No-Regret Bayesian Optimization with Finite-Library Input-Warped Kernels

Gaussian-process Bayesian optimization (GP-BO) excels at black-box optimization of costly functions, e.g., hyperparameter optimization (HPO) and multi-agent system (MAS) design. Convergence-rate guarantees exist for select methods, notably GP upper confidence bound (GP-UCB), but require a fixed kernel. Critically, the kernel encodes how input proximity affects objective value similarity. When raw coordinates poorly match this geometry - as with log-scaled hyperparameters or localized peaks - input warping can greatly improve sample efficiency, yet known GP-UCB proofs require a fixed kernel. We propose Finite-Library Input-Warped Bayesian Optimization (FLIWBO), which selects warps from a finite library of smooth input maps by any history-dependent rule. It adapts the input geometry to accelerate learning while retaining high-probability convergence guarantees under mild hypotheses, with an explicit $\sqrt(N_\varepsilon)$ library-size cost. Controlled diagnostics show that finite-library warping repairs planted geometry mismatches and identify FLIWBO failure cases. Across four repeated benchmarks - warped synthetic objectives, a confidence-fence trap, and Fashion-MNIST HPO - FLIWBO-UCB beats raw-coordinate GP-UCB under misspecified geometry, escapes traps that defeat even oracle-warp expected improvement, and recovers much of the gain from manual log scaling, while leading the tested methods that admit a matching regret guarantee. A 20-dimensional MAS design study further shows feasibility under costly noisy evaluations. Code for experiments is available: https://github.com/edvin-ketabati/bogp-paper-experiments.

Tabular Deep Learning for Algorithmic Trading: Cross-Regime Bayesian Optimisation for Equity Signal Generation

Algorithmic trading now represents a market exceeding $20 billion, where even marginal gains in signal robustness can translate into economically significant returns. Existing evaluations of equity prediction models do not explicitly target regime robustness during hyperparameter selection. Five model classes are trained on daily observations from approximately 300 large-cap US equities over eleven years, with Bayesian optimisation configured to target trading performance across three statistically different market regimes. Regime-robust hyperparameter selection is associated with out-of-sample generalisation, as signal precision remains above the random baseline across all four quarters of the test period, and portfolio performance slowly degrades under simulated input noise before collapsing beyond a defined threshold. No individual tabular deep learning architecture outperforms gradient-boosted trees, but combining XGBoost and TabNet using rank aggregation produces a Hybrid ensemble with an annualised return of 51.26%, a Sharpe ratio of 2.44, and a statistically significant CAPM alpha of 0.423 (p = 0.011). A near-zero beta indicates this outperformance is driven by stock selection, not market exposure. Alternative data plays a secondary role once technical and fundamental features are accounted for, as well as contributing more strongly on the short side than the long, and varies by model class. An interactive application makes these results explorable in real time, with live data integration the remaining step toward practical deployment.

Enhancing Bayesian Optimization and Active Learning Through Kernel Diversity

Hyperparameter selection remains a key challenge in Bayesian optimization (BO) and Bayesian active learning (AL), as model misspecification can lead to suboptimal performance, while more accurate fully Bayesian treatments typically rely on computationally expensive MCMC sampling. This paper proposes a unified framework, KENDO (Kernel ENsemble Disagreement-aware Operator), that integrates Ensemble Gaussian Processes (EGP) with disagreement-aware acquisition strategies. The central idea is to replace hyperparameter sampling with a kernel ensemble and adaptive Bayesian weighting, combined with disagreement-aware acquisition strategies. Within this unified framework, we instantiate KENDO-BO for BO and KENDO-AL for Bayesian AL, demonstrating that both arise from a common self-correcting mechanism with task-specific acquisition objectives. We further extend the approach to multi-objective optimization via random scalarization that preserves the single-optimizer conditioning structure. Thorough numerical tests on synthetic and real-world benchmarks across single-objective optimization, multi-objective optimization, and active learning demonstrate that (i) KENDO-BO achieves competitive or superior optimization performance compared to state-of-the-art methods while reducing computational overhead by up to $5\times$ and (ii) KENDO-AL achieves superior predictive calibration over MCMC-based active learning baselines with up to $27\times$ speedup.

