Despite the increasing effectiveness of language models, their reasoning capabilities remain underdeveloped. In particular, causal reasoning through counterfactual question answering is lacking. This work aims to bridge this gap. We first derive novel metrics that balance accuracy in factual and counterfactual questions, capturing a more complete view of the reasoning abilities of language models than traditional factual-only based metrics. Second, we propose several fine-tuning approaches that aim to elicit better reasoning mechanisms, in the sense of the proposed metrics. Finally, we evaluate the performance of the fine-tuned language models in a variety of realistic scenarios. In particular, we investigate to what extent our fine-tuning approaches systemically achieve better generalization with respect to the base models in several problems that require, among others, inductive and deductive reasoning capabilities.

Causal reasoning and compositional reasoning are two core aspirations in AI. Measuring the extent of these behaviors requires principled evaluation methods. We explore a unified perspective that considers both behaviors simultaneously, termed compositional causal reasoning (CCR): the ability to infer how causal measures compose and, equivalently, how causal quantities propagate through graphs. We instantiate a framework for the systematic evaluation of CCR for the average treatment effect and the probability of necessity and sufficiency. As proof of concept, we demonstrate CCR evaluation for language models in the Llama, Phi, and GPT families. On a math word problem, our framework revealed a range of taxonomically distinct error patterns. CCR errors increased with the complexity of causal paths for all models except o1.

Recent Large Language Models (LLMs) have reported high accuracy on reasoning benchmarks. However, it is still unclear whether the observed results arise from true “reasoning” or from statistical recall of the training set. Inspired by the ladder of causation (Pearl, 2009) and its three levels (associations, interventions and counterfactuals), this paper introduces RE-IMAGINE: a framework to characterize a hierarchy of reasoning ability in LLMs, alongside an automated pipeline to generate problem variations at different levels of the hierarchy. By altering problems in an intermediate symbolic representation, RE-IMAGINE generates arbitrarily many problems that are not solvable using memorization alone. Moreover, the framework is general and can work across reasoning domains, including math, code, and logic. We demonstrate our framework on four widely-used benchmarks to evaluate several families of LLMs, and observe reductions in performance when the models are queried with problem variations. These assessments indicate a degree of reliance on statistical recall for past performance, and open the door to further research targeting skills across the reasoning hierarchy.

Recent advances in AI have been significantly driven by the capabilities of large language models (LLMs) to solve complex problems in ways that resemble human thinking. However, there is an ongoing debate about the extent to which LLMs are capable of actual reasoning. Central to this debate are two key probabilistic concepts that are essential for connecting causes to their effects: the probability of necessity (PN) and the probability of sufficiency (PS). This paper introduces a framework that is both theoretical and practical, aimed at assessing how effectively LLMs are able to replicate real-world reasoning mechanisms using these probabilistic measures. By viewing LLMs as abstract machines that process information through a natural language interface, we examine the conditions under which it is possible to compute suitable approximations of PN and PS. Our research marks an important step towards gaining a deeper understanding of when LLMs are capable of reasoning, as illustrated by a series of math examples.

Predicting the impact of treatments from ob- servational data only still represents a major challenge despite recent significant advances in time series modeling. Treatment assignments are usually correlated with the predictors of the response, resulting in a lack of data support for counterfactual predictions and therefore in poor quality estimates. Developments in causal inference have lead to methods addressing this confounding by requiring a minimum level of overlap. However, overlap is difficult to assess and usually not satisfied in practice. In this work, we propose Counterfactual ODE (CF-ODE), a novel method to predict the impact of treatments continuously over time using Neural Ordinary Differential Equations equipped with uncertainty estimates. This allows to specifically assess which treatment outcomes can be reliably predicted. We demonstrate over several longitudinal datasets that CF-ODE provides more accurate predictions and more reliable uncertainty estimates than previously available methods.

