In addition to the work on noisy convex optimization, the current paper is also thematically related to works in learning theory and complexity where the goal is to reconstruct simple classes of functions under outlier noise. This includes work on reconstruction of low-degree polynomials [4, 14, 15]. In particular, [15] gave an efficient algorithm whose error tolerance matches the information theoretic limits. In addition, recently, [9] achieved similar algorithmic guarantees for functions which are sparse in the Fourier space. While similar in spirit, the model in these works differ from the current paper in one crucial way - namely, while we only put a bound on the volume of the outlier locations, they, in addition, assume that the outlier locations are also uniformly distributed in the domain. At a more technical level, the results in [4, 14, 15, 9] crucially rely on techniques originating from coding theory such as the Goldreich-Levin theorem [13] and the Berlekamp-Welch algorithm [6].

The Gaussian homotopy (GH) method is a popular approach to finding better stationary points for non-convex optimization problems by gradually reducing a parameter value t, which changes the problem to be solved from an almost convex one to the original target one. Existing GH-based methods repeatedly call an iterative optimization solver to find a stationary point every time t is updated, which incurs high computational costs. We propose a novel single loop framework for GH methods (SLGH) that updates the parameter tand the optimization decision variables at the same. Computational complexity analysis is performed on the SLGH algorithm under various situations: either a gradient or gradient-free oracle of a GH function can be obtained for both deterministic and stochastic settings. The convergence rate of SLGH with a tuned hyperparameter becomes consistent with the convergence rate of gradient descent, even though the problem to be solved is gradually changed due to t. In numerical experiments, our SLGH algorithms show faster convergence than an existing double loop GH method while outperforming gradient descent-based methods in terms of finding a better solution.

We say a real-valued random variable X is -sub-Gaussian if it its mean is zero and for all " 2 R we have E[exp("X)] exp Such assumptions on the noise variables are frequently used in bandit optimization. Typically, in kernelized bandits, we assume that unknown f 2F k(D;B)= {f 2H k(D): kfkk B}, where Hk(D) is the reproducing kernel Hilbert space of functions associated with the given positive-definite kernel function. Typically, the learner knows Fk(D;B), meaning that both k(,) and B are considered as input to the learner's algorithm. We outline some commonly used kernel functions k: D D! R, that we also consider: Linear kernel: klin(x,x0)= xTx0, Squared exponential kernel: kSE(x,x0)=exp kx x0k2 2l2, Matérn kernel: kMat(x,x0)= 2 Maximum information gain is a kernel-dependent quantity that measures the complexity of the given function class. It has first been introduced in [40], and since then it has been used in numerous works on Gaussian process bandits.

Algorithm 2 details the pseudocode for the partition function used in LaMCTS, which we use in LaP3 as well. Algorithm 2 Partition Function 1: Input: Input Space Ω, Samples St, Node partition threshold Nthres, Partitioning Latent Model s(x) 2: Set V0 = {Ω} 3: Set Vqueue = {Ω} 4: while Vqueue 6= do 5: Ωp Vqueue.pop(0) It is clear that Fk(y) is a monotonically decreasing function with Fk(0) = 1 and limy + Fk(y) = 0. Here we assume it is strictly decreasing so that Fk(y) has a well-defined inverse function F 1k . In the following, we will omit the subscript k for brevity. P[f(xi) g y|xi Ωk] (4) = 1 Fntk (y) (5) Note that 1 is due to the fact that all samples x1,...,xnt are independently drawn within the region Ωk.

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The goal of this paper is to characterize Gaussian-Process optimization in the setting where the function domain is large relative to the number of admissible function evaluations, i.e., where it is impossible to find the global optimum. We provide upper bounds on the suboptimality (Bayesian simple regret) of the solution found by optimization strategies that are closely related to the widely used expected improvement (EI) and upper confidence bound (UCB) algorithms. These regret bounds illuminate the relationship between the number of evaluations, the domain size (i.e.

Asynchronous Bayesian optimization is widely used for gradient-free optimization in domains with independent parallel experiments and varying evaluation times. Existing methods posit that standard acquisitions lead to redundant and repeated queries, proposing complex solutions to enforce diversity in queries. Challenging this fundamental premise, we show that methods, like the Upper Confidence Bound, can in fact achieve theoretical guarantees essentially equivalent to those of sequential Thompson sampling. A conceptual analysis of asynchronous Bayesian optimization reveals that existing works neglect intermediate posterior updates, which we find to be generally sufficient to avoid redundant queries. Further investigation shows that by penalizing busy locations, diversity-enforcing methods can over-explore in asynchronous settings, reducing their performance. Our extensive experiments demonstrate that simple standard acquisition functions match or outperform purpose-built asynchronous methods across synthetic and real-world tasks.