Physics-informed neural networks (PINNs) have lately received great attention thanks to their flexibility in tackling a wide range of forward and inverse problems involving partial differential equations. However, despite their noticeable empirical success, little is known about how such constrained neural networks behave during their training via gradient descent. More importantly, even less is known about why such models sometimes fail to train at all. In this work, we aim to investigate these questions through the lens of the Neural Tangent Kernel (NTK); a kernel that captures the behavior of fully-connected neural networks in the infinite width limit during training via gradient descent. Specifically, we derive the NTK of PINNs and prove that, under appropriate conditions, it converges to a deterministic kernel that stays constant during training in the infinite-width limit. This allows us to analyze the training dynamics of PINNs through the lens of their limiting NTK and find a remarkable discrepancy in the convergence rate of the different loss components contributing to the total training error. To address this fundamental pathology, we propose a novel gradient descent algorithm that utilizes the eigenvalues of the NTK to adaptively calibrate the convergence rate of the total training error. Finally, we perform a series of numerical experiments to verify the correctness of our theory and the practical effectiveness of the proposed algorithms. The data and code accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/PINNsNTK}.

Deep Neural Networks (DNNs) with sparse input features have been widely used in recommender systems in industry. These models have large memory requirements and need a huge amount of training data. The large model size usually entails a cost, in the range of millions of dollars, for storage and communication with the inference services. In this paper, we propose a hybrid hashing method to combine frequency hashing and double hashing techniques for model size reduction, without compromising performance. We evaluate the proposed models on two product surfaces. In both cases, experiment results demonstrated that we can reduce the model size by around 90 % while keeping the performance on par with the original baselines.

Membership inference attacks are one of the simplest forms of privacy leakage for machine learning models: given a data point and model, determine whether the point was used to train the model. Existing membership inference attacks exploit models' abnormal confidence when queried on their training data. These attacks do not apply if the adversary only gets access to models' predicted labels, without a confidence measure. In this paper, we introduce label-only membership inference attacks. Instead of relying on confidence scores, our attacks evaluate the robustness of a model's predicted labels under perturbations to obtain a fine-grained membership signal. These perturbations include common data augmentations or adversarial examples. We empirically show that our label-only membership inference attacks perform on par with prior attacks that required access to model confidences. We further demonstrate that label-only attacks break multiple defenses against membership inference attacks that (implicitly or explicitly) rely on a phenomenon we call confidence masking. These defenses modify a model's confidence scores in order to thwart attacks, but leave the model's predicted labels unchanged. Our label-only attacks demonstrate that confidence-masking is not a viable defense strategy against membership inference. Finally, we investigate worst-case label-only attacks, that infer membership for a small number of outlier data points. We show that label-only attacks also match confidence-based attacks in this setting. We find that training models with differential privacy and (strong) L2 regularization are the only known defense strategies that successfully prevents all attacks. This remains true even when the differential privacy budget is too high to offer meaningful provable guarantees.

Stochastic Gradient Descent (SGD) and its variants are the most used algorithms in machine learning applications. In particular, SGD with adaptive learning rates and momentum is the industry standard to train deep networks. Despite the enormous success of these methods, our theoretical understanding of these variants in the nonconvex setting is not complete, with most of the results only proving convergence in expectation and with strong assumptions on the stochastic gradients. In this paper, we present a high probability analysis for adaptive and momentum algorithms, under weak assumptions on the function, stochastic gradients, and learning rates. We use it to prove for the first time the convergence of the gradients to zero in high probability in the smooth nonconvex setting for Delayed AdaGrad with momentum.

We propose a robust parameter estimation method for dynamical systems based on Statistical Learning techniques which aims to estimate a set of parameters that well fit the dynamics in order to obtain robust evidences about the qualitative behaviour of its trajectory. The method is quite general and flexible, since it dos not rely on any specific property of the dynamical system, and represents a mathematical formalisation of the procedure consisting of sampling and testing parameters, in which evolutions generated by candidate parameters are tested against observed data to assess goodness-of-fit. The Statistical Learning framework introduces a mathematically rigorous scheme to this general approach for parameter estimation, adding to the great field of parameter estimation in dynamical systems. The method is specially useful for estimating parameters in epidemiological compartmental models. We illustrate it in simulated and real data about COVID-19 spread in the US in order to assess qualitatively the peak of deaths by the disease.

