A remarkable recent paper by Rubinfeld and Vasilyan (2022) initiated the study of \emph{testable learning}, where the goal is to replace hard-to-verify distributional assumptions (such as Gaussianity) with efficiently testable ones and to require that the learner succeed whenever the unknown distribution passes the corresponding test. In this model, they gave an efficient algorithm for learning halfspaces under testable assumptions that are provably satisfied by Gaussians. In this paper we give a powerful new approach for developing algorithms for testable learning using tools from moment matching and metric distances in probability. We obtain efficient testable learners for any concept class that admits low-degree \emph{sandwiching polynomials}, capturing most important examples for which we have ordinary agnostic learners. We recover the results of Rubinfeld and Vasilyan as a corollary of our techniques while achieving improved, near-optimal sample complexity bounds for a broad range of concept classes and distributions. Surprisingly, we show that the information-theoretic sample complexity of testable learning is tightly characterized by the Rademacher complexity of the concept class, one of the most well-studied measures in statistical learning theory. In particular, uniform convergence is necessary and sufficient for testable learning. This leads to a fundamental separation from (ordinary) distribution-specific agnostic learning, where uniform convergence is sufficient but not necessary.
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We study the paradoxical aspects of closed time-like curves and their impact on the theory of computation. After introducing the $\text{TM}_\text{CTC}$, a classical Turing machine benefiting CTCs for backward time travel, Aaronson et al. proved that $\text{P} = \text{PSPACE}$ and the $\Delta_2$ sets, such as the halting problem, are computable within this computational model. Our critical view is the physical consistency of this model, which leads to proposing the strong axiom, explaining that every particle rounding on a CTC will be destroyed before returning to its starting time, and the weak axiom, describing the same notion, particularly for Turing machines. We claim that in a universe containing CTCs, the two axioms must be true; otherwise, there will be an infinite number of any particle rounding on a CTC in the universe. An immediate result of the weak axiom is the incapability of Turing machines to convey information for a full round on a CTC, leading to the proposed $\text{TM}_\text{CTC}$ programs for the aforementioned corollaries failing to function. We suggest our solution for this problem as the data transferring hypothesis, which applies another $\text{TM}_\text{CTC}$ as a means for storing data. A prerequisite for it is the existence of the concept of Turing machines throughout time, which makes it appear infeasible in our universe. Then, we discuss possible physical conditions that can be held for a universe containing CTCs and conclude that if returning to an approximately equivalent universe by a CTC was conceivable, the above corollaries would be valid.
Low-rank approximation of tensors has been widely used in high-dimensional data analysis. It usually involves singular value decomposition (SVD) of large-scale matrices with high computational complexity. Sketching is an effective data compression and dimensionality reduction technique applied to the low-rank approximation of large matrices. This paper presents two practical randomized algorithms for low-rank Tucker approximation of large tensors based on sketching and power scheme, with a rigorous error-bound analysis. Numerical experiments on synthetic and real-world tensor data demonstrate the competitive performance of the proposed algorithms.
Quantum computing has received significant amounts of interest from many different research communities over the last few years. Although there are many introductory texts that focus on the algorithmic parts of quantum computing, there is a dearth of publications that describe the modeling, calibration and operation of current quantum computing devices. One aim of this report is to fill that void by providing a case study that walks through the entire procedure from the characterization and optimal control of a qudit device at Lawrence Livermore National Laboratory (LLNL) to the validation of the results. A goal of the report is to provide an introduction for students and researchers, especially computational mathematicians, who are interested in but new to quantum computing. Both experimental and mathematical aspects of this procedure are discussed. We present a description of the LLNL QuDIT testbed, the mathematical models that are used to describe it, and the numerical methods that are used to to design optimal controls. We also present experimental and computational methods that can be used to characterize a quantum device. Finally, an experimental validation of an optimized control pulse is presented, which relies on the accuracy of the characterization and the optimal control methodologies.
This PhD thesis contains several contributions to the field of statistical causal modeling. Statistical causal models are statistical models embedded with causal assumptions that allow for the inference and reasoning about the behavior of stochastic systems affected by external manipulation (interventions). This thesis contributes to the research areas concerning the estimation of causal effects, causal structure learning, and distributionally robust (out-of-distribution generalizing) prediction methods. We present novel and consistent linear and non-linear causal effects estimators in instrumental variable settings that employ data-dependent mean squared prediction error regularization. Our proposed estimators show, in certain settings, mean squared error improvements compared to both canonical and state-of-the-art estimators. We show that recent research on distributionally robust prediction methods has connections to well-studied estimators from econometrics. This connection leads us to prove that general K-class estimators possess distributional robustness properties. We, furthermore, propose a general framework for distributional robustness with respect to intervention-induced distributions. In this framework, we derive sufficient conditions for the identifiability of distributionally robust prediction methods and present impossibility results that show the necessity of several of these conditions. We present a new structure learning method applicable in additive noise models with directed trees as causal graphs. We prove consistency in a vanishing identifiability setup and provide a method for testing substructure hypotheses with asymptotic family-wise error control that remains valid post-selection. Finally, we present heuristic ideas for learning summary graphs of nonlinear time-series models.
