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math.OC

math.OC: explore 248 source-linked works published from 2017 to 2026, with original documents and citations.

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Includes records with this source-supplied label or an explicit phrase match in their metadata. Matches indicate a mention, not proof that a paper uses a method or tests a material. Source versions are consolidated by DOI.

Sources: arxiv. Collection updated 2026-09-16. Counts describe this index, not the complete source archives.

arXiv2026

Characterizations and Complexity of Minimum Forward and Integer Cycle Bases

The cycle space of a directed graph is generated by a cycle basis, where, in general, cycles are allowed to have both forward and backward arcs. In a forward cycle, all arcs must follow the given direction. Several open questions remain regarding the complexity of the minimum cycle basis problem, in particular the minimum-weight integral cycle basis problem, and the minimum-weight weakly and strictly fundamental forward cycle basis problems. In this paper, we address these open questions. First, we study the existence, structure, and computational complexity of minimum-weight forward cycle bases. We give a complete structural characterization of digraphs that admit weakly fundamental (and hence integral) forward cycle bases. We further provide a characterization when a strongly connected digraph admits a forward fundamental cycle basis, proving that such a basis exists if and only if the set of directed cycles has cardinality equal to the cycle rank; in this case, the basis is unique. Lastly, we show that while minimum-weight forward fundamental cycle bases can be found in polynomial time whenever they exist, the minimum-weight forward weakly fundamental cycle basis problem is APX-hard via an L-reduction from the minimum-weight weakly fundamental cycle basis problem on digraphs with metric weights. Second, we introduce opt-in graphs, i.e., the family of graphs for which minimum cycle bases are integral for any weight function. We show that this family is minor-closed and hence, by the Robertson-Seymour theorem, is characterized by a finite set of forbidden minors, so that the opt-in recognition problem is solvable in polynomial time. Lastly, we present an algorithm to check whether a graph is opt-in, and if not, to identify which of its minors belong to the set of forbidden minors. Applying this algorithm, we show that the complete graph $K_n$ is opt-in if and only if $n \leq 7$.

math.OC↗

arXiv2026

A simple derivation of the Kalman filter

In this lecture note, we present a concise and self-contained derivation of the discrete-time Kalman filter equations that requires only a basic understanding of least squares estimation. The treatment is designed to minimize mathematical overhead while preserving both rigor and generality.

math.OC↗

arXiv2026

On the invariance of risk-sensitive LQR gain under input randomization

This paper shows that the optimal gain of the risk-sensitive linear quadratic regulator (LQR) problem is invariant under input randomization, i.e., when the controller deliberately injects noise into the nominal control input. This appears counterintuitive at first glance because certainty equivalence does not hold for risk-sensitive LQR and input randomization inflates the effective process noise. Nonetheless, the gain is preserved because the input noise enters not only the system dynamics but also the cost functional, and its total effect on the gain eventually vanishes. Consequently, the optimal gain and its associated Riccati recursion need not be recomputed, and the increment in the optimal cost can be readily evaluated in closed form. This result facilitates the use of risk-sensitive LQR in applications that employ input randomization for privacy or exploration, such as watermarking for replay attack detection, differential privacy, and path integral control.

math.OC↗

arXiv2026

Reduced order model for parametric Boltzmann equation and its application to inverse problems

