We propose Gradient-Informed Neural Networks (gradinn s), a methodology that can be used to efficiently approximate a wide range of functions in low-data regimes, when only general prior beliefs are available, a condition that is often encountered in complex engineering problems. gradinn s incorporate prior beliefs about the first-order derivatives of the target function to constrain the behavior of its gradient, thus implicitly shaping it, without requiring explicit access to the target function's derivatives. This is achieved by using two Neural Networks: one modeling the target function and a second, auxiliary network expressing the prior beliefs about the first-order derivatives (e.g., smoothness, oscillations, etc.). A customized loss function enables the training of the first network while enforcing gradient constraints derived from the auxiliary network; at the same time, it allows these constraints to be relaxed in accordance with the training data. Numerical experiments demonstrate the advantages of gradinn s, particularly in low-data regimes, with results showing strong performance compared to standard Neural Networks across the tested scenarios, including synthetic benchmark functions and real-world engineering tasks.

Mehran Kazemi, Bahare Fatemi, Hritik Bansal, John Palowitch, Chrysovalantis Anastasiou, Sanket Vaibhav Mehta, Lalit K Jain, Virginia Aglietti, Disha Jindal, Peter Chen, Nishanth Dikkala, Gladys Tyen, Xin Liu, Uri Shalit, Silvia Chiappa, Kate Olszewska, Yi Tay, Vinh Q. Tran, Quoc V Le, Orhan Firat. Proceedings of the 63rd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). 2025.

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 study the problem of globally optimizing the causal effect on a target variable of an unknown causal graph in which interventions can be performed. This problem arises in many areas of science including biology, operations research and healthcare. We propose Causal Entropy Optimization (CEO), a framework that generalizes Causal Bayesian Optimization (CBO) to account for all sources of uncertainty, including the one arising from the causal graph structure. CEO incorporates the causal structure uncertainty both in the surrogate models for the causal effects and in the mechanism used to select interventions via an information-theoretic acquisition function. The resulting algorithm automatically trades-off structure learning and causal effect optimization, while naturally accounting for observation noise. For various synthetic and real-world structural causal models, CEO achieves faster convergence to the global optimum compared with CBO while also learning the graph. Furthermore, our joint approach to structure learning and causal optimization improves upon sequential, structure-learning-first approaches.

We propose constrained causal Bayesian optimization (cCBO), an approach for finding interventions in a known causal graph that optimize a target variable under some constraints. cCBO first reduces the search space by exploiting the graph structure and, if available, an observational dataset; and then solves the restricted optimization problem by modelling target and constraint quantities using Gaussian processes and by sequentially selecting interventions via a constrained expected improvement acquisition function. We propose different surrogate models that enable to integrate observational and interventional data while capturing correlation among effects with increasing levels of sophistication. We evaluate cCBO on artificial and real-world causal graphs showing successful trade off between fast convergence and percentage of feasible interventions.

We explore algorithms to select actions in the causal bandit setting where the learner can choose to intervene on a set of random variables related by a causal graph, and the learner sequentially chooses interventions and observes a sample from the interventional distribution. The learner’s goal is to quickly find the intervention, among all interventions on observable variables, that maximizes the expectation of an outcome variable. We depart from previous literature by assuming no knowledge of the causal graph except that latent confounders between the outcome and its ancestors are not present. We first show that the unknown graph problem can be exponentially hard in the parents of the outcome. To remedy this, we adopt an additional additive assumption on the outcome which allows us to solve the problem by casting it as an additive combinatorial linear bandit problem with full-bandit feedback. We propose a novel action-elimination algorithm for this setting, show how to apply this algorithm to the causal bandit problem, provide sample complexity bounds, and empirically validate our findings on a suite of randomly generated causal models, effectively showing that one does not need to explicitly learn the parents of the outcome to identify the best intervention.

We propose functional causal Bayesian optimization (fCBO), a method for finding interventions that optimize a target variable in a known causal graph. fCBO extends the CBO family of methods to enable functional interventions, which set a variable to be a deterministic function of other variables in the graph. fCBO models the unknown objectives with Gaussian processes whose inputs are defined in a reproducing kernel Hilbert space, thus allowing to compute distances among vector-valued functions. In turn, this enables to sequentially select functions to explore by maximizing an expected improvement acquisition functional while keeping the typical computational tractability of standard BO settings. We introduce graphical criteria that establish when considering functional interventions allows attaining better target effects, and conditions under which selected interventions are also optimal for conditional target effects. We demonstrate the benefits of the method in a synthetic and in a real-world causal graph.

