For clarity, we decompose Lemma 2 into three lemmas (Lemmas 4-6) and prove each of them. For clarity, we decompose Lemma 3 into three lemmas (Lemmas 7-9) and prove each of them. The proof of Lemma 10 is similar to that of Lemma 1. From the above process, it can be easily seen that Condition 1-3 in the lemma are satisfied. Theorem 3 can also be used to prove Theorem 2, simply by setting p = 1 . By similar reasoning with the proof of Theorem 1 (Appendix A.3), we get It can be easily verified that the social network monitoring problem considered in Section 5 is a non-monotone submodular maximization problem subject to a partition matroid constraint.

Pretrained diffusion models (DMs) have recently been popularly used in solving inverse problems (IPs). The existing methods mostly interleave iterative steps in the reverse diffusion process and iterative steps to bring the iterates closer to satisfying the measurement constraint. However, such interleaving methods struggle to produce final results that look like natural objects of interest (i.e., manifold feasibility) and fit the measurement (i.e., measurement feasibility), especially for nonlinear IPs. Moreover, their capabilities to deal with noisy IPs with unknown types and levels of measurement noise are unknown. In this paper, we advocate viewing the reverse process in DMs as a function and propose a novel plug-in method for solving IPs using pretrained DMs, dubbed DMPlug. DMPlug addresses the issues of manifold feasibility and measurement feasibility in a principled manner, and also shows great potential for being robust to unknown types and levels of noise. Through extensive experiments across various IP tasks, including two linear and three nonlinear IPs, we demonstrate that DMPlug consistently outperforms state-of-the-art methods, often by large margins especially for nonlinear IPs.

Drawbacks of ignoring the causal mechanisms when performing imitation learning have recently been acknowledged. Several approaches both to assess the feasibility of imitation and to circumvent causal confounding and causal misspecifications have been proposed in the literature.However, the potential benefits of the incorporation of additional information about the underlying causal structure are left unexplored.An example of such overlooked information is context-specific independence (CSI), i.e., independence that holds only in certain contexts.We consider the problem of causal imitation learning when CSI relations are known.We prove that the decision problem pertaining to the feasibility of imitation in this setting is NP-hard.Further, we provide a necessary graphical criterion for imitation learning under CSI and show that under a structural assumption, this criterion is also sufficient.Finally, we propose a sound algorithmic approach for causal imitation learning which takes both CSI relations and data into account.

In contrast to the advances in characterizing the sample complexity for solving Markov decision processes (MDPs), the optimal statistical complexity for solving constrained MDPs (CMDPs) remains unknown. We resolve this question by providing minimax upper and lower bounds on the sample complexity for learning near-optimal policies in a discounted CMDP with access to a generative model (simulator). In particular, we design a model-based algorithm that addresses two settings: (i) relaxed feasibility, where small constraint violations are allowed, and (ii) strict feasibility, where the output policy is required to satisfy the constraint. For (i), we prove that our algorithm returns an $\epsilon$-optimal policy with probability $1 - \delta$, by making $\tilde{O}\left(\frac{S A \log(1/\delta)}{(1 - \gamma)^3 \epsilon^2}\right)$ queries to the generative model, thus matching the sample-complexity for unconstrained MDPs.

We propose a reduced-space formulation for optimizing over trained neural networks where the network's outputs and derivatives are evaluated on a GPU. To do this, we treat the neural network as a "gray box" where intermediate variables and constraints are not exposed to the optimization solver. Compared to the full-space formulation, in which intermediate variables and constraints are exposed to the optimization solver, the reduced-space formulation leads to faster solves and fewer iterations in an interior point method. We demonstrate the benefits of this method on two optimization problems: Adversarial generation for a classifier trained on MNIST images and security-constrained optimal power flow with transient feasibility enforced using a neural network surrogate.

Learning to Optimize (L2O) is a subfield of machine learning (ML) in which ML models are trained to solve parametric optimization problems. The general goal is to learn a fast approximator of solutions to constrained optimization problems, as a function of their defining parameters. Prior L2O methods focus almost entirely on single-level programs, in contrast to the bilevel programs, whose constraints are themselves expressed in terms of optimization subproblems. Bilevel programs have numerous important use cases but are notoriously difficult to solve, particularly under stringent time demands. This paper proposes a framework for learning to solve a broad class of challenging bilevel optimization problems, by leveraging modern techniques for differentiation through optimization problems. The framework is illustrated on an array of synthetic bilevel programs, as well as challenging control system co-design problems, showing how neural networks can be trained as efficient approximators of parametric bilevel optimization.

