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We study the problem of robust multivariate polynomial regression: let $p\colon\mathbb{R}^n\to\mathbb{R}$ be an unknown $n$-variate polynomial of degree at most $d$ in each variable. We are given as input a set of random samples $(\mathbf{x}_i,y_i) \in [-1,1]^n \times \mathbb{R}$ that are noisy versions of $(\mathbf{x}_i,p(\mathbf{x}_i))$. More precisely, each $\mathbf{x}_i$ is sampled independently from some distribution $\chi$ on $[-1,1]^n$, and for each $i$ independently, $y_i$ is arbitrary (i.e., an outlier) with probability at most $\rho < 1/2$, and otherwise satisfies $|y_i-p(\mathbf{x}_i)|\leq\sigma$. The goal is to output a polynomial $\hat{p}$, of degree at most $d$ in each variable, within an $\ell_\infty$-distance of at most $O(\sigma)$ from $p$. Kane, Karmalkar, and Price [FOCS'17] solved this problem for $n=1$. We generalize their results to the $n$-variate setting, showing an algorithm that achieves a sample complexity of $O_n(d^n\log d)$, where the hidden constant depends on $n$, if $\chi$ is the $n$-dimensional Chebyshev distribution. The sample complexity is $O_n(d^{2n}\log d)$, if the samples are drawn from the uniform distribution instead. The approximation error is guaranteed to be at most $O(\sigma)$, and the run-time depends on $\log(1/\sigma)$. In the setting where each $\mathbf{x}_i$ and $y_i$ are known up to $N$ bits of precision, the run-time's dependence on $N$ is linear. We also show that our sample complexities are optimal in terms of $d^n$. Furthermore, we show that it is possible to have the run-time be independent of $1/\sigma$, at the cost of a higher sample complexity.

We present variational inference with sequential sample-average approximation (VISA), a method for approximate inference in computationally intensive models, such as those based on numerical simulations. VISA extends importance-weighted forward-KL variational inference by employing a sequence of sample-average approximations, which are considered valid inside a trust region. This makes it possible to reuse model evaluations across multiple gradient steps, thereby reducing computational cost. We perform experiments on high-dimensional Gaussians, Lotka-Volterra dynamics, and a Pickover attractor, which demonstrate that VISA can achieve comparable approximation accuracy to standard importance-weighted forward-KL variational inference with computational savings of a factor two or more for conservatively chosen learning rates.

In the context of an increasing popularity of data-driven models to represent dynamical systems, many machine learning-based implementations of the Koopman operator have recently been proposed. However, the vast majority of those works are limited to deterministic predictions, while the knowledge of uncertainty is critical in fields like meteorology and climatology. In this work, we investigate the training of ensembles of models to produce stochastic outputs. We show through experiments on real remote sensing image time series that ensembles of independently trained models are highly overconfident and that using a training criterion that explicitly encourages the members to produce predictions with high inter-model variances greatly improves the uncertainty quantification of the ensembles.

Stochastic gradient descent method and its variants constitute the core optimization algorithms that achieve good convergence rates for solving machine learning problems. These rates are obtained especially when these algorithms are fine-tuned for the application at hand. Although this tuning process can require large computational costs, recent work has shown that these costs can be reduced by line search methods that iteratively adjust the step length. We propose an alternative approach to stochastic line search by using a new algorithm based on forward step model building. This model building step incorporates second-order information that allows adjusting not only the step length but also the search direction. Noting that deep learning model parameters come in groups (layers of tensors), our method builds its model and calculates a new step for each parameter group. This novel diagonalization approach makes the selected step lengths adaptive. We provide convergence rate analysis, and experimentally show that the proposed algorithm achieves faster convergence and better generalization in well-known test problems. More precisely, SMB requires less tuning, and shows comparable performance to other adaptive methods.

We consider a binary decision aggregation problem in the presence of both truthful and adversarial experts. The truthful experts will report their private signals truthfully with proper incentive, while the adversarial experts can report arbitrarily. The decision maker needs to design a robust aggregator to forecast the true state of the world based on the reports of experts. The decision maker does not know the specific information structure, which is a joint distribution of signals, states, and strategies of adversarial experts. We want to find the optimal aggregator minimizing regret under the worst information structure. The regret is defined by the difference in expected loss between the aggregator and a benchmark who makes the optimal decision given the joint distribution and reports of truthful experts. We prove that when the truthful experts are symmetric and adversarial experts are not too numerous, the truncated mean is optimal, which means that we remove some lowest reports and highest reports and take averaging among the left reports. Moreover, for many settings, the optimal aggregators are in the family of piecewise linear functions. The regret is independent of the total number of experts but only depends on the ratio of adversaries. We evaluate our aggregators by numerical experiment in an ensemble learning task. We also obtain some negative results for the aggregation problem with adversarial experts under some more general information structures and experts' report space.

