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Stochastic gradient algorithms are widely used for both optimization and sampling in large-scale learning and inference problems. However, in practice, tuning these algorithms is typically done using heuristics and trial-and-error rather than rigorous, generalizable theory. To address this gap between theory and practice, we novel insights into the effect of tuning parameters by characterizing the large-sample behavior of iterates of a very general class of preconditioned stochastic gradient algorithms with fixed step size. In the optimization setting, our results show that iterate averaging with a large fixed step size can result in statistically efficient approximation of the (local) M-estimator. In the sampling context, our results show that with appropriate choices of tuning parameters, the limiting stationary covariance can match either the Bernstein--von Mises limit of the posterior, adjustments to the posterior for model misspecification, or the asymptotic distribution of the MLE; and that with a naive tuning the limit corresponds to none of these. Moreover, we argue that an essentially independent sample from the stationary distribution can be obtained after a fixed number of passes over the dataset. We validate our asymptotic results in realistic finite-sample regimes via several experiments using simulated and real data. Overall, we demonstrate that properly tuned stochastic gradient algorithms with constant step size offer a computationally efficient and statistically robust approach to obtaining point estimates or posterior-like samples.

Context: Differential testing is a useful approach that uses different implementations of the same algorithms and compares the results for software testing. In recent years, this approach was successfully used for test campaigns of deep learning frameworks. Objective: There is little knowledge on the application of differential testing beyond deep learning. Within this article, we want to close this gap for classification algorithms. Method: We conduct a case study using Scikit-learn, Weka, Spark MLlib, and Caret in which we identify the potential of differential testing by considering which algorithms are available in multiple frameworks, the feasibility by identifying pairs of algorithms that should exhibit the same behavior, and the effectiveness by executing tests for the identified pairs and analyzing the deviations. Results: While we found a large potential for popular algorithms, the feasibility seems limited because often it is not possible to determine configurations that are the same in other frameworks. The execution of the feasible tests revealed that there is a large amount of deviations for the scores and classes. Only a lenient approach based on statistical significance of classes does not lead to a huge amount of test failures. Conclusions: The potential of differential testing beyond deep learning seems limited for research into the quality of machine learning libraries. Practitioners may still use the approach if they have deep knowledge about implementations, especially if a coarse oracle that only considers significant differences of classes is sufficient.

We consider stochastic gradient descent and its averaging variant for binary classification problems in a reproducing kernel Hilbert space. In the traditional analysis using a consistency property of loss functions, it is known that the expected classification error converges more slowly than the expected risk even when assuming a low-noise condition on the conditional label probabilities. Consequently, the resulting rate is sublinear. Therefore, it is important to consider whether much faster convergence of the expected classification error can be achieved. In recent research, an exponential convergence rate for stochastic gradient descent was shown under a strong low-noise condition but provided theoretical analysis was limited to the squared loss function, which is somewhat inadequate for binary classification tasks. In this paper, we show an exponential convergence of the expected classification error in the final phase of the stochastic gradient descent for a wide class of differentiable convex loss functions under similar assumptions. As for the averaged stochastic gradient descent, we show that the same convergence rate holds from the early phase of training. In experiments, we verify our analyses on the $L_2$-regularized logistic regression.

We consider a causal inference model in which individuals interact in a social network and they may not comply with the assigned treatments. Estimating causal parameters is challenging in the presence of network interference of unknown form, as each individual may be influenced by both close individuals and distant ones in complex ways. Noncompliance with treatment assignment further complicates this problem, and prior methods dealing with network spillovers but disregarding the noncompliance issue may underestimate the effect of the treatment receipt on the outcome. To estimate meaningful causal parameters, we introduce a new concept of exposure mapping, which summarizes potentially complicated spillover effects into a fixed dimensional statistic of instrumental variables. We investigate identification conditions for the intention-to-treat effect and the average causal effect for compliers, while explicitly considering the possibility of misspecification of exposure mapping. Based on our identification results, we develop nonparametric estimation procedures via inverse probability weighting. Their asymptotic properties, including consistency and asymptotic normality, are investigated using an approximate neighborhood interference framework, which is convenient for dealing with unknown forms of spillovers between individuals. For an empirical illustration, we apply our method to experimental data on the anti-conflict intervention school program.

Learning controllers from data for stabilizing dynamical systems typically follows a two step process of first identifying a model and then constructing a controller based on the identified model. However, learning models means identifying generic descriptions of the dynamics of systems, which can require large amounts of data and extracting information that are unnecessary for the specific task of stabilization. The contribution of this work is to show that if a linear dynamical system has dimension (McMillan degree) $n$, then there always exist $n$ states from which a stabilizing feedback controller can be constructed, independent of the dimension of the representation of the observed states and the number of inputs. By building on previous work, this finding implies that any linear dynamical system can be stabilized from fewer observed states than the minimal number of states required for learning a model of the dynamics. The theoretical findings are demonstrated with numerical experiments that show the stabilization of the flow behind a cylinder from less data than necessary for learning a model.

