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Optimal margin Distribution Machine (ODM) is a newly proposed statistical learning framework rooting in the novel margin theory, which demonstrates better generalization performance than the traditional large margin based counterparts. Nonetheless, it suffers from the ubiquitous scalability problem regarding both computation time and memory as other kernel methods. This paper proposes a scalable ODM, which can achieve nearly ten times speedup compared to the original ODM training method. For nonlinear kernels, we propose a novel distribution-aware partition method to make the local ODM trained on each partition be close and converge fast to the global one. When linear kernel is applied, we extend a communication efficient SVRG method to accelerate the training further. Extensive empirical studies validate that our proposed method is highly computational efficient and almost never worsen the generalization.

Query re-optimization is an adaptive query processing technique that re-invokes the optimizer at certain points in query execution. The goal is to dynamically correct the cardinality estimation errors using the statistics collected at runtime to adjust the query plan to improve the overall performance. We identify a key weakness in existing re-optimization algorithms: their subquery division and re-optimization trigger strategies rely heavily on the optimizer's initial plan, which can be far away from optimal. We, therefore, propose QuerySplit, a novel re-optimization algorithm that skips the potentially misleading global plan and instead generates subqueries directly from the logical plan as the basic re-optimization units. By developing a cost function that prioritizes the execution of less "damaging" subqueries, QuerySplit successfully postpones (sometimes avoids) the execution of complex large joins to maximize their probability of having smaller input sizes. We implemented QuerySplit in PostgreSQL and compared our solution against four state-of-the-art re-optimization algorithms using the Join Order Benchmark. Our experiments show that QuerySplit reduces the benchmark execution time by 35% compared to the second-best alternative. The performance gap between QuerySplit and an optimal optimizer is within 4%.

Treatment effect estimation under unconfoundedness is a fundamental task in causal inference. In response to the challenge of analyzing high-dimensional datasets collected in substantive fields such as epidemiology, genetics, economics, and social sciences, many methods for treatment effect estimation with high-dimensional nuisance parameters (the outcome regression and the propensity score) have been developed in recent years. However, it is still unclear what is the necessary and sufficient sparsity condition on the nuisance parameters for the treatment effect to be $\sqrt{n}$-estimable. In this paper, we propose a new Double-Calibration strategy that corrects the estimation bias of the nuisance parameter estimates computed by regularized high-dimensional techniques and demonstrate that the corresponding Doubly-Calibrated estimator achieves $1 / \sqrt{n}$-rate as long as one of the nuisance parameters is sparse with sparsity below $\sqrt{n} / \log p$, where $p$ denotes the ambient dimension of the covariates, whereas the other nuisance parameter can be arbitrarily complex and completely misspecified. The Double-Calibration strategy can also be applied to settings other than treatment effect estimation, e.g. regression coefficient estimation in the presence of diverging number of controls in a semiparametric partially linear model.

Rough path theory provides one with the notion of signature, a graded family of tensors which characterise, up to a negligible equivalence class, and ordered stream of vector-valued data. In the last few years, use of the signature has gained traction in time-series analysis, machine learning , deep learning and more recently in kernel methods. In this article, we lay down the theoretical foundations for a connection between signature asymptotics, the theory of empirical processes, and Wasserstein distances, opening up the landscape and toolkit of the second and third in the study of the first. Our main contribution is to show that the Hambly-Lyons limit can be reinterpreted as a statement about the asymptotic behaviour of Wasserstein distances between two independent empirical measures of samples from the same underlying distribution. In the setting studied here, these measures are derived from samples from a probability distribution which is determined by geometrical properties of the underlying path. The general question of rates of convergence for these objects has been studied in depth in the recent monograph of Bobkov and Ledoux. By using these results, we generalise the original result of Hambly and Lyons from $C^3$ curves to a broad class of $C^2$ ones. We conclude by providing an explicit way to compute the limit in terms of a second-order differential equation.

