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Optimization · Optimization and Learning under Uncertainty

Zhuangzhuang Jia, Yijie Wang, Roy Dong, Grani A. Hanasusanto

In performative stochastic optimization, decisions can influence the distribution of random parameters, rendering the data-generating process itself decision-dependent. In practice, decision-makers rarely have access to the true distribution map and must instead rely on imperfect surrogate models, which can lead to severely suboptimal solutions under misspecification. Data scarcity or costly collection further exacerbates these challenges in real-world settings. To address these challenges, we propose a distributionally robust framework for performative optimization that explicitly accounts for ambiguity in the decision-dependent distribution. Our framework introduces three modeling paradigms that capture a broad range of applications in machine learning and decision-making under uncertainty. This latter setting has not previously been explored in the performative optimization literature. To tackle the intractability of the resulting nonconvex objectives, we develop an iterative algorithm named repeated robust risk minimization, which alternates between solving a decision-independent distributionally robust optimization problem and updating the ambiguity set based on the previous decision. This decoupling ensures computational tractability at each iteration while enhancing robustness to model uncertainty. We provide reformulations compatible with off-the-shelf solvers and establish theoretical guarantees on convergence and suboptimality. Extensive numerical experiments in strategic classification, revenue management, and portfolio optimization demonstrate significant performance gains over state-of-the-art baselines, highlighting the practical value of our approach.

Theory · Learning Theory

Hongrui Chen, Jihao Long, Lei Wu

"We consider the problem of learning functions within the $\mathcal{F}_{p,\pi}$ and Barron spaces, which play crucial roles in understanding random feature models (RFMs), two-layer neural networks, as well as kernel methods. Leveraging tools from information-based complexity (IBC), we establish a dual equivalence between approximation and estimation, and then apply it to study the learning of the preceding function spaces. The duality allows us to focus on the more tractable problem between approximation and estimation. To showcase the efficacy of our duality framework, we delve into two important but under-explored problems: \begin{itemize} \item \textbf{Random feature learning beyond kernel regime:} We derive sharp bounds for learning $\cF_{p,\pi}$ using RFMs. Notably, the learning is efficient without the curse of dimensionality for $p>1$. This underscores the extended applicability of RFMs beyond the traditional kernel regime, since $\mathcal{F}_{p,\pi}$ with $p<2$ is strictly larger than the corresponding reproducing kernel Hilbert space (RKHS) where $p=2$. \item \textbf{The $L^\infty$ learning of RKHS:} We establish sharp, spectrum-dependent characterizations for the convergence of $L^\infty$ learning error in both noiseless and noisy settings. Surprisingly, we show that popular kernel ridge regression can achieve near-optimal performance in $L^\infty$ learning, despite it primarily minimizing square loss. \end{itemize} To establish the aforementioned duality, we introduce a type of IBC, termed $I$-complexity, to measure the size of a function class. Notably, $I$-complexity offers a tight characterization of learning in noiseless settings, yields lower bounds comparable to Le Cam's in noisy settings, and is versatile in deriving upper bounds. We believe that our duality framework holds potential for broad application in learning analysis across more scenarios."

Tony Bonnaire, Raphaël Urfin, Giulio Biroli, Marc Mezard

Diffusion models have achieved remarkable success across a wide range of generative tasks. A key challenge is understanding the mechanisms that prevent their memorization of training data and allow generalization. In this work, we investigate the role of the training dynamics in the transition from generalization to memorization. Through extensive experiments and theoretical analysis, we identify two distinct timescales: an early time $\tau_\mathrm{gen}$ at which models begin to generate high-quality samples, and a later time $\tau_\mathrm{mem}$ beyond which memorization emerges. Crucially, we find that $\tau_\mathrm{mem}$ increases linearly with the training set size $n$, while $\tau_\mathrm{gen}$ remains constant. This creates a growing window of training times with $n$ where models generalize effectively, despite showing strong memorization if training continues beyond it. It is only when $n$ becomes larger than a model-dependent threshold that overfitting disappears at infinite training times. These findings reveal a form of implicit dynamical regularization in the training dynamics, which allow to avoid memorization even in highly overparameterized settings. Our results are supported by numerical experiments with standard U-Net architectures on realistic and synthetic datasets, and by a theoretical analysis using a tractable random features model studied in the high-dimensional limit.

