Gaussian Processes

We study the Gaussian process (GP) bandit problem, whose goal is to minimize regret under an unknown reward function lying in some reproducing kernel Hilbert space (RKHS). The maximum posterior variance analysis is vital in analyzing near-optimal GP bandit algorithms such as maximum variance reduction (MVR) and phased elimination (PE). Therefore, we first show the new upper bound of the maximum posterior variance, which improves the dependence of the noise variance parameters of the GP. By leveraging this result, we refine the MVR and PE to obtain (i) a nearly optimal regret upper bound in the noiseless setting and (ii) regret upper bounds that are optimal with respect to the RKHS norm of the reward function. Furthermore, as another application of our proposed bound, we analyze the GP bandit under the time-varying noise variance setting, which is the kernelized extension of the linear bandit with heteroscedastic noise. For this problem, we show that MVR and PE-based algorithms achieve noise variance-dependent regret upper bounds, which matches our regret lower bound.

Deep learning-based classifiers are known to be vulnerable to adversarial attacks. Existing methods for defending against such attacks require adding a defense mechanism or modifying the learning procedure (e.g., by adding adversarial examples). This paper shows that for certain data distributions one can learn a provably robust classifier using standard learning methods and without adding a defense mechanism. More specifically, this paper addresses the problem of finding a robust classifier for a binary classification problem in which the data comes from an isotropic mixture of Gaussians with orthonormal cluster centers. First, we characterize the largest $\ell_2$-attack any classifier can defend against while maintaining high accuracy, and show the existence of optimal robust classifiers achieving this maximum $\ell_2$-robustness. Next, we show that given data from the orthonormal Gaussian mixture model, gradient flow on a two-layer network with a polynomial ReLU activation and without adversarial examples provably finds an optimal robust classifier.

We consider the problem of clustering grouped data for which the observations may include group-specific variables in addition to the variables that are shared across groups. This type of data is quite common; for example, in cancer genomic studies, molecular information is available for all cancers whereas cancer-specific clinical information may only be available for certain cancers. Existing grouped clustering methods only consider the shared variables but ignore valuable information from the group-specific variables. To allow for these group-specific variables to aid in the clustering, we propose a novel Bayesian nonparametric approach, termed global-local (GLocal) Dirichlet process, that models the "global-local" structure of the observations across groups. We characterize the GLocal Dirichlet process using the stick-breaking representation and the representation as a limit of a finite mixture model. We theoretically quantify the approximation errors of the truncated prior, the corresponding finite mixture model, and the associated posterior distribution. We develop a fast variational Bayes algorithm for scalable posterior inference, which we illustrate with extensive simulations and a TCGA pan-gastrointestinal cancer dataset.

In this work, we design quantum algorithms that are more efficient than classical algorithms to solve time-dependent and finite-horizon Markov Decision Processes (MDPs) in two distinct settings: (1) In the exact dynamics setting, where the agent has full knowledge of the environment's dynamics (i.e., transition probabilities), we prove that our **Quantum Value Iteration (QVI)** algorithm **QVI-1** achieves a quadratic speedup in the size of the action space $(A)$ compared with the classical value iteration algorithm for computing the optimal policy ($\pi^{\ast}$) and the optimal V-value function ($V_{0}^{\ast}$). Furthermore, our algorithm **QVI-2** provides an additional speedup in the size of the state space $(S)$ when obtaining near-optimal policies and V-value functions. Both **QVI-1** and **QVI-2** achieve quantum query complexities that provably improve upon classical lower bounds, particularly in their dependences on $S$ and $A$. (2) In the generative model setting, where samples from the environment are accessible in quantum superposition, we prove that our algorithms **QVI-3** and **QVI-4** achieve improvements in sample complexity over the state-of-the-art (SOTA) classical algorithm in terms of $A$, estimation error $(\epsilon)$, and time horizon $(H)$. More importantly, we prove quantum lower bounds to show that **QVI-3** and **QVI-4** are asymptotically optimal, up to logarithmic factors, assuming a constant time horizon.

Gaussian process regression (GPR) or kernel ridge regression is a widely used and powerful tool for nonlinear prediction. Therefore, active learning (AL) for GPR, which actively collects data labels to achieve an accurate prediction with fewer data labels, is an important problem. However, existing AL methods do not theoretically guarantee prediction accuracy for target distribution. Furthermore, as discussed in the distributionally robust learning literature, specifying the target distribution is often difficult. Thus, this paper proposes two AL methods that effectively reduce the worst-case expected error for GPR, which is the worst-case expectation in target distribution candidates. We show an upper bound of the worst-case expected squared error, which suggests that the error will be arbitrarily small by a finite number of data labels under mild conditions. Finally, we demonstrate the effectiveness of the proposed methods through synthetic and real-world datasets.

