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arXiv2026

A Theory of Information Architecture for Networked Decisions: Freshness, Locality, and Coordination

Networked systems face a tradeoff between the scope of the information behind a decision and its freshness: a broader view of the system supports better coordination, but assembling and communicating it takes time, so it arrives older. We study this tradeoff for a team of agents that repeatedly choose actions to minimize a shared quadratic cost driven by an environment that evolves on its own, unaffected by the agents' actions. An information architecture specifies what each agent observes, from where, and with what delay; we measure an architecture by the cost it loses relative to a decision made with complete, current information. We first compare fresh local observations with a complete but delayed global view. When every component of the environment decorrelates at a common exponential rate, this comparison reduces to a closed-form threshold on the ratio of delay to coherence time. We then study intermediate architectures in which each agent acts on a time-aligned, and therefore older, snapshot of a wider neighborhood. The optimal neighborhood radius occurs where the marginal value of added scope equals the marginal cost of lost freshness. In a canonical spatial model, this radius is set by the spatial-correlation, decision-relevance, and temporal-propagation lengths. Throughout, architecture performance is governed by the predictability of the optimal decision rather than of the raw state.

eess.SY↗

arXiv2026

Adaptive stabilization of a leaderless bearing-constrained formation with disturbances

In this paper, we consider the problem of regulating and maintaining a target formation characterized by a set of bidirectional bearing constraints under disturbances. The agents in the formation are modeled by single integrators with bounded continuous disturbances of which the upper bound is unavailable for the control design. Due to the time-varying disturbances, the target formation is time-varying. We propose adaptive sliding mode control laws to uniformly globally asymptotically stabilizes the moving target formation and reject the matched disturbances. In addition, to alleviate chattering phenomena from sliding mode control, smooth adaptive control laws are then designed to guarantee uniform global boundedness of the desired formation. Finally, simulation results are given to support the analysis.

cs.MA↗

arXiv2026

From Tsallis to KL: Convergence and Error Estimates for Tsallis-Regularized Optimal Transport

We study the Tsallis-to-Kullback--Leibler (KL) limit for entropy-regularized optimal transport with nonnegative bounded continuous costs. Fixing the regularization parameter $\varepsilon > 0$, we first derive an exact variational reformulation of Tsallis-regularized optimal transport in terms of the Tsallis information projection onto the set of couplings. The formula isolates an explicit correction term and thereby explains why, unlike in the KL case, the regularized transport problem and the corresponding information projection problem do not coincide exactly. We also establish existence and uniqueness for the Tsallis information projection. We then prove, with respect to the narrow topology, the $Γ$-convergence of the Tsallis-regularized functionals to the KL-regularized functional as $q\downarrow1$, together with narrow convergence of their unique minimizers. Finally, we obtain explicit error estimates of order $O(q-1)$ for both the regularized optimal transport values and the associated information projection values. These results quantify the passage from Tsallis regularization to the classical KL setting and clarify the relation between entropic regularization and information projection for $1 < q \leq 2$.

cs.IT↗

arXiv2026

Bi-HYCO: Bi-Objective Cooperative Learning for PDE Parameter Identification under Fragmented Observations

Physical and synthetic models may describe complementary aspects of the same PDE-governed system while receiving different, possibly fragmented, observations. We propose Bi-Objective HYCO (Bi-HYCO), a cooperative framework that retains both representations and their local observational objectives while coupling their predicted states at unlabeled interaction points. These points contain no measurements and do not augment the data; they provide a communication mechanism in the common state space. The two criteria form a vector-valued objective, and weighted scalarizations provide computational realizations. For the deterministic shared-observation algorithm with fixed interaction points, we prove sufficient decrease and finite length of the whole alternating sequence, which converges to a mixed critical point under the stated Kurdyka-Lojasiewicz-type assumptions. Elliptic transmission and two-dimensional Navier-Stokes experiments assess parameter and state reconstruction, noise and scalarization effects, and PINN/XPINN references. Ablations show that removing state interaction while retaining aggregation deteriorates parameter recovery in the tested configurations, particularly for Navier-Stokes.

