We present a novel machine learning-based approach to solving bilevel programs that involve a large number of independent followers, which as a special case include two-stage stochastic programming. We propose an optimization model that explicitly considers a sampled subset of followers and exploits a machine learning model to estimate the objective values of unsampled followers. Unlike existing approaches, we embed machine learning model training into the optimization problem, which allows us to employ general follower features that can not be represented using leader decisions. We prove bounds on the optimality gap of the generated leader decision as measured by the original objective function that considers the full follower set. We then develop follower sampling algorithms to tighten the bounds and a representation learning approach to learn follower features, which can be used as inputs to the embedded machine learning model. Using synthetic instances of a cycling network design problem, we compare the computational performance of our approach versus baseline methods. Our approach provides more accurate predictions for follower objective values, and more importantly, generates leader decisions of higher quality. Finally, we perform a real-world case study on cycling infrastructure planning, where we apply our approach to solve a network design problem with over one million followers. Our approach presents favorable performance compared to the current cycling network expansion practices.

The discrepancy between in-distribution (ID) and out-of-distribution (OOD) samples can lead to \textit{distributional vulnerability} in deep neural networks, which can subsequently lead to high-confidence predictions for OOD samples. This is mainly due to the absence of OOD samples during training, which fails to constrain the network properly. To tackle this issue, several state-of-the-art methods include adding extra OOD samples to training and assign them with manually-defined labels. However, this practice can introduce unreliable labeling, negatively affecting ID classification. The distributional vulnerability presents a critical challenge for non-IID deep learning, which aims for OOD-tolerant ID classification by balancing ID generalization and OOD detection. In this paper, we introduce a novel \textit{supervision adaptation} approach to generate adaptive supervision information for OOD samples, making them more compatible with ID samples. Firstly, we measure the dependency between ID samples and their labels using mutual information, revealing that the supervision information can be represented in terms of negative probabilities across all classes. Secondly, we investigate data correlations between ID and OOD samples by solving a series of binary regression problems, with the goal of refining the supervision information for more distinctly separable ID classes. Our extensive experiments on four advanced network architectures, two ID datasets, and eleven diversified OOD datasets demonstrate the efficacy of our supervision adaptation approach in improving both ID classification and OOD detection capabilities.

Time-series causal discovery (TSCD) is a fundamental problem of machine learning. However, existing synthetic datasets cannot properly evaluate or predict the algorithms' performance on real data. This study introduces the CausalTime pipeline to generate time-series that highly resemble the real data and with ground truth causal graphs for quantitative performance evaluation. The pipeline starts from real observations in a specific scenario and produces a matching benchmark dataset. Firstly, we harness deep neural networks along with normalizing flow to accurately capture realistic dynamics. Secondly, we extract hypothesized causal graphs by performing importance analysis on the neural network or leveraging prior knowledge. Thirdly, we derive the ground truth causal graphs by splitting the causal model into causal term, residual term, and noise term. Lastly, using the fitted network and the derived causal graph, we generate corresponding versatile time-series proper for algorithm assessment. In the experiments, we validate the fidelity of the generated data through qualitative and quantitative experiments, followed by a benchmarking of existing TSCD algorithms using these generated datasets. CausalTime offers a feasible solution to evaluating TSCD algorithms in real applications and can be generalized to a wide range of fields. For easy use of the proposed approach, we also provide a user-friendly website, hosted on www.causaltime.cc.

Large models and enormous data are essential driving forces of the unprecedented successes achieved by modern algorithms, especially in scientific computing and machine learning. Nevertheless, the growing dimensionality and model complexity, as well as the non-negligible workload of data pre-processing, also bring formidable costs to such successes in both computation and data aggregation. As the deceleration of Moore's Law slackens the cost reduction of computation from the hardware level, fast heuristics for expensive classical routines and efficient algorithms for exploiting limited data are increasingly indispensable for pushing the limit of algorithm potency. This thesis explores some of such algorithms for fast execution and efficient data utilization. From the computational efficiency perspective, we design and analyze fast randomized low-rank decomposition algorithms for large matrices based on "matrix sketching", which can be regarded as a dimension reduction strategy in the data space. These include the randomized pivoting-based interpolative and CUR decomposition discussed in Chapter 2 and the randomized subspace approximations discussed in Chapter 3. From the sample efficiency perspective, we focus on learning algorithms with various incorporations of data augmentation that improve generalization and distributional robustness provably. Specifically, Chapter 4 presents a sample complexity analysis for data augmentation consistency regularization where we view sample efficiency from the lens of dimension reduction in the function space. Then in Chapter 5, we introduce an adaptively weighted data augmentation consistency regularization algorithm for distributionally robust optimization with applications in medical image segmentation.

We conduct a comprehensive investigation into the dynamics of gradient descent using large-order constant step-sizes in the context of quadratic regression models. Within this framework, we reveal that the dynamics can be encapsulated by a specific cubic map, naturally parameterized by the step-size. Through a fine-grained bifurcation analysis concerning the step-size parameter, we delineate five distinct training phases: (1) monotonic, (2) catapult, (3) periodic, (4) chaotic, and (5) divergent, precisely demarcating the boundaries of each phase. As illustrations, we provide examples involving phase retrieval and two-layer neural networks employing quadratic activation functions and constant outer-layers, utilizing orthogonal training data. Our simulations indicate that these five phases also manifest with generic non-orthogonal data. We also empirically investigate the generalization performance when training in the various non-monotonic (and non-divergent) phases. In particular, we observe that performing an ergodic trajectory averaging stabilizes the test error in non-monotonic (and non-divergent) phases.