Abstract--Large models (LMs), such as ChatGPT, have made a significant impact across diverse domains and hold great potential to facilitate the evolution of network intelligence. Wireless-native multi-modal large models (WMLMs) can sense and understand the physical world through multi-modal data, serving as a key enabler that integrates communication, sensing, and intelligence, and thus they can boost various smart services to billions of users. However, research on WMLMs remains in its infancy, and the construction of domain-specific multi-modal large models for wireless networks is still underexplored. In this paper, we outlines the key characteristics of WMLMs and summarizes existing methods, on the basis of which a wireless-native multimodal training paradigm is proposed. Specifically, we constructed a GPT -style WMLM model and trained it on a real-world large-scale dataset, leveraging wireless signals as an anchor modality for contrastive learning. Our approach demonstrates outstanding performance compared with existing small-scale models and large multi-modal models, validating the feasibility of using wireless signals as a universal modality and highlighting WMLM's potential to emerge as a new paradigm for future wireless networks. The advent of large AI models (LMs) such as ChatGPT has propelled network intelligence into a new evolutionary phase. These remarkable enablers are poised to revolutionize future wireless networks through their advanced performance and generalization capability.

Abstract--This paper presents a novel trajectory planning pipeline for complex driving scenarios like autonomous lane changing, by integrating risk-aware planning with guaranteed collision avoidance into a unified optimization framework. We first construct a dynamic risk fields (DRF) that captures both the static and dynamic collision risks from surrounding vehicles. Then, we develop a rigorous strategy for generating time-varying convex feasible spaces that ensure kinematic feasibility and safety requirements. The trajectory planning problem is formulated as a finite-horizon optimal control problem and solved using a constrained iterative Linear Quadratic Regulator (iLQR) algorithm that jointly optimizes trajectory smoothness, control effort, and risk exposure while maintaining strict feasibility. Extensive simulations demonstrate that our method outperforms traditional approaches in terms of safety and efficiency, achieving collision-free trajectories with shorter lane-changing distances (28.59 m) and times (2.84 s) while maintaining smooth and comfortable acceleration patterns. In dense roundabout environments the planner further demonstrates robust adaptability, producing larger safety margins, lower jerk, and superior curvature smoothness compared with APF, MPC, and RRT based baselines. These results confirm that the integrated DRF with convex feasible space and constrained iLQR solver provides a balanced solution for safe, efficient, and comfortable trajectory generation in dynamic and interactive traffic scenarios.

We study the problem of computing an optimal large language model (LLM) policy for the constrained alignment problem, where the goal is to maximize a primary reward objective while satisfying constraints on secondary utilities. Despite the popularity of Lagrangian-based LLM policy search in constrained alignment, iterative primal-dual methods often fail to converge, and non-iterative dual-based methods do not achieve optimality in the LLM parameter space. To address these challenges, we employ Lagrangian duality to develop an iterative dual-based alignment method that alternates between updating the LLM policy via Lagrangian maximization and updating the dual variable via dual descent. In theory, we characterize the primal-dual gap between the primal value in the distribution space and the dual value in the LLM parameter space. We further quantify the optimality gap of the learned LLM policies at near-optimal dual variables with respect to both the objective and the constraint functions. These results prove that dual-based alignment methods can find an optimal constrained LLM policy, up to an LLM parametrization gap. We demonstrate the effectiveness and merits of our approach through extensive experiments conducted on the PKU-SafeRLHF and Anthropic HH-RLHF datasets.

Deep learning models are increasingly deployed in safety-critical tasks where predictions must satisfy hard constraints, such as physical laws, fairness requirements, or safety limits. However, standard architectures lack built-in mechanisms to enforce such constraints, and existing approaches based on regularization or projection are often limited to simple constraints, computationally expensive, or lack feasibility guarantees. This paper proposes a model-agnostic framework for enforcing input-dependent linear equality and inequality constraints on neural network outputs. The architecture combines a task network trained for prediction accuracy with a safe network trained using decision rules from the stochastic and robust optimization literature to ensure feasibility across the entire input space. The final prediction is a convex combination of the two subnetworks, guaranteeing constraint satisfaction during both training and inference without iterative procedures or runtime optimization. We prove that the architecture is a universal approximator of constrained functions and derive computationally tractable formulations based on linear decision rules. Empirical results on benchmark regression tasks show that our method consistently satisfies constraints while maintaining competitive accuracy and low inference latency.