Limit Analysis of Graph Neural Networks with Wireless Conflict Graphs

Graph Neural Networks (GNNs) have emerged as a powerful tool for wireless resource allocation that leverages the underlying graph structure of communication networks. Their transferability property enables models trained on small-scale graphs to generalize to large-scale deployments with little performance deterioration, a desirable property for currently growing networks. Wireless networks are sparse regimes, where a single node is connected to a small number of other users. This work establishes theoretical results for transferability of GNNs over graphs derived from sparse Random Geometric Graphs (RGGs). In particular, we focus on conflict graphs of RGGs used to model interference among links. Our approach considers the closeness between RGGs and Deterministic Grid Graphs (DGG) to establish bounds in the performance loss when a model is transferred across scales. We validate our theoretical findings through the problem of link scheduling, demonstrating that our learned policies consistently outperform existing benchmarks at scale. Finally, we examine the impact of our theoretical assumptions on empirical performance.

Everywhere Learning: Artificial Intelligence with Pointwise Constraints

Everywhere learning is a new paradigm whereby Artificial Intelligence (AI) systems are trained to satisfy loss constraints with probability one over the data distribution. This is in contrast to the standard paradigm of training AI systems to minimize average losses. We develop an approximate duality theory to substantiate a generalization analysis that establishes the proximity between solutions of empirical and statistical everywhere learning problems. Our results show that dual variables reweigh the data distribution towards points in which loss constraints are more difficult to satisfy and that generalization is controlled by the mismatch between the concentration of mass of the data distribution and the concentration of mass on points where constraints are more difficult to satisfy. We further show that we can control generalization with a sparse L1 penalty on constraint relaxations. We illustrate the merits of everywhere learning with an experiment in agentic classification for language model tasks.

Unlearning in Diffusion Models: A Unified Framework with KL Divergence and Likelihood Constraints

Unlearning in diffusion models aims to remove undesirable data or concepts while preserving the utility of pretrained models -- two fundamentally conflicting objectives. We propose a principled constrained optimization framework that formulates unlearning as minimizing the deviation from a pretrained model, subject to explicit separation constraints from the unlearning distributions. Specifically, we formulate three constrained optimization problems based on reverse and forward KL divergences, and likelihood constraints. The first two generalize existing approaches for concept and data unlearning, while the third offers a novel and natural formulation for unlearning. Despite the nonconvexity of the KL constraints, we establish strong duality for all three problems, enabling us to explicitly characterize their optimal solutions as unlearning targets and develop primal-dual algorithms for each formulation. Experimental results demonstrate that our KL-constrained approach achieves superior retention-unlearning tradeoffs compared to weight-based baselines for concept and data unlearning, and that our likelihood-based approach matches unlearning effectiveness while better preserving retained concepts compared to baselines.

Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves

Modern deep learning architectures increasingly contend with sophisticated signals that are natively infinite-dimensional, such as time series, probability distributions, or operators, and are defined over irregular domains. Yet, a unified learning theory for these settings has been lacking. To start addressing this gap, we introduce a novel convolutional learning framework for possibly infinite-dimensional signals supported on a manifold. Namely, we use the connection Laplacian associated with a Hilbert bundle as a convolutional operator, and we derive filters and neural networks, dubbed as \textit{HilbNets}. We make HilbNets and, more generally, the convolution operation, implementable via a two-stage sampling procedure. First, we show that sampling the manifold induces a Hilbert Cellular Sheaf, a generalized graph structure with Hilbert feature spaces and edge-wise coupling rules, and we prove that its sheaf Laplacian converges in probability to the underlying connection Laplacian as the sampling density increases. Notably, this result is a generalization to the infinite-dimensional bundle setting of the Belkin \& Niyogi \cite{BELKIN20081289} convergence result for the graph Laplacian to the manifold Laplacian, a theoretical cornerstone of geometric learning methods. Second, we discretize the signals and prove that the discretized (implementable) HilbNets converge to the underlying continuous architectures and are transferable across different samplings of the same bundle, providing consistency for learning. Finally, we validate our framework on synthetic and real-world tasks. Overall, our results broaden the scope of geometric learning as a whole by lifting classical Laplacian-based frameworks to settings where the signal at each point lives in its own Hilbert space.

Graph Signal Diffusion Models for Wireless Resource Allocation

We consider constrained ergodic resource optimization in wireless networks with graph-structured interference. We train a diffusion model policy to match expert conditional distributions over resource allocations. By leveraging a primal-dual (expert) algorithm, we generate primal iterates that serve as draws from the corresponding expert conditionals for each training network instance. We view the allocations as stochastic graph signals supported on known channel state graphs. We implement the diffusion model architecture as a U-Net hierarchy of graph neural network (GNN) blocks, conditioned on the channel states and additional node states. At inference, the learned generative model amortizes the iterative expert policy by directly sampling allocation vectors from the near-optimal conditional distributions. In a power-control case study, we show that time-sharing the generated power allocations achieves near-optimal ergodic sum-rate utility and near-feasible ergodic minimum-rates, with strong generalization and transferability across network states.

