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Machine learning algorithms, especially Neural Networks (NNs), are a valuable tool used to approximate non-linear relationships, like the AC-Optimal Power Flow (AC-OPF), with considerable accuracy -- and achieving a speedup of several orders of magnitude when deployed for use. Often in power systems literature, the NNs are trained with a fixed dataset generated prior to the training process. In this paper, we show that adapting the NN training dataset during training can improve the NN performance and substantially reduce its worst-case violations. This paper proposes an algorithm that identifies and enriches the training dataset with critical datapoints that reduce the worst-case violations and deliver a neural network with improved worst-case performance guarantees. We demonstrate the performance of our algorithm in four test power systems, ranging from 39-buses to 162-buses.

Top-N recommendation aims to recommend each consumer a small set of N items from a large collection of items, and its accuracy is one of the most common indexes to evaluate the performance of a recommendation system. While a large number of algorithms are proposed to push the Top-N accuracy by learning the user preference from their history purchase data, a predictability question is naturally raised - whether there is an upper limit of such Top-N accuracy. This work investigates such predictability by studying the degree of regularity from a specific set of user behavior data. Quantifying the predictability of Top-N recommendations requires simultaneously quantifying the limits on the accuracy of the N behaviors with the highest probability. This greatly increases the difficulty of the problem. To achieve this, we firstly excavate the associations among N behaviors with the highest probability and describe the user behavior distribution based on the information theory. Then, we adopt the Fano inequality to scale and obtain the Top-N predictability. Extensive experiments are conducted on the real-world data where significant improvements are observed compared to the state-of-the-art methods. We have not only completed the predictability calculation for N targets but also obtained predictability that is much closer to the true value than existing methods. We expect our results to assist these research areas where the quantitative requirement of Top-N predictability is required.

In this paper, we consider distributed optimization problems where $n$ agents, each possessing a local cost function, collaboratively minimize the average of the local cost functions over a connected network. To solve the problem, we propose a distributed random reshuffling (D-RR) algorithm that invokes the random reshuffling (RR) update in each agent. We show that D-RR inherits favorable characteristics of RR for both smooth strongly convex and smooth nonconvex objective functions. In particular, for smooth strongly convex objective functions, D-RR achieves $\mathcal{O}(1/T^2)$ rate of convergence (where $T$ counts epoch number) in terms of the squared distance between the iterate and the global minimizer. When the objective function is assumed to be smooth nonconvex, we show that D-RR drives the squared norm of gradient to $0$ at a rate of $\mathcal{O}(1/T^{2/3})$. These convergence results match those of centralized RR (up to constant factors) and outperform the distributed stochastic gradient descent (DSGD) algorithm if we run a relatively large number of epochs. Finally, we conduct a set of numerical experiments to illustrate the efficiency of the proposed D-RR method on both strongly convex and nonconvex distributed optimization problems.

This paper addresses the optimization problem to maximize the total costs that can be shared among a group of agents, while maintaining stability in the sense of the core constraints of a cooperative transferable utility game, or TU game. This means that all subsets of agents have an outside option at a certain cost, and stability requires that the cost shares are defined so that none of the outside options is preferable. When maximizing total shareable costs, the cost shares must satisfy all constraints that define the core of a TU game, except for being budget balanced. The paper gives a fairly complete picture of the computational complexity of this optimization problem, in relation to classical computational problems on the core. We also show that, for games with an empty core, the problem is equivalent to computing minimal core relaxations for several relaxations that have been proposed earlier. As an example for a class of cost sharing games with non-empty core, we address minimum cost spanning tree games. While it is known that cost shares in the core can be found efficiently, we show that the computation of maximal cost shares is NP-hard for minimum cost spanning tree games. We also derive a 2-approximation algorithm. Our work opens several directions for future work.