The Boltzmann equation plays an important role in modeling mesoscopic behavior in a wide range of scientific and engineering applications. However, its numerical solution is computationally expensive due to the high dimensionality of the model and the nonlinear nonlocal collision operator, especially for steady-state problems that require iterative solvers. This cost becomes prohibitive for inverse problems, where the induced optimization problem requires repeated forward solves. In this work, we propose a reduced-order model (ROM) for the parametric Boltzmann equation to address this computational challenge. The ROM constructs a low-dimensional approximation space for the parameter-induced solution manifold through a residual-based greedy strategy, and the reduced solution is then obtained via residual minimization over the reduced space, subject to mass conservation. The overall efficiency of the ROM is achieved by exploiting the quadratic structure of the collision operator and a precomputed separable approximation of the collision kernel. The resulting ROM is further applied to a thermally-driven inverse problem for reconstructing collision parameters from the observed macroscopic temperature data. This is accomplished either by directly replacing the PDE constraint with the ROM, leading to a bilevel optimization formulation, or by reformulating the task as a single-level optimization problem through the Karush--Kuhn--Tucker (KKT) conditions. Numerical experiments in both collision-dominated and transport-dominated cases are performed to demonstrate the efficiency and accuracy of the proposed ROM and its effectiveness in inverse problems. In particular, the resulting inverse problem is computationally much more tractable, achieving speedups of several orders of magnitude over that based on the full-order model while maintaining comparable accuracy.

math.NA↗

arXiv2026

Momentum in large-batch training: Polyak enlarges the critical batch size, Nesterov improves data efficiency

We study when and how momentum improves large-batch training in the one-pass regime, using power-law kernel regression as a tractable setting. We first characterize risk stability through the critical learning rate, defined as the largest learning rate for stable training, and obtain $η_{\mathrm{SGD}}^{\mathrm{crit}}\eqsim 1$, $η_{\mathrm{Polyak}}^{\mathrm{crit}}\eqsim \min\{1,B(1-ρ)\}$, and $η_{\mathrm{Nesterov}}^{\mathrm{crit}}\eqsim \min\{1,B^β(1-ρ)\}$, where $B$ is the batch size, $ρ$ is the momentum factor, and $β>1$ is the capacity exponent. Within this admissible region, we derive scaling laws for the full risk dynamics, capturing the progression from an early transient, through power-law decay, to a noise floor. We then minimize the final-step risk over the admissible learning rates and momentum factors under a fixed data budget, yielding a three-regime batch-size phase diagram that reveals how the role of momentum changes with batch size. Notably, Polyak enlarges the critical batch size, the largest batch size preserving the best small-batch data-scaling exponent, thereby enabling greater parallelism without sacrificing data efficiency. In contrast, Nesterov achieves better data efficiency in the large-batch regime because its look-ahead mechanism suppresses noise accumulation. Numerical experiments validate the predicted stability boundaries, risk dynamics, and batch-size phase diagram.

stat.ML↗

arXiv2026

No-Regret Bayesian Optimization with Finite-Library Input-Warped Kernels

Gaussian-process Bayesian optimization (GP-BO) excels at black-box optimization of costly functions, e.g., hyperparameter optimization (HPO) and multi-agent system (MAS) design. Convergence-rate guarantees exist for select methods, notably GP upper confidence bound (GP-UCB), but require a fixed kernel. Critically, the kernel encodes how input proximity affects objective value similarity. When raw coordinates poorly match this geometry - as with log-scaled hyperparameters or localized peaks - input warping can greatly improve sample efficiency, yet known GP-UCB proofs require a fixed kernel. We propose Finite-Library Input-Warped Bayesian Optimization (FLIWBO), which selects warps from a finite library of smooth input maps by any history-dependent rule. It adapts the input geometry to accelerate learning while retaining high-probability convergence guarantees under mild hypotheses, with an explicit $\sqrt(N_\varepsilon)$ library-size cost. Controlled diagnostics show that finite-library warping repairs planted geometry mismatches and identify FLIWBO failure cases. Across four repeated benchmarks - warped synthetic objectives, a confidence-fence trap, and Fashion-MNIST HPO - FLIWBO-UCB beats raw-coordinate GP-UCB under misspecified geometry, escapes traps that defeat even oracle-warp expected improvement, and recovers much of the gain from manual log scaling, while leading the tested methods that admit a matching regret guarantee. A 20-dimensional MAS design study further shows feasibility under costly noisy evaluations. Code for experiments is available: https://github.com/edvin-ketabati/bogp-paper-experiments.