cs.LG↗

arXiv2026

Stochastic Nonconvex Bilevel Optimization: Improved Rates Without Rare-Visit Assumption

We investigate stochastic simple bilevel optimization with smooth and possibly nonconvex upper- and lower-level objectives. Existing stochastic extensions of dynamic barrier gradient descent (DBGD) either obtain fast convergence under an unverifiable trajectory-dependent ``rare-visit'' assumption, or remove this assumption at a substantially higher oracle cost. We show that a simple denominator-only regularization of the DBGD multiplier eliminates the need for such an assumption while preserving fast convergence rates. Specifically, our method achieves $(\varepsilon, \varepsilon)$-stationarity in $O(\varepsilon^{-2})$ iterations using $O(\varepsilon^{-4})$ upper-level and $O(\varepsilon^{-7})$ lower-level stochastic gradients, which improves upon the best assumption-free complexities. We additionally derive anytime parameter schedules.

math.OC↗

arXiv2026

Convex optimization on moment polytopes: Hadamard mirror descent and efficient algorithms for quantum functionals and other tensor parameters

Convex optimization on polytopes arises in many areas of science. When the polytope is given implicitly or has exponentially many vertices and facets, standard methods may not apply or be ineffective. This is the case for moment polytopes, such as the entanglement polytopes, which play a foundational role in quantum information and algebraic complexity. They give rise to important entanglement measures and tensor parameters such as the quantum functionals, yet general effective methods for computing these quantities have been elusive. In this paper we address this challenge. We develop a first-order framework called Hadamard mirror descent to optimize suitable convex functions over moment polytopes and, more generally, the gradient sets of geodesically convex functions. It operates locally and does not rely on any explicit description of the polytope. Our framework extends mirror descent, an effective and widely used framework for convex optimization, from the Euclidean setting to Hadamard manifolds, and is motivated by a recent work by Hirai, which we interpret as a Hadamard version of mirror flow. Applying the framework to entanglement polytopes yields the first efficient first-order algorithms to compute the quantum functionals, the symmetric quantum functional, and the G-stable ranks, as well as a new direct algorithm for the non-commutative rank.

cs.CC↗

arXiv2026

Spectral Koopman-Hopf Formula for Reachability with Adversary

This paper develops a spectral Koopman-Hopf framework for adversarial reachability analysis of nonlinear systems. By lifting the nonlinear drift dynamics into Koopman eigenfunction coordinates, the proposed approach transforms the original state-dependent Hamilton-Jacobi-Isaacs (HJI) equation into an approximate state-independent optimization problem in spectral coordinates. The transformed control and disturbance directions are approximated using least-squares spectral projections, and upper and lower bounds on the transformed Hamiltonian are used to derive corresponding bounds on the value function and backward reachable sets. Koopman eigenfunctions are computed using a path-integral-based Galerkin framework that avoids spatial discretization of the associated eigenfunction PDEs. Numerical examples demonstrate tractable approximation of backward reachable sets for nonlinear adversarial systems.

eess.SY↗

arXiv2026

Collision-Free Velocity Scheduling for Multi-Agent Systems on Predefined Routes via Inexact-Projection ADMM

In structured multi-agent transportation systems, agents often must follow predefined routes, making spatial rerouting undesirable or impossible. This paper addresses route-constrained multi-agent coordination by optimizing waypoint passage times while preserving each agent's assigned waypoint order and nominal route assignment. A differentiable surrogate trajectory model maps waypoint timings to smooth position profiles and captures first-order tracking lag, enabling pairwise safety to be encoded through distance-based penalties evaluated on a dense temporal grid spanning the mission horizon. The resulting nonlinear and nonconvex velocity-scheduling problem is solved using an inexact-projection Alternating Direction Method of Multipliers (ADMM) algorithm that combines structured timing updates with gradient-based collision-correction steps and avoids explicit integer sequencing variables. Numerical experiments on random-crossing, bottleneck, and graph-based network scenarios show that the proposed method computes feasible and time-efficient schedules across a range of congestion levels and yields shorter mission completion times than a representative hierarchical baseline in the tested bottleneck cases.

