Multi-fidelity surrogate modeling aims to learn an accurate surrogate at the highest fidelity level by combining data from multiple sources. Traditional methods relying on Gaussian processes can hardly scale to high-dimensional data. Deep learning approaches utilize neural network based encoders and decoders to improve scalability. These approaches share encoded representations across fidelities without including corresponding decoder parameters. At the highest fidelity, the representations are decoded with different parameters, making the shared information inherently inaccurate. This hinders inference performance, especially in out-of-distribution scenarios when the highest fidelity data has limited domain coverage. To address these limitations, we propose Multi-fidelity Residual Neural Processes (MFRNP), a novel multi-fidelity surrogate modeling framework. MFRNP optimizes lower fidelity decoders for accurate information sharing by aggregating lower fidelity surrogate outputs and models residual between the aggregation and ground truth on the highest fidelity. We show that MFRNP significantly outperforms current state-of-the-art in learning partial differential equations and a real-world climate modeling task.
Addressing real-world optimization problems becomes particularly challenging when analytic objective functions or constraints are unavailable. While numerous studies have addressed the issue of unknown objectives, limited research has focused on scenarios where feasibility constraints are not given explicitly. Overlooking these constraints can lead to spurious solutions that are unrealistic in practice. To deal with such unknown constraints, we propose to perform optimization within the data manifold using diffusion models. To constrain the optimization process to the data manifold, we reformulate the original optimization problem as a sampling problem from the product of the Boltzmann distribution defined by the objective function and the data distribution learned by the diffusion model. To enhance sampling efficiency, we propose a two-stage framework that begins with a guided diffusion process for warm-up, followed by a Langevin dynamics stage for further correction. Theoretical analysis shows that the initial stage results in a distribution focused on feasible solutions, thereby providing a better initialization for the later stage. Comprehensive experiments on a synthetic dataset, six real-world black-box optimization datasets, and a multi-objective optimization dataset show that our method achieves better or comparable performance with previous state-of-the-art baselines.
We address the problem of learning Granger causality from asynchronous, interdependent, multi-type event sequences. In particular, we are interested in discovering instance-level causal structures in an unsupervised manner. Instance-level causality identifies causal relationships among individual events, providing more fine-grained information for decision-making. Existing work in the literature either requires strong assumptions, such as linearity in the intensity function, or heuristically defined model parameters that do not necessarily meet the requirements of Granger causality. We propose Instance-wise Self-Attentive Hawkes Processes (ISAHP), a novel deep learning framework that can directly infer the Granger causality at the event instance level. ISAHP is the first neural point process model that meets the requirements of Granger causality. It leverages the self-attention mechanism of the transformer to align with the principles of Granger causality. We empirically demonstrate that ISAHP is capable of discovering complex instance-level causal structures that cannot be handled by classical models. We also show that ISAHP achieves state-of-the-art performance in proxy tasks involving type-level causal discovery and instance-level event type prediction.
Recent studies in reinforcement learning (RL) have made significant progress by leveraging function approximation to alleviate the sample complexity hurdle for better performance. Despite the success, existing provably efficient algorithms typically rely on the accessibility of immediate feedback upon taking actions. The failure to account for the impact of delay in observations can significantly degrade the performance of real-world systems due to the regret blow-up. In this work, we tackle the challenge of delayed feedback in RL with linear function approximation by employing posterior sampling, which has been shown to empirically outperform the popular UCB algorithms in a wide range of regimes. We first introduce Delayed-PSVI, an optimistic value-based algorithm that effectively explores the value function space via noise perturbation with posterior sampling. We provide the first analysis for posterior sampling algorithms with delayed feedback in RL and show our algorithm achieves $\widetilde{O}(\sqrt{d^3H^3 T} + d^2H^2 E[\tau])$ worst-case regret in the presence of unknown stochastic delays. Here $E[\tau]$ is the expected delay. To further improve its computational efficiency and to expand its applicability in high-dimensional RL problems, we incorporate a gradient-based approximate sampling scheme via Langevin dynamics for Delayed-LPSVI, which maintains the same order-optimal regret guarantee with $\widetilde{O}(dHK)$ computational cost. Empirical evaluations are performed to demonstrate the statistical and computational efficacy of our algorithms.