cs.LG↗

arXiv2026

Discrete Gromov-Wasserstein Duality: Algorithms and Isomorphism Testing

The Gromov-Wasserstein (GW) distance provides a principled framework for aligning metric measure (mm) spaces based solely on their intrinsic structure. Its ability to identify isomorphic representations of distributions across spaces renders it valuable for comparing data where equality up to isomorphism occurs naturally such as in graphs or, more generally, distributions on graphs. Recently, a type of dual form for the GW distance between Euclidean distributions with the squared Euclidean or inner product costs was derived, spurring the development of new statistical and algorithmic results for this setting. This work furnishes a novel duality result for GW distances with and without entropic regularization that is applicable to all finitely supported mm spaces. Leveraging this result, we derive the sample complexity of empirical GW distances between finite mm spaces, as well as limit distributions under proper centering and scaling. Furthermore, we propose new algorithms for solving the regularized GW problem which are subject to formal convergence guarantees. These statistical and algorithmic advancements give rise to a principled and efficient framework for testing whether two distributions on the set of graphs with a fixed number of nodes are isomorphic based on samples.

math.ST↗

arXiv2026

Structural Compatibility and Uniform Stability of Temporally Degenerate Parabolic Systems

Modern feedback design for distributed parameter systems presupposes that the closed-loop dynamics define a well-posed evolution problem. This presupposition becomes nontrivial for temporally degenerate parabolic systems, where temporal degeneracy affects not only the analytical properties of the evolution equation but also the mathematical formulation of the feedback interconnection itself. It is shown that admissible feedback interconnections for temporally degenerate parabolic systems are completely characterized by an operator compatibility condition linking the singular reaction operator with the actuator and observation operators. This characterization removes the singular component of the closed-loop dynamics and reduces the degenerate evolution equation to a regular evolution equation. Building upon this regularized formulation, a critical--residual decomposition yields a uniform exponential stability certificate, which is subsequently extended to the original infinite-dimensional evolution through a finite-to-infinite lifting theorem. A constructive static output feedback synthesis is finally obtained as a consequence of these results. Numerical experiments illustrate the regularization mechanism, validate the stability certificate, and confirm the finite-to-infinite lifting.

math.OC↗

arXiv2026

Generalized Regret Analysis of Thompson Sampling using Fractional Posteriors

Thompson sampling (TS) is one of the most popular and earliest algorithms to solve stochastic multi-armed bandit problems. We consider a variant of TS, named $α$-TS, where we use a fractional or $α$-posterior ($α\in(0,1)$) instead of the standard posterior distribution. To compute an $α$-posterior, the likelihood in the definition of the standard posterior is tempered with a factor $α$. For $α$-TS we obtain both instance-dependent $\mathcal{O}\left(\sum_{k \neq i^*} Δ_k\left(\frac{\log(T)}{C(α)Δ_k^2} + \frac{1}{2} \right)\right)$ and instance-independent $\mathcal{O}(\sqrt{KT\log K})$ frequentist regret bounds under very mild conditions on the prior and reward distributions, where $Δ_k$ is the gap between the true mean rewards of the $k^{th}$ and the best arms, and $C(α)$ is a known constant. Both the sub-Gaussian and exponential family models satisfy our general conditions on the reward distribution. Our conditions on the prior distribution can be easily satisfied by a density that is positive, continuous, and bounded. We also establish another instance-dependent regret upper bound that matches (up to constants) to that of improved UCB [Auer and Ortner, 2010]. Our regret analysis carefully adapts and combines recent theoretical developments in the non-asymptotic concentration analysis and Bernstein-von Mises type results for the $α$-posterior distribution. Moreover, our analysis does not require additional structural properties such as closed-form posteriors or conjugate priors.

stat.ML↗

arXiv2026

A Feedback Linearized Model Predictive Control Strategy for Input-Constrained Self-Driving Cars