cs.RO↗

arXiv2026

Observability conditions for neural state-space models with eigenvalues and their roots of unity

We operate through the lens of ordinary differential equations and control theory to study the concept of observability in the context of neural state-space models and the Mamba architecture. We develop strategies to enforce observability, which are tailored to a learning context, specifically where the hidden states are learnable at initial time, in conjunction to over its continuum, and high-dimensional. We also highlight our methods emphasize eigenvalues, roots of unity, or both. Our methods effectuate computational efficiency when enforcing observability, sometimes at great scale. We formulate observability conditions in machine learning based on classical control theory and discuss their computational complexity. Our nontrivial results are fivefold. We discuss observability through the use of permutations in neural applications with learnable matrices without high precision. We present two results built upon the Fourier transform that effect observability with high probability up to the randomness in the learning. These results are worked with the interplay of representations in Fourier space and their eigenstructure, nonlinear mappings, and the observability matrix. We present a result for Mamba that is similar to a Hautus-type condition, but instead employs an argument using a Vandermonde matrix instead of eigenvectors. Our final result is a shared-parameter construction of the Mamba system, which is computationally efficient in high exponentiation. We develop a training algorithm with this coupling, showing it satisfies a Robbins-Monro condition under certain orthogonality, while a more classical training procedure fails to satisfy a contraction with high Lipschitz constant.

cs.LG↗

arXiv2026

Incremental Peak-to-Peak Gain Computation and Regulation of Discrete-Time Positive Luré Systems using Linear Programming

Incremental gains provide a quantitative measure of robustness for trajectories of a dynamical system. Finite incremental gains ensure that all pairs of trajectories will together converge to a fixed point or will diverge together in the absence of an applied input. This work approaches the problem of computing the incremental peak-to-peak gains for positive linear systems in Luré feedback with static memoryless nonlinearities, and regulating the incremental $\ell_\infty$ gain through the design of a state-feedback controller. Upper-bounds on these incremental gains can be computed through linear programming. Computation and state-feedback regulation of the $\ell_\infty$ incremental gain is verified by numerical examples.

math.OC↗

arXiv2026

Distributed Time-Varying Optimization via Unbiased Extremum Seeking

This paper proposes a novel distributed optimization framework that addresses time-varying optimization problems without requiring explicit derivative information of the objective functions. Traditional distributed methods often rely on derivative computations, limiting their applicability when only real-time objective function measurements are available. Leveraging unbiased extremum seeking, we develop continuous-time algorithms that utilize local measurements and neighbor-shared data to collaboratively track time-varying optima. Key advancements include compatibility with directed communication graphs, customizable convergence rates (asymptotic, exponential, or prescribed-time), and the ability to handle dynamically evolving objectives. By integrating chirpy probing signals with time-varying frequencies, our unified framework achieves accelerated convergence while maintaining stability under mild assumptions. Theoretical guarantees are established through Lie bracket averaging and Lyapunov-based analysis, with linear matrix inequality conditions ensuring rigorous convergence. Numerical simulations validate the effectiveness of the algorithms.

math.OC↗

arXiv2026

A New Kernel Regularity Condition for Distributed Mirror Descent: Broader Coverage and Simpler Analysis

Existing convergence analyses of distributed optimization methods in non-Euclidean geometries typically rely on kernel assumptions: (i) global Lipschitz smoothness and (ii) bi-convexity of the associated Bregman divergence function. Unfortunately, these conditions are violated by nearly all kernels used in practice, leaving a huge theory-practice gap. This work closes this gap by developing a unified analytical tool that guarantees convergence under mild conditions. Specifically, we introduce Hessian relative uniform continuity (HRUC), a regularity satisfied by nearly all standard kernels. Importantly, HRUC is closed under concatenation, positive scaling, composition, and various kernel combinations. Leveraging the geometric structure induced by HRUC, we derive convergence guarantees for mirror-descent-based gradient tracking without imposing any restrictive assumptions. More broadly, our analysis techniques extend seamlessly to other decentralized optimization methods in genuinely non-Euclidean and non-Lipschitz settings.

