ICML 2026 Papers

Many real-world optimization problems are guided by complex, subjective preferences that are difficult to express as explicit closed-form objectives. In response, we introduce Language-in-the-Loop Optimization (LILO), a Bayesian optimization (BO) framework that employs a large language model (LLM) to translate free-form natural language feedback and prior knowledge from a decision maker into structured preference signals, going beyond the restrictive scalar or pairwise feedback formats typically assumed in preferential BO. The LLM-derived preferences are integrated by a Gaussian process proxy model, enabling principled acquisition-driven exploration with calibrated uncertainty. By placing the LLM in a supporting role rather than as the optimizer itself, LILO preserves the sample efficiency and stability of BO while providing a flexible and expressive feedback interface. Across synthetic and real-world benchmarks, LILO consistently outperforms both conventional preference-based BO methods and LLM-only optimizers, with particularly strong gains in feedback-limited regimes. The code for reproducing our experimental results is available at: https://github.com/facebookresearch/lilo.

Large language models are typically optimized for accuracy and, therefore, will guess even when uncertain about their predictions. This problem becomes especially pronounced when the model is fine-tuned on small datasets, which often causes overfitting and results in a tendency toward miscalibration. In this work, we introduce Bayesian-LoRA, which reformulates the deterministic LoRA update as a probabilistic low-rank representation inspired by Sparse Gaussian Processes (SGP). We identify a structural isomorphism between LoRA's factorization and Kronecker-factored SGP posteriors, and show that LoRA emerges as a limiting case when posterior uncertainty collapses. We conduct extensive experiments on various LLM architectures across commonsense reasoning, language modeling, and mathematical reasoning benchmarks. With only approximately 0.42M additional parameters and ${\approx}1.2{\times}$ training cost relative to standard LoRA, Bayesian-LoRA significantly improves calibration across models from 7B up to 30B, achieving up to 84\% Expected Calibration Error (ECE) and 76\% Negative Log-Likelihood (NLL) reduction while maintaining competitive accuracy for both in-distribution and out-of-distribution (OoD) evaluations.

Electronic health records (EHR) pose large-scale multi-disease modeling problems in which many outcomes are rare and strongly influenced by shared risk factors. While modern approaches achieve strong predictive performance, they often treat diseases independently or rely on black-box architectures, offering limited insight into how risk factors organize disease risk and little principled uncertainty quantification. We introduce a Bayesian hypergraph inference framework that reframes multi-disease modeling around **latent, risk-factor-modulated disease pathways**. Risk factors act on hyperedges, latent disease subsets with shared risk patterns, allowing diseases to participate in multiple distinct pathways and enabling interpretable, higher-order structure beyond pairwise associations. A repulsion prior encourages parsimonious and identifiable structure, while posterior inference provides calibrated uncertainty over both disease groupings and risk-factor influence. To enable scalable inference on large EHR datasets, we develop a structured variational inference algorithm that preserves logical dependencies among hyperedge existence, disease membership, and pathway-level effects. Experiments on simulated data and UK Biobank demonstrate stable and interpretable disease pathway structure, well-calibrated uncertainty, improved estimation for rare diseases, and competitive predictive performance.

Offline reinforcement learning (RL) aims to optimize policies from pre-collected datasets. A bottleneck of this paradigm is managing epistemic uncertainty, which arises from limited data coverage (sample-level) and the ambiguity in identifying transition dynamics from finite data (model-level). To provide a unified quantification of these uncertainties, Bayesian RL has been proposed by treating the dynamics model as a random variable and maintaining a corresponding belief. Despite its theoretical appeal, policy optimization in Bayesian RL remains computationally challenging as it requires solving composite objectives with expectations. Prior methods either employ search-based techniques with poor computational scalability or impose restrictive posterior assumptions that sacrifice the adaptability of Bayesian RL. To address these limitations, we propose Posterior Hybrid Bayesian Belief (PhyB), which reformulates the expectation as a convex combination over a subset of dynamics models. Theoretical analysis demonstrates that the objective discrepancy induced by this approximation remains bounded. Based on PhyB, we develop an iterative regularized policy optimization algorithm that provides metric-agnostic guarantees for monotonic improvement until convergence. Empirical results demonstrate that PhyB achieves state-of-the-art performance on various benchmarks.