The vast majority of techniques to train fair models require access to the protected attribute (e.g., race, gender), either at train time or in production. However, in many important applications this protected attribute is largely unavailable. In this paper, we develop methods for measuring and reducing fairness violations in a setting with limited access to protected attribute labels. Specifically, we assume access to protected attribute labels on a small subset of the dataset of interest, but only probabilistic estimates of protected attribute labels (e.g., via Bayesian Improved Surname Geocoding) for the rest of the dataset. With this setting in mind, we propose a method to estimate bounds on common fairness metrics for an existing model, as well as a method for training a model to limit fairness violations by solving a constrained non-convex optimization problem. Unlike similar existing approaches, our methods take advantage of contextual information -- specifically, the relationships between a model's predictions and the probabilistic prediction of protected attributes, given the true protected attribute, and vice versa -- to provide tighter bounds on the true disparity. We provide an empirical illustration of our methods using voting data. First, we show our measurement method can bound the true disparity up to 5.5x tighter than previous methods in these applications. Then, we demonstrate that our training technique effectively reduces disparity while incurring lesser fairness-accuracy trade-offs than other fair optimization methods with limited access to protected attributes.

Generalized cross-validation (GCV) is a widely-used method for estimating the squared out-of-sample prediction risk that employs a scalar degrees of freedom adjustment (in a multiplicative sense) to the squared training error. In this paper, we examine the consistency of GCV for estimating the prediction risk of arbitrary ensembles of penalized least squares estimators. We show that GCV is inconsistent for any finite ensemble of size greater than one. Towards repairing this shortcoming, we identify a correction that involves an additional scalar correction (in an additive sense) based on degrees of freedom adjusted training errors from each ensemble component. The proposed estimator (termed CGCV) maintains the computational advantages of GCV and requires neither sample splitting, model refitting, or out-of-bag risk estimation. The estimator stems from a finer inspection of ensemble risk decomposition and two intermediate risk estimators for the components in this decomposition. We provide a non-asymptotic analysis of the CGCV and the two intermediate risk estimators for ensembles of convex penalized estimators under Gaussian features and a linear response model. In the special case of ridge regression, we extend the analysis to general feature and response distributions using random matrix theory, which establishes model-free uniform consistency of CGCV.

We introduce a new model for multivariate probabilistic time series prediction, designed to flexibly address a range of tasks including forecasting, interpolation, and their combinations. Building on copula theory, we propose a simplified objective for the recently-introduced transformer-based attentional copulas (TACTiS), wherein the number of distributional parameters now scales linearly with the number of variables instead of factorially. The new objective requires the introduction of a training curriculum, which goes hand-in-hand with necessary changes to the original architecture. We show that the resulting model has significantly better training dynamics and achieves state-of-the-art performance across diverse real-world forecasting tasks, while maintaining the flexibility of prior work, such as seamless handling of unaligned and unevenly-sampled time series.

This paper considers the task of linear regression with shuffled labels, i.e., $\mathbf Y = \mathbf \Pi \mathbf X \mathbf B + \mathbf W$, where $\mathbf Y \in \mathbb R^{n\times m}, \mathbf Pi \in \mathbb R^{n\times n}, \mathbf X\in \mathbb R^{n\times p}, \mathbf B \in \mathbb R^{p\times m}$, and $\mathbf W\in \mathbb R^{n\times m}$, respectively, represent the sensing results, (unknown or missing) corresponding information, sensing matrix, signal of interest, and additive sensing noise. Given the observation $\mathbf Y$ and sensing matrix $\mathbf X$, we propose a one-step estimator to reconstruct $(\mathbf \Pi, \mathbf B)$. From the computational perspective, our estimator's complexity is $O(n^3 + np^2m)$, which is no greater than the maximum complexity of a linear assignment algorithm (e.g., $O(n^3)$) and a least square algorithm (e.g., $O(np^2 m)$). From the statistical perspective, we divide the minimum $snr$ requirement into four regimes, e.g., unknown, hard, medium, and easy regimes; and present sufficient conditions for the correct permutation recovery under each regime: $(i)$ $snr \geq \Omega(1)$ in the easy regime; $(ii)$ $snr \geq \Omega(\log n)$ in the medium regime; and $(iii)$ $snr \geq \Omega((\log n)^{c_0}\cdot n^{{c_1}/{srank(\mathbf B)}})$ in the hard regime ($c_0, c_1$ are some positive constants and $srank(\mathbf B)$ denotes the stable rank of $\mathbf B$). In the end, we also provide numerical experiments to confirm the above claims.

Recent work has proposed Wasserstein k-means (Wk-means) clustering as a powerful method to identify regimes in time series data, and one-dimensional asset returns in particular. In this paper, we begin by studying in detail the behaviour of the Wasserstein k-means clustering algorithm applied to synthetic one-dimensional time series data. We study the dynamics of the algorithm and investigate how varying different hyperparameters impacts the performance of the clustering algorithm for different random initialisations. We compute simple metrics that we find are useful in identifying high-quality clusterings. Then, we extend the technique of Wasserstein k-means clustering to multidimensional time series data by approximating the multidimensional Wasserstein distance as a sliced Wasserstein distance, resulting in a method we call `sliced Wasserstein k-means (sWk-means) clustering'. We apply the sWk-means clustering method to the problem of automated regime detection in multidimensional time series data, using synthetic data to demonstrate the validity of the approach. Finally, we show that the sWk-means method is effective in identifying distinct market regimes in real multidimensional financial time series, using publicly available foreign exchange spot rate data as a case study. We conclude with remarks about some limitations of our approach and potential complementary or alternative approaches.