Julius Broomfield, Tom Gibbs, George Ingebretsen, Ethan Kosak-Hine, Tia Nasir, Jason Zhang, Reihaneh Iranmanesh, Sara Pieri, Reihaneh Rabbany, Kellin Pelrine

LLM jailbreaks are a widespread safety challenge. Given this problem has not yet been tractable, we suggest targeting a key failure mechanism: the failure of safety to generalize across semantically equivalent inputs. We further focus the target by requiring desirable tractability properties of attacks to study: explainability, transferability between models, and transferability between goals. We perform red-teaming within this framework by uncovering new vulnerabilities to multi-turn, multi-image, and translation-based attacks. These attacks are semantically equivalent by our design to their single-turn, single-image, or untranslated counterparts, enabling systematic comparisons; we show that the different structures yield different safety outcomes. We then demonstrate the potential for this framework to enable new defenses by proposing a Structure Rewriting Guardrail, which converts an input to a structure more conducive to safety assessment. This guardrail significantly improves refusal of harmful inputs, without over-refusing benign ones. Thus, by framing this intermediate challenge—more tractable than universal defenses but essential for long-term safety—we highlight a critical milestone for AI safety research.

Loïc Fosse, Frederic Bechet, Benoit Favre, Géraldine Damnati, Gwénolé Lecorvé, Maxime Darrin, Philippe Formont, Pablo Piantanida

Tasks are central in machine learning, as they are the most natural objects to assess the capabilities of current models. The trend is to build general models able to address any task. Even though transfer learning and multitask learning try to leverage the underlying task space, no well-founded tools are available to study its structure. This study proposes a theoretically grounded setup to define the notion of task and to compute the inclusion between two tasks from a statistical deficiency point of view. We propose a tractable proxy as information sufficiency to estimate the degree of inclusion between tasks, show its soundness on synthetic data, and use it to reconstruct empirically the classic NLP pipeline.

Probabilistic Methods · Graphical Models

Honghua Zhang, Meihua Dang, Benjie Wang, Stefano Ermon, Nanyun Peng, Guy Van den Broeck

Probabilistic Circuits (PCs) are tractable representations of probability distributions allowing for exact and efficient computation of likelihoods and marginals. Recent advancements have improved the scalability of PCs either by leveraging their sparse properties or through the use of tensorized operations for better hardware utilization. However, no existing method fully exploits both aspects simultaneously. In this paper, we propose a novel sparse and structured parameterization for the sum blocks in PCs. By replacing dense matrices with sparse Monarch matrices, we significantly reduce the memory and computation costs, enabling unprecedented scaling of PCs. From a theory perspective, our construction arises naturally from circuit multiplication; from a practical perspective, compared to previous efforts on scaling up tractable probabilistic models, our approach not only achieves state-of-the-art generative modeling performance on challenging benchmarks like Text8, LM1B and ImageNet, but also demonstrates superior scaling behavior, achieving the same performance with substantially less compute as measured by the number of floating-point operations (FLOPs) during training.

Deep Learning · Generative Models and Autoencoders

Anji Liu, Xuejie Liu, Dayuan Zhao, Mathias Niepert, Yitao Liang, Guy Van den Broeck

Non-autoregressive (NAR) generative models are valuable because they can handle diverse conditional generation tasks in a more principled way than their autoregressive (AR) counterparts, which are constrained by sequential dependency requirements. Recent advancements in NAR models, such as diffusion language models, have demonstrated superior performance in unconditional generation compared to AR models (e.g., GPTs) of similar sizes. However, such improvements do not always lead to improved conditional generation performance. We show that a key reason for this gap is the difficulty in generalizing to conditional probability queries unseen during training. As a result, strong unconditional generation performance does not guarantee high-quality conditional generation. This paper proposes Tractable Transformers (Tracformer), a Transformer-based generative model that is more robust to different conditional generation tasks. Unlike existing models that rely solely on global contextual features derived from full inputs, Tracformers incorporate a sparse Transformer encoder to capture both local and global contextual information. This information is routed through a decoder for conditional generation. Empirical results demonstrate that Tracformers achieve state-of-the-art conditional generation performance on text modeling compared to recent diffusion and AR model baselines.

Theory · Reinforcement Learning and Planning

Joongkyu Lee, Min-hwan Oh

In this paper, we consider combinatorial reinforcement learning with preference feedback, where a learning agent sequentially offers an action—an assortment of multiple items—to a user, whose preference feedback follows a multinomial logistic (MNL) model. This framework allows us to model real-world scenarios, particularly those involving long-term user engagement, such as in recommender systems and online advertising. However, this framework faces two main challenges: (1) the unknown value of each item, unlike traditional MNL bandits that only address single-step preference feedback, and (2) the difficulty of ensuring optimism while maintaining tractable assortment selection in the combinatorial action space with unknown values. In this paper, we assume a contextual MNL preference model, where the mean utilities are linear, and the value of each item is approximated by a general function. We propose an algorithm, MNL-VQL, that addresses these challenges, making it both computationally and statistically efficient. As a special case, for linear MDPs (with the MNL preference feedback), we establish the first regret lower bound in this framework and show that MNL-VQL achieves near-optimal regret. To the best of our knowledge, this is the first work to provide statistical guarantees in combinatorial RL with preference feedback.