Bayesian neural networks promise calibrated uncertainty but require $O(mn)$ parameters for standard mean-field Gaussian posteriors. We argue this cost is often unnecessary, particularly when weight matrices exhibit fast singular value decay. By parameterizing weights as $W = AB^{\top}$ with $A \in \mathbb{R}^{m \times r}$, $B \in \mathbb{R}^{n \times r}$, we induce a posterior that is \emph{singular} with respect to the Lebesgue measure, concentrating on the rank-$r$ manifold. This singularity captures structured weight correlations through shared latent factors, geometrically distinct from mean-field's independence assumption. We derive PAC-Bayes generalization bounds whose complexity term scales as $\sqrt{r(m+n)}$ instead of $\sqrt{m n}$, and prove loss bounds that decompose the error into optimization and rank-induced bias using the Eckart-Young-Mirsky theorem. We further adapt recent Gaussian complexity bounds for low-rank deterministic networks to Bayesian predictive means. Empirically, across MLPs, LSTMs, and Transformers on standard benchmarks, our method achieves competitive predictive performance while using up to $33\times$ fewer parameters than 5-member Deep Ensembles. It substantially improves OOD detection and often improves calibration relative to mean-field and perturbation baselines, while Deep Ensembles can still be stronger on in-distribution likelihood-based metrics.

Low-rank tensor decomposition (TD) is usually effective on clean, fully observed data, but it often degrades under severe missingness or noise. Low-rankness is itself a useful but limited structural prior, and additional handcrafted priors (e.g., sparsity or smoothness) still fall short of capturing the rich statistics of real-world data. To compensate for this weak inductive bias under heavy corruption, one would like to inject a learned, data-driven prior; however, the state-of-the-art diffusion models are not readily compatible with current TD and tractable posterior inference. To address these challenges, we introduce DiffBCP, a hybrid-prior Bayesian CP decomposition framework that couples a cumulative shrinkage process prior over the CP factors for automatic rank selection with an off-the-shelf pre-trained diffusion model as an implicit data prior on the reconstructed tensor. To make posterior inference tractable despite the coupling among the likelihood, low-rank constraint, and diffusion prior, we develop a split Gibbs sampler: CP factors admit conjugate updates, while the diffusion block is sampled via low-rank-guided denoising. A noise-adaptive coupling schedule further reduces sensitivity to hand-tuned annealing. Experiments on image inpainting and denoising, including high-resolution out-of-distribution images, show consistent gains over Bayesian, nonlinear, and plug-and-play TD baselines.

Epistemic uncertainty is often viewed as a reducible uncertainty that vanishes with increasing data. This perspective implicitly assumes parameter identifiability and equates epistemic uncertainty with predictive variability. In overparametrized neural networks, however, model parameters are typically non-identifiable due to symmetries and redundant representations. As a consequence, substantial parameter uncertainty can persist even when the underlying function is fully identified. In this work, we analyze epistemic uncertainty through the lens of non-identifiability and characterize both discrete and continuous sources of residual uncertainty. Focusing on one-hidden-layer ReLU networks, we thoroughly analyze the resulting posterior structure and validate our theoretical insights through empirical studies.

Bayesian inference provides principled uncertainty quantification but is often limited by the challenges of prior and likelihood elicitation. The martingale posterior (MGP) (Fong et al., 2023) offers an alternative by replacing these requirements with a predictive rule. In addition, the MGP focuses inference on parameters defined through a loss function. This framework is especially resonant in the era of foundation transformers; practitioners increasingly leverage models like TabPFN for their state-of-the-art capabilities, yet often require epistemic uncertainty for a scientific estimand $\theta$ that need not parameterise the implicit latent model. The MGP provides a mechanism to recover these posterior distributions. We introduce TabMGP, an MGP built on TabPFN for tabular data. TabMGP produces credible sets with near-nominal coverage and often outperforms both handcrafted MGP constructions and standard Bayesian baselines.

Stochastic-gradient MCMC methods enable scalable Bayesian posterior sampling but often suffer from sensitivity to minibatch size and gradient noise. To address this, we propose Stochastic Gradient Lattice Random Walk (SGLRW), an extension of the Lattice Random Walk discretisation. Unlike conventional Stochastic Gradient Langevin Dynamics (SGLD), SGLRW introduces stochastic noise only through the off-diagonal elements of the update covariance; this yields greater robustness to minibatch size while retaining asymptotic correctness. Furthermore, as a comparison we analyse a natural analogue of SGLD utilising gradient clipping. Experimental validation on Bayesian regression and classification demonstrates that SGLRW remains stable in regimes where SGLD fails, including in the presence of heavy-tailed gradient noise, and matches or improves predictive performance.

