Regression on function spaces is typically limited to models with Gaussian process priors. We introduce the notion of universal functional regression, in which we aim to learn a prior distribution over non-Gaussian function spaces that remains mathematically tractable for functional regression. To do this, we develop Neural Operator Flows (OpFlow), an infinite-dimensional extension of normalizing flows. OpFlow is an invertible operator that maps the (potentially unknown) data function space into a Gaussian process, allowing for exact likelihood estimation of functional point evaluations. OpFlow enables robust and accurate uncertainty quantification via drawing posterior samples of the Gaussian process and subsequently mapping them into the data function space. We empirically study the performance of OpFlow on regression and generation tasks with data generated from Gaussian processes with known posterior forms and non-Gaussian processes, as well as real-world earthquake seismograms with an unknown closed-form distribution.
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Scoring rules are an established way of comparing predictive performances across model classes. In the context of survival analysis, they require adaptation in order to accommodate censoring. This work investigates using scoring rules for model training rather than evaluation. Doing so, we establish a general framework for training survival models that is model agnostic and can learn event time distributions parametrically or non-parametrically. In addition, our framework is not restricted to any specific scoring rule. While we focus on neural network-based implementations, we also provide proof-of-concept implementations using gradient boosting, generalized additive models, and trees. Empirical comparisons on synthetic and real-world data indicate that scoring rules can be successfully incorporated into model training and yield competitive predictive performance with established time-to-event models.
Federated Graph Learning (FGL) is tasked with training machine learning models, such as Graph Neural Networks (GNNs), for multiple clients, each with its own graph data. Existing methods usually assume that each client has both node features and graph structure of its graph data. In real-world scenarios, however, there exist federated systems where only a part of the clients have such data while other clients (i.e. graphless clients) may only have node features. This naturally leads to a novel problem in FGL: how to jointly train a model over distributed graph data with graphless clients? In this paper, we propose a novel framework FedGLS to tackle the problem in FGL with graphless clients. In FedGLS, we devise a local graph learner on each graphless client which learns the local graph structure with the structure knowledge transferred from other clients. To enable structure knowledge transfer, we design a GNN model and a feature encoder on each client. During local training, the feature encoder retains the local graph structure knowledge together with the GNN model via knowledge distillation, and the structure knowledge is transferred among clients in global update. Our extensive experiments demonstrate the superiority of the proposed FedGLS over five baselines.
The combination of the Bayesian game and learning has a rich history, with the idea of controlling a single agent in a system composed of multiple agents with unknown behaviors given a set of types, each specifying a possible behavior for the other agents. The idea is to plan an agent's own actions with respect to those types which it believes are most likely to maximize the payoff. However, the type beliefs are often learned from past actions and likely to be incorrect. With this perspective in mind, we consider an agent in a game with type predictions of other components, and investigate the impact of incorrect beliefs to the agent's payoff. In particular, we formally define a tradeoff between risk and opportunity by comparing the payoff obtained against the optimal payoff, which is represented by a gap caused by trusting or distrusting the learned beliefs. Our main results characterize the tradeoff by establishing upper and lower bounds on the Pareto front for both normal-form and stochastic Bayesian games, with numerical results provided.
We introduce a continuous domain for function spaces over topological spaces which are not core-compact. Notable examples of such topological spaces include the real line with the upper limit topology, which is used in solution of initial value problems with temporal discretization, and various infinite dimensional Banach spaces which are ubiquitous in functional analysis and solution of partial differential equations. If a topological space $\mathbb{X}$ is not core-compact and $\mathbb{D}$ is a non-singleton bounded-complete domain, the function space $[\mathbb{X} \to \mathbb{D}]$ is not a continuous domain. To construct a continuous domain, we consider a spectral compactification $\mathbb{Y}$ of $\mathbb{X}$ and relate $[\mathbb{X} \to \mathbb{D}]$ with the continuous domain $[\mathbb{Y} \to \mathbb{D}]$ via a Galois connection. This allows us to perform computations in the native structure $[\mathbb{X} \to \mathbb{D}]$ while computable analysis is performed in the continuous domain $[\mathbb{Y} \to \mathbb{D}]$, with the left and right adjoints used for moving between the two function spaces.