On the Brittleness of Maximum Likelihood Estimation for Gaussian Process Hyperparameter Optimization

Machine learning (ML) has become an indispensable part of modern engineering design workflows. A crucial step in training an ML model is the selection of the loss function which can be systematically formulated via various techniques such as maximum likelihood estimation (MLE) and cross-validation . While MLE is one of the most popular, effective, and intuitive mechanisms for training ML models, it is brittle: if the assumptions underpinning it are not met, the trained ML model may generalize poorly. This brittleness affects even Gaussian processes (GPs) which are widely used in engineering design and are often (incorrectly) presumed to be very robust to overfitting. In this paper, we fundamentally evaluate the brittleness of MLE in the context of training GPs for probabilistic regression or classification tasks. We compare theoretically grounded metrics against MLE and propose practical solutions. Our extensive studies demonstrate the effectiveness of our solutions in downstream design tasks such as Bayesian optimization and provide a blueprint for practitioners to build accurate and robust GPs that can even outperform tabular foundation models in terms of prediction accuracy, uncertainty quantification, and inference cost. Our contributions are publicly available via GitHub at https://github.com/Bostanabad-Research-Group/GP-vs-TabPFN-vs-GPyTorch.

Adaptive KappaSharp: Condition-Number Shaping for Preferential Bayesian Optimization

Preferential Bayesian optimization (PBO) optimizes objectives accessible only through pairwise user comparisons. The standard approach fits a Gaussian process surrogate for observed pairwise comparisons (PairwiseGP) using the Laplace approximation and selects queries with the Expected Utility of Best Option (EUBO) acquisition function. EUBO queries new candidates at each step, producing pairs that share no candidates with previous queries. Each such pair forms an isolated component in the comparison graph, removing one degree of freedom from the likelihood Hessian and making it rank-deficient. This deficiency is structural and cannot be resolved by changing the surrogate modeling approach. Existing approaches to remedy this issue either waste query budget by forcing comparisons to stay connected, or apply uniform regularization that also perturbs directions already well-constrained by the observed comparisons. We propose KappaSharp that enables a diagonal correction to the Hessian to reduce its condition number, with larger corrections where the prior uncertainty is higher. The correction is only applied in the model fitting step, not query selection. An adaptive variant of KappaSharp is also presented that activates the correction only when the surrogate is confident about recent comparisons, avoiding unnecessary corrections when the problem is well-conditioned. On 11 benchmarks (5--20 dimensions), including a 16-dimensional controller tuning problem in plasma medicine, Adaptive KappaSharp outperforms the standard PBO baseline, with up to +10.9% ($p{=}0.003$).

Simple-regret rates and minimax optimality of fixed-prior expected improvement in Matérn and squared-exponential RKHSs

We study the expected improvement (EI) policy for minimizing a deterministic objective function $f$ on a nonempty compact set $\mathcal X \subset\mathbb R^d$. We assume that $f$ belongs to the RKHS $\mathcal H_k$ of a continuous positive-semidefinite kernel $k$ on $\mathcal X$. Function values are observed exactly, and EI is computed from a fixed zero-mean Gaussian-process model with covariance $σ^2k$. After an initial design, the policy queries a point whose EI is at least a fixed positive fraction of its maximum. We identify the normalized posterior standard deviation at a candidate point $x$ with the norm of the corresponding innovation in the canonical feature space, namely the component of $k(x,\cdot)$ orthogonal to the span of the preceding evaluation representers. Sequential separation radii bound the ranked innovation norms along arbitrary query sequences. We estimate these radii using Gram determinants and Kolmogorov widths for subspaces of different dimensions, then combine the estimates with a one-step regret inequality to obtain finite-budget bounds for simple regret. After $N$ post-initial queries, simple regret is $O(N^{-ν/d})$ for isotropic Matérn kernels of smoothness $ν>0$. For the isotropic squared-exponential kernel, simple regret is $O(\exp[-c_1\min\{N, N^{1/d}\log(eN)\}])$ for some $c_1>0$. With exact EI maximization, it is $O(\exp[-c_2N^{1/d} \log(eN)])$ for some $c_2>0$. For every fixed $B\geq0$, these bounds are uniform over the RKHS ball of radius $B$. If $\mathcal X$ has nonempty interior and $B>0$, then, among deterministic methods whose final recommendation may be any point of $\mathcal X$, the exact EI policy is minimax-rate optimal over the RKHS ball of radius $B$ for Matérn kernels and minimax-rate optimal up to constants in the exponent for squared-exponential kernels.

