Dose-finding design based on level set estimation in phase I cancer clinical trials Keiichiro Seno 1 a, Kota Matsui 2b, Shogo Iwazaki 3, Yu Inatsu 4, Shion Takeno 5, 6 and Shigeyuki Matsui 2, 7 1 Department of Biostatistics, Nagoya University 2 Department of Biostatistics, Kyoto University 3 MI-6 Ltd. 4 Department of Computer Science, Nagoya Institute of Technology 5 Department of Mechanical Systems Engineering, Nagoya University 6 Center for Advanced Intelligence Project, RIKEN 7 Research Center for Medical and Health Data Science, The Institute of Statistical Mathematics Abstract The primary objective of phase I cancer clinical trials is to evaluate the safety of a new experimental treatment and to find the maximum tolerated dose (MTD). We show that the MTD estimation problem can be regarded as a level set estimation (LSE) problem whose objective is to determine the regions where an unknown function value is above or below a given threshold. Then, we propose a novel ...

We study the noise-free Gaussian Process (GP) bandits problem, in which the learner seeks to minimize regret through noise-free observations of the black-box objective function lying on the known reproducing kernel Hilbert space (RKHS). Gaussian process upper confidence bound (GP-UCB) is the well-known GP-bandits algorithm whose query points are adaptively chosen based on the GP-based upper confidence bound score. Although several existing works have reported the practical success of GP-UCB, the current theoretical results indicate its suboptimal performance. However, GP-UCB tends to perform well empirically compared with other nearly optimal noise-free algorithms that rely on a non-adaptive sampling scheme of query points. This paper resolves this gap between theoretical and empirical performance by showing the nearly optimal regret upper bound of noise-free GP-UCB. Specifically, our analysis shows the first constant cumulative regret in the noise-free settings for the squared exponential kernel and Mat\'ern kernel with some degree of smoothness.

The Gaussian process (GP) bandits [Srinivas et al., 2010] is a powerful framework for sequential decision-making tasks to minimize regret defined by a black-box reward function, which belongs to known reproducing kernel Hilbert space (RKHS). The applications include many fileds such as robotics Berkenkamp et al. [2021], experimental design Lei et al. [2021], and hyperparameter tuning task Snoek et al. [2012]. Many existing studies have been conducted to obtain the theoretical guarantee for the regret. Establised work by Srinivas et al. [2010] has shown the upper bounds of the cumulative regret for the GP upper confidence bound (GP-UCB) algorithm. Furthermore, Valko et al. [2013] have shown the tighter regret upper bound for the SupKernelUCB algorithm. Scarlett et al. [2017] have shown the lower bound of the regret, which implies that the regret upper bound from [Valko et al., 2013] is near-optimal; that is, the regret upper bound matches the lower bound except for the poly-logarithmic factor. Then, several studies further tackled obtaining a near-optimal GP-bandit algorithm. Vakili et al. [2021] have proposed maximum variance reduction (MVR), which is shown to be near-optimal for the simple regret incurred by the last recommended action.

Kernelized bandit (KB) problem [Srinivas et al., 2010], also called Gaussian process bandit or Bayesian optimization, is one of the important sequential decision-making problems where one seeks to minimize the regret under an unknown reward function via sequentially acquiring function evaluations. As the name suggests, in the KB problem, the underlying reward function is assumed to be an element of reproducing kernel Hilbert space (RKHS) induced by a known fixed kernel function. KB has been applied in many applications, such as materials discovery [Ueno et al., 2016], drug discovery [Korovina et al., 2020], and robotics [Berkenkamp et al., 2023]. In addition, the near-optimal KB algorithms, whose regret upper bound matches the regret lower bound derived in Scarlett et al. [2017], have been shown [Camilleri et al., 2021, Salgia et al., 2021, Li and Scarlett, 2022, Salgia et al., 2024]. Non-stationary KB [Bogunovic et al., 2016] considers the optimization under a non-stationary environment; that is, the reward function may change over time within some RKHS. This modification is crucial in many practical applications where an objective function varies over time, such as financial markets [Heaton and Lucas, 1999] and recommender systems [Hariri et al., 2015]. For example, Zhou and Shroff [2021], Deng et al. [2022] have proposed upper confidence bound (UCB)-based algorithms for the non-stationary KB problem and derived the upper bound of the cumulative regret. Recently, Hong et al. [2023] have proposed an optimization-based KB

Complex processes in science and engineering are often formulated as multi-stage decision-making problems. In this paper, we consider a type of multi-stage decision-making process called a cascade process. A cascade process is a multi-stage process in which the output of one stage is used as an input for the next stage. When the cost of each stage is expensive, it is difficult to search for the optimal controllable parameters for each stage exhaustively. To address this problem, we formulate the optimization of the cascade process as an extension of Bayesian optimization framework and propose two types of acquisition functions (AFs) based on credible intervals and expected improvement. We investigate the theoretical properties of the proposed AFs and demonstrate their effectiveness through numerical experiments. In addition, we consider an extension called suspension setting in which we are allowed to suspend the cascade process at the middle of the multi-stage decision-making process that often arises in practical problems. We apply the proposed method in the optimization problem of the solar cell simulator, which was the motivation for this study.

