Real-time state estimation and forecasting is critical for efficient operation of power grids. In this paper, a physics-informed Gaussian process regression (PhI-GPR) method is presented and used for probabilistic forecasting and estimating the phase angle, angular speed, and wind mechanical power of a three-generator power grid system using sparse measurements. In standard data-driven Gaussian process regression (GPR), parameterized models for the prior statistics are fit by maximizing the marginal likelihood of observed data, whereas in PhI-GPR, we compute the prior statistics by solving stochastic equations governing power grid dynamics. The short-term forecast of a power grid system dominated by wind generation is complicated by the stochastic nature of the wind and the resulting uncertain mechanical wind power. Here, we assume that the power-grid dynamic is governed by the swing equations, and we treat the unknown terms in the swing equations (specifically, the mechanical wind power) as random processes, which turns these equations into stochastic differential equations. We solve these equations for the mean and variance of the power grid system using the Monte Carlo simulations method. We demonstrate that the proposed PhI-GPR method can accurately forecast and estimate both observed and unobserved states, including the mean behavior and associated uncertainty. For observed states, we show that PhI-GPR provides a forecast comparable to the standard data-driven GPR, with both forecasts being significantly more accurate than the autoregressive integrated moving average (ARIMA) forecast. We also show that the ARIMA forecast is much more sensitive to observation frequency and measurement errors than the PhI-GPR forecast.

We propose a few-shot learning method for spatial regression. Although Gaussian processes (GPs) have been successfully used for spatial regression, they require many observations in the target task to achieve a high predictive performance. Our model is trained using spatial datasets on various attributes in various regions, and predicts values on unseen attributes in unseen regions given a few observed data. With our model, a task representation is inferred from given small data using a neural network. Then, spatial values are predicted by neural networks with a GP framework, in which task-specific properties are controlled by the task representations. The GP framework allows us to analytically obtain predictions that are adapted to small data. By using the adapted predictions in the objective function, we can train our model efficiently and effectively so that the test predictive performance improves when adapted to newly given small data. In our experiments, we demonstrate that the proposed method achieves better predictive performance than existing meta-learning methods using spatial datasets.

The goal of clustering is to group similar objects into meaningful partitions. This process is well understood when an explicit similarity measure between the objects is given. However, far less is known when this information is not readily available and, instead, one only observes ordinal comparisons such as "object i is more similar to j than to k." In this paper, we tackle this problem using a two-step procedure: we estimate a pairwise similarity matrix from the comparisons before using a clustering method based on semi-definite programming (SDP). We theoretically show that our approach can exactly recover a planted clustering using a near-optimal number of passive comparisons. We empirically validate our theoretical findings and demonstrate the good behaviour of our method on real data.

As local explanations of black box models are increasingly being employed to establish model credibility in high stakes settings, it is important to ensure that these explanations are accurate and reliable. However, local explanations generated by existing techniques are often prone to high variance. Further, these techniques are computationally inefficient, require significant hyper-parameter tuning, and provide little insight into the quality of the resulting explanations. We identify lack of uncertainty modeling as a main cause of these challenges and develop a novel set of tools for analyzing explanation uncertainty in a Bayesian framework. In particular, we estimate credible intervals (CIs) that capture the uncertainty associated with each feature importance in local explanations. These credible intervals are tight when we have high confidence in the feature importances of a local explanation. The CIs are also informative both for estimating how many perturbations we need to sample -- sampling can proceed until the CIs are sufficiently narrow -- and where to sample -- sampling in regions with high predictive uncertainty leads to faster convergence. We instantiate this framework to generate Bayesian versions of LIME and KernelSHAP. Experimental evaluation with multiple real world datasets and user studies demonstrate the efficacy of our framework and the resulting explanations.

Meta continual learning algorithms seek to train a model when faced with similar tasks observed in a sequential manner. Despite promising methodological advancements, there is a lack of theoretical frameworks that enable analysis of learning challenges such as generalization and catastrophic forgetting. To that end, we develop a new theoretical approach for meta continual learning~(MCL) where we mathematically model the learning dynamics using dynamic programming, and we establish conditions of optimality for the MCL problem. Moreover, using the theoretical framework, we derive a new dynamic-programming-based MCL method that adopts stochastic-gradient-driven alternating optimization to balance generalization and catastrophic forgetting. We show that, on MCL benchmark data sets, our theoretically grounded method achieves accuracy better than or comparable to that of existing state-of-the-art methods.