Deep Learning · Theory

Mateusz Piotrowski, Paul Riechers, Daniel Filan, Adam Shai

What computational structures emerge in transformers trained on next-token prediction? In this work, we provide evidence that transformers implement constrained Bayesian belief updating---a parallelized version of partial Bayesian inference shaped by architectural constraints. We integrate the model-agnostic theory of optimal prediction with mechanistic interpretability to analyze transformers trained on a tractable family of hidden Markov models that generate rich geometric patterns in neural activations. Our primary analysis focuses on single-layer transformers, revealing how the first attention layer implements these constrained updates, with extensions to multi-layer architectures demonstrating how subsequent layers refine these representations. We find that attention carries out an algorithm with a natural interpretation in the probability simplex, and create representations with distinctive geometric structure. We show how both the algorithmic behavior and the underlying geometry of these representations can be theoretically predicted in detail---including the attention pattern, OV-vectors, and embedding vectors---by modifying the equations for optimal future token predictions to account for the architectural constraints of attention. Our approach provides a principled lens on how architectural constraints shape the implementation of optimal prediction, revealing why transformers develop specific intermediate geometric structures.

Deep Learning · Generative Models and Autoencoders

Omer Ronen, Ahmed Imtiaz Humayun, Richard Baraniuk, Randall Balestriero, Bin Yu

We develop Latent Exploration Score (LES) to mitigate over-exploration in Latent Space Optimization (LSO), a popular method for solving black-box discrete optimization problems. LSO utilizes continuous optimization within the latent space of a Variational Autoencoder (VAE) and is known to be susceptible to over-exploration, which manifests in unrealistic solutions that reduce its practicality. LES leverages the trained decoder’s approximation of the data distribution, and can be employed with any VAE decoder–including pretrained ones–without additional training, architectural changes or access to the training data. Our evaluation across five LSO benchmark tasks and twenty-two VAE models demonstrates that LES always enhances the quality of the solutions while maintaining high objective values, leading to improvements over existing solutions in most cases. We believe that new avenues to LSO will be opened by LES’ ability to identify out of distribution areas, differentiability, and computational tractability.

Reinforcement Learning · Multi-agent

Cassidy Laidlaw, Eli Bronstein, Timothy Guo, Dylan Feng, Lukas Berglund, Justin Svegliato, Stuart Russell, Anca Dragan

Assistance games are a promising alternative to reinforcement learning from human feedback (RLHF) for training AI assistants. Assistance games resolve key drawbacks of RLHF, such as incentives for deceptive behavior, by explicitly modeling the interaction between assistant and user as a two-player game where the assistant cannot observe their shared goal. Despite their potential, assistance games have only been explored in simple settings. Scaling them to more complex environments is difficult because it requires both solving intractable decision-making problems under uncertainty and accurately modeling human users' behavior. We present the first scalable approach to solving assistance games and apply it to a new, challenging Minecraft-based assistance game with over $10^{400}$ possible goals. Our approach, AssistanceZero, extends AlphaZero with a neural network that predicts human actions and rewards, enabling it to plan under uncertainty. We show that AssistanceZero outperforms model-free RL algorithms and imitation learning in the Minecraft-based assistance game. In a human study, our AssistanceZero-trained assistant significantly reduces the number of actions participants take to complete building tasks in Minecraft. Our results suggest that assistance games are a tractable framework for training effective AI assistants in complex environments. Code and videos are available at https://anonymous.4open.science/w/scalably-solving-assistance-games/.

Deep Learning · Generative Models and Autoencoders

Leo Brunswic, Mateo Clémente, Rui Heng Yang, Adam Sigal, Amir Rasouli, Yinchuan Li

Generative Flow Networks (GFNs) were initially introduced on directed non-acyclic graphs to sample from an unnormalized distribution density. Recent works have extended the theoretical framework for generative methods allowing more flexibility and enhancing application range. However, many challenges remain in training GFNs in continuous settings and for imitation learning (IL), including intractability of flow-matching loss, limited tests of non-acyclic training, and the need for a separate reward model in imitation learning. The present work proposes a family of generative flows called Ergodic Generative Flows (EGFs) which are used to address the aforementioned issues. First, we leverage ergodicity to build simple generative flows with finitely many globally defined transformations (diffeomorphisms) with universality guarantees and tractable flow-matching loss (FM loss). Second, we introduce a new loss involving cross-entropy coupled to weak flow-matching control, coined KL-weakFM loss. It is designed for IL training without a separate reward model. We evaluate IL-EGFs on toy 2D tasks and real-world datasets from NASA on the sphere, using the KL-weakFM loss. Additionally, we conduct toy 2D reinforcement learning experiments with a target reward, using the FM loss.