This paper proposes a novel real-time affordable solution to the trajectory tracking control problem for self-driving cars subject to longitudinal and steering angular velocity constraints. To this end, we develop a dual-mode Model Predictive Control (MPC) solution starting from an input-output feedback linearized description of the vehicle kinematics. First, we derive the state-dependent input constraints acting on the linearized model and characterize their worst-case time-invariant inner approximation. Then, a dual-mode MPC is derived to be real-time affordable and ensuring, by design, constraints fulfillment, recursive feasibility, and uniformly ultimate boundedness of the tracking error in an ad-hoc built robust control invariant region. The approach's effectiveness and performance are experimentally validated via laboratory experiments on a Quanser Qcar. The obtained results show that the proposed solution is computationally affordable and with tracking capabilities that outperform two alternative control schemes.

eess.SY↗

arXiv2026

Infrequent Resolving Algorithm for Online Linear Programming

Online linear programming (OLP) has gained significant attention from both researchers and practitioners due to its extensive applications such as online auctions, network revenue management, order fulfillment and advertising. Existing OLP algorithms fall into two categories: LP-based algorithms and LP-free algorithms. The former typically guarantees better performance but requires solving a large number of LPs, which could be computationally expensive. In contrast, LP-free algorithms only require first-order computations but induce a worse performance. In this work, we bridge the gap between these two extremes by proposing a well-performing algorithm that solves LPs at a few selected time points and conducts first-order computations at other time points. Specifically, for the case where the inputs are drawn from an unknown finite-support distribution, the proposed algorithm achieves a constant regret (even for the hard "degenerate" case) while solving LPs only $O(\log\log T)$ times over the time horizon $T$. Moreover, when we are allowed to solve LPs only $M$ times, we design the corresponding schedule such that the proposed algorithm can guarantee a nearly $O\left(T^{(1/2)^{M-1}}\right)$ regret. Our work highlights the value of resolving both at the beginning and the end of the selling horizon, and provides a novel framework to prove the performance guarantee of the proposed policy under different infrequent resolving schedules. Numerical experiments are conducted to demonstrate the efficiency of the proposed algorithms. The code is available at https://github.com/guokai-li/infrequent_resolving_olp.

cs.DS↗

arXiv2026

A Compositional Kernel Model for Feature Learning

We study a compositional variant of kernel ridge regression in which the predictor is applied to a coordinate-wise reweighting of the inputs. Formulated as a variational problem, this model provides a tractable setting for studying feature learning in compositional architectures. From the perspective of variable selection, we show how relevant variables are recovered while noise variables are eliminated. We prove that both global minimizers and stationary points discard noise coordinates when the noise variables are Gaussian distributed. A central finding is that $\ell_1$-type kernels, such as the Laplace kernel, succeed in recovering features contributing to nonlinear effects at stationary points, whereas Gaussian kernels recover only linear ones.

cs.LG↗

arXiv2026

Nonlinear Dynamics In Optimization Landscape of Shallow Neural Networks with Tunable Leaky ReLU

In this work, we study the nonlinear dynamics of a shallow neural network trained with mean-squared loss and leaky ReLU activation. Under Gaussian inputs and equal layer width k, (1) we establish, based on the equivariant gradient degree, a theoretical framework, applicable to any number of neurons k>= 4, to detect bifurcation of critical points with associated symmetries from global minimum as leaky parameter $α$ varies. Typically, our analysis reveals that a multi-mode degeneracy consistently occurs at the critical number 0, independent of k. (2) As a by-product, we further show that such bifurcations are width-independent, arise only for nonnegative $α$ and that the global minimum undergoes no further symmetry-breaking instability throughout the engineering regime $α$ in range (0,1). An explicit example with k=5 is presented to illustrate the framework and exhibit the resulting bifurcation together with their symmetries.

math.OC↗

arXiv2026

Uniform a priori bounds and error analysis for the Adam stochastic gradient descent optimization method