For supervised classification problems, this paper considers estimating the query's label probability through local regression using observed covariates. Well-known nonparametric kernel smoother and $k$-nearest neighbor ($k$-NN) estimator, which take label average over a ball around the query, are consistent but asymptotically biased particularly for a large radius of the ball. To eradicate such bias, local polynomial regression (LPoR) and multiscale $k$-NN (MS-$k$-NN) learn the bias term by local regression around the query and extrapolate it to the query itself. However, their theoretical optimality has been shown for the limit of the infinite number of training samples. For correcting the asymptotic bias with fewer observations, this paper proposes a \emph{local radial regression (LRR)} and its logistic regression variant called \emph{local radial logistic regression~(LRLR)}, by combining the advantages of LPoR and MS-$k$-NN. The idea is quite simple: we fit the local regression to observed labels by taking only the radial distance as the explanatory variable and then extrapolate the estimated label probability to zero distance. The usefulness of the proposed method is shown theoretically and experimentally. We prove the convergence rate of the $L^2$ risk for LRR with reference to MS-$k$-NN, and our numerical experiments, including real-world datasets of daily stock indices, demonstrate that LRLR outperforms LPoR and MS-$k$-NN.

Topological data analysis (TDA) is a tool from data science and mathematics that is beginning to make waves in environmental science. In this work, we seek to provide an intuitive and understandable introduction to a tool from TDA that is particularly useful for the analysis of imagery, namely persistent homology. We briefly discuss the theoretical background but focus primarily on understanding the output of this tool and discussing what information it can glean. To this end, we frame our discussion around a guiding example of classifying satellite images from the Sugar, Fish, Flower, and Gravel Dataset produced for the study of mesocale organization of clouds by Rasp et. al. in 2020 (arXiv:1906:01906). We demonstrate how persistent homology and its vectorization, persistence landscapes, can be used in a workflow with a simple machine learning algorithm to obtain good results, and explore in detail how we can explain this behavior in terms of image-level features. One of the core strengths of persistent homology is how interpretable it can be, so throughout this paper we discuss not just the patterns we find, but why those results are to be expected given what we know about the theory of persistent homology. Our goal is that a reader of this paper will leave with a better understanding of TDA and persistent homology, be able to identify problems and datasets of their own for which persistent homology could be helpful, and gain an understanding of results they obtain from applying the included GitHub example code.

Pre-trained deep neural network language models such as ELMo, GPT, BERT and XLNet have recently achieved state-of-the-art performance on a variety of language understanding tasks. However, their size makes them impractical for a number of scenarios, especially on mobile and edge devices. In particular, the input word embedding matrix accounts for a significant proportion of the model's memory footprint, due to the large input vocabulary and embedding dimensions. Knowledge distillation techniques have had success at compressing large neural network models, but they are ineffective at yielding student models with vocabularies different from the original teacher models. We introduce a novel knowledge distillation technique for training a student model with a significantly smaller vocabulary as well as lower embedding and hidden state dimensions. Specifically, we employ a dual-training mechanism that trains the teacher and student models simultaneously to obtain optimal word embeddings for the student vocabulary. We combine this approach with learning shared projection matrices that transfer layer-wise knowledge from the teacher model to the student model. Our method is able to compress the BERT_BASE model by more than 60x, with only a minor drop in downstream task metrics, resulting in a language model with a footprint of under 7MB. Experimental results also demonstrate higher compression efficiency and accuracy when compared with other state-of-the-art compression techniques.

Extreme multi-label text classification (XMC) aims to tag each input text with the most relevant labels from an extremely large label set, such as those that arise in product categorization and e-commerce recommendation. Recently, pretrained language representation models such as BERT achieve remarkable state-of-the-art performance across a wide range of NLP tasks including sentence classification among small label sets (typically fewer than thousands). Indeed, there are several challenges in applying BERT to the XMC problem. The main challenges are: (i) the difficulty of capturing dependencies and correlations among labels, whose features may come from heterogeneous sources, and (ii) the tractability to scale to the extreme label setting as the model size can be very large and scale linearly with the size of the output space. To overcome these challenges, we propose X-BERT, the first feasible attempt to finetune BERT models for a scalable solution to the XMC problem. Specifically, X-BERT leverages both the label and document text to build label representations, which induces semantic label clusters in order to better model label dependencies. At the heart of X-BERT is finetuning BERT models to capture the contextual relations between input text and the induced label clusters. Finally, an ensemble of the different BERT models trained on heterogeneous label clusters leads to our best final model. Empirically, on a Wiki dataset with around 0.5 million labels, X-BERT achieves new state-of-the-art results where the precision@1 reaches 67:80%, a substantial improvement over 32.58%/60.91% of deep learning baseline fastText and competing XMC approach Parabel, respectively. This amounts to a 11.31% relative improvement over Parabel, which is indeed significant since the recent approach SLICE only has 5.53% relative improvement.

Graph convolutional networks (GCNs) have been successfully applied in node classification tasks of network mining. However, most of these models based on neighborhood aggregation are usually shallow and lack the "graph pooling" mechanism, which prevents the model from obtaining adequate global information. In order to increase the receptive field, we propose a novel deep Hierarchical Graph Convolutional Network (H-GCN) for semi-supervised node classification. H-GCN first repeatedly aggregates structurally similar nodes to hyper-nodes and then refines the coarsened graph to the original to restore the representation for each node. Instead of merely aggregating one- or two-hop neighborhood information, the proposed coarsening procedure enlarges the receptive field for each node, hence more global information can be learned. Comprehensive experiments conducted on public datasets demonstrate the effectiveness of the proposed method over the state-of-art methods. Notably, our model gains substantial improvements when only a few labeled samples are provided.