math.OC↗

arXiv2026

Finite-time Reachability for Constrained, Partially Uncontrolled Nonlinear Systems

This paper presents a technique to drive the state of a constrained nonlinear system to a specified target state in finite time, when the system suffers a partial loss in control authority. Our technique builds on a recent method to control constrained nonlinear systems by building a simple, linear driftless approximation at the initial state. We construct a partition of the finite time horizon into successively smaller intervals, and design controlled inputs based on the approximate dynamics in each partition. Under conditions that bound the length of the time horizon, we prove that these inputs result in bounded error from the target state in the original nonlinear system. As successive partitions of the time horizon become shorter, the error reduces to zero despite the effect of uncontrolled inputs. A simulation example on the model of a fighter jet demonstrates that the designed sequence of controlled inputs achieves the target state despite the system suffering a loss of control authority over one of its inputs.

math.OC↗

arXiv2026

Representation Costs in Data Science: Foundations and the Quasi-Banach Spaces of Deep Neural Networks

We develop a general framework for analyzing representation costs induced by parameter-space regularizers in data-fitting methods. For an arbitrary parametric method, we define its representation cost and native function space, prove existence, and identify conditions under which parameter-space and function-space problems have equal infimal values and minimizers transfer between them. This framework yields representer theorems and recovers classical formulations---including kernel methods and RKHSs, wavelets and Besov spaces, and shallow neural networks and variation spaces---as special cases. Our main new results concern depth-$L$ feedforward ReLU networks with weight-decay regularization. For these networks, we prove that the representation cost is a power of a quasi-seminorm and that, under suitable hypotheses, the native space is a quasi-Banach space with nonconvex unit ball when $L > 2$. These results identify a novel depth-dependent quasi-Banach geometry induced by weight decay.

math.FA↗

arXiv2026

Mean-Field PhiBE: Continuous-Time Mean-Field Reinforcement Learning from Discrete-Time Data

This paper develops a model-free framework for continuous-time mean-field control when the population evolves according to unknown controlled McKean--Vlasov dynamics and only discrete-time transition data are available. Model-based mean-field control requires the continuous-time drift and diffusion coefficients, which are not directly observed from fixed-step transitions, while a direct reduction to a discrete-time Bellman equation loses the continuous-time generator structure. To bridge these two viewpoints, we introduce a Mean-Field-PhiBE (MF-PhiBE), which incorporates discrete-time transition information into a continuous-time PDE on the Wasserstein space. The MF-PhiBE replaces the unknown infinitesimal drift and covariance in the policy-evaluation equation by one-step estimators computed from data, while preserving the generator structure of the McKean-Vlasov HJB equation. We also derive a policy-gradient theorem for entropy-regularized randomized feedback policies, expressing the actor direction through an action-wise infinitesimal advantage and the score of the policy. Combining these two ingredients yields a model-free actor-critic method. We prove a first-order consistency estimate showing that the value induced by an optimal MF-PhiBE policy approximates the optimal continuous-time value as the observation time step vanishes. For entropy-regularized LQR, we establish first-order policy convergence and second-order value convergence; under suitable conditions, the population-averaged feedback means coincide exactly. Numerical experiments on an LQR benchmark and a crowd-aversion problem illustrate the proposed framework.