Deep Learning · Algorithms

Quan Xiao, Hui Yuan, A F M Saif, Gaowen Liu, Ramana Kompella, Mengdi Wang, Tianyi Chen

Diffusion models, which iteratively denoise data samples to synthesize high-quality outputs, have achieved empirical success across domains. However, optimizing these models for downstream tasks often involves nested bilevel structures, such as tuning hyperparameters for fine-tuning tasks or noise schedules in training dynamics, where traditional bilevel methods fail due to the infinite-dimensional probability space and prohibitive sampling costs. We formalize this challenge as a generative bilevel optimization problem and address two key scenarios: (1) fine-tuning pre-trained models via an inference-only lower-level solver paired with a sample-efficient gradient estimator for the upper level, and (2) training diffusion model from scratch with noise schedule optimization by reparameterizing the lower-level problem and designing a computationally tractable gradient estimator. Our first-order bilevel framework overcomes the incompatibility of conventional bilevel methods with diffusion processes, offering theoretical grounding and computational practicality. Experiments demonstrate that our method outperforms existing fine-tuning and hyperparameter search baselines.

Deep Learning · Generative Models and Autoencoders

Rafał Karczewski, Markus Heinonen, Vikas Garg

Diffusion models have emerged as a powerful class of generative models, capable of producing high-quality images by mapping noise to a data distribution. However, recent findings suggest that image likelihood does not align with perceptual quality: high-likelihood samples tend to be smooth, while lower-likelihood ones are more detailed. Controlling sample density is thus crucial for balancing realism and detail. In this paper, we analyze an existing technique, Prior Guidance, which scales the latent code to influence image detail. We introduce score alignment, a condition that explains why this method works and show that it can be tractably checked for any continuous normalizing flow model. We then propose Density Guidance, a principled modification of the generative ODE that enables exact log-density control during sampling. Finally, we extend Density Guidance to stochastic sampling, ensuring precise log-density control while allowing controlled variation in structure or fine details. Our experiments demonstrate that these techniques provide fine-grained control over image detail without compromising sample quality. Code is available at https://github.com/Aalto-QuML/density-guidance.

General Machine Learning · Everything Else

Paulius Rauba, Qiyao Wei, Mihaela van der Schaar

Consider the problem of testing whether the outputs of a large language model (LLM) system change under an arbitrary intervention, such as an input perturbation or changing the model variant. We cannot simply compare two LLM outputs since they might differ due to the stochastic nature of the system, nor can we compare the entire output distribution due to computational intractability. While existing methods for analyzing text-based outputs exist, they focus on fundamentally different problems, such as measuring bias or fairness. To this end, we introduce distribution-based perturbation analysis, a framework that reformulates LLM perturbation analysis as a frequentist hypothesis testing problem. We construct empirical null and alternative output distributions within a low-dimensional semantic similarity space via Monte Carlo sampling, enabling tractable inference without restrictive distributional assumptions. The framework is (i) model-agnostic, (ii) supports the evaluation of arbitrary input perturbations on any black-box LLM, (iii) yields interpretable p-values; (iv) supports multiple perturbations via controlled error rates; and (v) provides scalar effect sizes. We demonstrate the usefulness of the framework across multiple case studies, showing how we can quantify response changes, measure true/false positive rates, and evaluate alignment with reference models. Above all, we see this as a reliable frequentist hypothesis testing framework for LLM auditing.

Theory · Deep Learning

Joonkyu Kim, Yejin Kim, Jy-yong Sohn

In continual learning scenarios, catastrophic forgetting of previously learned tasks is a critical issue, making it essential to effectively measure such forgetting. Recently, there has been growing interest in focusing on representation forgetting, the forgetting measured at the hidden layer. In this paper, we provide the first theoretical analysis of representation forgetting and use this analysis to better understand the behavior of continual learning. First, we introduce a new metric called representation discrepancy, which measures the difference between representation spaces constructed by two snapshots of a model trained through continual learning. We demonstrate that our proposed metric serves as an effective surrogate for the representation forgetting while remaining analytically tractable. Second, through mathematical analysis of our metric, we derive several key findings about the dynamics of representation forgetting: the forgetting occurs more rapidly to a higher degree as the layer index increases, while increasing the width of the network slows down the forgetting process. Third, we support our theoretical findings through experiments on real image datasets, including Split-CIFAR100 and ImageNet1K.