While Contrastive Learning (CL) has revolutionized self-supervised representation learning, its latent representations remain highly entangled and opaque, limiting their interpretability in safety-critical applications. We identify that a fundamental cause of this entanglement is the reliance on deterministic similarity measures, which treat all feature dimensions equally. In compositional scenes, this creates an Optimization Conflict: common background features, such as, "blue sky", are encouraged to align in positive pairs but simultaneously repelled in negative pairs, causing gradient oscillations that hinder precise semantic disentanglement. To address this, we propose **BayesNCL** (Bayesian Gated Non-Negative Contrastive Learning). Unlike standard approaches, BayesNCL introduces a probabilistic gating mechanism that dynamically filters out task-irrelevant, high-frequency common features while selectively retaining discriminative semantics. By formalizing feature selection as a variational inference problem with a sparse Bernoulli prior, our method effectively resolves the optimization conflict. Empirical experimental results on Imagenet-100 demonstrate that BayesNCL achieves a remarkable 142.1\% improvement in semantic consistency compared to state-of-the-art baselines, yielding highly interpretable representations without compromising downstream task performance. Code is available at https://github.com/Cui-Peng-624/BayesNCL.

Bayesian optimization (BO) for high-dimensional constrained problems remains a significant challenge due to the curse of dimensionality. We propose **L**ocal **C**onstrained **B**ayesian **O**ptimization (LCBO), a novel framework tailored for such settings. Unlike trust-region methods that are prone to premature shrinking when confronting tight or complex constraints, LCBO leverages the differentiable landscape of constraint-penalized surrogates to alternate between rapid local descent and uncertainty-driven exploration. Theoretically, we prove that LCBO achieves a convergence rate for the Karush-Kuhn-Tucker (KKT) residual that depends polynomially on the dimension $d$ for common kernels under mild assumptions, offering a rigorous alternative to global BO where regret bounds typically scale exponentially. Extensive evaluations on high-dimensional benchmarks (up to 100D) demonstrate that LCBO consistently outperforms state-of-the-art baselines.

Weight-space learning studies neural architectures that operate directly on the parameters of other neural networks. Motivated by the growing availability of pretrained models, recent work has demonstrated the effectiveness of weight-space networks across a wide range of tasks. SOTA weight-space networks rely on permutation-equivariant designs to improve generalization. However, this may negatively affect expressive power, warranting theoretical investigation. Importantly, unlike other structured domains, weight-space learning targets maps operating on both weight and function spaces, making expressivity analysis particularly subtle. While a few prior works provide partial expressivity results, a comprehensive characterization is still missing. In this work, we address this gap by developing a systematic theory for expressivity of weight-space networks. We first prove that all prominent permutation-equivariant networks are equivalent in expressive power. We then establish universality in both weight- and function-space settings under mild, natural assumptions on the input weights, and characterize the edge-case regimes where universality no longer holds. Guided by our theoretical results, we show that slight modifications to existing weight-space models yield a 34\% improvement over prior SOTA, demonstrating the practical relevance of our framework.

Bayesian Last Layers (BLLs) provide a convenient and computationally efficient way to estimate uncertainty in neural networks. However, they underestimate epistemic uncertainty because they apply a Bayesian treatment only to the final layer, ignoring uncertainty induced by earlier layers. We propose a method that improves BLLs by leveraging a projection of Neural Tangent Kernel (NTK) features onto the space spanned by the last-layer features. This enables posterior inference that accounts for variability of the full network while retaining the low computational cost of inference of a standard BLL. We show that our method yields posterior variances that are provably greater or equal to those of a standard BLL, correcting its tendency to underestimate epistemic uncertainty. To further reduce computational cost, we introduce a uniform subsampling scheme for estimating the projection matrix and for posterior inference. We derive approximation bounds for both types of subsampling. Empirical evaluations on UCI regression, contextual bandits, image classification, and out-of-distribution detection tasks in image and tabular datasets, demonstrate improved calibration and uncertainty estimates compared to standard BLLs and competitive baselines, while reducing computational cost.

Vanilla variational inference finds an optimal approximation to the Bayesian posterior distribution, but even the exact Bayesian posterior is often not meaningful under model misspecification. We propose predictive variational inference (PVI): a general inference framework that seeks and samples from an optimal posterior density such that the resulting posterior predictive distribution is as close to the true data generating process as possible, while this closeness is measured by multiple scoring rules. By optimizing the objective, the predictive variational inference is generally not the same as, or even attempting to approximate, the Bayesian posterior, even asymptotically. Rather, we interpret it as implicit hierarchical expansion. Further, the learned posterior uncertainty detects heterogeneity of parameters among the population, enabling automatic model diagnosis. This framework applies to both likelihood-exact and likelihood-free models. We demonstrate its application in real data examples.

