Julien Martinelli

Petrus Mikkola, Julien Martinelli, Louis Filstroff et al.

Bayesian optimization (BO) is a powerful framework for optimizing black-box, expensive-to-evaluate functions. Over the past decade, many algorithms have been proposed to integrate cheaper, lower-fidelity approximations of the objective function into the optimization process, with the goal of converging towards the global optimum at a reduced cost. This task is generally referred to as multi-fidelity Bayesian optimization (MFBO). However, MFBO algorithms can lead to higher optimization costs than their vanilla BO counterparts, especially when the low-fidelity sources are poor approximations of the objective function, therefore defeating their purpose. To address this issue, we propose rMFBO (robust MFBO), a methodology to make any GP-based MFBO scheme robust to the addition of unreliable information sources. rMFBO comes with a theoretical guarantee that its performance can be bound to its vanilla BO analog, with high controllable probability. We demonstrate the effectiveness of the proposed methodology on a number of numerical benchmarks, outperforming earlier MFBO methods on unreliable sources. We expect rMFBO to be particularly useful to reliably include human experts with varying knowledge within BO processes.

Sabina J. Sloman, Ayush Bharti, Julien Martinelli et al.

In many settings, such as scientific inference, optimization, and transfer learning, the learner has a well-defined objective, which can be treated as estimation of a target parameter, and no intrinsic interest in characterizing the entire data-generating process. Usually, the learner must also contend with additional sources of uncertainty or variables -- with nuisance parameters. Bayesian active learning, or sequential optimal experimental design, can straightforwardly accommodate the presence of nuisance parameters, and so is a natural active learning framework for such problems. However, the introduction of nuisance parameters can lead to bias in the Bayesian learner's estimate of the target parameters, a phenomenon we refer to as negative interference. We characterize the threat of negative interference and how it fundamentally changes the nature of the Bayesian active learner's task. We show that the extent of negative interference can be extremely large, and that accurate estimation of the nuisance parameters is critical to reducing it. The Bayesian active learner is confronted with a dilemma: whether to spend a finite acquisition budget in pursuit of estimation of the target or of the nuisance parameters. Our setting encompasses Bayesian transfer learning as a special case, and our results shed light on the phenomenon of negative transfer between learning environments.

Julien Martinelli, Maksim Sinelnikov, Harri Lähdesmäki et al.

Dynamical modelling is central to many scientific domains, including pharmacometrics, systems biology, physiology, and epidemiology. In these settings, heterogeneity is often intrinsic: different subjects or units follow related but distinct continuous-time dynamics. Classical nonlinear mixed-effects Ordinary Differential Equation (ODE) models address this by combining population-level structure with subject-specific effects, but they rely on a parametric vector field and are therefore vulnerable to structural misspecification and unmodelled mechanisms. This motivates nonparametric approaches that can retain principled uncertainty quantification, yet existing nonparametric ODE methods typically assume a single shared dynamical system rather than an explicit mixed-effect hierarchy over subject-specific dynamics. We propose MEGPODE, a Bayesian nonparametric mixed-effect ODE model in which each subject's vector field is decomposed into a shared population component and a subject-specific deviation, both endowed with Gaussian process (GP) priors. To avoid repeated ODE solves per subject during training, we combine state-space GP trajectory priors with virtual collocation observations, yielding Kalman-smoothing trajectory updates and closed-form regressions for the vector fields. Across controlled heterogeneous ODE benchmarks spanning oscillatory, biomedical systems, MEGPODE improves population-field recovery and subject-level trajectory prediction relative to strong baselines.

Marshal Arijona Sinaga, Julien Martinelli, Teemu Turpeinen et al.

Bayesian optimization (BO) is a widely used framework for optimizing expensive black-box functions, commonly based on Gaussian process (GP) surrogate models. Its effectiveness relies on uncertainty quantification that is both sharp (informative) and well-calibrated along the BO trajectory. In practice, GP kernel hyperparameters are unknown and are refit online from sequentially collected (non-i.i.d.) data, which can yield miscalibrated or overly conservative uncertainty and lies outside the fixed-kernel assumptions of standard BO regret theory. We propose Online Sharp-Calibrated Bayesian Optimization (OSCBO), a BO algorithm that adaptively balances GP sharpness and calibration by casting hyperparameter selection as a constrained online-learning problem. We also show that OSCBO preserves sublinear regret bounds by leveraging the theoretical guarantees of the underlying online learning algorithm. Empirically, OSCBO performs competitively across synthetic and real-world benchmarks, ranking among the strongest methods in final simple regret while maintaining robust cumulative-regret behavior.