RKHS Representation of Algebraic Convolutional Filters with Integral Operators

Integral operators play a central role in signal processing, underpinning classical convolution, and filtering on continuous network models such as graphons. While these operators are traditionally analyzed through spectral decompositions, their connection to reproducing kernel Hilbert spaces (RKHS) has not been systematically explored within the algebraic signal processing framework. In this paper, we develop a comprehensive theory showing that the range of integral operators naturally induces RKHS convolutional signal models whose reproducing kernels are determined by a box product of the operator symbols. We characterize the algebraic and spectral properties of these induced RKHS and show that polynomial filtering with integral operators corresponds to iterated box products, giving rise to a unital kernel algebra. This perspective yields pointwise RKHS representations of filters via the reproducing property, providing an alternative to operator-based implementations. Our results establish precise connections between eigendecompositions and RKHS representations in graphon signal processing, extend naturally to directed graphons, and enable novel spatial--spectral localization results. Furthermore, we show that when the spectral domain is a subset of the original domain of the signals, optimal filters for regularized learning problems admit finite-dimensional RKHS representations, providing a principled foundation for learnable filters in integral-operator-based neural architectures.

Size Transferability of Graph Transformers with Convolutional Positional Encodings

Transformers have achieved remarkable success across domains, motivating the rise of Graph Transformers (GTs) as attention-based architectures for graph-structured data. A key design choice in GTs is the use of Graph Neural Network (GNN)-based positional encodings to incorporate structural information. In this work, we study GTs through the lens of manifold limit models for graph sequences and establish a theoretical connection between GTs with GNN positional encodings and Manifold Neural Networks (MNNs). Building on transferability results for GNNs under manifold convergence, we show that GTs inherit transferability guarantees from their positional encodings. In particular, GTs trained on small graphs provably generalize to larger graphs under mild assumptions. We complement our theory with extensive experiments on standard graph benchmarks, demonstrating that GTs exhibit scalable behavior on par with GNNs. To further show the efficiency in a real-world scenario, we implement GTs for shortest path distance estimation over terrains to better illustrate the efficiency of the transferable GTs. Our results provide new insights into the understanding of GTs and suggest practical directions for efficient training of GTs in large-scale settings.

Learning Policy Representations for Steerable Behavior Synthesis

Given a Markov decision process (MDP), we seek to learn representations for a range of policies to facilitate behavior steering at test time. As policies of an MDP are uniquely determined by their occupancy measures, we propose modeling policy representations as expectations of state-action feature maps with respect to occupancy measures. We show that these representations can be approximated uniformly for a range of policies using a set-based architecture. Our model encodes a set of state-action samples into a latent embedding, from which we decode both the policy and its value functions corresponding to multiple rewards. We use variational generative approach to induce a smooth latent space, and further shape it with contrastive learning so that latent distances align with differences in value functions. This geometry permits gradient-based optimization directly in the latent space. Leveraging this capability, we solve a novel behavior synthesis task, where policies are steered to satisfy previously unseen value function constraints without additional training.

Unrolled Neural Networks for Constrained Optimization

In this paper, we develop unrolled neural networks to solve constrained optimization problems, offering accelerated, learnable counterparts to dual ascent (DA) algorithms. Our framework, termed constrained dual unrolling (CDU), comprises two coupled neural networks that jointly approximate the saddle point of the Lagrangian. The primal network emulates an iterative optimizer that finds a stationary point of the Lagrangian for a given dual multiplier, sampled from an unknown distribution. The dual network generates trajectories towards the optimal multipliers across its layers while querying the primal network at each layer. Departing from standard unrolling, we induce DA dynamics by imposing primal-descent and dual-ascent constraints through constrained learning. We formulate training the two networks as a nested optimization problem and propose an alternating procedure that updates the primal and dual networks in turn, mitigating uncertainty in the multiplier distribution required for primal network training. We numerically evaluate the framework on mixed-integer quadratic programs (MIQPs) and power allocation in wireless networks. In both cases, our approach yields near-optimal near-feasible solutions and exhibits strong out-of-distribution (OOD) generalization.

A Constrained Optimization Perspective of Unrolled Transformers

We introduce a constrained optimization framework for training transformers that behave like optimization descent algorithms. Specifically, we enforce layerwise descent constraints on the objective function and replace standard empirical risk minimization (ERM) with a primal-dual training scheme. This approach yields models whose intermediate representations decrease the loss monotonically in expectation across layers. We apply our method to both unrolled transformer architectures and conventional pretrained transformers on tasks of video denoising and text classification. Across these settings, we observe constrained transformers achieve stronger robustness to perturbations and maintain higher out-of-distribution generalization, while preserving in-distribution performance.