The sample efficiency of Bayesian optimization algorithms depends on carefully crafted acquisition functions (AFs) guiding the sequential collection of function evaluations. The best-performing AFs can vary significantly across optimization problems, often requiring ad-hoc and problem-specific choices. This work tackles the challenge of designing novel AFs that perform well across a variety of experimental settings. Based on FunSearch, a recent work using Large Language Models (LLMs) for discovery in mathematical sciences, we propose FunBO, an LLM-based method that can be used to learn new AFs written in computer code by leveraging access to a number of evaluations for a limited set of objective functions. We provide the analytic expression of all discovered AFs and evaluate them on various global optimization benchmarks and hyperparameter optimization tasks. We show how FunBO identifies AFs that generalize well both in and out of the training distribution of functions, thus outperforming established general-purpose AFs and achieving competitive performance against AFs that are customized to specific function types and are learned via transfer-learning algorithms.

We address the problem of causal discovery in the two-variable case given a sample from their joint distribution. The proposed method is based on a known assumption that, if X -> Y (X causes Y), the marginal distribution of the cause, P(X), contains no information about the conditional distribution P(Y|X). Consequently, estimating P(Y|X) from P(X) should not be possible. However, estimating P(X|Y) based on P(Y) may be possible. This paper employs this asymmetry to propose CURE, a causal discovery method which decides upon the causal direction by comparing the accuracy of the estimations of P(Y|X) and P(X|Y). To this end, we propose a method for estimating a conditional from samples of the corresponding marginal, which we call unsupervised inverse GP regression. We evaluate CURE on synthetic and real data. On the latter, our method outperforms existing causal inference methods.

We analyze a family of methods for statistical causal inference from sample under the so-called Additive Noise Model. While most work on the subject has concentrated on establishing the soundness of the Additive Noise Model, the statistical consistency of the resulting inference methods has received little attention. We derive general conditions under which the given family of inference methods consistently infers the causal direction in a nonparametric setting.