Publications
My papers study learning-to-defer and model routing: statistical guarantees, online learning, and robustness, with applications to non-stationary time series and query allocation among large language models.
* indicates equal contribution.
2026
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A Query Is Not a Commitment: Learning to Correct Expert Answers in Online Deferral.
Yannis Montreuil, Axel Carlier, Lai Xing Ng, Wei Tsang Ooi.TL;DR: We study online deferral when the learner chooses an expert and a correction before buying the expert's answer, and show how to account for uncertainty in the learned correction when routing.
Keywords: Online learning · Expert correction · Model routing · Confidence bounds · Regret guaranteesPotential applications: Time-series forecasting · Sensor fusion · LLM scoring
An inaccurate expert can still provide useful information after correction. We study online learning to defer in which the learner chooses an expert and fixes a correction function before purchasing its answer, then applies that function to the answer received. The difficulty is that observed losses reflect both expert quality and an unfinished correction: early errors can discourage queries that would be valuable after learning. We propose ORUCB, which pools shared and expert-specific polynomial responses. A bound on cumulative response-learning error calibrates confidence-weighted risk regression and exploration, allowing the router to account for this error when deciding which answers to buy. Under bounded residuals and disagreements, a fixed feasible model of optimal responses, and linear models of free and optimal queried risk, the calibrated algorithm achieves high-probability pseudo-regret $O(\sqrt{T}\log(T+1))$ over $T$ rounds for fixed problem parameters. The guarantee permits singular answer distributions and misspecified shared responses; optimality is relative to the bounded response class. On four test streams, the selected cubic policy has lower fee-inclusive cost than seven baselines that deploy answers unchanged. Comparisons with a common correction learner examine routing, while six-price comparisons measure cost and query rates.
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Beyond Augmented-Action Surrogates for Multi-Expert Learning-to-Defer.
Yannis Montreuil, Axel Carlier, Lai Xing Ng, Wei Tsang Ooi.TL;DR: We show that although existing surrogates are statistically consistent, their shared augmented-action design leads to harmful training dynamics, and we introduce a decoupled surrogate that avoids these pathologies.
Keywords: Multi-expert deferral · Training dynamics · Statistical consistency · Decoupled surrogatesPotential applications: LLMs · Vision · Time-series classification · Tabular prediction
Existing multi-expert learning-to-defer surrogates are statistically consistent, yet they can underfit, suppress useful experts, or degrade as the expert pool grows. We trace these failures to a shared architectural choice: casting classes and experts as actions inside one augmented prediction geometry. Consistency governs the population target; it says nothing about how the surrogate distributes gradient mass during training. We analyze five surrogates along both axes and show that each trades a fix on one for a failure on the other. We then introduce a decoupled surrogate that estimates the class posterior with a softmax and each expert utility with an independent sigmoid. It admits an $\mathcal{H}$-consistency bound whose constant is $J$-independent for fixed per-expert weight $\beta{=}\lambda/J$, and its gradients are free of the amplification, starvation, and coupling pathologies of the augmented family. Experiments on synthetic benchmarks, CIFAR-10, CIFAR-10H, and Covertype confirm that the decoupled surrogate is the only method that avoids amplification under redundancy, preserves rare specialists, and consistently improves over a standalone classifier across all settings.
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Consistent Learning-to-Defer with Expert-Conditional Advice.
Yannis Montreuil, Leina Montreuil, Axel Carlier, Lai Xing Ng, Wei Tsang Ooi.TL;DR: We show how learning-to-defer should jointly choose both the expert and the extra advice that expert receives.
Keywords: Expert-conditioned advice · Joint routing · Surrogate consistency · Information acquisitionPotential applications: LLMs · Retrieval and tool use · Multimodal models
Learning-to-Defer routes each input to the expert that minimizes expected cost, but it assumes that the information available to every expert is fixed at decision time. Many modern systems violate this assumption: after selecting an expert, one may also choose what additional information that expert should receive, such as retrieved documents, tool outputs, or escalation context. We study this problem and call it Learning-to-Defer with advice. We show that a broad family of natural separated surrogates, which learn routing and advice with distinct heads, are inconsistent even in the smallest non-trivial setting. We then introduce an augmented surrogate that operates on the composite expert–advice action space and prove an H-consistency guarantee together with an excess-risk transfer bound, yielding recovery of the Bayes-optimal policy in the limit. Experiments on tabular, LLMs, and multi-modal tasks show that the resulting method improves over standard Learning-to-Defer while adapting its advice-acquisition behavior to the cost regime.