This paper focuses on stochastic methods for solving smooth non-convex strongly-concave min-max problems, which have received increasing attention due to their potential applications in deep learning (e.g., deep AUC maximization). However, most of the existing algorithms are slow in practice, and their analysis revolves around the convergence to a nearly stationary point. We consider leveraging the Polyak-\L ojasiewicz (PL) condition to design faster stochastic algorithms with stronger convergence guarantee. Although PL condition has been utilized for designing many stochastic minimization algorithms, their applications for non-convex min-max optimization remains rare. In this paper, we propose and analyze proximal epoch-based methods, and establish fast convergence in terms of both {\bf the primal objective gap and the duality gap}. Our analysis is interesting in threefold: (i) it is based on a novel Lyapunov function that consists of the primal objective gap and the duality gap of a regularized function; (ii) it only requires a weaker PL condition for establishing the primal objective convergence than that required for the duality gap convergence; (iii) it yields the optimal dependence on the accuracy level $\epsilon$, i.e., $O(1/\epsilon)$. We also make explicit the dependence on the problem parameters and explore regions of weak convexity parameter that lead to improved dependence on condition numbers. Experiments on deep AUC maximization demonstrate the effectiveness of our methods. Our method (MaxAUC) achieved an AUC of 0.922 on private testing set on {\bf CheXpert competition}.

Neural Network (NN) classifiers can assign extreme probabilities to samples that have not appeared during training (out-of-distribution samples) resulting in erroneous and unreliable predictions. One of the causes for this unwanted behaviour lies in the use of the standard softmax operator which pushes the posterior probabilities to be either zero or unity hence failing to model uncertainty. The statistical derivation of the softmax operator relies on the assumption that the distributions of the latent variables for a given class are Gaussian with known variance. However, it is possible to use different assumptions in the same derivation and attain from other families of distributions as well. This allows derivation of novel operators with more favourable properties. Here, a novel operator is proposed that is derived using $t$-distributions which are capable of providing a better description of uncertainty. It is shown that classifiers that adopt this novel operator can be more robust to out of distribution samples, often outperforming NNs that use the standard softmax operator. These enhancements can be reached with minimal changes to the NN architecture.

Quasi-Maximum Likelihood (QML) procedures are theoretically appealing and widely used for statistical inference. While there are extensive references on QML estimation in batch settings, the QML estimation in streaming settings has attracted little attention until recently. An investigation of the convergence properties of the QML procedure in a general conditionally heteroscedastic time series model is conducted, and the classical batch optimization routines extended to the framework of streaming and large-scale problems. An adaptive recursive estimation routine for GARCH models named AdaVol is presented. The AdaVol procedure relies on stochastic approximations combined with the technique of Variance Targeting Estimation (VTE). This recursive method has computationally efficient properties, while VTE alleviates some convergence difficulties encountered by the usual QML estimation due to a lack of convexity. Empirical results demonstrate a favorable trade-off between AdaVol's stability and the ability to adapt to time-varying estimates for real-life data.

Hierarchical Multi-Agent Systems provide a convenient and relevant way to analyze, model, and simulate complex systems in which a large number of entities are interacting at different levels of abstraction. In this paper, we introduce HAMLET (Hierarchical Agent-based Machine LEarning plaTform), a platform based on hierarchical multi-agent systems, to facilitate the research and democratization of machine learning entities distributed geographically or locally. This is carried out by firstly modeling the machine learning solutions as a hypergraph and then autonomously setting up a multi-level structure composed of heterogeneous agents based on their innate capabilities and learned skills. HAMLET aids the design and management of machine learning systems and provides analytical capabilities for the research communities to assess the existing and/or new algorithms/datasets through flexible and customizable queries. The proposed platform does not assume restrictions on the type of machine learning algorithms/datasets and is theoretically proven to be sound and complete with polynomial computational requirements. Additionally, it is examined empirically on 120 training and four generalized batch testing tasks performed on 24 machine learning algorithms and 9 standard datasets. The experimental results provided not only establish confidence in the platform's consistency and correctness but also demonstrates its testing and analytical capacity.

When developing models for regulated decision making, sensitive features like age, race and gender cannot be used and must be obscured from model developers to prevent bias. However, the remaining features still need to be tested for correlation with sensitive features, which can only be done with the knowledge of those features. We resolve this dilemma using a fully homomorphic encryption scheme, allowing model developers to train linear regression and logistic regression models and test them for possible bias without ever revealing the sensitive features in the clear. We demonstrate how it can be applied to leave-one-out regression testing, and show using the adult income data set that our method is practical to run.