Base Models for Parabolic Partial Differential Equations

Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often necessary to compute the solutions or a function of the solutions to a parametric PDE in multiple scenarios corresponding to different parameters of this PDE. This process often requires resolving the PDEs from scratch, which is time-consuming. To better employ existing simulations for the PDEs, we propose a framework for finding solutions to parabolic PDEs across different scenarios by meta-learning an underlying base distribution. We build upon this base distribution to propose a method for computing solutions to parametric PDEs under different parameter settings. Finally, we illustrate the application of the proposed methods through extensive experiments in generative modeling, stochastic control, and finance. The empirical results suggest that the proposed approach improves generalization to solving PDEs under new parameter regimes.

Robust Score-Based Quickest Change Detection

Methods in the field of quickest change detection rapidly detect in real-time a change in the data-generating distribution of an online data stream. Existing methods have been able to detect this change point when the densities of the pre- and post-change distributions are known. Recent work has extended these results to the case where the pre- and post-change distributions are known only by their score functions. This work considers the case where the pre- and post-change score functions are known only to correspond to distributions in two disjoint sets. This work employs a pair of "least-favorable" distributions to robustify the existing score-based quickest change detection algorithm, the properties of which are studied. This paper calculates the least-favorable distributions for specific model classes and provides methods of estimating the least-favorable distributions for common constructions. Simulation results are provided demonstrating the performance of our robust change detection algorithm.

RASPNet: A Benchmark Dataset for Radar Adaptive Signal Processing Applications

This work presents a large-scale dataset for radar adaptive signal processing (RASP) applications, aimed at supporting the development of data-driven models within the radar community. The dataset, called RASPNet, consists of 100 realistic scenarios compiled over a variety of topographies and land types from across the contiguous United States, designed to reflect a diverse array of real-world environments. Within each scenario, RASPNet consists of 10,000 clutter realizations from an airborne radar setting, which can be utilized for radar algorithm development and evaluation. RASPNet intends to fill a prominent gap in the availability of a large-scale, realistic dataset that standardizes the evaluation of adaptive radar processing techniques. We describe its construction, organization, and several potential applications, which includes a transfer learning example to demonstrate how RASPNet can be leveraged for realistic adaptive radar processing scenarios.

ColA: Collaborative Adaptation with Gradient Learning

A primary function of back-propagation is to compute both the gradient of hidden representations and parameters for optimization with gradient descent. Training large models requires high computational costs due to their vast parameter sizes. While Parameter-Efficient Fine-Tuning (PEFT) methods aim to train smaller auxiliary models to save computational space, they still present computational overheads, especially in Fine-Tuning as a Service (FTaaS) for numerous users. We introduce Collaborative Adaptation (ColA) with Gradient Learning (GL), a parameter-free, model-agnostic fine-tuning approach that decouples the computation of the gradient of hidden representations and parameters. In comparison to PEFT methods, ColA facilitates more cost-effective FTaaS by offloading the computation of the gradient to low-cost devices. We also provide a theoretical analysis of ColA and experimentally demonstrate that ColA can perform on par or better than existing PEFT methods on various benchmarks.

Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion Processes

McKean-Vlasov stochastic differential equations (MV-SDEs) provide a mathematical description of the behavior of an infinite number of interacting particles by imposing a dependence on the particle density. As such, we study the influence of explicitly including distributional information in the parameterization of the SDE. We propose a series of semi-parametric methods for representing MV-SDEs, and corresponding estimators for inferring parameters from data based on the properties of the MV-SDE. We analyze the characteristics of the different architectures and estimators, and consider their applicability in relevant machine learning problems. We empirically compare the performance of the different architectures and estimators on real and synthetic datasets for time series and probabilistic modeling. The results suggest that explicitly including distributional dependence in the parameterization of the SDE is effective in modeling temporal data with interaction under an exchangeability assumption while maintaining strong performance for standard It\^o-SDEs due to the richer class of probability flows associated with MV-SDEs.