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Learning to Defer in Non-Stationary Time Series via Switching State-Space Models.
Yannis Montreuil, Letian Yu, Axel Carlier, Lai Xing Ng, Wei Tsang Ooi.TL;DR: We model deferral in non-stationary time series with changing experts and partial feedback.
Keywords: Model routing · Non-stationarity · Partial feedback · State-space models · Expert availabilityPotential applications: Time series · Sensor fusion · Predictive maintenance
We study Learning to Defer for non-stationary time series with partial feedback and time-varying expert availability. At each time step, the router selects an available expert, observes the target, and sees only the queried expert’s prediction. We model signed expert residuals using L2D-SLDS, a factorized switching linear-Gaussian state-space model with context-dependent regime transitions, a shared global factor enabling cross-expert information transfer, and per-expert idiosyncratic states. The model supports expert entry and pruning via a dynamic registry. Using one-step-ahead predictive beliefs, we propose an IDS-inspired routing rule that trades off predicted cost against information gained about the latent regime and shared factor. Experiments show improvements over contextual-bandit baselines and a no-shared-factor ablation.
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Transition Affinity: A Unified Notion of Geometry for LLMs and Text Corpora.
Maxime Meyer*, Yannis Montreuil*, Zongmin Li*, Caroline Chaux, Vincent Y. F. Tan.TL;DR: We show that the geometry of a transformer's next-token argmax partition tracks how often tokens follow one another, giving a shared representation for comparing LLMs and corpora and for detecting distillation.
Keywords: Transformer interpretability · Embedding geometry · Argmax partitions · Token transition probabilitiesPotential applications: LLM comparison · Corpus analysis · Distillation detection · Model provenance
Transformer next-token predictions induce an argmax partition of the embedding space, where each cell consists of representations for which a given vocabulary token has maximal logit. We first argue that this representation is a useful way to interpret a transformer. We then show that the geometry of this partition is governed by a phenomenon we call transition affinity: the size of the shared region between two cells is highly correlated to the probability of the two corresponding tokens following one another. This yields a common geometric representation, allowing us to compute correlations between two LLMs, two corpora, or an LLM and a corpus. We conclude by showing how transition affinity can be used to detect that a model has been distilled.
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Why Ask One When You Can Ask k? Learning-to-Defer to the Top-k Experts.
Yannis Montreuil, Axel Carlier, Lai Xing Ng, Wei Tsang Ooi.TL;DR: We generalize learning-to-defer from sending each query to one expert to sending it to the best top-k experts, with a consistent learning objective and better accuracy-cost trade-offs.
Keywords: Top-k deferral · Adaptive expert selection · Surrogate consistency · Selective predictionPotential applications: LLMs · Vision · Time series · Model cascades
Existing Learning-to-Defer (L2D) frameworks are limited to single-expert deferral, forcing each query to rely on only one expert and preventing the use of collective expertise. We introduce the first framework for Top-$k$ Learning-to-Defer, which allocates queries to the $k$ most cost-effective entities. Our formulation unifies and strictly generalizes prior approaches, including the one-stage and two-stage regimes, selective prediction, and classical cascades. In particular, it recovers the usual Top-1 deferral rule as a special case while enabling principled collaboration with multiple experts when $k > 1$. We further propose Top-$k(x)$ Learning-to-Defer, an adaptive variant that learns the optimal number of experts per query based on input difficulty, expert quality, and consultation cost. To enable practical learning, we develop a novel surrogate loss that is Bayes-consistent, $\mathcal{H}_h$-consistent in the one-stage setting, and $(\mathcal{H}_r,\mathcal{H}_g)$-consistent in the two-stage setting. Crucially, this surrogate is independent of $k$, allowing a single policy to be learned once and deployed flexibly across $k$. Experiments across both regimes show that Top-$k$ and Top-$k(x)$ deliver superior accuracy–cost trade-offs, opening a new direction for multi-expert deferral in L2D.