We introduce a novel kernel learning framework toward efficiently solving nonlinear partial differential equations (PDEs). In contrast to the state-of-the-art kernel solver that embeds differential operators within kernels, posing challenges with a large number of collocation points, our approach eliminates these operators from the kernel. We model the solution using a standard kernel interpolation form and differentiate the interpolant to compute the derivatives. Our framework obviates the need for complex Gram matrix construction between solutions and their derivatives, allowing for a straightforward implementation and scalable computation. As an instance, we allocate the collocation points on a grid and adopt a product kernel, which yields a Kronecker product structure in the interpolation. This structure enables us to avoid computing the full Gram matrix, reducing costs and scaling efficiently to a large number of collocation points. We provide a proof of the convergence and rate analysis of our method under appropriate regularity assumptions. In numerical experiments, we demonstrate the advantages of our method in solving several benchmark PDEs.

Flow matching (FM) is a family of training algorithms for fitting continuous normalizing flows (CNFs). Conditional flow matching (CFM) exploits the fact that the marginal vector field of a CNF can be learned by fitting least-squares regression to the conditional vector field specified given one or both ends of the flow path. In this paper, we extend the CFM algorithm by defining conditional probability paths along "streams'', instances of latent stochastic paths that connect data pairs of source and target, which are modeled with Gaussian process (GP) distributions. The unique distributional properties of GPs help preserve the ``simulation-free'' nature of CFM training. We show that this generalization of the CFM can effectively reduce the variance in the estimated marginal vector field at a moderate computational cost, thereby improving the quality of the generated samples under common metrics. Additionally, adopting the GP on the streams allows for flexibly linking multiple correlated training data points (e.g., time series). We empirically validate our claim through both simulations and applications to image and neural time series data.

Neural Processes (NPs) are deep probabilistic models that represent stochastic processes by conditioning their prior distributions on a set of context points. Despite their advantages in uncertainty estimation for complex distributions, NPs enforce parameterization coupling between the conditional prior model and the posterior model. We show that this coupling amounts to prior misspecification and revisit the NP objective to address this issue. More specifically, we propose Rényi Neural Processes (RNP), a method that replaces the standard KL divergence with the Rényi divergence, dampening the effects of the misspecified prior during posterior updates. We validate our approach across multiple benchmarks including regression and image inpainting tasks, and show significant performance improvements of RNPs in real-world problems. Our extensive experiments show consistently better log-likelihoods over state-of-the-art NP models.

Understanding and constructing brain communications that capture dynamic communications across multiple regions is fundamental to modern system neuroscience, yet current methods struggle to find time-varying region-level communications or scale to large neural datasets with long recording durations. We present a novel framework using Markovian Gaussian Processes to learn brain communications with time-varying temporal delays from multi-region neural recordings, named Adaptive Delay Model (ADM). Our method combines Gaussian Processes with State Space Models and employs parallel scan inference algorithms, enabling efficient scaling to large datasets while identifying concurrent communication patterns that evolve over time. This time-varying approach captures how brain region interactions shift dynamically during cognitive processes. Validated on synthetic and multi-region neural recordings datasets, our approach discovers both the directionality and temporal dynamics of neural communication. This work advances our understanding of distributed neural computation and provides a scalable tool for analyzing dynamic brain networks. Code is available at https://github.com/BRAINML-GT/Adaptive-Delay-Model.

Correlation analysis is a fundamental step in uncovering meaningful insights from complex datasets. In this paper, we study the problem of detecting correlations between two random graphs following the Gaussian Wigner model with unlabeled vertices. Specifically, the task is formulated as a hypothesis testing problem: under the null hypothesis, the two graphs are independent, while under the alternative hypothesis, they are edge-correlated through a latent vertex permutation, yet maintain the same marginal distributions as under the null. We focus on the scenario where two induced subgraphs, each with a fixed number of vertices, are sampled. We determine the optimal rate for the sample size required for correlation detection, derived through an analysis of the conditional second moment. Additionally, we propose an efficient approximate algorithm that significantly reduces running time.

