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We study online learning problems in which a decision maker wants to maximize their expected reward without violating a finite set of $m$ resource constraints. By casting the learning process over a suitably defined space of strategy mixtures, we recover strong duality on a Lagrangian relaxation of the underlying optimization problem, even for general settings with non-convex reward and resource-consumption functions. Then, we provide the first best-of-many-worlds type framework for this setting, with no-regret guarantees under stochastic, adversarial, and non-stationary inputs. Our framework yields the same regret guarantees of prior work in the stochastic case. On the other hand, when budgets grow at least linearly in the time horizon, it allows us to provide a constant competitive ratio in the adversarial case, which improves over the best known upper bound bound of $O(\log m \log T)$. Moreover, our framework allows the decision maker to handle non-convex reward and cost functions. We provide two game-theoretic applications of our framework to give further evidence of its flexibility. In doing so, we show that it can be employed to implement budget-pacing mechanisms in repeated first-price auctions.

To infer the treatment effect for a single treated unit using panel data, synthetic control methods construct a linear combination of control units' outcomes that mimics the treated unit's pre-treatment outcome trajectory. This linear combination is subsequently used to impute the counterfactual outcomes of the treated unit had it not been treated in the post-treatment period, and used to estimate the treatment effect. Existing synthetic control methods rely on correctly modeling certain aspects of the counterfactual outcome generating mechanism and may require near-perfect matching of the pre-treatment trajectory. Inspired by proximal causal inference, we obtain two novel nonparametric identifying formulas for the average treatment effect for the treated unit: one is based on weighting, and the other combines models for the counterfactual outcome and the weighting function. We introduce the concept of covariate shift to synthetic controls to obtain these identification results conditional on the treatment assignment. We also develop two treatment effect estimators based on these two formulas and the generalized method of moments. One new estimator is doubly robust: it is consistent and asymptotically normal if at least one of the outcome and weighting models is correctly specified. We demonstrate the performance of the methods via simulations and apply them to evaluate the effectiveness of a Pneumococcal conjugate vaccine on the risk of all-cause pneumonia in Brazil.

Estimating probability distributions which describe where an object is likely to be from camera data is a task with many applications. In this work we describe properties which we argue such methods should conform to. We also design a method which conform to these properties. In our experiments we show that our method produces uncertainties which correlate well with empirical errors. We also show that the mode of the predicted distribution outperform our regression baselines. The code for our implementation is available online.

In this paper, we consider the problem of phase retrieval, which consists of recovering an $n$-dimensional real vector from the magnitude of its $m$ linear measurements. We propose a mirror descent (or Bregman gradient descent) algorithm based on a wisely chosen Bregman divergence, hence allowing to remove the classical global Lipschitz continuity requirement on the gradient of the non-convex phase retrieval objective to be minimized. We apply the mirror descent for two random measurements: the \iid standard Gaussian and those obtained by multiple structured illuminations through Coded Diffraction Patterns (CDP). For the Gaussian case, we show that when the number of measurements $m$ is large enough, then with high probability, for almost all initializers, the algorithm recovers the original vector up to a global sign change. For both measurements, the mirror descent exhibits a local linear convergence behaviour with a dimension-independent convergence rate. Our theoretical results are finally illustrated with various numerical experiments, including an application to the reconstruction of images in precision optics.

In a dual weighted residual method based on the finite element framework, the Galerkin orthogonality is an issue that prevents solving the dual equation in the same space as the one for the primal equation. In the literature, there have been two popular approaches to constructing a new space for the dual problem, i.e., refining mesh grids ($h$-approach) and raising the order of approximate polynomials ($p$-approach). In this paper, a novel approach is proposed for the purpose based on the multiple-precision technique, i.e., the construction of the new finite element space is based on the same configuration as the one for the primal equation, except for the precision in calculations. The feasibility of such a new approach is discussed in detail in the paper. In numerical experiments, the proposed approach can be realized conveniently with C++ \textit{template}. Moreover, the new approach shows remarkable improvements in both efficiency and storage compared with the $h$-approach and the $p$-approach. It is worth mentioning that the performance of our approach is comparable with the one through a higher order interpolation ($i$-approach) in the literature. The combination of these two approaches is believed to further enhance the efficiency of the dual weighted residual method.