The adaptive moment estimation (Adam) optimizer proposed by Kingma & Ba (2014) is presumably the most popular stochastic gradient descent (SGD) optimization method for the training of deep neural networks (DNNs) in artificial intelligence (AI) systems. Despite its groundbreaking success in the training of AI systems, it still remains an open research problem to provide a complete error analysis of Adam, not only for optimizing DNNs but even when applied to strongly convex stochastic optimization problems (SOPs). Previous error analysis results for strongly convex SOPs in the literature provide conditional convergence analyses that rely on the assumption that Adam does not diverge to infinity but remains uniformly bounded. It is the key contribution of this work to establish uniform a priori bounds for Adam and, thereby, to provide -- for the first time -- an unconditional error analysis for Adam for a large class of strongly convex SOPs.

cs.LG↗

arXiv2026

Adjoint DSMC Method for Spatially Inhomogeneous Boltzmann Equation with General Boundary Conditions

We develop adjoint Direct Simulation Monte Carlo (DSMC) formulations for the spatially inhomogeneous Boltzmann equation with periodic, specular reflecting, diffuse thermal, and prescribed inflow boundary conditions. Periodic and specular boundaries are treated using a pathwise particle adjoint conditional on the realized event history. For diffuse thermal boundaries, we introduce a randomized-time regularization of wall-crossing events and use score-function terms to differentiate the resulting boundary probabilities. Reparameterization of the outgoing half-Maxwellian samples provides sensitivities with respect to wall temperatures and tangential wall velocities. Prescribed inflow requires a different construction because perturbations of incoming particles affect subsequent cell populations, local collision frequencies, and collision schedules. We therefore derive an ensemble adjoint based on the locally linearized Boltzmann collision operator and evaluate boundary sensitivities using local inflow-source scores, with injection counts held fixed. For a scalar objective, the dominant adjoint cost is largely independent of the number of parameters. Numerical experiments for Maxwell molecules validate the formulations against centered finite differences for thermal, mixed thermal-specular, two-sided inflow, and high-Mach Couette-flow configurations. The results demonstrate consistent gradient estimates, Monte Carlo convergence, stability with respect to the thermal regularization parameter, and accurate sensitivity calculation in a regime with limited relative statistical noise.

math.NA↗

arXiv2026

The Endogeneity of Miscalibration: Impossibility and Escape in Scored Reporting

An agent's probability report is paid for twice: by a strictly proper scoring rule, and by an approval rule for the decision it triggers. In this classical decision-coupled setting, non-affine approval is known to defeat truthful reporting. We show the conflict is endogenous: when feasible, the welfare-maximizing approval rule is never affine. The distortion, however, is predictable and can be designed around. There is a reserve report at which pretending to be the marginal type costs exactly the approval prize. Approving at or above the reserve screens types perfectly under every strictly proper score, and the reserve does not depend on the type distribution. A Lipschitz rule with a single kink attains first-best exactly; under strict feasibility no continuously differentiable rule does. The binding constraint is steepness, not smoothness. First-best is attainable within a slope budget if and only if the budget is at least the critical slope: the steepest chord of the pretending cost up to the reserve. Below it the welfare loss is cubic in the shortfall. Where the pretending cost is convex up to the reserve, as for Brier, log and power scores, the critical slope is closed-form. The instances are AI-agent oversight and marketplace operation.

cs.GT↗

arXiv2026

Universal Approximation of Nonlinear Operators and Their Derivatives

We show that Universal Approximation (UA) of nonlinear operators and their derivatives via Operator Learning (OL) architectures fails in ${C^k_F}$ (Fréchet) compact-open topologies and in Fréchet--Sobolev norms (i.e. under operator norms). We solve this obstruction by restoring UA in natural weaker topologies: $C^k_B$ (Bastiani) compact-open topologies and (novel) weighted Bastiani--Sobolev spaces for general finite input measures. In full Banach-space generality, these are the first complete generalizations of the corresponding influential classical results in [Hornik, 1991] to infinite-dimensional spaces and OL. Based on our UATs, we formulate Bastiani--Sobolev training in DIOL. These results launch Derivative-Informed Operator Learning (DIOL) (i.e. learning nonlinear operators and their derivatives) on general Banach spaces. We parameterize nonlinear operators via Encoder-Decoder Architectures, classical OL architectures available in general Banach spaces; these include DeepONets, Deep-H-ONets, and PCA-Nets, which our UATs cover. A key mathematical result is that our new weighted Bastiani--Sobolev spaces generalize classical Gaussian (Malliavin) Sobolev spaces on Banach spaces. Open frontiers where DIOL and our UATs find applications are: high-order accuracy in OL; fast constrained optimization in Banach spaces (e.g. optimal control of PDEs, inverse problems) via Learn-Then-Optimize; numerical methods for infinite-dimensional PDEs (e.g. HJB PDEs on Banach spaces from infinite-dimensional optimal control via Optimize-Then-Learn, such as optimal control of PDEs, SPDEs, path-dependent systems, partially observed systems, mean-field control).