Large Deviation Analysis of Score-based Hypothesis Testing

Score-based statistical models play an important role in modern machine learning, statistics, and signal processing. For hypothesis testing, a score-based hypothesis test is proposed in \cite{wu2022score}. We analyze the performance of this score-based hypothesis testing procedure and derive upper bounds on the probabilities of its Type I and II errors. We prove that the exponents of our error bounds are asymptotically (in the number of samples) tight for the case of simple null and alternative hypotheses. We calculate these error exponents explicitly in specific cases and provide numerical studies for various other scenarios of interest.

Data-Driven Target Localization: Benchmarking Gradient Descent Using the Cramér-Rao Bound

In modern radar systems, precise target localization using azimuth and velocity estimation is paramount. Traditional unbiased estimation methods have leveraged gradient descent algorithms to reach the theoretical limits of the Cram\'er Rao Bound (CRB) for the error of the parameter estimates. In this study, we present a data-driven neural network approach that outperforms these traditional techniques, demonstrating improved accuracies in target azimuth and velocity estimation. Using a representative simulated scenario, we show that our proposed neural network model consistently achieves improved parameter estimates due to its inherently biased nature, yielding a diminished mean squared error (MSE). Our findings underscore the potential of employing deep learning methods in radar systems, paving the way for more accurate localization in cluttered and dynamic environments.

Random Linear Projections Loss for Hyperplane-Based Optimization in Regression Neural Networks

Despite their popularity across a wide range of domains, regression neural networks are prone to overfitting complex datasets. In this work, we propose a loss function termed Random Linear Projections (RLP) loss, which is empirically shown to mitigate overfitting. With RLP loss, the distance between sets of hyperplanes connecting fixed-size subsets of the neural network's feature-prediction pairs and feature-label pairs is minimized. The intuition behind this loss derives from the notion that if two functions share the same hyperplanes connecting all subsets of feature-label pairs, then these functions must necessarily be equivalent. Our empirical studies, conducted across benchmark datasets and representative synthetic examples, demonstrate the improvements of the proposed RLP loss over mean squared error (MSE). Specifically, neural networks trained with the RLP loss achieve better performance while requiring fewer data samples and are more robust to additive noise. We provide theoretical analysis supporting our empirical findings.

Counterfactual Data Augmentation with Contrastive Learning

Statistical disparity between distinct treatment groups is one of the most significant challenges for estimating Conditional Average Treatment Effects (CATE). To address this, we introduce a model-agnostic data augmentation method that imputes the counterfactual outcomes for a selected subset of individuals. Specifically, we utilize contrastive learning to learn a representation space and a similarity measure such that in the learned representation space close individuals identified by the learned similarity measure have similar potential outcomes. This property ensures reliable imputation of counterfactual outcomes for the individuals with close neighbors from the alternative treatment group. By augmenting the original dataset with these reliable imputations, we can effectively reduce the discrepancy between different treatment groups, while inducing minimal imputation error. The augmented dataset is subsequently employed to train CATE estimation models. Theoretical analysis and experimental studies on synthetic and semi-synthetic benchmarks demonstrate that our method achieves significant improvements in both performance and robustness to overfitting across state-of-the-art models.

Off-Policy Evaluation for Human Feedback

Off-policy evaluation (OPE) is important for closing the gap between offline training and evaluation of reinforcement learning (RL), by estimating performance and/or rank of target (evaluation) policies using offline trajectories only. It can improve the safety and efficiency of data collection and policy testing procedures in situations where online deployments are expensive, such as healthcare. However, existing OPE methods fall short in estimating human feedback (HF) signals, as HF may be conditioned over multiple underlying factors and is only sparsely available; as opposed to the agent-defined environmental rewards (used in policy optimization), which are usually determined over parametric functions or distributions. Consequently, the nature of HF signals makes extrapolating accurate OPE estimations to be challenging. To resolve this, we introduce an OPE for HF (OPEHF) framework that revives existing OPE methods in order to accurately evaluate the HF signals. Specifically, we develop an immediate human reward (IHR) reconstruction approach, regularized by environmental knowledge distilled in a latent space that captures the underlying dynamics of state transitions as well as issuing HF signals. Our approach has been tested over two real-world experiments, adaptive in-vivo neurostimulation and intelligent tutoring, as well as in a simulation environment (visual Q&A). Results show that our approach significantly improves the performance toward estimating HF signals accurately, compared to directly applying (variants of) existing OPE methods.