apply gronwall inequalities

This work addresses the failure of Wormald’s differential equation method when only an upper bound on the expected one-step change is available—lacking a lower bound or tight estimate—and establishes, for the first time, a theoretical framework under one-sided constraints. Methodologically, it integrates martingale inequalities, coupling techniques, and discrete dynamical systems analysis to derive a general one-sided concentration theorem. Theoretically, it rigorously proves that an upper bound alone suffices to guarantee, with high probability, that the trajectory of a random process stays close to the solution of the associated deterministic differential equation; moreover, the original Wormald method emerges naturally as a special (non-degenerate) case. Practically, this significantly lowers technical barriers, providing a more flexible and robust analytical tool for settings lacking symmetric estimates—such as greedy algorithm analysis and stochastic graph evolution processes.

This work addresses the problem of estimating the drift function in stochastic differential equations when the diffusion coefficient is known, framing it as a denoising task amenable to diffusion-based modeling. By leveraging conditional score matching, the method recovers the drift function from discrete observations across multiple sample trajectories. The study establishes, for the first time, an explicit time-averaged mean squared error risk bound for this class of estimators. The theoretical analysis elucidates the interplay among four key sources of error: Euler–Maruyama discretization, score approximation, noise initialization, and sampling variance. The resulting risk decomposition provides a sharp characterization of how various hyperparameters influence estimation accuracy, thereby offering rigorous theoretical guarantees for drift estimation powered by diffusion models.

This work presents the first fully AI-generated, rigorous explicit lower bounds for the advection-diffusion equation across three distinct flow regimes—non-diffusive shear flows, diffusive shear flows, and rapidly oscillating time-periodic flows—without any human intervention. Leveraging the multi-agent automated reasoning system QED, which integrates partial differential equation analysis with formal verification, the study derives data-dependent explicit constants: a polynomial Ḣ⁻¹ decay bound for non-diffusive shear flows, a uniform positive lower bound on mixing scales for diffusive shear flows, and an exponential L² decay bound for rapidly oscillating flows. This achievement marks a breakthrough in the application of artificial intelligence to complex mathematical proofs.

This work investigates the non-asymptotic convergence of Particle Gradient Descent (PGD) for maximum likelihood estimation in large latent-variable models. Addressing free energy functional optimization, we introduce a unified generalization of logarithmic Sobolev and Polyak–Łojasiewicz-type conditions, establishing their first equivalence with Talagrand’s inequality and quadratic growth—thereby extending the Bakry–Émery theory. Leveraging tools from optimal transport, information geometry, and stochastic differential equations, we prove that under strong concavity of the log-likelihood, PGD—implemented as a discrete-time approximation of the free energy gradient flow—achieves exponential convergence. Moreover, we derive the first tight non-asymptotic upper bound on the discretization error. These results provide foundational theoretical guarantees for PGD in latent-variable modeling, marking a key advance in the rigorous analysis of particle-based variational inference methods.

This paper investigates the asymptotic efficiency bound for estimating the average treatment effect (ATE) in binary and multi-treatment sequential experiments under covariate-dependent assignment mechanisms—such as stratification and adaptive designs. Methodologically, it integrates asymptotic statistical theory, stochastic process modeling, and semiparametric efficiency analysis. The key contribution is the first rigorous proof that, under covariate-adaptive allocation, no estimator can achieve first-order asymptotic efficiency exceeding Hahn’s (1998) classical bound. This establishes a unified upper efficiency bound encompassing multi-treatment settings, constrained experimental designs, and covariate-driven single-outcome sampling. The result provides the first general theoretical benchmark for assessing optimality in experimental design and reveals the fundamental theoretical ceiling for design optimization.

This work addresses the looseness of existing non-asymptotic error bounds for Langevin Monte Carlo methods under strongly log-concave distributions, which overly rely on global smoothness constants and consequently deteriorate in high-dimensional or correlated-covariate settings. To overcome this limitation, the paper introduces a coordinate-wise averaged smoothness condition to characterize the potential function and combines synchronous coupling with Wasserstein distance analysis to derive substantially tighter bounds. The key innovation lies in replacing the global smoothness constant with an average coordinate-wise counterpart and employing a trace-type third-order smoothness quantity to weaken the Hessian-Lipschitz assumption. These improvements are extended to variable step sizes, Laplacian-smooth potentials, and finite-sum structures such as SGLD. Notably, the resulting bounds exhibit significantly improved dimension dependence in high-dimensional generalized linear models, especially under covariate correlation, offering broader theoretical applicability and outperforming current state-of-the-art results.

本文研究了在基于采样的随机模型预测控制中，使用高阶方法（如Milstein离散化和Lévy-area模拟）来提高退出时间的数值估计精度，从而提升控制器的安全性。

This work addresses the challenge that conventional surrogate models struggle to provide rigorous error guarantees for path-dependent observables—such as first-passage times—when learning stochastic dynamical systems, often leading to inaccurate predictions of critical statistics. To overcome this limitation, the authors propose a goal-oriented learning framework that, for the first time, establishes general error bounds for observables defined on path space and leverages these bounds to construct a variational loss function. The approach accommodates complex functionals over unbounded time horizons, including first-passage times, and derives analytical gradients via Fréchet derivatives to enable efficient optimization with stochastic gradient descent. Numerical experiments demonstrate that the resulting surrogate models achieve significantly improved accuracy in predicting first-passage time statistics and exhibit enhanced robustness under distributional shifts in the data.

This work establishes rigorous theoretical bounds on the sampling error of Denoising Diffusion Probabilistic Models (DDPMs) measured in the 2-Wasserstein distance. Departing from the conventional view of DDPM samplers as discretized reverse Ornstein–Uhlenbeck processes, the paper introduces a novel perspective by modeling them as discretizations of Föllmer processes. Under general Lipschitz-type assumptions on the score function and across various variance schedules—including the cosine schedule—it derives non-asymptotic Wasserstein error bounds. The key contributions include proving that the Lipschitz condition implies both a logarithmic Sobolev inequality and a quadratic transportation-cost inequality, and demonstrating that even when the target distribution fails to satisfy the latter, dimension- and step-optimal Wasserstein error bounds can still be achieved. Furthermore, existing KL divergence bounds are extended to the Wasserstein setting.

This study addresses the challenge of transient instability induced by finite-time disturbances in nonlinear stochastic flight dynamics, where conventional asymptotic or mean-square stability criteria fail to ensure short-term safety. The work presents the first extension of logarithmic norms to nonlinear Itô stochastic systems, establishing a finite-time transient stability analysis framework based on matrix measures of Lipschitz nonlinear mappings. By leveraging Itô calculus, it derives explicit bounds on the growth of state mean and variance, elucidating the fundamental distinction between expected-value stability and sample-path stability. Furthermore, it introduces a trade-off mechanism between estimation consistency and transient robustness under data injection. Experiments on lunar lander–like telemetry data demonstrate that trajectories with identical mean behavior can exhibit markedly different transient responses, with mission failure strongly correlated to cumulative transient instability during critical short intervals, thereby offering a novel finite-time probabilistic stability metric for autonomous systems.