Calculating or accurately estimating log-determinants of large positive semi-definite matrices is of fundamental importance in many machine learning tasks. While its cubic computational complexity can already be prohibitive, in modern applications even storing the matrices themselves can pose a memory bottleneck. To address this, we derive a novel hierarchical algorithm based on block-wise computation of the LDL decomposition for large-scale log-determinant calculation in memory-constrained settings. In extreme cases where matrices are highly ill-conditioned, accurately computing the full matrix itself may be infeasible. This is particularly relevant when considering kernel matrices at scale, including the empirical Neural Tangent Kernel (NTK) of neural networks trained on large datasets. Under the assumption of neural scaling laws in the test error, we show that the ratio of pseudo-determinants satisfies a power-law relationship, enabling the derivation of corresponding scaling laws. This allows for accurate estimation of NTK log-determinants from a tiny fraction of the full dataset; in our experiments, this results in a $\sim$100,000$\times$ speedup with improved accuracy to other state-of-the-art approaches. Using these techniques, we successfully estimate log-determinants for dense matrices of extreme sizes, which were previously deemed intractable and inaccessible due to their enormous scale and computational requirements.

Deep neural networks learn structured features from complex, non-Gaussian inputs, but the mechanisms behind this process remain poorly understood. Our work is motivated by the observation that the first-layer filters learnt by deep convolutional neural networks from natural images resemble those learnt by independent component analysis (ICA), a simple unsupervised method that seeks the most non-Gaussian projections of its inputs. This similarity suggests that ICA provides a simple, yet principled model for studying feature learning. Here, we leverage this connection to investigate the interplay between data structure and optimisation in feature learning for the most popular ICA algorithm, FastICA, and stochastic gradient descent (SGD), which is used to train deep networks. We rigorously establish that FastICA requires at least $n\gtrsim d^4$ samples to recover a single non-Gaussian direction from $d$-dimensional inputs on a simple synthetic data model. We show that vanilla online SGD outperforms FastICA, and prove that the optimal sample complexity $n\gtrsim d^2$ can be reached by smoothing the loss, albeit in a data-dependent way. We finally demonstrate the existence of a search phase for FastICA on ImageNet, and discuss how the strong non-Gaussianity of said images compensates for the poor sample complexity of FastICA.

Although transformers have demonstrated impressive capabilities for in-context learning (ICL) in practice, theoretical understanding of the underlying mechanism that allows transformers to perform ICL is still in its infancy. This work aims to theoretically study the training dynamics of transformers for in-context classification tasks. We demonstrate that, for in-context classification of Gaussian mixtures under certain assumptions, a single-layer transformer trained via gradient descent converges to a globally optimal model at a linear rate. We further quantify the impact of the training and testing prompt lengths on the ICL inference error of the trained transformer. We show that when the lengths of training and testing prompts are sufficiently large, the prediction of the trained transformer approaches the ground truth distribution of the labels. Experimental results corroborate the theoretical findings.

The goal in thinning is to summarize a dataset using a small set of representative points. Remarkably, sub-Gaussian thinning algorithms like Kernel Halving and Compress can match the quality of uniform subsampling while substantially reducing the number of summary points. However, existing guarantees cover only a restricted range of distributions and kernel-based quality measures and suffer from pessimistic dimension dependence. To address these deficiencies, we introduce a new low-rank analysis of sub-Gaussian thinning that applies to any distribution and any kernel, guaranteeing high-quality compression whenever the kernel or data matrix is approximately low-rank. To demonstrate the broad applicability of the techniques, we design practical sub-Gaussian thinning approaches that improve upon the best known guarantees for approximating attention in transformers, accelerating stochastic gradient training through reordering, and distinguishing distributions in near-linear time.

Data imbalance remains a fundamental challenge in real-world machine learning. However, most existing work has focused on classification, leaving imbalanced regression underexplored despite its importance in many applications. To address this gap, we propose PRIME, a framework that leverages learnable proxies to construct a balanced and well-ordered feature space for imbalanced regression. At its core, PRIME arranges proxies to be uniformly distributed in the feature space while preserving the ordinal structure of regression targets, and then aligns each sample feature to its corresponding proxy. By using proxies as reference points, PRIME induces the desired structure of learned representations, promoting better generalization, especially in underrepresented target regions. Moreover, since proxy-based alignment resembles classification, PRIME enables the seamless application of class imbalance techniques to regression, facilitating more balanced feature learning. Extensive experiments demonstrate the effectiveness and broad applicability of PRIME, achieving state-of-the-art performance on four real-world regression benchmark datasets across diverse target domains.