Federated learning (FL) is an emerging, privacy-preserving machine learning paradigm, drawing tremendous attention in both academia and industry. A unique characteristic of FL is heterogeneity, which resides in the various hardware specifications and dynamic states across the participating devices. Theoretically, heterogeneity can exert a huge influence on the FL training process, e.g., causing a device unavailable for training or unable to upload its model updates. Unfortunately, these impacts have never been systematically studied and quantified in existing FL literature. In this paper, we carry out the first empirical study to characterize the impacts of heterogeneity in FL. We collect large-scale data from 136k smartphones that can faithfully reflect heterogeneity in real-world settings. We also build a heterogeneity-aware FL platform that complies with the standard FL protocol but with heterogeneity in consideration. Based on the data and the platform, we conduct extensive experiments to compare the performance of state-of-the-art FL algorithms under heterogeneity-aware and heterogeneity-unaware settings. Results show that heterogeneity causes non-trivial performance degradation in FL, including up to 9.2% accuracy drop, 2.32x lengthened training time, and undermined fairness. Furthermore, we analyze potential impact factors and find that device failure and participant bias are two potential factors for performance degradation. Our study provides insightful implications for FL practitioners. On the one hand, our findings suggest that FL algorithm designers consider necessary heterogeneity during the evaluation. On the other hand, our findings urge system providers to design specific mechanisms to mitigate the impacts of heterogeneity.
The notion of uncertainty is of major importance in machine learning and constitutes a key element of machine learning methodology. In line with the statistical tradition, uncertainty has long been perceived as almost synonymous with standard probability and probabilistic predictions. Yet, due to the steadily increasing relevance of machine learning for practical applications and related issues such as safety requirements, new problems and challenges have recently been identified by machine learning scholars, and these problems may call for new methodological developments. In particular, this includes the importance of distinguishing between (at least) two different types of uncertainty, often refereed to as aleatoric and epistemic. In this paper, we provide an introduction to the topic of uncertainty in machine learning as well as an overview of hitherto attempts at handling uncertainty in general and formalizing this distinction in particular.
Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this achievement, the design and training of neural networks are still challenging and unpredictable procedures. To lower the technical thresholds for common users, automated hyper-parameter optimization (HPO) has become a popular topic in both academic and industrial areas. This paper provides a review of the most essential topics on HPO. The first section introduces the key hyper-parameters related to model training and structure, and discusses their importance and methods to define the value range. Then, the research focuses on major optimization algorithms and their applicability, covering their efficiency and accuracy especially for deep learning networks. This study next reviews major services and toolkits for HPO, comparing their support for state-of-the-art searching algorithms, feasibility with major deep learning frameworks, and extensibility for new modules designed by users. The paper concludes with problems that exist when HPO is applied to deep learning, a comparison between optimization algorithms, and prominent approaches for model evaluation with limited computational resources.
Many tasks in natural language processing can be viewed as multi-label classification problems. However, most of the existing models are trained with the standard cross-entropy loss function and use a fixed prediction policy (e.g., a threshold of 0.5) for all the labels, which completely ignores the complexity and dependencies among different labels. In this paper, we propose a meta-learning method to capture these complex label dependencies. More specifically, our method utilizes a meta-learner to jointly learn the training policies and prediction policies for different labels. The training policies are then used to train the classifier with the cross-entropy loss function, and the prediction policies are further implemented for prediction. Experimental results on fine-grained entity typing and text classification demonstrate that our proposed method can obtain more accurate multi-label classification results.
Small data challenges have emerged in many learning problems, since the success of deep neural networks often relies on the availability of a huge amount of labeled data that is expensive to collect. To address it, many efforts have been made on training complex models with small data in an unsupervised and semi-supervised fashion. In this paper, we will review the recent progresses on these two major categories of methods. A wide spectrum of small data models will be categorized in a big picture, where we will show how they interplay with each other to motivate explorations of new ideas. We will review the criteria of learning the transformation equivariant, disentangled, self-supervised and semi-supervised representations, which underpin the foundations of recent developments. Many instantiations of unsupervised and semi-supervised generative models have been developed on the basis of these criteria, greatly expanding the territory of existing autoencoders, generative adversarial nets (GANs) and other deep networks by exploring the distribution of unlabeled data for more powerful representations. While we focus on the unsupervised and semi-supervised methods, we will also provide a broader review of other emerging topics, from unsupervised and semi-supervised domain adaptation to the fundamental roles of transformation equivariance and invariance in training a wide spectrum of deep networks. It is impossible for us to write an exclusive encyclopedia to include all related works. Instead, we aim at exploring the main ideas, principles and methods in this area to reveal where we are heading on the journey towards addressing the small data challenges in this big data era.
Learning from a few examples remains a key challenge in machine learning. Despite recent advances in important domains such as vision and language, the standard supervised deep learning paradigm does not offer a satisfactory solution for learning new concepts rapidly from little data. In this work, we employ ideas from metric learning based on deep neural features and from recent advances that augment neural networks with external memories. Our framework learns a network that maps a small labelled support set and an unlabelled example to its label, obviating the need for fine-tuning to adapt to new class types. We then define one-shot learning problems on vision (using Omniglot, ImageNet) and language tasks. Our algorithm improves one-shot accuracy on ImageNet from 87.6% to 93.2% and from 88.0% to 93.8% on Omniglot compared to competing approaches. We also demonstrate the usefulness of the same model on language modeling by introducing a one-shot task on the Penn Treebank.
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