We revisit widely used preferential Gaussian processes (PGP) by Chu and Ghahramani [2005] and challenge their modelling assumption that imposes rankability of data items via latent utility function values. We propose a generalisation of PGP which can capture more expressive latent preferential structures in the data and thus be used to model inconsistent preferences, i.e. where transitivity is violated, or to discover clusters of comparable items via spectral decomposition of the learned preference functions. We also consider the properties of associated covariance kernel functions and its reproducing kernel Hilbert Space (RKHS), giving a simple construction that satisfies universality in the space of preference functions. Finally, we provide an extensive set of numerical experiments on simulated and real-world datasets showcasing the competitiveness of our proposed method with state-of-the-art. Our experimental findings support the conjecture that violations of rankability are ubiquitous in real-world preferential data.

Feature attribution for kernel methods is often heuristic and not individualised for each prediction. To address this, we turn to the concept of Shapley values (SV), a coalition game theoretical framework that has previously been applied to different machine learning model interpretation tasks, such as linear models, tree ensembles and deep networks. By analysing SVs from a functional perspective, we propose RKHS-SHAP, an attribution method for kernel machines that can efficiently compute both Interventional and Observational Shapley values using kernel mean embeddings of distributions. We show theoretically that our method is robust with respect to local perturbations - a key yet often overlooked desideratum for consistent model interpretation. Further, we propose Shapley regulariser, applicable to a general empirical risk minimisation framework, allowing learning while controlling the level of specific feature's contributions to the model. We demonstrate that the Shapley regulariser enables learning which is robust to covariate shift of a given feature and fair learning which controls the SVs of sensitive features.

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.

While causal models are becoming one of the mainstays of machine learning, the problem of uncertainty quantification in causal inference remains challenging. In this paper, we study the causal data fusion problem, where data arising from multiple causal graphs are combined to estimate the average treatment effect of a target variable. As data arises from multiple sources and can vary in quality and sample size, principled uncertainty quantification becomes essential. To that end, we introduce \emph{Bayesian Causal Mean Processes}, the framework which combines ideas from probabilistic integration and kernel mean embeddings to represent interventional distributions in the reproducing kernel Hilbert space, while taking into account the uncertainty within each causal graph. To demonstrate the informativeness of our uncertainty estimation, we apply our method to the Causal Bayesian Optimisation task and show improvements over state-of-the-art methods.

This paper describes a general-purpose extension of max-value entropy search, a popular approach for Bayesian Optimisation (BO). A novel approximation is proposed for the information gain -- an information-theoretic quantity central to solving a range of BO problems, including noisy, multi-fidelity and batch optimisations across both continuous and highly-structured discrete spaces. Previously, these problems have been tackled separately within information-theoretic BO, each requiring a different sophisticated approximation scheme, except for batch BO, for which no computationally-lightweight information-theoretic approach has previously been proposed. GIBBON (General-purpose Information-Based Bayesian OptimisatioN) provides a single principled framework suitable for all the above, out-performing existing approaches whilst incurring substantially lower computational overheads. In addition, GIBBON does not require the problem's search space to be Euclidean and so is the first high-performance yet computationally light-weight acquisition function that supports batch BO over general highly structured input spaces like molecular search and gene design. Moreover, our principled derivation of GIBBON yields a natural interpretation of a popular batch BO heuristic based on determinantal point processes. Finally, we analyse GIBBON across a suite of synthetic benchmark tasks, a molecular search loop, and as part of a challenging batch multi-fidelity framework for problems with controllable experimental noise.

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.

We consider the problem of optimising functions in the reproducing kernel Hilbert space (RKHS) of a Matérn kernel with smoothness parameter $u$ over the domain $[0,1]^d$ under noisy bandit feedback. Our contribution, the $\pi$-GP-UCB algorithm, is the first practical approach with guaranteed sublinear regret for all $u>1$ and $d \geq 1$. Empirical validation suggests better performance and drastically improved computational scalablity compared with its predecessor, Improved GP-UCB.