This paper describes BUT-FIT's submission at SemEval-2020 Task 5: Modelling Causal Reasoning in Language: Detecting Counterfactuals. The challenge focused on detecting whether a given statement contains a counterfactual (Subtask 1) and extracting both antecedent and consequent parts of the counterfactual from the text (Subtask 2). We experimented with various state-of-the-art language representation models (LRMs). We found RoBERTa LRM to perform the best in both subtasks. We achieved the first place in both exact match and F1 for Subtask 2 and ranked second for Subtask 1.

Neural networks achieve outstanding accuracy in classification and regression tasks. However, understanding their behavior still remains an open challenge that requires questions to be addressed on the robustness, explainability and reliability of predictions. We answer these questions by computing reachable sets of neural networks, i.e. sets of outputs resulting from continuous sets of inputs. We provide two efficient approaches that lead to over- and under-approximations of the reachable set. This principle is highly versatile, as we show. First, we analyze and enhance the robustness properties of both classifiers and regression models. This is in contrast to existing works, which only handle classification. Specifically, we verify (non-)robustness, propose a robust training procedure, and show that our approach outperforms adversarial attacks as well as state-of-the-art methods of verifying classifiers for non-norm bound perturbations. We also provide a technique of distinguishing between reliable and non-reliable predictions for unlabeled inputs, quantify the influence of each feature on a prediction, and compute a feature ranking.

Most of the existing coastal flood Forecast and Early-Warning Systems do not model the flood, but instead, rely on the prediction of hydrodynamic conditions at the coast and on expert judgment. Recent scientific contributions are now capable to precisely model flood events, even in situations where wave overtopping plays a significant role. Such models are nevertheless costly-to-evaluate and surrogate ones need to be exploited for substantial computational savings. For the latter models, the hydro-meteorological forcing conditions (inputs) or flood events (outputs) are conveniently parametrised into scalar representations. However, they neglect the fact that inputs are actually functions (more precisely, time series), and that floods spatially propagate inland. Here, we introduce a multi-output Gaussian process model accounting for both criteria. On various examples, we test its versatility for both learning spatial maps and inferring unobserved ones. We demonstrate that efficient implementations are obtained by considering tensor-structured data and/or sparse-variational approximations. Finally, the proposed framework is applied on a coastal application aiming at predicting flood events. We conclude that accurate predictions are obtained in the order of minutes rather than the couples of days required by dedicated hydrodynamic simulators.

Stochastic gradient descent (SGD) and its variants have been the dominating optimization methods in machine learning. Compared with small batch training, SGD with large batch training can better utilize the computational power of current multi-core systems like GPUs and can reduce the number of communication rounds in distributed training. Hence, SGD with large batch training has attracted more and more attention. However, existing empirical results show that large batch training typically leads to a drop of generalization accuracy. As a result, large batch training has also become a challenging topic. In this paper, we propose a novel method, called stochastic normalized gradient descent with momentum (SNGM), for large batch training. We theoretically prove that compared to momentum SGD (MSGD) which is one of the most widely used variants of SGD, SNGM can adopt a larger batch size to converge to the $\epsilon$-stationary point with the same computation complexity (total number of gradient computation). Empirical results on deep learning also show that SNGM can achieve the state-of-the-art accuracy with a large batch size.

While modern large-scale datasets often consist of heterogeneous subpopulations---for example, multiple demographic groups or multiple text corpora---the standard practice of minimizing average loss fails to guarantee uniformly low losses across all subpopulations. We propose a convex procedure that controls the worst-case performance over all subpopulations of a given size. Our procedure comes with finite-sample (nonparametric) convergence guarantees on the worst-off subpopulation. Empirically, we observe on lexical similarity, wine quality, and recidivism prediction tasks that our worst-case procedure learns models that do well against unseen subpopulations.