Expert-Guided Forecast Editing for Time-Series Foundation Models

Time-series foundation models can forecast across heterogeneous domains without task-specific training, but their forecasts are fixed once produced and cannot directly incorporate task-specific expert feedback. We study expert-guided forecast editing: a frozen foundation model generates candidate future trajectories, and an expensive expert evaluator scores them to guide forecast revision. Under a tight query budget, two natural strategies sit at opposite ends: best-of-$N$ purely exploits the foundation model's predictive distribution, while optimization approaches mostly explore the forecast horizon as an unstructured high-dimensional vector. Each extreme is individually sub-optimal. We introduce \textbf{DEFT}, an expert-guided forecast editing framework that balances the two by first exploiting the foundation model's predictive samples in a decomposed trend--seasonal space, then exploring around them via component-wise refinement. DEFT queries the expert only on complete trajectories, then reuses scores for the trend and seasonal components that appeared in the queried recombinations. This lets each expert query provide structured component-level feedback while keeping the foundation model frozen. We compare DEFT against direct search approaches, including best-of-$N$, cross-entropy methods, and Bayesian optimization, under matched expert-query budgets. Across two forecasting benchmarks consisting of 78 datasets, three time-series foundation models, four feedback types, and seven query budgets, DEFT consistently improves the effectiveness of expert guidance. A molecular-dynamics case study further suggests that the same principle extends to more physically grounded feedback, supporting the hypothesis that sparse test-time guidance should be spent balancing prior exploitation with structured exploration.

Modernizing HEBO: a robust Bayesian optimization baseline for practical heteroskedastic and non-stationary problems

Bayesian optimization is increasingly used to guide data-efficient experimentation in chemistry, materials science, and related laboratory settings, but its practical performance depends strongly on how well surrogate-model assumptions match the geometry and noise structure of the underlying objective. We introduce tidyHEBO, a robust Bayesian optimization model inspired by heteroskedastic evolutionary Bayesian optimization (HEBO) for single-objective, sequential optimization. tidyHEBO reconstructs the HEBO design philosophy in BoTorch and revises surrogate training, output-warping selection, acquisition function evaluation, and Pareto-front search. We benchmarked tidyHEBO on synthetic functions, Olympus emulators, fully experimental reaction-optimization datasets, needle-in-a-haystack (NIAH) materials problems, and Bayesmark hyperparameter optimization tasks. On these tasks tidyHEBO achieved competitive to superior performance and improvement in robustness across repeated optimization runs. We therefore propose tidyHEBO as a practical tool for sequential experimentations and a strong general-purpose benchmark for future Bayesian optimization research.

Pitfalls and Remedies for Multi-Task Bayesian Optimization

Bayesian optimization routinely warm-starts a target experiment with data from related source tasks, and the multi-task Gaussian process is the textbook surrogate for the job. We revisit this default in a controlled setting and find that it misestimates the cross-task correlation even in the simplest non-trivial case, affinely related source and target tasks, where a working transfer learning method should obviously succeed. We trace the failure to two independent structural mechanisms. Per-task standardization, the textbook fix for the affine slice ambiguity, propagates a finite-sample alignment error into the recovered correlation. The marginal likelihood itself identifies the correlation only at a per-sample rate that a Gaussian process at non-overlapping designs further dilutes. We propose three conservative remedies that follow from the analysis: promoting per-task means and scales to model parameters, restricting the task covariance to non-negative correlations, and co-locating part of the source and target designs. Across synthetic multi-task problems and surrogate-based hyperparameter tuning transfer, these remedies recover the target-only baseline on the simple instances, while the broader failure persists on harder instances and across most rank-based and latent-context variants.