In fields such as finance, climate science, and neuroscience, inferring causal relationships from time series data poses a formidable challenge. While contemporary techniques can handle nonlinear relationships between variables and flexible noise distributions, they rely on the simplifying assumption that data originates from the same underlying causal model. In this work, we relax this assumption and perform causal discovery from time series data originating from mixtures of different causal models. We infer both the underlying structural causal models and the posterior probability for each sample belonging to a specific mixture component. Our approach employs an end-to-end training process that maximizes an evidence-lower bound for data likelihood. Through extensive experimentation on both synthetic and real-world datasets, we demonstrate that our method surpasses state-of-the-art benchmarks in causal discovery tasks, particularly when the data emanates from diverse underlying causal graphs. Theoretically, we prove the identifiability of such a model under some mild assumptions.
In multi-objective optimization, a single decision vector must balance the trade-offs between many objectives. Solutions achieving an optimal trade-off are said to be Pareto optimal: these are decision vectors for which improving any one objective must come at a cost to another. But as the set of Pareto optimal vectors can be very large, we further consider a more practically significant Pareto-constrained optimization problem, where the goal is to optimize a preference function constrained to the Pareto set. We investigate local methods for solving this constrained optimization problem, which poses significant challenges because the constraint set is (i) implicitly defined, and (ii) generally non-convex and non-smooth, even when the objectives are. We define notions of optimality and stationarity, and provide an algorithm with a last-iterate convergence rate of $O(K^{-1/2})$ to stationarity when the objectives are strongly convex and Lipschitz smooth.
Thompson sampling (TS) is widely used in sequential decision making due to its ease of use and appealing empirical performance. However, many existing analytical and empirical results for TS rely on restrictive assumptions on reward distributions, such as belonging to conjugate families, which limits their applicability in realistic scenarios. Moreover, sequential decision making problems are often carried out in a batched manner, either due to the inherent nature of the problem or to serve the purpose of reducing communication and computation costs. In this work, we jointly study these problems in two popular settings, namely, stochastic multi-armed bandits (MABs) and infinite-horizon reinforcement learning (RL), where TS is used to learn the unknown reward distributions and transition dynamics, respectively. We propose batched $\textit{Langevin Thompson Sampling}$ algorithms that leverage MCMC methods to sample from approximate posteriors with only logarithmic communication costs in terms of batches. Our algorithms are computationally efficient and maintain the same order-optimal regret guarantees of $\mathcal{O}(\log T)$ for stochastic MABs, and $\mathcal{O}(\sqrt{T})$ for RL. We complement our theoretical findings with experimental results.
Modern machine learning paradigms, such as deep learning, occur in or close to the interpolation regime, wherein the number of model parameters is much larger than the number of data samples. In this work, we propose a regularity condition within the interpolation regime which endows the stochastic gradient method with the same worst-case iteration complexity as the deterministic gradient method, while using only a single sampled gradient (or a minibatch) in each iteration. In contrast, all existing guarantees require the stochastic gradient method to take small steps, thereby resulting in a much slower linear rate of convergence. Finally, we demonstrate that our condition holds when training sufficiently wide feedforward neural networks with a linear output layer.
Variational inference has recently emerged as a popular alternative to the classical Markov chain Monte Carlo (MCMC) in large-scale Bayesian inference. The core idea of variational inference is to trade statistical accuracy for computational efficiency. It aims to approximate the posterior, reducing computation costs but potentially compromising its statistical accuracy. In this work, we study this statistical and computational trade-off in variational inference via a case study in inferential model selection. Focusing on Gaussian inferential models (a.k.a. variational approximating families) with diagonal plus low-rank precision matrices, we initiate a theoretical study of the trade-offs in two aspects, Bayesian posterior inference error and frequentist uncertainty quantification error. From the Bayesian posterior inference perspective, we characterize the error of the variational posterior relative to the exact posterior. We prove that, given a fixed computation budget, a lower-rank inferential model produces variational posteriors with a higher statistical approximation error, but a lower computational error; it reduces variances in stochastic optimization and, in turn, accelerates convergence. From the frequentist uncertainty quantification perspective, we consider the precision matrix of the variational posterior as an uncertainty estimate. We find that, relative to the true asymptotic precision, the variational approximation suffers from an additional statistical error originating from the sampling uncertainty of the data. Moreover, this statistical error becomes the dominant factor as the computation budget increases. As a consequence, for small datasets, the inferential model need not be full-rank to achieve optimal estimation error. We finally demonstrate these statistical and computational trade-offs inference across empirical studies, corroborating the theoretical findings.