New algorithms tackle RKHS bandits with reduced complexity and improved performance.
problem Adversarial and stochastic RKHS bandit problems with high computational complexity.
method Combining approximation theory with misspecified linear bandit methods.
result First general algorithm for adversarial RKHS bandit problem.
Improved GP bandit algorithms for noiseless, varying noise, and RKHS norms.
problem Minimizing regret in Gaussian process bandits with unknown reward functions.
method New upper bound on maximum posterior variance, refined MVR and PE algorithms.
result Optimal regret bounds for noiseless, varying noise, and RKHS norms.
New algorithm optimizes functions in Matérn kernel RKHS with noisy feedback.
problem Optimizing functions in RKHS of Matérn kernel with noisy bandit feedback.
method π-GP-UCB algorithm with guaranteed sublinear regret for all ν > 1 and d ≥ 1.
result First practical approach with guaranteed sublinear regret for all ν > 1 and d ≥ 1.
Kernel ε ε ε -Greedy optimizes multi-armed bandits with covariates for sub-linear regret.
problem Optimizing multi-armed bandits with covariates in a reproducing kernel Hilbert space.
method Online weighted kernel ridge regression estimator for mean reward function estimation.
result Achieves sub-linear regret rate and optimal T \sqrt{T} T regret rate under margin condition. Research aims to improve confidence intervals for RKHS elements in online learning.
problem Improper confidence intervals lead to suboptimal regret bounds in kernel-based bandit and reinforcement learning.
method Formalizes the open problem of online confidence intervals in RKHS and reviews existing results.
result Identifies the online nature of observation points as the main challenge for tight confidence intervals.
Optimal simple regret bound for Gaussian Process bandits.
problem Sequential optimization of expensive-to-evaluate functions.
method Proved a bound on simple regret for pure exploration algorithms.
result Order optimal bound on simple regret for Gaussian Process bandits.
Study on adaptivity to kernel regularity in bandit problems.
problem Adaptation to unknown kernel regularity in continuum-armed bandit problems.
method Derive adaptivity lower bound and verify with minimax non-adaptive kernelised bandit algorithms.
result Impossibility of achieving optimal cumulative regret in different RKHSs with varying regularities.
Unified analysis of kernel-based and locally adaptive bandit optimization methods.
problem Performance of bandit optimization algorithms in RKHS functions.
method Investigates the relationship between kernel regularity and algorithmic performance, characterizing spectral properties of various kernels.
result Unified framework for analyzing kernel-based and locally adaptive bandit algorithms, deriving explicit regret bounds.
Paper establishes lower bounds for non-stationary kernelized bandits.
problem Optimizing functions with noisy observations in non-stationary scenarios.
method Develops algorithm-independent lower bounds for time-varying functions under total variation constraints.
result First algorithm-independent lower bounds for time-varying kernelized bandits.
We tackle the problem of online reward maximisation over a large finite set of actions described by their contexts. We focus on the case when the number of actions is too big to sample all of them even once. However we assume that we have access to the similarities between actions' contexts and that the expected reward…
Unified framework for multi-user bandits using Laplacian kernels.
problem Multi-user contextual bandits with graph-related users and non-linear rewards.
method Joint penalty combining graph smoothness and individual roughness in a unified RKHS.
result Unified multi-user RKHS and effective dimension for regret bounds.
Algorithm optimizes cascaded functions with known structure.
problem Optimizing a function network with known structure.
method GPN-UCB algorithm with upper confidence bounds and theoretical regret bounds.
result Near-optimal cumulative and simple regret bounds.
New method clusters items with bandit feedback without parametric assumptions.
problem Clustering with noisy observations from unknown distributions.
method Kernel-based approach to nonparametric clustering with bandit feedback.
result Adaptive algorithm with theoretical guarantees for unknown signal-to-noise ratio.
Meta-KeL learns kernels from offline data to improve sequential decision-making.
problem Adaptive confidence sets for prediction functions in sequential decision-making tasks.
method Meta-KeL: meta-learning a kernel from offline data; structured sparsity estimator for unknown kernel combinations.
result Valid confidence sets that become as tight as those given the true unknown kernel with increasing offline data.
This paper tackles open problem of tight bounds for KBs with Bernoulli rewards.
problem Open problem of tight bounds for Kernelized Bandits with Bernoulli rewards.
method Focus on Bernoulli model, not subgaussian noise, and optimize function in RKHS.
result Open problem remains unsolved in this context.
New algorithms estimate function levels with near-optimal efficiency.
problem Estimating points where an unknown function exceeds a given threshold.
method Relates to adaptive experimental design methods for linear bandits in RKHS.
result Proves nearly optimal sample complexity bounds for level set estimation.