Tuan A. Vu, Julien Martinelli, Harri Lähdesmäki

Latent-space Bayesian optimization (LSBO) extends Bayesian optimization to structured domains, such as molecular design, by searching in the continuous latent space of a generative model. However, most LSBO methods assume a fixed objective, whereas real design campaigns often face temporal drift (e.g., evolving preferences or shifting targets). Bringing time-varying BO into LSBO is nontrivial: drift can affect not only the surrogate, but also the latent search space geometry induced by the representation. We propose Time-Aware Latent-space Bayesian Optimization (TALBO), which incorporates time in both the surrogate and the learned generative representation via a GP-prior variational autoencoder, yielding a latent space aligned as objectives evolve. To evaluate timevarying LSBO systematically, we adapt widely used molecular design tasks to drifting multi-property objectives and introduce metrics tailored to changing targets. Across these benchmarks, TALBO consistently outperforms strong LSBO baselines and remains robust across drift speeds and design choices, while remaining competitive under actually time-invariant objectives.

Nicolas Samuel Blumer, Julien Martinelli, Samuel Kaski

Black-box optimization in science and engineering often comes with side information: experts, simulators, pretrained predictors, or heuristics can suggest which candidates look promising. This information can accelerate search, but it can also be biased, input-dependent, or misleading. Feedback-aware BO methods typically handle one task at a time, limiting their ability to generalize over multiple sources of feedback. In-context optimizers address cross-task adaptation, but usually assume that optimization history is the only available signal at test time. We study feedback-informed in-context black-box optimization (FICBO), where a pretrained optimizer conditions on both the observed history and cheap auxiliary feedback for the current candidate set. We introduce a structured feedback prior that models how feedback sources vary in their access, relevance, and distortion relative to the true objective, and use it to pretrain a feedback-aware transformer. At test time, the model estimates source reliability in context by comparing observed objective values with auxiliary signals, improving query selection. On synthetic and real-world tasks, FICBO effectively exploits informative feedback while remaining robust to weak or misleading sources, improving over other baselines. Empirical investigations further illustrate how the model perceives test-time sources, offering insights into its interpretability and decision-making process.

Xinyu Zhang, Julien Martinelli, ST John

We review generalized additive models as a type of ``transparent'' model that has recently seen renewed interest in the deep learning community as neural additive models. We highlight multiple types of nonidentifiability in this model class and discuss challenges in interpretability, arguing for restraint when claiming ``interpretability'' or ``suitability for safety-critical applications'' of such models.

Xinyu Zhang, Daolang Huang, Samuel Kaski et al.

Preferential Bayesian Optimization (PBO) is a sample-efficient method to learn latent user utilities from preferential feedback over a pair of designs. It relies on a statistical surrogate model for the latent function, usually a Gaussian process, and an acquisition strategy to select the next candidate pair to get user feedback on. Due to the non-conjugacy of the associated likelihood, every PBO step requires a significant amount of computations with various approximate inference techniques. This computational overhead is incompatible with the way humans interact with computers, hindering the use of PBO in real-world cases. Building on the recent advances of amortized BO, we propose to circumvent this issue by fully amortizing PBO, meta-learning both the surrogate and the acquisition function. Our method comprises a novel transformer neural process architecture, trained using reinforcement learning and tailored auxiliary losses. On a benchmark composed of synthetic and real-world datasets, our method is several orders of magnitude faster than the usual Gaussian process-based strategies and often outperforms them in accuracy.

