one-out tools originating in the literature on high-dimensional $M$-estimation and provides an avenue for sharply analyzing higher-order iterative algorithms from a random initialization in other high-dimensional optimization problems with random data.

This paper presents the findings of an exploratory study on the continuously generating Big Data on Twitter related to the sharing of information, news, views, opinions, ideas, feedback, and experiences about the COVID-19 pandemic, with a specific focus on the Omicron variant, which is the globally dominant variant of SARS-CoV-2 at this time. A total of 12028 tweets about the Omicron variant were studied, and the specific characteristics of tweets that were analyzed include - sentiment, language, source, type, and embedded URLs. The findings of this study are manifold. First, from sentiment analysis, it was observed that 50.5% of tweets had a neutral emotion. The other emotions - bad, good, terrible, and great were found in 15.6%, 14.0%, 12.5%, and 7.5% of the tweets, respectively. Second, the findings of language interpretation showed that 65.9% of the tweets were posted in English. It was followed by Spanish, French, Italian, and other languages. Third, the findings from source tracking showed that Twitter for Android was associated with 35.2% of tweets. It was followed by Twitter Web App, Twitter for iPhone, Twitter for iPad, and other sources. Fourth, studying the type of tweets revealed that retweets accounted for 60.8% of the tweets, it was followed by original tweets and replies that accounted for 19.8% and 19.4% of the tweets, respectively. Fifth, in terms of embedded URL analysis, the most common domain embedded in the tweets was found to be twitter.com, which was followed by biorxiv.org, nature.com, and other domains. Finally, to support similar research in this field, we have developed a Twitter dataset that comprises more than 500,000 tweets about the SARS-CoV-2 omicron variant since the first detected case of this variant on November 24, 2021.

Classifiers are biased when trained on biased datasets. As a remedy, we propose Learning to Split (ls), an algorithm for automatic bias detection. Given a dataset with input-label pairs, ls learns to split this dataset so that predictors trained on the training split cannot generalize to the testing split. This performance gap suggests that the testing split is under-represented in the dataset, which is a signal of potential bias. Identifying non-generalizable splits is challenging since we have no annotations about the bias. In this work, we show that the prediction correctness of each example in the testing split can be used as a source of weak supervision: generalization performance will drop if we move examples that are predicted correctly away from the testing split, leaving only those that are mis-predicted. ls is task-agnostic and can be applied to any supervised learning problem, ranging from natural language understanding and image classification to molecular property prediction. Empirical results show that ls is able to generate astonishingly challenging splits that correlate with human-identified biases. Moreover, we demonstrate that combining robust learning algorithms (such as group DRO) with splits identified by ls enables automatic de-biasing. Compared to previous state-of-the-art, we substantially improve the worst-group performance (23.4% on average) when the source of biases is unknown during training and validation.

This work in progress aims to provide a unified introduction to statistical learning, building up slowly from classical models like the GMM and HMM to modern neural networks like the VAE and diffusion models. There are today many internet resources that explain this or that new machine-learning algorithm in isolation, but they do not (and cannot, in so brief a space) connect these algorithms with each other or with the classical literature on statistical models, out of which the modern algorithms emerged. Also conspicuously lacking is a single notational system which, although unfazing to those already familiar with the material (like the authors of these posts), raises a significant barrier to the novice's entry. Likewise, I have aimed to assimilate the various models, wherever possible, to a single framework for inference and learning, showing how (and why) to change one model into another with minimal alteration (some of them novel, others from the literature). Some background is of course necessary. I have assumed the reader is familiar with basic multivariable calculus, probability and statistics, and linear algebra. The goal of this book is certainly not completeness, but rather to draw a more or less straight-line path from the basics to the extremely powerful new models of the last decade. The goal then is to complement, not replace, such comprehensive texts as Bishop's \emph{Pattern Recognition and Machine Learning}, which is now 15 years old.

An important unresolved challenge in the theory of regularization is to set the regularization coefficients of popular techniques like the ElasticNet with general provable guarantees. We consider the problem of tuning the regularization parameters of Ridge regression, LASSO, and the ElasticNet across multiple problem instances, a setting that encompasses both cross-validation and multi-task hyperparameter optimization. We obtain a novel structural result for the ElasticNet which characterizes the loss as a function of the tuning parameters as a piecewise-rational function with algebraic boundaries. We use this to bound the structural complexity of the regularized loss functions and show generalization guarantees for tuning the ElasticNet regression coefficients in the statistical setting. We also consider the more challenging online learning setting, where we show vanishing average expected regret relative to the optimal parameter pair. We further extend our results to tuning classification algorithms obtained by thresholding regression fits regularized by Ridge, LASSO, or ElasticNet. Our results are the first general learning-theoretic guarantees for this important class of problems that avoid strong assumptions on the data distribution. Furthermore, our guarantees hold for both validation and popular information criterion objectives.