ALAS: Additive Learnable Alpha-Stable Kernels for Flexible Bayesian Optimization

Bayesian Optimization is widely used for expensive black-box optimization, yet its success often depends on choosing a kernel that matches the objective's unknown structure. In this work, we propose ALAS, a flexible Gaussian Process kernel family built from symmetric $α$-stable spectral components. By learning the stability parameter $α$, ALAS adapts its effective smoothness from data, capturing both smooth trends and sharp irregularities. We present two parameterizations: ALAS, a single stationary component with joint spectral modulation, and ALAS-Sep, a separable variant that learns dimension-wise tail behavior to improve robustness on approximately decomposable objectives. Experiments on standard benchmarks and real-world surrogates demonstrate strong and robust performance across diverse settings.

B3O: Scalable Boltzmann Batch Bayesian Optimization

Modern engineering workflows increasingly rely on massive parallel simulation, driving the need for scalable, large-batch Bayesian Optimization (BO). Existing batch BO methods, however, incur large computational cost or rely on approximations that erode batch diversity. We propose B3O (Boltzmann Batch Bayesian Optimization), a framework that reframes batch generation as a pure sampling problem: drawing samples directly from the Boltzmann distribution defined by the acquisition function avoids the bottlenecks of existing large-batch methods. Theoretically, we prove that queries sampled from this distribution incur only negligible additional regret. Empirically, B3O outperforms existing batch BO methods on standard synthetic benchmarks and adapts robustly across complex applied tasks, including multi-objective electrode design and mixed-variable race car configuration.

Efficient Adaptive Data Acquisition via Pretrained Belief Representations

Learning effective policies for adaptive data acquisition remains challenging: posterior-based methods rely on surrogate models and posterior approximations that can be misspecified or biased, while direct policy-learning methods map from historical observations and fail to exploit available model representations, making learning harder. We introduce policy learning with belief representations (POLAR), based on the insight that optimal data acquisition depends on the observation history only through a sufficient belief state. Specifically, POLAR decouples representation learning from policy learning by leveraging pretrained predictive foundation models as belief-state encoders, training a policy head on top of their representations. This yields a simple, unified amortised policy learning framework for Bayesian experimental design, Bayesian optimisation, and active learning, differing only in the task-specific utility used to train the policy. Empirically, we find that POLAR outperforms state-of-the-art amortised methods across diverse tasks while requiring far fewer training samples, demonstrating a significant step in the scalability and efficiency of amortised data acquisition.

Target-confidence Recourse Using tSeTlin machines: TRUST

Counterfactual explanations are widely used to provide algorithmic recourse in high-stakes decision-making systems. Most existing methods seek the smallest change to an input that flips a model's decision. However, decision-makers often rely not only on predicted labels but also on confidence thresholds and risk margins. Counterfactuals that barely cross a decision boundary can be fragile and unstable under noise or model variation. In this paper, we propose Target-confidence Recourse Using tSeTlin machines (TRUST), a framework in which users explicitly specify the desired prediction confidence for recourse. Rather than generating counterfactuals and evaluating confidence afterward, TRUST directly searches for minimal changes that satisfy a user-defined confidence target, enabling comparison of recourse options in terms of cost, confidence, and robustness. We instantiate TRUST using a Probabilistic Tsetlin Machine (PTM) combined with Bayesian optimization. The probabilistic clause-based structure of PTM links prediction confidence to the stability of decision rules. We show that counterfactuals satisfying the same rules can still differ substantially in reliability depending on how securely they satisfy those rules, revealing whether decisions are supported by robust or fragile clause activations. Experiments on synthetic and real-world datasets demonstrate that target-confidence counterfactuals produce more robust and interpretable recourse than conventional boundary-based approaches. Across multiple benchmarks, TRUST achieves perfect robustness while maintaining low recourse cost, including an L2 distance of 0.10 on the Haberman dataset at 0.92 confidence. By explicitly controlling confidence and exposing rule-level stability, TRUST provides actionable recourse for high-stakes decision support.

Improving Bayesian Optimization via Training-Aware Conditional Diffusion Models

Bayesian optimization (BO) is a widely used approach for black-box optimization that uses a Gaussian process (GP) as a surrogate and guides sequential evaluations via an acquisition function, with the ultimate goal of locating the global optimum $\mathbf{x}^{\star}$. To align with this goal, information-based acquisition functions such as Predictive Entropy Search (PES) model $\mathbf{x}^{\star}$ as a random variable and reduce the entropy of its distribution, but approximating this distribution via traditional GP posterior sampling is computationally expensive. To address this limitation, we leverage Conditional Diffusion Models (CDMs) to efficiently approximate the distribution of $\mathbf{x}^{\star}$ and develop BO-inherent training strategies for CDMs. Motivated by the structural properties of the CDM-learned distribution, we further develop an acquisition strategy termed Diffusion-based Mode Seeking (DMS) to guide the sequential evaluation. We establish a sub-optimality guarantee for the CDM-learned distribution and demonstrate through extensive experiments that DMS outperforms standard BO baselines.