A new algorithm for differential privacy in kernelized contextual bandits reduces error rate.
problem Joint differential privacy in kernelized contextual bandits.
method Proposes a novel algorithm with a specific error rate and privacy parameter dependence.
result Achieves an error rate of $\mathcal{O}\left(\sqrt{\frac{γ_T}{T}} + \frac{γ_T}{T \varepsilon}
ight)$ after T T T queries. Kernel method improves cooperative decision-making among agents.
problem Cooperative multi-agent decision making with contextual information.
method Proposed extsc{Coop-KernelUCB} algorithm for near-optimal per-agent regret.
result Near-optimal bounds on per-agent regret with efficient computation and communication.
Kernelized bandit algorithm tackles adaptive contextual bandits with single-index models.
problem Adaptive contextual bandits with single-index models and unknown link functions.
method Kernelized ε-greedy algorithm combining Stein-based index estimation and kernel ridge regression for reward functions.
result Unified framework for simultaneous learning and inference in single-index contextual bandits.
Robust algorithm optimizes corrupted Gaussian process bandits.
problem Sequential optimization of corrupted, expensive reward functions.
method Robust GP Phased Elimination (RGP-PE) algorithm.
result Algorithm balances robustness to corruptions with exploration and exploitation.
A batched Gaussian Process bandit optimization method achieves near-optimal regret bounds.
problem Black-box optimization with limited function evaluations.
method Batched Gaussian Process bandit optimization algorithm.
result Achieves near-optimal cumulative regret bound of O ∗ ( T γ T ) O^\ast(\sqrt{Tγ_T}) O ∗ ( T γ T ) using O ( log log T ) O(\log\log T) O ( log log T ) batches. Paper establishes lower bounds for Gaussian process bandit optimization under various perturbation models.
problem Lower bounds for Gaussian process bandit optimization in noisy and robust settings.
method Novel proof techniques for standard and robust settings, including deterministic strategies.
result Demonstrates inevitable joint dependence of cumulative regret on corruption level and time horizon in robust settings.
New algorithm optimizes Hölder smooth functions in RKHS with tighter regret bounds.
problem Optimizing Hölder smooth functions in RKHS with bounded norm.
method Proposes a new algorithm ( exttt{LP-GP-UCB}) using Local Polynomial (LP) estimators and multi-scale UCB.
result Derives high probability bounds on simple and cumulative regret, matching optimal performance for SE kernel and uniformly tighter bounds for Matérn kernels.
New algorithm optimizes noisy, potentially corrupted functions.
problem Optimizing unknown functions with noisy bandit feedback, especially when evaluations are corrupted.
method Fast-Slow GP-UCB algorithm, combining robust and non-robust evaluations, enlarged confidence bounds.
result Theoretical analysis upper bounds cumulative regret, showing dependencies on corruption level and kernel.
New algorithms for optimizing functions with noisy feedback, even when the model is misspecified.
problem Optimizing a black-box function with noisy bandit feedback, especially when the model is misspecified.
method Developed two algorithms based on Gaussian process methods: EC-GP-UCB and Phased GP Uncertainty Sampling.
result Achieved optimal dependence on misspecification error without prior knowledge, and effective in stochastic contextual settings.
We study the regret minimization problem in the novel setting of generalized kernelized bandits (GKBs), where we optimize an unknown function f ∗ f^* f ∗ belonging to a reproducing kernel Hilbert space (RKHS) having access to samples generated by an exponential family (EF) reward model whose mean is a non-linear function $μ(…
New algorithm reduces regret for kernelized bandits by adapting to specific problem instances.
problem Efficiently learning the optimizer of an unknown function in RKHS with noisy oracle.
method Instance-dependent regret analysis and a new minimax near-optimal algorithm.
result New algorithm achieves better performance on specific problem instances.
In this paper, we consider the problem of sequentially optimizing a black-box function f f f based on noisy samples and bandit feedback. We assume that f f f is smooth in the sense of having a bounded norm in some reproducing kernel Hilbert space (RKHS), yielding a commonly-considered non-Bayesian form of Gaussian process …
Efficient global optimization is the problem of minimizing an unknown function f, using as few evaluations f(x) as possible. It can be considered as a continuum-armed bandit problem, with noiseless data and simple regret. Expected improvement is perhaps the most popular method for solving this problem; the algorithm pe…
No-regret optimization for time-varying functions using uncertainty injection.
problem Optimizing time-varying functions with no-regret in bandit feedback.
method W-SparQ-GP-UCB, incorporating uncertainty injection and additional queries.
result Achieves no-regret with a vanishing number of additional queries per iteration.
Constructs non-asymptotic confidence regions for unknown functions in RKHS.
problem Global probabilistic confidence regions for unknown functions in RKHS.
method Reduces confidence region construction to estimating RKHS norm.
result Valid confidence regions can be constructed non-asymptotically.