Decision trees are widely used for their low computational cost, good predictive performance, and ability to assess the importance of features. Though often used in practice for feature selection, the theoretical guarantees of these methods are not well understood. We here obtain a tight finite sample bound for the feature selection problem in linear regression using single-depth decision trees. We examine the statistical properties of these "decision stumps" for the recovery of the $s$ active features from $p$ total features, where $s \ll p$. Our analysis provides tight sample performance guarantees on high-dimensional sparse systems which align with the finite sample bound of $O(s \log p)$ as obtained by Lasso, improving upon previous bounds for both the median and optimal splitting criteria. Our results extend to the non-linear regime as well as arbitrary sub-Gaussian distributions, demonstrating that tree based methods attain strong feature selection properties under a wide variety of settings and further shedding light on the success of these methods in practice. As a byproduct of our analysis, we show that we can provably guarantee recovery even when the number of active features $s$ is unknown. We further validate our theoretical results and proof methodology using computational experiments.

As autonomous agents enter complex environments, it becomes more difficult to adequately model the interactions between the two. Agents must therefore cope with greater ambiguity (e.g., unknown environments, underdefined models, and vague problem definitions). Despite the consequences of ignoring ambiguity, tools for decision making under ambiguity are understudied. The general approach has been to avoid ambiguity (exploit known information) using robust methods. This work contributes ambiguity attitude graph search (AAGS), generalizing robust methods with ambiguity attitudes--the ability to trade-off between seeking and avoiding ambiguity in the problem. AAGS solves online decision making problems with limited budget to learn about their environment. To evaluate this approach AAGS is tasked with path planning in static and dynamic environments. Results demonstrate that appropriate ambiguity attitudes are dependent on the quality of information from the environment. In relatively certain environments, AAGS can readily exploit information with robust policies. Conversely, model complexity reduces the information conveyed by individual samples; this allows the risks taken by optimistic policies to achieve better performance.

Unsupervised domain adaptation has recently emerged as an effective paradigm for generalizing deep neural networks to new target domains. However, there is still enormous potential to be tapped to reach the fully supervised performance. In this paper, we present a novel active learning strategy to assist knowledge transfer in the target domain, dubbed active domain adaptation. We start from an observation that energy-based models exhibit free energy biases when training (source) and test (target) data come from different distributions. Inspired by this inherent mechanism, we empirically reveal that a simple yet efficient energy-based sampling strategy sheds light on selecting the most valuable target samples than existing approaches requiring particular architectures or computation of the distances. Our algorithm, Energy-based Active Domain Adaptation (EADA), queries groups of targe data that incorporate both domain characteristic and instance uncertainty into every selection round. Meanwhile, by aligning the free energy of target data compact around the source domain via a regularization term, domain gap can be implicitly diminished. Through extensive experiments, we show that EADA surpasses state-of-the-art methods on well-known challenging benchmarks with substantial improvements, making it a useful option in the open world. Code is available at //github.com/BIT-DA/EADA.

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.

Meta-learning extracts the common knowledge acquired from learning different tasks and uses it for unseen tasks. It demonstrates a clear advantage on tasks that have insufficient training data, e.g., few-shot learning. In most meta-learning methods, tasks are implicitly related via the shared model or optimizer. In this paper, we show that a meta-learner that explicitly relates tasks on a graph describing the relations of their output dimensions (e.g., classes) can significantly improve the performance of few-shot learning. This type of graph is usually free or cheap to obtain but has rarely been explored in previous works. We study the prototype based few-shot classification, in which a prototype is generated for each class, such that the nearest neighbor search between the prototypes produces an accurate classification. We introduce "Gated Propagation Network (GPN)", which learns to propagate messages between prototypes of different classes on the graph, so that learning the prototype of each class benefits from the data of other related classes. In GPN, an attention mechanism is used for the aggregation of messages from neighboring classes, and a gate is deployed to choose between the aggregated messages and the message from the class itself. GPN is trained on a sequence of tasks from many-shot to few-shot generated by subgraph sampling. During training, it is able to reuse and update previously achieved prototypes from the memory in a life-long learning cycle. In experiments, we change the training-test discrepancy and test task generation settings for thorough evaluations. GPN outperforms recent meta-learning methods on two benchmark datasets in all studied cases.