$α$-PFN: Fast Entropy Search via In-Context Learning

Information-theoretic acquisition functions such as Entropy Search (ES) offer a principled exploration-exploitation framework for Bayesian optimization (BO). However, their practical implementation relies on complicated and slow approximations, i.e., a Monte Carlo estimation of the information gain. This complexity can introduce numerical errors and requires specialized, hand-crafted implementations. We propose a two-stage amortization strategy that learns to approximate entropy search-based acquisition functions using Prior-data Fitted Networks (PFNs) in a single forward pass. A first PFN is trained to be conditioned on information about the optima; second, the $α$-PFN is trained to predict the expected information gain by training on information gains measured with the first PFN. The $α$-PFN offers a flexible learned approximation, which replaces the complex heuristic approximations with a single forward pass per candidate, enabling rapid and extensible acquisition evaluation. Empirically, our approach is competitive with state-of-the-art entropy search implementations on synthetic and real-world benchmarks, while accelerating the different entropy search variants across all our experiments, with speed ups over 50x. Source code: https://github.com/automl/AlphaPFN.

Optuna Constrained Tree-Structured Parzen Estimator Is a Joint Density Generalization of c-TPE

Constrained hyperparameter optimization (HPO) is common in practice, yet Optuna's widely used constrained TPE lacks algorithmic analysis. While c-TPE proposes an expected constrained improvement (ECI) approach assuming independence between the objective and constraints, Optuna uses a single joint density over both. We show that Optuna's constrained TPE is joint c-TPE -- the same ECI acquisition function using a joint likelihood. We demonstrate joint c-TPE is invariant to constraint duplication whereas independent c-TPE degrades as the product accumulates duplicated factors. We outline practical tradeoffs between the formulations and directions for future study.

Regret-Based $(ε,δ)$-optimal Stopping Criteria for Bayesian Optimization

Bayesian optimization (BO) is a widely used iterative black-box optimization method that utilizes Gaussian process (GP) surrogate models. In practice, BO is typically terminated after a fixed evaluation budget is exhausted, which can incur unnecessary cost and provides no optimality guarantee on solution quality. Recent research in developing a practical stopping criterion has made empirical progress, yet a theoretically sound stopping criterion remains a work in progress. In this work, we present provably tighter instantaneous regret bounds for GP upper confidence bound (GP-UCB) at any given iteration. Then, we propose stopping criteria for GP-UCB based on this tighter bound that ensures an $ε$-optimal solution with high probability $1-δ$ upon termination. Numerical experiments are performed to validate and demonstrate the effectiveness and efficiency of our stopping criteria.

Sequential Inference for Gaussian Processes: A Signal Processing Perspective

The proliferation of capable and efficient machine learning (ML) models marks one of the strongest methodological shifts in signal processing (SP) in its nearly 100-year history. ML models support the development of SP systems that represent complex, nonlinear relationships with high predictive accuracy. Adapting these models often requires sequential inference, which differs both theoretically and methodologically from the usual paradigm of ML, where data are often assumed independent and identically distributed. Gaussian processes (GPs) are a flexible yet principled framework for modeling random functions, and they have become increasingly relevant to SP as statistical and ML methods assume a more prominent role. We provide a self-contained, tutorial-style overview of GPs, with a particular focus on recent methodological advances in sequential, incremental, or streaming inference. We introduce these techniques from a signal-processing perspective while bridging them to recent advances in ML. Many of the developments we survey have direct applications to state-space modeling, sequential regression and forecasting, anomaly detection in time series, sequential Bayesian optimization, adaptive and active sensing, and sequential detection and decision-making. By organizing these advances from a signal-processing perspective, we intend to equip practitioners with practical tools and a coherent roadmap for deploying sequential GP models in real-world systems.