The paper refines and extends batched kernelized bandits, improving regret bounds and introducing a robust setting.
problem Optimizing black-box functions with noisy batches in Reproducing Kernel Hilbert Space.
method Refined and extended existing regret bounds, including adaptive batch sizes and robust optimization.
result Improved regret bounds for batched kernelized bandits, showing optimal number of batches and adaptive batch sizes.
Reconstruction of a function from noisy data is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution describes the observed data and has a small RKHS norm. When the data fit is measured using a quadratic loss, this estimator has a know…
Deep neural kernels and Laplace kernel have equivalent RKHS on spheres.
problem Comparing RKHS of deep neural tangent and Laplace kernels.
method Proof of RKHS equivalence using sphere restrictions and kernel properties.
result RKHS of deep neural tangent kernel and Laplace kernel are the same on S d − 1 \mathbb{S}^{d-1} S d − 1 . Kernel methods are powerful tools to capture nonlinear patterns behind data. They implicitly learn high (even infinite) dimensional nonlinear features in the Reproducing Kernel Hilbert Space (RKHS) while making the computation tractable by leveraging the kernel trick. Classic kernel methods learn a single layer of nonl…
This paper develops a general framework for metric learning in RKHS with theoretical guarantees.
problem Learning a metric in RKHS from triplet comparisons.
method Develops a general RKHS framework for metric learning with theoretical guarantees.
result Provides novel generalization guarantees and sample complexity bounds for metric learning in RKHS.
New framework assesses regularization norms in ill-posed problems, revealing L2 instability and proposing adaptive fractional RKHS solutions.
problem Comparative analysis of regularization norms in ill-posed problems.
method Small noise analysis framework for Tikhonov and RKHS regularizations.
result Optimal convergence rates achieved with adaptive fractional RKHS, but hyper-parameters decay too fast.
Enhances DGPs with adaptive RKHS Fourier features for better non-stationary pattern modeling.
problem Capturing complex non-stationary patterns in non-linear dynamical systems.
method Integrates ODE-based RKHS Fourier features into DGPs using convolution operations for adaptive amplitude and phase modulation. Uses a doubly stochastic variational inference framework.
result Improved predictive performance across various regression tasks.
Paper introduces RKHM for more explicit variable structures analysis.
problem Explicitly analyzing structures among variables.
method Orthonormal systems in Hilbert C ∗ C^* C ∗ -modules, RKHM. result Theoretical and practical procedures for RKHM orthonormalization.
This work analyzes the role of data augmentation in self-supervised learning using RKHS approximation and regression.
problem Limited theoretical understanding of the role of data augmentation in self-supervised learning.
method Geometric characterization of the target function given by augmentation, proving generalization bounds.
result Two generalization bounds are derived, one free of model complexity, the other specific to near-optimal encoders.
We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with…
Neural networks can approximate functionals on RKHS with error bounds.
problem Approximating functionals on RKHS using neural networks.
method Interpolating orthogonal projections in RKHS using point evaluations.
result Explicit error bounds for various kernels (inverse multiquadric, Gaussian, Sobolev).
Uniform bounds for neural networks' generalization error in overparameterized settings.
problem Generalization error in overparameterized neural networks.
method Neural Tangent kernel theory and Mercer decomposition of the NT kernel in spherical harmonics.
result Uniform generalization bounds for overparameterized neural networks in RKHS.
This paper generalizes regularized regression problems in a hyper-reproducing kernel Hilbert space (hyper-RKHS), illustrates its utility for kernel learning and out-of-sample extensions, and proves asymptotic convergence results for the introduced regression models in an approximation theory view. Algorithmically, we c…
New approach to supervised learning in RKHS and vvRKHS using C ∗ C^* C ∗ -algebras.
problem Traditional supervised learning in RKHS and vvRKHS.
method Generalizing supervised learning to RKHM using C ∗ C^* C ∗ -algebras. result Constructing RKHMs with enhanced representation power.
Study evaluates RKHS choices for assessing graph models using KSD tests.
problem Effect of RKHS choice on KSD tests for graph model assessment.
method Investigated power performance and computational runtime of KSD tests for ERGMs and synthetic graph generators.
result Different RKHS choices affect KSD test performance and computational runtime.
A new method for learning function parameters in operators using data-adaptive RKHS.
problem Learning function parameters in operators with robustness to noise and numerical error.
method Data Adaptive RKHS Tikhonov Regularization (DARTR) method.
result DARTR leads to an accurate estimator robust to noise and numerical error, converging at a consistent rate as data refines.
New analysis shows a gap between Gaussian RKHS and neural networks on unbounded domains.
problem Understanding the function space bias of neural networks compared to Gaussian RKHS.
method Infinite-center asymptotic analysis of neural network Banach space and Gaussian RKHS on unbounded domains.
result Certain functions in Gaussian RKHS have infinite norm in neural network Banach space on unbounded domains.