Gaussian processes have become a promising tool for various safety-critical settings, since the posterior variance can be used to directly estimate the model error and quantify risk. However, state-of-the-art techniques for safety-critical settings hinge on the assumption that the kernel hyperparameters are known, which does not apply in general. To mitigate this, we introduce robust Gaussian process uniform error bounds in settings with unknown hyperparameters. Our approach computes a confidence region in the space of hyperparameters, which enables us to obtain a probabilistic upper bound for the model error of a Gaussian process with arbitrary hyperparameters. We do not require to know any bounds for the hyperparameters a priori, which is an assumption commonly found in related work. Instead, we are able to derive bounds from data in an intuitive fashion. We additionally employ the proposed technique to derive performance guarantees for a class of learning-based control problems. Experiments show that the bound performs significantly better than vanilla and fully Bayesian Gaussian processes.

Node classification on graphs is an important task in many practical domains. It usually requires labels for training, which can be difficult or expensive to obtain in practice. Given a budget for labelling, active learning aims to improve performance by carefully choosing which nodes to label. Previous graph active learning methods learn representations using labelled nodes and select some unlabelled nodes for label acquisition. However, they do not fully utilize the representation power present in unlabelled nodes. We argue that the representation power in unlabelled nodes can be useful for active learning and for further improving performance of active learning for node classification. In this paper, we propose a latent space clustering-based active learning framework for node classification (LSCALE), where we fully utilize the representation power in both labelled and unlabelled nodes. Specifically, to select nodes for labelling, our framework uses the K-Medoids clustering algorithm on a latent space based on a dynamic combination of both unsupervised features and supervised features. In addition, we design an incremental clustering module to avoid redundancy between nodes selected at different steps. Extensive experiments on five datasets show that our proposed framework LSCALE consistently and significantly outperforms the stateof-the-art approaches by a large margin.

As learning machines increase their influence on decisions concerning human lives, analyzing their fairness properties becomes a subject of central importance. Yet, our best tools for measuring the fairness of learning systems are rigid fairness metrics encapsulated as mathematical one-liners, offer limited power to the stakeholders involved in the prediction task, and are easy to manipulate when we exhort excessive pressure to optimize them. To advance these issues, we propose to shift focus from shaping fairness metrics to curating the distributions of examples under which these are computed. In particular, we posit that every claim about fairness should be immediately followed by the tagline "Fair under what examples, and collected by whom?". By highlighting connections to the literature in domain generalization, we propose to measure fairness as the ability of the system to generalize under multiple stress tests -- distributions of examples with social relevance. We encourage each stakeholder to curate one or multiple stress tests containing examples reflecting their (possibly conflicting) interests. The machine passes or fails each stress test by falling short of or exceeding a pre-defined metric value. The test results involve all stakeholders in a discussion about how to improve the learning system, and provide flexible assessments of fairness dependent on context and based on interpretable data. We provide full implementation guidelines for stress testing, illustrate both the benefits and shortcomings of this framework, and introduce a cryptographic scheme to enable a degree of prediction accountability from system providers.

The seminal paper by Mazumdar and Saha \cite{MS17a} introduced an extensive line of work on clustering with noisy queries. Yet, despite significant progress on the problem, the proposed methods depend crucially on knowing the exact probabilities of errors of the underlying fully-random oracle. In this work, we develop robust learning methods that tolerate general semi-random noise obtaining qualitatively the same guarantees as the best possible methods in the fully-random model. More specifically, given a set of $n$ points with an unknown underlying partition, we are allowed to query pairs of points $u,v$ to check if they are in the same cluster, but with probability $p$, the answer may be adversarially chosen. We show that information theoretically $O\left(\frac{nk \log n} {(1-2p)^2}\right)$ queries suffice to learn any cluster of sufficiently large size. Our main result is a computationally efficient algorithm that can identify large clusters with $O\left(\frac{nk \log n} {(1-2p)^2}\right) + \text{poly}\left(\log n, k, \frac{1}{1-2p} \right)$ queries, matching the guarantees of the best known algorithms in the fully-random model. As a corollary of our approach, we develop the first parameter-free algorithm for the fully-random model, answering an open question by \cite{MS17a}.

Compressed Stochastic Gradient Descent (SGD) algorithms have been recently proposed to address the communication bottleneck in distributed and decentralized optimization problems, such as those that arise in federated machine learning. Existing compressed SGD algorithms assume the use of non-adaptive step-sizes(constant or diminishing) to provide theoretical convergence guarantees. Typically, the step-sizes are fine-tuned in practice to the dataset and the learning algorithm to provide good empirical performance. Such fine-tuning might be impractical in many learning scenarios, and it is therefore of interest to study compressed SGD using adaptive step-sizes. Motivated by prior work on adaptive step-size methods for SGD to train neural networks efficiently in the uncompressed setting, we develop an adaptive step-size method for compressed SGD. In particular, we introduce a scaling technique for the descent step in compressed SGD, which we use to establish order-optimal convergence rates for convex-smooth and strong convex-smooth objectives under an interpolation condition and for non-convex objectives under a strong growth condition. We also show through simulation examples that without this scaling, the algorithm can fail to converge. We present experimental results on deep neural networks for real-world datasets, and compare the performance of our proposed algorithm with previously proposed compressed SGD methods in literature, and demonstrate improved performance on ResNet-18, ResNet-34 and DenseNet architectures for CIFAR-100 and CIFAR-10 datasets at various levels of compression.