Probabilistic Methods · Gaussian Processes

Michalis Titsias

Sparse variational Gaussian processes (GPs) construct tractable posterior approximations to GP models. At the core of these methods is the assumption that the true posterior distribution over training function values ${\bf f}$ and inducing variables ${\bf u}$ is approximated by a variational distribution that incorporates the conditional GP prior $p({\bf f} | {\bf u})$ in its factorization. While this assumption is considered as fundamental, we show that for model training we can relax it through the use of a more general variational distribution $q({\bf f} | {\bf u} )$ that depends on $N$ extra parameters, where $N$ is the number of training examples. In GP regression, we can analytically optimize the evidence lower bound over the extra parameters and express a tractable collapsed bound that is tighter than the previous bound. The new bound is also amenable to stochastic optimization and its implementation requires minor modifications to existing sparse GP code. Further, we also describe extensions to non-Gaussian likelihoods. On several datasets we demonstrate that our method can reduce bias when learning the hyperparameters and can lead to better predictive performance.

Applications · Computer Vision

Zixuan Hu, Yichun Hu, Xiaotong Li, SHIXIANG TANG, LINGYU DUAN

Wild Test-Time Adaptation (WTTA) is proposed to adapt a source model to unseen domains under extreme data scarcity and multiple shifts. Previous approaches mainly focused on sample selection strategies, while overlooking the fundamental problem on underlying optimization. Initially, we critically analyze the widely-adopted entropy minimization framework in WTTA and uncover its significant limitations in noisy optimization dynamics that substantially hinder adaptation efficiency. Through our analysis, we identify region confidence as a superior alternative to traditional entropy, however, its direct optimization remains computationally prohibitive for real-time applications. In this paper, we introduce a novel region-integrated method **ReCAP** that bypasses the lengthy process. Specifically, we propose a probabilistic region modeling scheme that flexibly captures semantic changes in embedding space. Subsequently, we develop a finite-to-infinite asymptotic approximation that transforms the intractable region confidence into a tractable and upper-bounded proxy. These innovations significantly unlock the overlooked potential dynamics in local region in a concise solution. Our extensive experiments demonstrate the consistent superiority of ReCAP over existing methods across various datasets and wild scenarios. The source code will be available at https://github.com/hzcar/ReCAP.

Theory · Probabilistic Methods

Oliver Broadrick, Sanyam Agarwal, Guy Van den Broeck, Markus Bläser

Marginalization -- summing a function over all assignments to a subset of its inputs -- is a fundamental computational problem with applications from probabilistic inference to formal verification.Despite its computational hardness in general, there exist many classes of functions (e.g., probabilistic models) for which marginalization remains tractable, and they can all be commonly expressed by arithmetic circuits computing multilinear polynomials.This raises the question, can *all* functions with polynomial time marginalization algorithms be succinctly expressed by such circuits? We give a negative answer, exhibiting simple functions with tractable marginalization yet no efficient representation by known models, assuming $\\mathsf{FP} \\neq \\#\\mathsf{P}$ (an assumption implied by $\\mathsf{P} \\neq \\mathsf{NP}$). To this end, we identify a hierarchy of complexity classes corresponding to stronger forms of marginalization, all of which are efficiently computable on the known circuit models. We conclude with a completeness result, showing that whenever there is an efficient real RAM performing virtual evidence marginalization for a function, then there are small arithmetic circuits for that function's multilinear representation.

Optimization · Non-Convex

Omar Bennouna, Jiawei Zhang, Saurabh Amin, Asuman Ozdaglar

Contextual optimization problems are prevalent in decision-making applications where historical data and contextual features are used to learn predictive models that inform optimal actions. However, practical applications often suffer from model misspecification due to incomplete knowledge of the underlying data-generating process, leading to suboptimal decisions. Existing approaches primarily address the well-specified case, leaving a critical gap in handling misspecified models. In this paper, we propose a novel Integrated Learning and Optimization (ILO) framework that explicitly accounts for model misspecification by introducing a tractable surrogate loss function with strong theoretical guarantees on generalizability, tractability, and optimality. Our surrogate loss aligns with the true decision performance objective, ensuring robustness to misspecification without imposing restrictive assumptions. The proposed approach effectively mitigates the challenges of non-convexity and non-smoothness in the target loss function, leading to efficient optimization procedures. We provide rigorous theoretical analysis and experimental validation, demonstrating superior performance compared to state-of-the-art methods. Our work offers a principled solution to the practically relevant challenge of model misspecification in contextual optimization.