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Online Learning-to-Defer with Varying Experts.
Yannis Montreuil*, Duy Dang Hoang*, Maxime Meyer*, Axel Carlier, Lai Xing Ng, Wei Tsang Ooi.TL;DR: We introduce the first online learning-to-defer algorithm that handles bandit feedback and changing expert availability, with regret guarantees.
Keywords: Online learning · Bandit feedback · Dynamic expert pools · Regret boundsPotential applications: LLM classification · Vision · Online monitoring
Learning-to-Defer (L2D) methods route each query either to a predictive model or to external experts. While existing work studies this problem in batch settings, real-world deployments require handling streaming data, changing expert availability, and shifting expert distribution. We introduce the first online L2D algorithm for multiclass classification with bandit feedback and a dynamically varying pool of experts. Our method achieves regret guarantees of $O((n+n_e)T^{2/3})$ in general and $O((n+n_e)\sqrt{T})$ under a near-realizable condition, where $T$ is the time horizon, $n$ the number of labels, and $n_e$ the number of distinct experts observed across rounds. The analysis builds on novel $\mathcal{H}$-consistency bounds for the online framework, combined with first-order methods for online convex optimization. Experiments on synthetic and real-world datasets demonstrate that our approach effectively extends standard Learning-to-Defer to settings with varying expert availability and reliability.
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Adversarial Robustness in One-Stage Learning-to-Defer.
Yannis Montreuil*, Letian Yu*, Axel Carlier, Lai Xing Ng, Wei Tsang Ooi.TL;DR: We give the first robustness framework for one-stage learning-to-defer and design adversarially robust surrogates with theoretical guarantees.
Keywords: Adversarial robustness · Joint predictor–router training · Cost-sensitive surrogates · Consistency guaranteesPotential applications: LLMs · Vision · Time series
Learning-to-Defer (L2D) enables hybrid decision-making by routing inputs either to a predictor or to external experts. While promising, L2D is highly vulnerable to adversarial perturbations, which can not only flip predictions but also manipulate deferral decisions. Prior robustness analyses focus solely on two-stage settings, leaving open the end-to-end (one-stage) case where predictor and allocation are trained jointly. We introduce the first framework for adversarial robustness in one-stage L2D, covering both classification and regression. Our approach formalizes attacks, proposes cost-sensitive adversarial surrogate losses, and establishes theoretical guarantees including $\mathcal{H}$, $(\mathcal{R }, \mathcal{F})$, and Bayes consistency. Experiments on benchmark datasets confirm that our methods improve robustness against untargeted and targeted attacks while preserving clean performance.
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Optimal Query Allocation in Extractive QA with LLMs: A Learning-to-Defer Framework with Theoretical Guarantees.
Yannis Montreuil*, Yeo Shu Heng*, Axel Carlier, Lai Xing Ng, Wei Tsang Ooi.TL;DR: We cast extractive QA model routing as learning-to-defer and learn a theoretically grounded policy that improves the accuracy-efficiency trade-off across LLMs.
Keywords: LLM routing · Extractive question answering · Query allocation · Statistical decision theoryPotential applications: LLMs · Document question answering · On-device inference
Large Language Models (LLMs) excel at generative language tasks but remain unreliable for structured prediction—particularly in extractive question answering (EQA), where success hinges on precise span selection. These challenges are magnified in resource-constrained environments, such as mobile or embedded systems, where deploying high-capacity models is often infeasible. We propose a Learning-to-Defer framework that routes EQA queries across a pool of models with varying capabilities and costs, balancing accuracy against efficiency. Our approach is grounded in statistical decision theory: we define a differentiable surrogate loss whose minimizer provably converges to the Bayes-optimal allocation policy. Experiments on SQuADv1, SQuADv2, and TriviaQA show that our method consistently improves accuracy-efficiency trade-offs relative to static baselines and prior routing heuristics. Our work provides a principled and scalable solution for EQA in both high-performance and on-device deployment settings.