We propose Log-Normal Multiplicative Dynamics (LMD), a new algorithm to stabilize low-precision deep learning. LMD is motivated by the robustness of biological neural networks where synaptic spine sizes follow a log-normal distribution and fluctuate via noisy multiplicative dynamics. So far, no algorithm has successfully incorporated all these synaptic properties to train large models. LMD incorporates these properties through a variational Bayesian formulation that uses a log-normal posterior over weights. We evaluate LMD with low-precision forward-pass matrix multiplications, including training a Vision Transformer and GPT-2 from scratch. Our findings demonstrate the utility of biologically inspired approaches for designing energy-efficient algorithms for stable low-precision deep learning.

While Bayesian inference provides a principled framework for reasoning under uncertainty, its widespread adoption is limited by the intractability of exact posterior computation, necessitating the use of approximate inference. However, existing methods are often computationally expensive, or demand costly retraining when priors change, limiting their utility, particularly in sequential inference problems such as real-time sensor fusion. To address these challenges, we introduce the Distribution Transformer---a novel architecture that can learn arbitrary distribution-to-distribution mappings. Our method can be trained to map a prior to the corresponding posterior, conditioned on some dataset---thus performing approximate Bayesian inference. Our novel architecture represents a prior distribution as a (universally-approximating) Gaussian Mixture Model (GMM), and transforms it into a GMM representation of the posterior. The components of the GMM attend to each other via self-attention, and to the datapoints via cross-attention. We demonstrate that Distribution Transformers both maintain flexibility to vary the prior, and significantly reduces computation times---from minutes to milliseconds---while achieving expected log-likelihood performance on par with or superior to existing approximate inference methods across tasks such as sequential inference, quantum system parameter inference, and Gaussian Process predictive posterior inference with hyperpriors.

Bayesian additive spanning tree (BAST) is an useful tool for interpretable, non-parametric regression on complex constrained domains. It improves upon the performance of Bayesian additive regression trees (BART) by replacing axis-aligned splits through binary tree components by cuts on a spanning tree components, enabling the formation of contiguous splits that respect the underlying complex structure. While BAST is effective for learning on constrained spaces, it still relies on hard partitions, albeit on spanning trees, which limits its ability to represent smoothly varying functions on constrained domains. We propose Probabilistically-routed Bayesian additive spanning trees (PR-BAST), a principled relaxation that replaces hard cuts on spanning tree components with probabilistic routing along spanning tree components. PR-BAST represents the regression surface as an additive ensemble of such spanning tree-aligned smooth components. Conditional on a fixed spanning tree, each component in PR-BAST induces a Gaussian random field with a sparse, tree-structured precision matrix, enabling scalable posterior computation via sparse linear algebra. We theoretically establish that PR-BAST yields strictly faster posterior contraction rates compared to BAST under graph-smooth truth. Experiments on synthetic and real datasets demonstrate that PR-BAST consistently improves accuracy over BAST and other baselines, while retaining the interpretability of tree-based models.

We are considering the problem of sampling from a posterior distribution related to Bayesian inverse problems arising in science, engineering, and imaging. Our method belongs to the family of independence Metropolis--Hastings (IMH) sampling algorithms. These are quite common in Bayesian inference. Relying on the existence of an approximate posterior distribution that is cheaper to sample from but can have significant bias, we introduce Proximal-IMH, a scheme that removes this bias: it corrects samples from the approximate posterior solving an auxiliary optimization problem, yielding a local adjustment that trades off adherence to the exact model against stability around the approximate reference point. For idealized settings, we prove that the proximal correction tightens the match between approximate and exact posteriors, and thereby improves acceptance rates and mixing. The new method works with both linear and nonlinear input-output operators and is especially suitable for inverse problems where exact posterior sampling is too expensive. We perform several numerical experiments that include multimodal and data-driven priors and nonlinear input-output operators. The results show that Proximal-IMH reliably outperforms existing IMH variants.

Foundation models are increasingly being deployed in contexts where understanding the uncertainty of their outputs is critical to ensuring responsible deployment. While Bayesian methods offer a principled approach to uncertainty quantification, their computational overhead renders their use impractical for training or inference at foundation model scale. State-of-the-art models achieve parameter counts in the trillions through carefully engineered sparsity including Mixture-of-Experts (MoE) layers. In this work, we demonstrate calibrated uncertainty at scale by introducing Variational Mixture-of-Experts Routing (VMoER), a structured Bayesian approach for modelling uncertainty in MoE layers. VMoER confines Bayesian inference to the expert-selection stage which is typically done by a deterministic routing network. We instantiate VMoER using two inference strategies: amortised variational inference over routing logits and inferring a temperature parameter for stochastic expert selection. Across tested foundation models, VMoER improves routing stability under noise by 38%, reduces calibration error by 94%, and increases out-of-distribution AUROC by 12%, while incurring less than 1% additional FLOPs. These results suggest VMoER offers a scalable path toward robust and uncertainty-aware foundation models.