Clémence Métayer, Annabelle Ballesta, Julien Martinelli

Recent technological advances have expanded the availability of high-throughput biological datasets, enabling the reliable design of digital twins of biomedical systems or patients. Such computational tools represent key reaction networks driving perturbation or drug response and can guide drug discovery and personalized therapeutics. Yet, their development still relies on laborious data integration by the human modeler, so that automated approaches are critically needed. The success of data-driven system discovery in Physics, rooted in clean datasets and well-defined governing laws, has fueled interest in applying similar techniques in Biology, which presents unique challenges. Here, we reviewed methodologies for automatically inferring digital twins from biological time series, which mostly involve symbolic or sparse regression. We evaluate algorithms according to eight biological and methodological challenges, associated to noisy/incomplete data, multiple conditions, prior knowledge integration, latent variables, high dimensionality, unobserved variable derivatives, candidate library design, and uncertainty quantification. Upon these criteria, sparse regression generally outperformed symbolic regression, particularly when using Bayesian frameworks. We further highlight the emerging role of deep learning and large language models, which enable innovative prior knowledge integration, though the reliability and consistency of such approaches must be improved. While no single method addresses all challenges, we argue that progress in learning digital twins will come from hybrid and modular frameworks combining chemical reaction network-based mechanistic grounding, Bayesian uncertainty quantification, and the generative and knowledge integration capacities of deep learning. To support their development, we further propose a benchmarking framework to evaluate methods across all challenges.

Sabina J. Sloman, Julien Martinelli, Samuel Kaski

Generalization outside the scope of one's training data requires leveraging prior knowledge about the effects that transfer, and the effects that don't, between different data sources. Transfer learning is a framework for specifying and refining this knowledge about sets of source (training) and target (prediction) data. A challenging open problem is addressing the empirical phenomenon of negative transfer, whereby the transfer learner performs worse on the target data after taking the source data into account than before. We first introduce a Bayesian perspective on negative transfer, and then a method to address it. The key insight from our formulation is that negative transfer can stem from misspecified prior information about non-transferable causes of the source data. Our proposed method, proxy-informed robust method for probabilistic transfer learning (PROMPT), does not require prior knowledge of the source data (the data sources may be "unknown"). PROMPT is thus applicable when differences between tasks are unobserved, such as in the presence of latent confounders. Moreover, the learner need not have access to observations in the target task (may not have the ability to "fine-tune"), and instead makes use of proxy (indirect) information. Our theoretical results show that the threat of negative transfer does not depend on the informativeness of the proxy information, highlighting the usefulness of PROMPT in cases where only noisy indirect information, such as human feedback, is available.

Marshal Arijona Sinaga, Julien Martinelli, Vikas Garg et al.

Preferential Bayesian optimization (PBO) is a sample-efficient framework for learning human preferences between candidate designs. PBO classically relies on homoscedastic noise models to represent human aleatoric uncertainty. Yet, such noise fails to accurately capture the varying levels of human aleatoric uncertainty, particularly when the user possesses partial knowledge among different pairs of candidates. For instance, a chemist with solid expertise in glucose-related molecules may easily compare two compounds from that family while struggling to compare alcohol-related molecules. Currently, PBO overlooks this uncertainty during the search for a new candidate through the maximization of the acquisition function, consequently underestimating the risk associated with human uncertainty. To address this issue, we propose a heteroscedastic noise model to capture human aleatoric uncertainty. This model adaptively assigns noise levels based on the distance of a specific input to a predefined set of reliable inputs known as anchors provided by the human. Anchors encapsulate partial knowledge and offer insight into the comparative difficulty of evaluating different candidate pairs. Such a model can be seamlessly integrated into the acquisition function, thus leading to candidate design pairs that elegantly trade informativeness and ease of comparison for the human expert. We perform an extensive empirical evaluation of the proposed approach, demonstrating a consistent improvement over homoscedastic PBO.

Xinyu Zhang, Conor Hassan, Julien Martinelli et al.