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Towards Robust Human-AI Decision-Making via Learning-to-Defer.
Yannis Montreuil.TL;DR: We study robust routing between models and experts when inputs may be adversarially perturbed.
Keywords: Human–AI decision-making · Adversarial attacks · Robust deferral · Surrogate consistencyPotential applications: Human–AI workflows · Clinical decision support · Vision
Learning-to-Defer (L2D) facilitates optimal task allocation between AI systems and decision-makers. Despite its potential, we show that current two-stage L2D frameworks are highly vulnerable to adversarial attacks, which can misdirect queries or overwhelm decision agents, significantly degrading system performance. This paper conducts the first comprehensive analysis of adversarial robustness in two-stage L2D frameworks. We introduce two novel attack strategies, untargeted and targeted, that exploit inherent structural vulnerabilities in these systems. To mitigate these threats, we propose SARD, a robust, convex deferral algorithm rooted in Bayes and $(\mathcal{R},\mathcal{G})$-consistency. Our approach guarantees optimal task allocation under adversarial perturbations for all surrogates in the cross-entropy family. Extensive experiments on classification, regression, and multi-task benchmarks validate the robustness of the approach.
2025
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Adversarial Robustness in Two-Stage Learning-to-Defer: Algorithms and Guarantees.
Yannis Montreuil, Axel Carlier, Lai Xing Ng, Wei Tsang Ooi.TL;DR: We analyze how adversarial perturbations break two-stage learning-to-defer and introduce a robust algorithm with consistency guarantees.
Keywords: Adversarial robustness · Two-stage deferral · Convex optimization · Surrogate consistencyPotential applications: LLMs · Vision · Forecasting
Two-stage Learning-to-Defer (L2D) enables optimal task delegation by assigning each input to either a fixed main model or one of several offline experts, supporting reliable decision-making in complex, multi-agent environments. However, existing L2D frameworks assume clean inputs and are vulnerable to adversarial perturbations that can manipulate query allocation, causing costly misrouting or expert overload. We present the first comprehensive study of adversarial robustness in two-stage L2D systems. We introduce two novel attack strategies, untargeted and targeted, which respectively disrupt optimal allocations or force queries to specific agents. To defend against such threats, we propose SARD, a convex learning algorithm built on a family of surrogate losses that are provably Bayes-consistent and $(\mathcal{R}, \mathcal{G})$-consistent. These guarantees hold across classification, regression, and multi-task settings. Empirical results demonstrate that SARD significantly improves robustness under adversarial attacks while maintaining strong clean performance, marking a critical step toward secure and trustworthy L2D deployment.
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A Two-Stage Learning-to-Defer Approach for Multi-Task Learning.
Yannis Montreuil*, Yeo Shu Heng*, Axel Carlier, Lai Xing Ng, Wei Tsang Ooi.TL;DR: We extend two-stage learning-to-defer to joint classification and regression tasks, with theoretical guarantees and multi-task experiments.
Keywords: Multi-task learning · Classification and regression · Shared representations · Consistency boundsPotential applications: Vision · Healthcare · Time series
The Two-Stage Learning-to-Defer (L2D) framework has been extensively studied for classification and, more recently, regression tasks. However, many real-world applications require solving both tasks jointly in a multi-task setting. We introduce a novel Two-Stage L2D framework for multi-task learning that integrates classification and regression through a unified deferral mechanism. Our method leverages a two-stage surrogate loss family, which we prove to be both Bayes-consistent and $(\mathcal{G}, \mathcal{R})$-consistent, ensuring convergence to the Bayes-optimal rejector. We derive explicit consistency bounds tied to the cross-entropy surrogate and the $L_1$-norm of agent-specific costs, and extend minimizability gap analysis to the multi-expert two-stage regime. We also make explicit how shared representation learning—commonly used in multi-task models—affects these consistency guarantees. Experiments on object detection and electronic health record analysis demonstrate the effectiveness of our approach and highlight the limitations of existing L2D methods in multi-task scenarios.