Bayesian variable selection methods are powerful techniques for fitting and inferring on sparse high-dimensional linear regression models. However, many are computationally intensive or require restrictive prior distributions on model parameters. In this paper, we proposed a computationally efficient and powerful Bayesian approach for sparse high-dimensional linear regression. Minimal prior assumptions on the parameters are used through the use of plug-in empirical Bayes estimates of hyperparameters. Efficient maximum a posteriori (MAP) estimation is completed through a Parameter-Expanded Expectation-Conditional-Maximization (PX-ECM) algorithm. The PX-ECM results in a robust computationally efficient coordinate-wise optimization, which adjusts for the impact of other predictor variables. The completion of the E-step uses an approach motivated by the popular two-groups approach to multiple testing. The result is a PaRtitiOned empirical Bayes Ecm (PROBE) algorithm applied to sparse high-dimensional linear regression, which can be completed using one-at-a-time or all-at-once type optimization. We compare the empirical properties of PROBE to comparable approaches with numerous simulation studies and an analysis of cancer cell lines drug response study. The proposed approach is implemented in the R package probe.

Motivated by an application from geodesy, we introduce a novel clustering problem which is a $k$-center (or k-diameter) problem with a side constraint. For the side constraint, we are given an undirected connectivity graph $G$ on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in $G$. We call the resulting problems the connected $k$-center problem and the connected $k$-diameter problem. We prove several results on the complexity and approximability of these problems. Our main result is an $O(\log^2{k})$-approximation algorithm for the connected $k$-center and the connected $k$-diameter problem. For Euclidean metrics and metrics with constant doubling dimension, the approximation factor of this algorithm improves to $O(1)$. We also consider the special cases that the connectivity graph is a line or a tree. For the line we give optimal polynomial-time algorithms and for the case that the connectivity graph is a tree, we either give an optimal polynomial-time algorithm or a $2$-approximation algorithm for all variants of our model. We complement our upper bounds by several lower bounds.

Bayesian optimization (BO) is a powerful tool for seeking the global optimum of black-box functions. While evaluations of the black-box functions can be highly costly, it is desirable to reduce the use of expensive labeled data. For the first time, we introduce a teacher-student model to exploit semi-supervised learning that can make use of large amounts of unlabelled data under the context of BO. Importantly, we show that the selection of the validation and unlabeled data is key to the performance of BO. To optimize the sampling of unlabeled data, we employ a black-box parameterized sampling distribution optimized as part of the employed bi-level optimization framework. Taking one step further, we demonstrate that the performance of BO can be further improved by selecting unlabeled data from a dynamically fitted extreme value distribution. Our BO method operates in a learned latent space with reduced dimensionality, making it scalable to high-dimensional problems. The proposed approach outperforms significantly the existing BO methods on several synthetic and real-world optimization tasks.

Most state-of-the-art machine learning techniques revolve around the optimisation of loss functions. Defining appropriate loss functions is therefore critical to successfully solving problems in this field. We present a survey of the most commonly used loss functions for a wide range of different applications, divided into classification, regression, ranking, sample generation and energy based modelling. Overall, we introduce 33 different loss functions and we organise them into an intuitive taxonomy. Each loss function is given a theoretical backing and we describe where it is best used. This survey aims to provide a reference of the most essential loss functions for both beginner and advanced machine learning practitioners.

Causal discovery and causal reasoning are classically treated as separate and consecutive tasks: one first infers the causal graph, and then uses it to estimate causal effects of interventions. However, such a two-stage approach is uneconomical, especially in terms of actively collected interventional data, since the causal query of interest may not require a fully-specified causal model. From a Bayesian perspective, it is also unnatural, since a causal query (e.g., the causal graph or some causal effect) can be viewed as a latent quantity subject to posterior inference -- other unobserved quantities that are not of direct interest (e.g., the full causal model) ought to be marginalized out in this process and contribute to our epistemic uncertainty. In this work, we propose Active Bayesian Causal Inference (ABCI), a fully-Bayesian active learning framework for integrated causal discovery and reasoning, which jointly infers a posterior over causal models and queries of interest. In our approach to ABCI, we focus on the class of causally-sufficient, nonlinear additive noise models, which we model using Gaussian processes. We sequentially design experiments that are maximally informative about our target causal query, collect the corresponding interventional data, and update our beliefs to choose the next experiment. Through simulations, we demonstrate that our approach is more data-efficient than several baselines that only focus on learning the full causal graph. This allows us to accurately learn downstream causal queries from fewer samples while providing well-calibrated uncertainty estimates for the quantities of interest.

Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.