We study the incorporation of equivariances into vector-valued GPs and more general classes of random field models. While kernels guaranteeing equivariances have been investigated previously, their evaluation is often computationally prohibitive due to required integrations over the involved groups. In this work, we provide a kernel characterization of stochastic equivariance for centred second-order vector-valued random fields and we construct integration-free equivariant kernels based on the notion of fundamental regions of group actions. We establish data-efficient and computationally lightweight GP models for velocity fields and molecular electric dipole moments and demonstrate that proposed integration-free kernels may also be leveraged to extract equivariant components from data.

We study the problem of learning the topology of a directed Gaussian Graphical Model under the equal-variance assumption, where the graph has $n$ nodes and maximum in-degree $d$. Prior work has established that $O(d \log n)$ samples are sufficient for this task. However, an important factor that is often overlooked in these analyses is the dependence on the condition number of the covariance matrix of the model. Indeed, all algorithms from prior work require a number of samples that grows polynomially with this condition number. In many cases this is unsatisfactory, since the condition number could grow polynomially with $n$, rendering these prior approaches impractical in high-dimensional settings. In this work, we provide an algorithm that recovers the underlying graph and prove that the number of samples required is independent of the condition number. Furthermore, we establish lower bounds that nearly match the upper bound up to a $d$-factor, thus providing an almost tight characterization of the true sample complexity of the problem. Moreover, under a further assumption that all the variances of the variables are bounded, we design a polynomial-time algorithm that recovers the underlying graph, at the cost of an additional polynomial dependence of the sample complexity on $d$. We complement our theoretical findings with simulations on synthetic datasets that confirm our predictions.

We consider the problem of *exact* recovery of a $k$-sparse binary vector from generalized linear measurements (such as *logistic regression*). We analyze the *linear estimation* algorithm (Plan, Vershynin, Yudovina, 2017), and also show information theoretic lower bounds on the number of required measurements. As a consequence of our results, for noisy one bit quantized linear measurements ($\mathsf{1bCS}$), we obtain a sample complexity of $O((k+\sigma^2)\log{n})$, where $\sigma^2$ is the noise variance. This is shown to be optimal due to the information theoretic lower bound. We also obtain tight sample complexity characterization for logistic regression. Since $\mathsf{1bCS}$ is a strictly harder problem than noisy linear measurements ($\mathsf{SparseLinearReg}$) because of added quantization, the same sample complexity is achievable for $\mathsf{SparseLinearReg}$. While this sample complexity can be obtained via the popular lasso algorithm, linear estimation is computationally more efficient. Our lower bound holds for any set of measurements for $\mathsf{SparseLinearReg}$ (similar bound was known for Gaussian measurement matrices) and is closely matched by the maximum-likelihood upper bound. For $\mathsf{SparseLinearReg}$, it was conjectured in Gamarnik and Zadik, 2017 that there is a statistical-computational gap and the number of measurements should be at least $(2k+\sigma^2)\log{n}$ for efficient algorithms to exist. It is worth noting that our results imply that there is no such statistical-computational gap for $\mathsf{1bCS}$ and logistic regression.

Nonparametric estimation using uniform-width binning is a standard approach for evaluating the calibration performance of machine learning models. However, existing theoretical analyses of the bias induced by binning are limited to binary classification, creating a significant gap with practical applications such as multiclass classification. Additionally, many parametric recalibration algorithms lack theoretical guarantees for their generalization performance. To address these issues, we conduct a generalization analysis of calibration error using the probably approximately correct Bayes framework. This approach enables us to derive the first optimizable upper bound for generalization error in the calibration context. On the basis of our theory, we propose a generalization-aware recalibration algorithm. Numerical experiments show that our algorithm enhances the performance of Gaussian process-based recalibration across various benchmark datasets and models.

High-dimensional time-series datasets are common in domains such as healthcare and economics. Variational autoencoder (VAE) models, where latent variables are modeled with a Gaussian process (GP) prior, have become a prominent model class to analyze such correlated datasets. However, their applications are challenged by the inherent cubic time complexity that requires specific GP approximation techniques, as well as the general challenge of modeling both shared and individual-specific correlations across time. Though inducing points enhance GP prior VAE scalability, optimizing them remains challenging, especially since discrete covariates resist gradient‑based methods. In this work, we propose a scalable basis function approximation technique for GP prior VAEs that mitigates these challenges and results in linear time complexity, with a global parametrization that eliminates the need for amortized variational inference and the associated amortization gap, making it well-suited for conditional generation tasks where accuracy and efficiency are crucial. Empirical evaluations on synthetic and real-world benchmark datasets demonstrate that our approach not only improves scalability and interpretability but also drastically enhances predictive performance.