Counting logics with a bounded number of variables form one of the central concepts in descriptive complexity theory. Although they restrict the number of variables that a formula can contain, the variables can be nested within scopes of quantified occurrences of themselves. In other words, the variables can be requantified. We study the fragments obtained from counting logics by restricting requantification for some but not necessarily all the variables. Similar to the logics without limitation on requantification, we develop tools to investigate the restricted variants. Specifically, we introduce a bijective pebble game in which certain pebbles can only be placed once and for all, and a corresponding two-parametric family of Weisfeiler-Leman algorithms. We show close correspondences between the three concepts. By using a suitable cops-and-robber game and adaptations of the Cai-F\"urer-Immerman construction, we completely clarify the relative expressive power of the new logics. We show that the restriction of requantification has beneficial algorithmic implications in terms of graph identification. Indeed, we argue that with regard to space complexity, non-requantifiable variables only incur an additive polynomial factor when testing for equivalence. In contrast, for all we know, requantifiable variables incur a multiplicative linear factor. Finally, we observe that graphs of bounded tree-depth and 3-connected planar graphs can be identified using no, respectively, only a very limited number of requantifiable variables.
Hysteresis modeling is crucial to comprehend the behavior of magnetic devices, facilitating optimal designs. Hitherto, deep learning-based methods employed to model hysteresis, face challenges in generalizing to novel input magnetic fields. This paper addresses the generalization challenge by proposing neural operators for modeling constitutive laws that exhibit magnetic hysteresis by learning a mapping between magnetic fields. In particular, three neural operators-deep operator network, Fourier neural operator, and wavelet neural operator-are employed to predict novel first-order reversal curves and minor loops, where novel means they are not used to train the model. In addition, a rate-independent Fourier neural operator is proposed to predict material responses at sampling rates different from those used during training to incorporate the rate-independent characteristics of magnetic hysteresis. The presented numerical experiments demonstrate that neural operators efficiently model magnetic hysteresis, outperforming the traditional neural recurrent methods on various metrics and generalizing to novel magnetic fields. The findings emphasize the advantages of using neural operators for modeling hysteresis under varying magnetic conditions, underscoring their importance in characterizing magnetic material based devices. The codes related to this paper are at github.com/chandratue/magnetic_hysteresis_neural_operator.
Markov chains are the de facto finite-state model for stochastic dynamical systems, and Markov decision processes (MDPs) extend Markov chains by incorporating non-deterministic behaviors. Given an MDP and rewards on states, a classical optimization criterion is the maximal expected total reward where the MDP stops after T steps, which can be computed by a simple dynamic programming algorithm. We consider a natural generalization of the problem where the stopping times can be chosen according to a probability distribution, such that the expected stopping time is T, to optimize the expected total reward. Quite surprisingly we establish inter-reducibility of the expected stopping-time problem for Markov chains with the Positivity problem (which is related to the well-known Skolem problem), for which establishing either decidability or undecidability would be a major breakthrough. Given the hardness of the exact problem, we consider the approximate version of the problem: we show that it can be solved in exponential time for Markov chains and in exponential space for MDPs.
Accurately modeling the correlation structure of errors is critical for reliable uncertainty quantification in probabilistic time series forecasting. While recent deep learning models for multivariate time series have developed efficient parameterizations for time-varying contemporaneous covariance, but they often assume temporal independence of errors for simplicity. However, real-world data often exhibit significant error autocorrelation and cross-lag correlation due to factors such as missing covariates. In this paper, we introduce a plug-and-play method that learns the covariance structure of errors over multiple steps for autoregressive models with Gaussian-distributed errors. To ensure scalable inference and computational efficiency, we model the contemporaneous covariance using a low-rank-plus-diagonal parameterization and capture cross-covariance through a group of independent latent temporal processes. The learned covariance matrix is then used to calibrate predictions based on observed residuals. We evaluate our method on probabilistic models built on RNNs and Transformer architectures, and the results confirm the effectiveness of our approach in improving predictive accuracy and uncertainty quantification without significantly increasing the parameter size.
In evolutionary computation, it is commonly assumed that a search algorithm acquires knowledge about a problem instance by sampling solutions from the search space and evaluating them with a fitness function. This is necessarily inefficient because fitness reveals very little about solutions -- yet they contain more information that can be potentially exploited. To address this observation in genetic programming, we propose EvoNUDGE, which uses a graph neural network to elicit additional knowledge from symbolic regression problems. The network is queried on the problem before an evolutionary run to produce a library of subprograms, which is subsequently used to seed the initial population and bias the actions of search operators. In an extensive experiment on a large number of problem instances, EvoNUDGE is shown to significantly outperform multiple baselines, including the conventional tree-based genetic programming and the purely neural variant of the method.
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.
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