Many cases exist in which a black-box function $f$ with high evaluation cost depends on two types of variables $\bm x$ and $\bm w$, where $\bm x$ is a controllable \emph{design} variable and $\bm w$ are uncontrollable \emph{environmental} variables that have random variation following a certain distribution $P$. In such cases, an important task is to find the range of design variables $\bm x$ such that the function $f(\bm x, \bm w)$ has the desired properties by incorporating the random variation of the environmental variables $\bm w$. A natural measure of robustness is the probability that $f(\bm x, \bm w)$ exceeds a given threshold $h$, which is known as the \emph{probability threshold robustness} (PTR) measure in the literature on robust optimization. However, this robustness measure cannot be correctly evaluated when the distribution $P$ is unknown. In this study, we addressed this problem by considering the \textit{distributionally robust PTR} (DRPTR) measure, which considers the worst-case PTR within given candidate distributions. Specifically, we studied the problem of efficiently identifying a reliable set $H$, which is defined as a region in which the DRPTR measure exceeds a certain desired probability $\alpha$, which can be interpreted as a level set estimation (LSE) problem for DRPTR. We propose a theoretically grounded and computationally efficient active learning method for this problem. We show that the proposed method has theoretical guarantees on convergence and accuracy, and confirmed through numerical experiments that the proposed method outperforms existing methods.

In the past few years, various approaches have been developed to explain and interpret deep neural network (DNN) representations, but it has been pointed out that these representations are sometimes unstable and not reproducible. In this paper, we interpret these representations as hypotheses driven by DNN (called DNN-driven hypotheses) and propose a method to quantify the reliability of these hypotheses in statistical hypothesis testing framework. To this end, we introduce Selective Inference (SI) framework, which has received much attention in the past few years as a new statistical inference framework for data-driven hypotheses. The basic idea of SI is to make conditional inferences on the selected hypotheses under the condition that they are selected. In order to use SI framework for DNN representations, we develop a new SI algorithm based on homotopy method which enables us to derive the exact (non-asymptotic) conditional sampling distribution of the DNN-driven hypotheses. We conduct experiments on both synthetic and real-world datasets, through which we offer evidence that our proposed method can successfully control the false positive rate, has decent performance in terms of computational efficiency, and provides good results in practical applications.

Decision making in an uncertain environment has been studied in various domains. For example, in financial engineering, the mean-variance analysis [1, 2, 3] has been introduced as a framework for making investment decisions, taking into account the tradeoff between the return (mean) and the risk (variance) of the investment. In this paper we study active learning (AL) in an uncertain environment. In many practical AL problems, there are two types of parameters called design parameters and environmental parameters. For example, in a product design, while the design parameters are fully controllable, the environmental parameters vary depending on the environment in which the product is used. In this paper, we examine AL problems under such an uncertain environment, where the goal is to efficiently find the optimal design parameters by properly taking into account the uncertainty of the environmental parameters. Concretely, let f(x, w) be a blackbox function indicating the performance of a product, where x X is the set of controllable design parameters and w Ω is the set of uncontrollable environmental parameters whose uncertainty is characterized by a probability distribution p(w).

In many product development problems, the performance of the product is governed by two types of parameters called design parameter and environmental parameter. While the former is fully controllable, the latter varies depending on the environment in which the product is used. The challenge of such a problem is to find the design parameter that maximizes the probability that the performance of the product will meet the desired requisite level given the variation of the environmental parameter. In this paper, we formulate this practical problem as active learning (AL) problems and propose efficient algorithms with theoretically guaranteed performance. Our basic idea is to use Gaussian Process (GP) model as the surrogate model of the product development process, and then to formulate our AL problems as Bayesian Quadrature Optimization problems for probabilistic threshold robustness (PTR) measure. We derive credible intervals for the PTR measure and propose AL algorithms for the optimization and level set estimation of the PTR measure. We clarify the theoretical properties of the proposed algorithms and demonstrate their efficiency in both synthetic and real-world product development problems.

When the cost of an operational test is expensive, it is desirable to Nagoya Institute of Technology † RIKEN Center for Advanced Intelligence Project ‡ National Institute for Materials Sciences § email:takeuchi.ichiro@nitech.ac.jp be able to identify the region of appropriate input conditions in as few operational tests as possible. If we regard the operational conditions as inputs and the results of the operational tests as outputs of a black-box function, this problem can be viewed as a type of active learning (AL) problem called Level Set Estimation (LSE) . LSE is defined as the problem of identifying the input region in which the outputs of a function are smaller/greater than a certain threshold. In the statistics and machine learning literature, many methods for the LSE problem have been proposed [Bryan et al., 2006, Gotovos et al., 2013, Zanette et al., 2018]. In practical manufacturing applications, since it is often difficult to accurately control the input conditions during the actual usage of the machine, there is a need to guarantee the performance of the machine after properly incorporating the possible variation of input conditions.