Spatiotemporal datasets, which consist of spatially-referenced time series, are ubiquitous in many scientific and business-intelligence applications, such as air pollution monitoring, disease tracking, and cloud-demand forecasting. As modern datasets continue to increase in size and complexity, there is a growing need for new statistical methods that are flexible enough to capture complex spatiotemporal dynamics and scalable enough to handle large prediction problems. This work presents the Bayesian Neural Field (BayesNF), a domain-general statistical model for inferring rich probability distributions over a spatiotemporal domain, which can be used for data-analysis tasks including forecasting, interpolation, and variography. BayesNF integrates a novel deep neural network architecture for high-capacity function estimation with hierarchical Bayesian inference for robust uncertainty quantification. By defining the prior through a sequence of smooth differentiable transforms, posterior inference is conducted on large-scale data using variationally learned surrogates trained via stochastic gradient descent. We evaluate BayesNF against prominent statistical and machine-learning baselines, showing considerable improvements on diverse prediction problems from climate and public health datasets that contain tens to hundreds of thousands of measurements. The paper is accompanied with an open-source software package (//github.com/google/bayesnf) that is easy-to-use and compatible with modern GPU and TPU accelerators on the JAX machine learning platform.

Quantum circuit compilation comprises many computationally hard reasoning tasks that nonetheless lie inside #$\mathbf{P}$ and its decision counterpart in $\mathbf{PP}$. The classical simulation of general quantum circuits is a core example. We show for the first time that a strong simulation of universal quantum circuits can be efficiently tackled through weighted model counting by providing a linear encoding of Clifford+T circuits. To achieve this, we exploit the stabilizer formalism by Knill, Gottesmann, and Aaronson and the fact that stabilizer states form a basis for density operators. With an open-source simulator implementation, we demonstrate empirically that model counting often outperforms state-of-the-art simulation techniques based on the ZX calculus and decision diagrams. Our work paves the way to apply the existing array of powerful classical reasoning tools to realize efficient quantum circuit compilation; one of the obstacles on the road towards quantum supremacy.

It is important to detect anomalous inputs when deploying machine learning systems. The use of larger and more complex inputs in deep learning magnifies the difficulty of distinguishing between anomalous and in-distribution examples. At the same time, diverse image and text data are available in enormous quantities. We propose leveraging these data to improve deep anomaly detection by training anomaly detectors against an auxiliary dataset of outliers, an approach we call Outlier Exposure (OE). This enables anomaly detectors to generalize and detect unseen anomalies. In extensive experiments on natural language processing and small- and large-scale vision tasks, we find that Outlier Exposure significantly improves detection performance. We also observe that cutting-edge generative models trained on CIFAR-10 may assign higher likelihoods to SVHN images than to CIFAR-10 images; we use OE to mitigate this issue. We also analyze the flexibility and robustness of Outlier Exposure, and identify characteristics of the auxiliary dataset that improve performance.

Recently, graph neural networks (GNNs) have revolutionized the field of graph representation learning through effectively learned node embeddings, and achieved state-of-the-art results in tasks such as node classification and link prediction. However, current GNN methods are inherently flat and do not learn hierarchical representations of graphs---a limitation that is especially problematic for the task of graph classification, where the goal is to predict the label associated with an entire graph. Here we propose DiffPool, a differentiable graph pooling module that can generate hierarchical representations of graphs and can be combined with various graph neural network architectures in an end-to-end fashion. DiffPool learns a differentiable soft cluster assignment for nodes at each layer of a deep GNN, mapping nodes to a set of clusters, which then form the coarsened input for the next GNN layer. Our experimental results show that combining existing GNN methods with DiffPool yields an average improvement of 5-10% accuracy on graph classification benchmarks, compared to all existing pooling approaches, achieving a new state-of-the-art on four out of five benchmark data sets.

We propose a new method for event extraction (EE) task based on an imitation learning framework, specifically, inverse reinforcement learning (IRL) via generative adversarial network (GAN). The GAN estimates proper rewards according to the difference between the actions committed by the expert (or ground truth) and the agent among complicated states in the environment. EE task benefits from these dynamic rewards because instances and labels yield to various extents of difficulty and the gains are expected to be diverse -- e.g., an ambiguous but correctly detected trigger or argument should receive high gains -- while the traditional RL models usually neglect such differences and pay equal attention on all instances. Moreover, our experiments also demonstrate that the proposed framework outperforms state-of-the-art methods, without explicit feature engineering.