cs.LG↗

arXiv2026

Independent Reinforcement Learning in Discounted Markov Games

In this work, we study radically uncoupled learning in discounted general-sum Markov games. Assuming ``$\mathsf{ETH}$ for $\mathsf{PPAD}$", we show that, for every fixed discount factor, there is no polynomial-time algorithm for computing inverse-polynomially accurate coarse correlated equilibria in discounted general-sum Markov games when players learn independently in decentralized settings. Complementing this hardness result, we provide what appears to be the first \emph{radically uncoupled} algorithm with sub-exponential convergence guarantees to coarse correlated equilibria in discounted general-sum Markov games without imposing any structural restrictions on the game. Our algorithm is a \emph{layered} variant of optimistic mirror descent with an increasing step-size schedule tailored to the multi-agent setting. Finally, we develop both full-feedback and partial feedback versions of the aforementioned algorithm and establish sub-exponential convergence guarantees for each case.

cs.GT↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Characterizations and Complexity of Minimum Forward and Integer Cycle Bases	2026-09-02	2609.02317	arxiv
A simple derivation of the Kalman filter	2026-09-02	2609.02332	arxiv
On the invariance of risk-sensitive LQR gain under input randomization	2026-09-02	2609.02363	arxiv
Reduced order model for parametric Boltzmann equation and its application to inverse problems	2026-09-02	2609.02578	arxiv
Momentum in large-batch training: Polyak enlarges the critical batch size, Nesterov improves data efficiency	2026-09-02	2609.02728	arxiv
No-Regret Bayesian Optimization with Finite-Library Input-Warped Kernels	2026-09-02	2609.02993	arxiv
Discrete Gromov-Wasserstein Duality: Algorithms and Isomorphism Testing	2026-09-02	2609.03094	arxiv
Structural Compatibility and Uniform Stability of Temporally Degenerate Parabolic Systems	2026-09-02	2609.03136	arxiv
Generalized Regret Analysis of Thompson Sampling using Fractional Posteriors	2026-09-01	2309.06349	arxiv
A Feedback Linearized Model Predictive Control Strategy for Input-Constrained Self-Driving Cars	2026-09-01	2405.01753	arxiv
Infrequent Resolving Algorithm for Online Linear Programming	2026-09-21	2408.00465	arxiv
A Compositional Kernel Model for Feature Learning	2026-09-01	2509.14158	arxiv
Nonlinear Dynamics In Optimization Landscape of Shallow Neural Networks with Tunable Leaky ReLU	2026-09-01	2510.25060	arxiv
Uniform a priori bounds and error analysis for the Adam stochastic gradient descent optimization method	2026-09-01	2603.18899	arxiv
Adjoint DSMC Method for Spatially Inhomogeneous Boltzmann Equation with General Boundary Conditions	2026-09-01	2603.20946	arxiv
The Endogeneity of Miscalibration: Impossibility and Escape in Scored Reporting	2026-10-05	2605.07671	arxiv
Universal Approximation of Nonlinear Operators and Their Derivatives	2026-10-01	2605.15285	arxiv
Independent Reinforcement Learning in Discounted Markov Games	2026-09-01	2609.00504	arxiv

These are bibliographic comparisons, not experimental rankings. Follow the original document for methods and conditions.