Balancing competing objectives is omnipresent across disciplines, from drug design to autonomous systems. Multi-objective Bayesian optimization is a promising solution for such expensive, black-box problems: it fits probabilistic surrogates and selects new designs via an acquisition function that balances exploration and exploitation. In practice, it requires tailored choices of surrogate and acquisition that rarely transfer to the next problem, is myopic when multi-step planning is often required, and adds refitting overhead, particularly in parallel or time-sensitive loops. We present TAMO, a fully amortized, universal policy for multi-objective black-box optimization. TAMO uses a transformer architecture that operates across varying input and objective dimensions, enabling pretraining on diverse corpora and transfer to new problems without retraining: at test time, the pretrained model proposes the next design with a single forward pass. We pretrain the policy with reinforcement learning to maximize cumulative hypervolume improvement over full trajectories, conditioning on the entire query history to approximate the Pareto frontier. Across synthetic benchmarks and real tasks, TAMO produces fast proposals, reducing proposal time by 50-1000x versus alternatives while matching or improving Pareto quality under tight evaluation budgets. These results show that transformers can perform multi-objective optimization entirely in-context, eliminating per-task surrogate fitting and acquisition engineering, and open a path to foundation-style, plug-and-play optimizers for scientific discovery workflows.

Julien Martinelli

Physics-Informed Machine Learning (PIML) has successfully integrated mechanistic understanding into machine learning, particularly in domains governed by well-known physical laws. This success has motivated efforts to apply PIML to biology, a field rich in dynamical systems but shaped by different constraints. Biological modeling, however, presents unique challenges: multi-faceted and uncertain prior knowledge, heterogeneous and noisy data, partial observability, and complex, high-dimensional networks. In this position paper, we argue that these challenges should not be seen as obstacles to PIML, but as catalysts for its evolution. We propose Biology-Informed Machine Learning (BIML): a principled extension of PIML that retains its structural grounding while adapting to the practical realities of biology. Rather than replacing PIML, BIML retools its methods to operate under softer, probabilistic forms of prior knowledge. We outline four foundational pillars as a roadmap for this transition: uncertainty quantification, contextualization, constrained latent structure inference, and scalability. Foundation Models and Large Language Models will be key enablers, bridging human expertise with computational modeling. We conclude with concrete recommendations to build the BIML ecosystem and channel PIML-inspired innovation toward challenges of high scientific and societal relevance.

Marshal Arijona Sinaga, Julien Martinelli, Samuel Kaski

Gaussian processes (GPs) are widely used for regression and optimization tasks such as Bayesian optimization (BO) due to their expressiveness and principled uncertainty estimates. However, in settings with large datasets corrupted by outliers, standard GPs and their sparse approximations struggle with computational tractability and robustness. We introduce Robust Computation-aware Gaussian Process (RCaGP), a novel GP model that jointly addresses these challenges by combining a principled treatment of approximation-induced uncertainty with robust generalized Bayesian updating. The key insight is that robustness and approximation-awareness are not orthogonal but intertwined: approximations can exacerbate the impact of outliers, and mitigating one without the other is insufficient. Unlike previous work that focuses narrowly on either robustness or approximation quality, RCaGP combines both in a principled and scalable framework, thus effectively managing both outliers and computational uncertainties introduced by approximations such as low-rank matrix multiplications. Our model ensures more conservative and reliable uncertainty estimates, a property we rigorously demonstrate. Additionally, we establish a robustness property and show that the mean function is key to preserving it, motivating a tailored model selection scheme for robust mean functions. Empirical results confirm that solving these challenges jointly leads to superior performance across both clean and outlier-contaminated settings, both on regression and high-throughput Bayesian optimization benchmarks.

Julien Martinelli, Ayush Bharti, Armi Tiihonen et al.

Contextual Bayesian Optimization (CBO) efficiently optimizes black-box functions with respect to design variables, while simultaneously integrating contextual information regarding the environment, such as experimental conditions. However, the relevance of contextual variables is not necessarily known beforehand. Moreover, contextual variables can sometimes be optimized themselves at an additional cost, a setting overlooked by current CBO algorithms. Cost-sensitive CBO would simply include optimizable contextual variables as part of the design variables based on their cost. Instead, we adaptively select a subset of contextual variables to include in the optimization, based on the trade-off between their relevance and the additional cost incurred by optimizing them compared to leaving them to be determined by the environment. We learn the relevance of contextual variables by sensitivity analysis of the posterior surrogate model while minimizing the cost of optimization by leveraging recent developments on early stopping for BO. We empirically evaluate our proposed Sensitivity-Analysis-Driven Contextual BO (SADCBO) method against alternatives on both synthetic and real-world experiments, together with extensive ablation studies, and demonstrate a consistent improvement across examples.