We study a general formulation of regularized Wasserstein barycenters that enjoys favorable regularity, approximation, stability and (grid-free) optimization properties. This barycenter is defined as the unique probability measure that minimizes the sum of entropic optimal transport (EOT) costs with respect to a family of given probability measures, plus an entropy term. We denote it $(\lambda,\tau)$-barycenter, where $\lambda$ is the inner regularization strength and $\tau$ the outer one. This formulation recovers several previously proposed EOT barycenters for various choices of $\lambda,\tau \geq 0$ and generalizes them. First, in spite of -- and in fact owing to -- being \emph{doubly} regularized, we show that our formulation is debiased for $\tau=\lambda/2$: the suboptimality in the (unregularized) Wasserstein barycenter objective is, for smooth densities, of the order of the strength $\lambda^2$ of entropic regularization, instead of $\max\{\lambda,\tau\}$ in general. We discuss this phenomenon for isotropic Gaussians where all $(\lambda,\tau)$-barycenters have closed form. Second, we show that for $\lambda,\tau>0$, this barycenter has a smooth density and is strongly stable under perturbation of the marginals. In particular, it can be estimated efficiently: given $n$ samples from each of the probability measures, it converges in relative entropy to the population barycenter at a rate $n^{-1/2}$. And finally, this formulation lends itself naturally to a grid-free optimization algorithm: we propose a simple \emph{noisy particle gradient descent} which, in the mean-field limit, converges globally at an exponential rate to the barycenter.

We review Quasi Maximum Likelihood estimation of factor models for high-dimensional panels of time series. We consider two cases: (1) estimation when no dynamic model for the factors is specified (Bai and Li, 2016); (2) estimation based on the Kalman smoother and the Expectation Maximization algorithm thus allowing to model explicitly the factor dynamics (Doz et al., 2012). Our interest is in approximate factor models, i.e., when we allow for the idiosyncratic components to be mildly cross-sectionally, as well as serially, correlated. Although such setting apparently makes estimation harder, we show, in fact, that factor models do not suffer of the curse of dimensionality problem, but instead they enjoy a blessing of dimensionality property. In particular, we show that if the cross-sectional dimension of the data, $N$, grows to infinity, then: (i) identification of the model is still possible, (ii) the mis-specification error due to the use of an exact factor model log-likelihood vanishes. Moreover, if we let also the sample size, $T$, grow to infinity, we can also consistently estimate all parameters of the model and make inference. The same is true for estimation of the latent factors which can be carried out by weighted least-squares, linear projection, or Kalman filtering/smoothing. We also compare the approaches presented with: Principal Component analysis and the classical, fixed $N$, exact Maximum Likelihood approach. We conclude with a discussion on efficiency of the considered estimators.

We study the corrupted bandit problem, i.e. a stochastic multi-armed bandit problem with $k$ unknown reward distributions, which are heavy-tailed and corrupted by a history-independent adversary or Nature. To be specific, the reward obtained by playing an arm comes from corresponding heavy-tailed reward distribution with probability $1-\varepsilon \in (0.5,1]$ and an arbitrary corruption distribution of unbounded support with probability $\varepsilon \in [0,0.5)$. First, we provide $\textit{a problem-dependent lower bound on the regret}$ of any corrupted bandit algorithm. The lower bounds indicate that the corrupted bandit problem is harder than the classical stochastic bandit problem with sub-Gaussian or heavy-tail rewards. Following that, we propose a novel UCB-type algorithm for corrupted bandits, namely HubUCB, that builds on Huber's estimator for robust mean estimation. Leveraging a novel concentration inequality of Huber's estimator, we prove that HubUCB achieves a near-optimal regret upper bound. Since computing Huber's estimator has quadratic complexity, we further introduce a sequential version of Huber's estimator that exhibits linear complexity. We leverage this sequential estimator to design SeqHubUCB that enjoys similar regret guarantees while reducing the computational burden. Finally, we experimentally illustrate the efficiency of HubUCB and SeqHubUCB in solving corrupted bandits for different reward distributions and different levels of corruptions.

In complex large-scale systems such as climate, important effects are caused by a combination of confounding processes that are not fully observable. The identification of sources from observations of system state is vital for attribution and prediction, which inform critical policy decisions. The difficulty of these types of inverse problems lies in the inability to isolate sources and the cost of simulating computational models. Surrogate models may enable the many-query algorithms required for source identification, but data challenges arise from high dimensionality of the state and source, limited ensembles of costly model simulations to train a surrogate model, and few and potentially noisy state observations for inversion due to measurement limitations. The influence of auxiliary processes adds an additional layer of uncertainty that further confounds source identification. We introduce a framework based on (1) calibrating deep neural network surrogates to the flow maps provided by an ensemble of simulations obtained by varying sources, and (2) using these surrogates in a Bayesian framework to identify sources from observations via optimization. Focusing on an atmospheric dispersion exemplar, we find that the expressive and computationally efficient nature of the deep neural network operator surrogates in appropriately reduced dimension allows for source identification with uncertainty quantification using limited data. Introducing a variable wind field as an auxiliary process, we find that a Bayesian approximation error approach is essential for reliable source inversion when uncertainty due to wind stresses the algorithm.

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.