We present BOFFIN TTS (Bayesian Optimization For FIne-tuning Neural Text To Speech), a novel approach for few-shot speaker adaptation. Here, the task is to fine-tune a pre-trained TTS model to mimic a new speaker using a small corpus of target utterances. We demonstrate that there does not exist a one-size-fits-all adaptation strategy, with convincing synthesis requiring a corpus-specific configuration of the hyper-parameters that control fine-tuning. By using Bayesian optimization to efficiently optimize these hyper-parameter values for a target speaker, we are able to perform adaptation with an average 30% improvement in speaker similarity over standard techniques. Results indicate, across multiple corpora, that BOFFIN TTS can learn to synthesize new speakers using less than ten minutes of audio, achieving the same naturalness as produced for the speakers used to train the base model.

Finite-horizon sequential experimental design (SED) arises naturally in many contexts, including hyperparameter tuning in machine learning among more traditional settings. Computing the optimal policy for such problems requires solving Bellman equations, which are generally intractable. Most existing work resorts to severely myopic approximations by limiting the decision horizon to only a single time-step, which can underweight exploration in favor of exploitation. We present BINOCULARS: Batch-Informed NOnmyopic Choices, Using Long-horizons for Adaptive, Rapid SED, a general framework for deriving efficient, nonmyopic approximations to the optimal experimental policy. Our key idea is simple and surprisingly effective: we first compute a one-step optimal batch of experiments, then select a single point from this batch to evaluate. We realize BINOCULARS for Bayesian optimization and Bayesian quadrature – two notable example problems with radically different objectives – and demonstrate that BINOCULARS significantly outperforms significantly outperforms myopic alternatives in real-world scenarios.

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.

This article develops a Bayesian optimization (BO) method which acts directly over raw strings, proposing the first uses of string kernels and genetic algorithms within BO loops. Recent applications of BO over strings have been hindered by the need to map inputs into a smooth and unconstrained latent space. Learning this projection is computationally and data-intensive. Our approach instead builds a powerful Gaussian process surrogate model based on string kernels, naturally supporting variable length inputs, and performs efficient acquisition function maximization for spaces with syntactic constraints. Experiments demonstrate considerably improved optimization over existing approaches across a broad range of constraints, including the popular setting where syntax is governed by a context-free grammar.

Abstract Differential privacy is a mathematical framework for privacy-preserving data analysis. Changing the hyperparameters of a differentially private algorithm allows one to trade off privacy and utility in a principled way. Quantifying this trade-off in advance is essential to decision-makers tasked with deciding how much privacy can be provided in a particular application while maintaining acceptable utility. Analytical utility guarantees offer a rigorous tool to reason about this tradeoff, but are generally only available for relatively simple problems. For more complex tasks, such as training neural networks under differential privacy, the utility achieved by a given algorithm can only be measured empirically. This paper presents a Bayesian optimization methodology for efficiently characterizing the privacy– utility trade-off of any differentially private algorithm using only empirical measurements of its utility. The versatility of our method is illustrated on a number of machine learning tasks involving multiple models, optimizers, and datasets.

Despite the recent progress in hyperparameter optimization (HPO), available benchmarks that resemble real-world scenarios consist of a few and very large problem instances that are expensive to solve. This blocks researchers and practitioners no only from systematically running large-scale comparisons that are needed to draw statistically significant results but also from reproducing experiments that were conducted before. This work proposes a method to alleviate these issues by means of a meta-surrogate model for HPO tasks trained on off-line generated data. The model combines a probabilistic encoder with a multi-task model such that it can generate inexpensive and realistic tasks of the class of problems of interest. We demonstrate that benchmarking HPO methods on samples of the generative model allows us to draw more coherent and statistically significant conclusions that can be reached orders of magnitude faster than using the original tasks. We provide evidence of our findings for various HPO methods on a wide class of problems.

We tackle the problem of optimizing a black-box objective function defined over a highly-structured input space. This problem is ubiquitous in science and engineering. In machine learning, inferring the structure of a neural network or the Automatic Statistician (AS), where the optimal kernel combination for a Gaussian process is selected, are two important examples. We use the \as as a case study to describe our approach, that can be easily generalized to other domains. We propose an Structure Generating Variational Auto-encoder (SG-VAE) to embed the original space of kernel combinations into some low-dimensional continuous manifold where Bayesian optimization (BO) ideas are used. This is possible when structural knowledge of the problem is available, which can be given via a simulator or any other form of generating potentially good solutions. The right exploration-exploitation balance is imposed by propagating into the search the uncertainty of the latent space of the SG-VAE, that is computed using variational inference. The key aspect of our approach is that the SG-VAE can be used to bias the search towards relevant regions, making it suitable for transfer learning tasks. Several experiments in various application domains are used to illustrate the utility and generality of the approach described in this work.