math.OC↗

arXiv2026

Learning-enabled Acceleration of Scenario-based Model Predictive Control

Scenario-based model predictive control (SBMPC) is a variant of model predictive control (MPC) that explicitly accounts for uncertainty by optimizing control actions over multiple predicted scenarios. However, its computational complexity increases rapidly with the number of scenarios and prediction horizon, limiting its applicability to real-time planning and control. This paper presents a learning-accelerated Alternating Direction Method of Multipliers (ADMM) algorithm for efficiently solving SBMPC problems by leveraging parallel computing and Moreau envelope learning, while maintaining high solution accuracy. We reformulate the SBMPC problems into consensus forms that can be decomposed via ADMM, separating the scenario-dependent dynamics from non-anticipativity constraints and enabling parallel updates across scenarios and time steps. Building on this decomposition, we utilize a learning-to-optimize scheme that leverages Moreau envelope learning of the cost function to accelerate the primal update in ADMM, thereby reducing computation time. The proposed framework is evaluated on a microgrid energy management problem subject to load and renewable generation uncertainties. Comparisons with IPOPT and MadNLP, two popular and modern nonlinear programming solvers, demonstrate substantial computational speedups while maintaining reliable closed-loop control performance.

math.OC↗

arXiv2026

Efficient Hessian-Free Methods for Multi-Objective Bilevel Optimization with Nonconvex Lower Level

Multi-objective bilevel optimization has wide applications in the AI area such as automated learning and multi-task meta-learning. Although recently some works have been begun to study the multi-objective bilevel optimization, the proposed methods rely on the (strongly) convex lower level problems. In fact, these multi-objective bilevel learning problems are generally nonconvex, and particularly their lower level problems are nonconvex. To fill this gap, we propose a class of Multi-Objective Moreau Envelope based Hessian-free Algorithms (MOMEHA) for the multi-objective bilevel learning problems with nonconvex lower level. Specifically, our method uses the Moreau envelope to relax the original problem into a multi-objective single-level optimization with an envelope constraint. In particular, our method retains computational advantages of being single-loop and Hessian-free in the multi-objective setting by incorporating a smooth weighted Tchebycheff scalarization. Furthermore, we propose a momentum-based variant of MOMEHA (i.e., MB-MOMEHA) method for the stochastic multi-objective bilevel learning problems. In theory, we provide the convergence properties of our algorithms under both deterministic and stochastic setting. Some experiments on few-shot meta-learning and neural architecture search demonstrate that our methods outperform the existing approaches in Pareto front, validating its effectiveness and robustness.

math.OC↗

arXiv2026

Anchored Regularized Direct Least Squares (ARDLS): Integrating Established Prioritization Operators for Priority Elicitation in the Analytic Hierarchy Process

Pairwise reciprocal matrices are fundamental to the Analytic Hierarchy Process (AHP),a decision-making model. While the Direct Least Squares (DLS) method provides an intuitive mechanism for deriving priority vectors without complex transformations, it is susceptible to solution non-uniqueness. Under high levels of inconsistency, such as severe cyclic contradictions, the DLS optimization landscape becomes non-convex, yielding multiple distinct global minima. Consequently, priority rankings become unstable and critically dependent on initial algorithmic guesses. Furthermore, established prioritization operators (POs), including normalization techniques, the Eigenvector method, Singular Value Decomposition, Cosine Maximization, and the Pseudo-Inverse Gram Matrix (the closed-form solution of Weighted Least Squares), frequently generate disparate outcomes. To overcome these structural deficiencies, this paper introduces the Anchored Regularized Direct Least Squares (ARDLS) optimization model as a harmonizing framework. ARDLS integrates uniquely determined established POs as theoretical anchors within a regularization penalty. This integration systematically breaks mathematical symmetries and tilts the optimization landscape to guarantee convergence upon a single, unique global minimum. By minimizing the root mean square variance (RMSV) of the initial baseline vectors, ARDLS effectively unifies these divergent solutions. Comprehensive numerical experiments validate that the framework successfully fine-tune the solution of established POs by reducing RMSV while ensuring strict mathematical uniqueness. The practical utility of the method is further demonstrated through a numerical case study resolving an innovation fund dilemma in FinTech project selection. The proposed ARDLS approach offers a robust alternative to classical AHP across a wide range of decision-making domains.

math.OC↗