We propose data-dependent test statistics based on a one-dimensional witness function, which we call witness two-sample tests (WiTS tests). We first optimize the witness function by maximizing an asymptotic test-power objective and then use as the test statistic the difference in means of the witness evaluated on two held-out test samples. When the witness function belongs to a reproducing kernel Hilbert space, we show that the optimal witness is given via kernel Fisher discriminant analysis, whose solution we compute in closed form. We show that the WiTS test based on a characteristic kernel is consistent against any fixed alternative. Our experiments demonstrate that the WiTS test can achieve higher test power than existing two-sample tests with optimized kernels, suggesting that learning a high- or infinite-dimensional representation of the data may not be necessary for two-sample testing. The proposed procedure works beyond kernel methods, allowing practitioners to apply it within their preferred machine learning framework.

The noise in stochastic gradient descent (SGD), caused by minibatch sampling, remains poorly understood despite its enormous practical importance in offering good training efficiency and generalization ability. In this work, we study the minibatch noise in SGD. Motivated by the observation that minibatch sampling does not always cause a fluctuation, we set out to find the conditions that cause minibatch noise to emerge. We first derive the analytically solvable results for linear regression under various settings, which are compared to the commonly used approximations that are used to understand SGD noise. We show that some degree of mismatch between model and data complexity is needed in order for SGD to "cause" a noise, and that such mismatch may be due to the existence of static noise in the labels, in the input, the use of regularization, or underparametrization. Our results motivate a more accurate general formulation to describe minibatch noise.

We consider nonnegative time series forecasting framework. Based on recent advances in Nonnegative Matrix Factorization (NMF) and Archetypal Analysis, we introduce two procedures referred to as Sliding Mask Method (SMM) and Latent Clustered Forecast (LCF). SMM is a simple and powerful method based on time window prediction using Completion of Nonnegative Matrices. This new procedure combines low nonnegative rank decomposition and matrix completion where the hidden values are to be forecasted. LCF is two stage: it leverages archetypal analysis for dimension reduction and clustering of time series, then it uses any black-box supervised forecast solver on the clustered latent representation. Theoretical guarantees on uniqueness and robustness of the solution of NMF Completion-type problems are also provided for the first time. Finally, numerical experiments on real-world and synthetic data-set confirms forecasting accuracy for both the methodologies.

We develop a computationally-efficient PAC active learning algorithm for $d$-dimensional homogeneous halfspaces that can tolerate Massart noise~\citep{massart2006risk} and Tsybakov noise~\citep{tsybakov2004optimal}. Specialized to the $\eta$-Massart noise setting, our algorithm achieves an information-theoretic optimal label complexity of $\tilde{O}\left( \frac{d}{(1-2\eta)^2} \mathrm{polylog}(\frac1\epsilon) \right)$ under a wide range of unlabeled data distributions (specifically, the family of "structured distributions" defined in~\citet{diakonikolas2020polynomial}). Under the more challenging Tsybakov noise condition, we identify two subfamilies of noise conditions, under which our algorithm achieves computational efficiency and provide label complexity guarantees strictly lower than passive learning algorithms.

Granger causal modeling is an emerging topic that can uncover Granger causal relationship behind multivariate time series data. In many real-world systems, it is common to encounter a large amount of multivariate time series data collected from different individuals with sharing commonalities. However, there are ongoing concerns regarding Granger causality's applicability in such large scale complex scenarios, presenting both challenges and opportunities for Granger causal structure reconstruction. Existing methods usually train a distinct model for each individual, suffering from inefficiency and over-fitting issues. To bridge this gap, we propose an Inductive GRanger cAusal modeling (InGRA) framework for inductive Granger causality learning and common causal structure detection on multivariate time series, which exploits the shared commonalities underlying the different individuals. In particular, we train one global model for individuals with different Granger causal structures through a novel attention mechanism, called prototypical Granger causal attention. The model can detect common causal structures for different individuals and infer Granger causal structures for newly arrived individuals. Extensive experiments, as well as an online A/B test on an E-commercial advertising platform, demonstrate the superior performances of InGRA.

Stochastic Gradient Descent (SGD) based methods have been widely used for training large-scale machine learning models that also generalize well in practice. Several explanations have been offered for this generalization performance, a prominent one being algorithmic stability [18]. However, there are no known examples of smooth loss functions for which the analysis can be shown to be tight. Furthermore, apart from the properties of the loss function, data distribution has also been shown to be an important factor in generalization performance. This raises the question: is the stability analysis of [18] tight for smooth functions, and if not, for what kind of loss functions and data distributions can the stability analysis be improved? In this paper we first settle open questions regarding tightness of bounds in the data-independent setting: we show that for general datasets, the existing analysis for convex and strongly-convex loss functions is tight, but it can be improved for non-convex loss functions. Next, we give a novel and improved data-dependent bounds: we show stability upper bounds for a large class of convex regularized loss functions, with negligible regularization parameters, and improve existing data-dependent bounds in the non-convex setting. We hope that our results will initiate further efforts to better understand the data-dependent setting under non-convex loss functions, leading to an improved understanding of the generalization abilities of deep networks.