Bayesian optimization (BO) has emerged during the last few years as an effective approach to optimize black-box functions where direct queries of the objective are expensive. We consider the case where direct access to the function is not possible, but information about user preferences is. Such scenarios arise in problems where human preferences are modeled, such as A/B tests or recommender systems. We present a new framework for this scenario that we call Preferential Bayesian Optimization (PBO) and that allows to find the optimum of a latent function that can only be queried through pairwise comparisons, so-called duels. PBO extend the applicability of standard BO ideas and generalizes previous discrete dueling approaches by modeling the probability of the the winner of each duel by means of Gaussian process model with a Bernoulli likelihood. The latent preference function is used to define a family of acquisition functions that extend usual policies used in BO. We illustrate the benefits of PBO in a variety of experiments in which we show how the way correlations are modeled is the key ingredient to drastically reduce the number of comparisons to find the optimum of the latent function of interest.

Bayesian optimization has been successfully used to optimize complex black-box functions whose evaluations are expensive. In many applications, like in deep learning and predictive analytics, the optimization domain is itself complex and structured. In this work, we focus on use cases where this domain exhibits a known dependency structure. The benefit of leveraging this structure is twofold: we explore the search space more efficiently and posterior inference scales more favorably with the number of observations than Gaussian Process-based approaches published in the literature. We introduce a novel surrogate model for Bayesian optimization which combines independent Gaussian Processes with a linear model that encodes a tree-based dependency structure and can transfer information between overlapping decision sequences. We also design a specialized two-step acquisition function that explores the search space more effectively. Our experiments on synthetic tree-structured functions and the tuning of feedforward neural networks trained on a range of binary classification datasets show that our method compares favorably with competing approaches.

The popularity of Bayesian optimization methods for efficient exploration of parameter spaces has lead to a series of papers applying Gaussian processes as surrogates in the optimization of functions. However, most proposed approaches only allow the exploration of the parameter space to occur sequentially. Often, it is desirable to simultaneously propose batches of parameter values to explore. This is particularly the case when large parallel processing facilities are available. These could either be computational or physical facets of the process being optimized. Batch methods, however, require the modeling of the interaction between the different evaluations in the batch, which can be expensive in complex scenarios. We investigate this issue and propose a highly effective heuristic based on an estimate of the function’s Lipschitz constant that captures the most important aspect of this interaction–local repulsion–at negligible computational overhead. A penalized acquisition function is used to collect batches of points minimizing the non-parallelizable computational effort. The resulting algorithm compares very well, in run-time, with much more elaborate alternatives.

We present GLASSES: Global optimisation with Look-Ahead through Stochastic Simulation and Expected-loss Search. The majority of global optimisation approaches in use are myopic, in only considering the impact of the next function value; the non-myopic approaches that do exist are able to consider only a handful of future evaluations. Our novel algorithm, GLASSES, permits the consideration of dozens of evaluations into the future. This is done by approximating the ideal look-ahead loss function, which is expensive to evaluate, by a cheaper alternative in which the future steps of the algorithm are simulated beforehand. An Expectation Propagation algorithm is used to compute the expected value of the loss. We show that the far-horizon planning thus enabled leads to substantive performance gains in empirical tests.

The Mahalanobis distance (MD) is a distance widely used in Statistics, Machine Learning and Pattern Recognition. When the data come from a Gaussian distribution, the MD uses the covariance matrix to evaluate the distance between a data point and the distribution mean. In this paper, we generalize the MD for general unimodal distributions, introducing a particular class of Mercer kernel, the density kernel, based on the underlying data density. Density kernels induce distances that generalize the MD and that are useful when data do not fit to the Gaussian distribution. We study the theoretical properties of the proposed distance and show its performance on a variety of artificial and real data analysis problems.