We study the problem of performing a sequence of optimal interventions in a dynamic causal system where both the target variable of interest, and the inputs, evolve over time. This problem arises in a variety of domains including healthcare, operational research and policy design. Our approach, which we call Dynamic Causal Bayesian Optimisation (DCBO), brings together ideas from decision making, causal inference and Gaussian process (GP) emulation. DCBO is useful in scenarios where the causal effects are changing over time. Indeed, at every time step, DCBO identifies a local optimal intervention by integrating both observational and past interventional data collected from the system. We give theoretical results detailing how one can transfer interventional information across time steps and define a dynamic causal GP model which can be used to find optimal interventions in practice. Finally, we demonstrate how DCBO identifies optimal interventions faster than competing approaches in multiple settings and applications.

This paper studies the problem of globally optimizing a variable of interest that is part of a causal model in which a sequence of interventions can be performed. This problem arises in biology, operational research, communications and, more generally, in all fields where the goal is to optimize an output metric of a system of interconnected nodes. Our approach combines ideas from causal inference, uncertainty quantification and sequential decision making. In particular, it generalizes Bayesian optimization, which treats the input variables of the objective function as independent, to scenarios where causal information is available. We show how knowing the causal graph significantly improves the ability to reason about optimal decision making strategies decreasing the optimization cost while avoiding suboptimal solutions. We propose a new algorithm called Causal Bayesian Optimization (CBO). CBO automatically balances two trade-offs: the classical exploration-exploitation and the new observation-intervention, which emerges when combining real interventional data with the estimated intervention effects computed via do-calculus. We demonstrate the practical benefits of this method in a synthetic setting and in two real-world applications.

This paper studies the problem of learning the correlation structure of a set of intervention functions defined on the directed acyclic graph (DAG) of a causal model. This is useful when we are interested in jointly learning the causal effects of interventions on different subsets of variables in a DAG, which is common in field such as healthcare or operations research. We propose the first multi-task causal Gaussian process (GP) model, which we call DAG-GP, that allows for information sharing across continuous interventions and across experiments on different variables. DAG-GP accommodates different assumptions in terms of data availability and captures the correlation between functions lying in input spaces of different dimensionality via a well-defined integral operator. We give theoretical results detailing when and how the DAG-GP model can be formulated depending on the DAG. We test both the quality of its predictions and its calibrated uncertainties. Compared to single-task models, DAG-GP achieves the best fitting performance in a variety of real and synthetic settings. In addition, it helps to select optimal interventions faster than competing approaches when used within sequential decision making frameworks, like active learning or Bayesian optimization.

We generalize the log Gaussian Cox process (LGCP) framework to model multiple correlated point data jointly. The observations are treated as realizations of multiple LGCPs, whose log intensities are given by linear combinations of latent functions drawn from Gaussian process priors. The combination coefficients are also drawn from Gaussian processes and can incorporate additional dependencies. We derive closed-form expressions for the moments of the intensity functions and develop an efficient variational inference algorithm that is orders of magnitude faster than competing deterministic and stochastic approximations of multivariate LGCPs, coregionalization models, and multi-task permanental processes. Our approach outperforms these benchmarks in multiple problems, offering the current state of the art in modeling multivariate point processes.

We propose a scalable framework for inference in a continuous sigmoidal Cox process that assumes the corresponding intensity function is given by a Gaussian process (GP) prior transformed with a scaled logistic sigmoid function. We present a tractable representation of the likelihood through augmentation with a superposition of Poisson processes. This view enables a structured variational approximation capturing dependencies across variables in the model. Our framework avoids discretization of the domain, does not require accurate numerical integration over the input space and is not limited to GPs with squared exponential kernels. We evaluate our approach on synthetic and real-world data showing that its benefits are particularly pronounced on multivariate input settings where it overcomes the limitations of mean-field methods and sampling schemes. We provide the state of-the-art in terms of speed, accuracy and uncertainty quantification trade-offs.