The problem of finding K-nearest neighbors in the given dataset for a given query point has been worked upon since several years. In very high dimensional spaces the K-nearest neighbor search (KNNS) suffers in terms of complexity in computation of high dimensional distances. With the issue of curse of dimensionality, it gets quite tedious to reliably bank on the results of variety approximate nearest neighbor search approaches. In this paper, we survey some novel K-Nearest Neighbor Search approaches that tackles the problem of Search from the perspectives of computations, the accuracy of approximated results and leveraging parallelism to speed-up computations. We attempt to derive a relationship between the true positive and false points for a given KNNS approach. Finally, in order to evaluate the robustness of a KNNS approach against adversarial points, we propose a generic Reinforcement Learning based framework for the same.

The knowledge of the label noise transition matrix, characterizing the probabilities of a training instance being wrongly annotated, is crucial to designing popular solutions to learning with noisy labels, including loss correction and loss reweighting approaches. Existing works heavily rely on the existence of "anchor points" or their approximates, defined as instances that belong to a particular class almost surely. Nonetheless, finding anchor points remains a non-trivial task, and the estimation accuracy is also often throttled by the number of available anchor points. In this paper, we propose an alternative option to the above task. Our main contribution is the discovery of an efficient estimation procedure based on a clusterability condition. We prove that with clusterable representations of features, using up to third-order consensuses of noisy labels among neighbor representations is sufficient to estimate a unique transition matrix. Compared with methods using anchor points, our approach uses substantially more instances and benefits from a much better sample complexity. We demonstrate the estimation accuracy and advantages of our estimates using both synthetic noisy labels (on CIFAR-10/100) and real human-level noisy labels (on Clothing1M and our self-collected human-annotated CIFAR-10).

Many methods now exist for conditioning model outputs on task instructions, retrieved documents, and user-provided explanations and feedback. Rather than relying solely on examples of task inputs and outputs, these approaches use valuable additional data for improving model correctness and aligning learned models with human priors. Meanwhile, a growing body of evidence suggests that some language models can (1) store a large amount of knowledge in their parameters, and (2) perform inference over tasks in textual inputs at test time. These results raise the possibility that, for some tasks, humans cannot explain to a model any more about the task than it already knows or could infer on its own. In this paper, we study the circumstances under which explanations of individual data points can (or cannot) improve modeling performance. In order to carefully control important properties of the data and explanations, we introduce a synthetic dataset for experiments, and we also make use of three existing datasets with explanations: e-SNLI, TACRED, and SemEval. We first give a formal framework for the available modeling approaches, in which explanation data can be used as model inputs, as targets, or as a prior. After arguing that the most promising role for explanation data is as model inputs, we propose to use a retrieval-based method and show that it solves our synthetic task with accuracies upwards of 95%, while baselines without explanation data achieve below 65% accuracy. We then identify properties of datasets for which retrieval-based modeling fails. With the three existing datasets, we find no improvements from explanation retrieval. Drawing on findings from our synthetic task, we suggest that at least one of six preconditions for successful modeling fails to hold with these datasets. Our code is publicly available at https://github.com/peterbhase/ExplanationRoles

The automation of probabilistic reasoning is one of the primary aims of machine learning. Recently, the confluence of variational inference and deep learning has led to powerful and flexible automatic inference methods that can be trained by stochastic gradient descent. In particular, normalizing flows are highly parameterized deep models that can fit arbitrarily complex posterior densities. However, normalizing flows struggle in highly structured probabilistic programs as they need to relearn the forward-pass of the program. Automatic structured variational inference (ASVI) remedies this problem by constructing variational programs that embed the forward-pass. Here, we combine the flexibility of normalizing flows and the prior-embedding property of ASVI in a new family of variational programs, which we named cascading flows. A cascading flows program interposes a newly designed highway flow architecture in between the conditional distributions of the prior program such as to steer it toward the observed data. These programs can be constructed automatically from an input probabilistic program and can also be amortized automatically. We evaluate the performance of the new variational programs in a series of structured inference problems. We find that cascading flows have much higher performance than both normalizing flows and ASVI in a large set of structured inference problems.