drift-variance regularizer

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.

本文提出一种非参数方差惩罚的演员-评论家框架，通过统计推断方法减少策略学习中的方差，提高了风险敏感强化学习的效率和稳定性。

We study long-horizon deployment of a frozen predictor under dynamic covariate shift. A time-domain Poincaré inequality reduces temporal risk volatility to derivative energy, and a Jacobian-velocity theorem identifies directional tangent energy along the deployment path as the governing quantity under explicit along-path regularity and domination assumptions. Under low-rank drift, that quantity reduces to directional Jacobian energy in the drift subspace, motivating drift-aligned tangent regularization (DTR) and a matched monitoring proxy. Rather than smoothing the network isotropically, DTR penalizes sensitivity only along estimated drift directions. We validate the theorem-to-method pipeline in four experiments: a synthetic benchmark for the time-domain inequality, a controlled synthetic comparison against isotropic Jacobian regularization, and two frozen-deployment studies on the UCI Air Quality and Tetouan power-consumption datasets. DTR reduces risk volatility and directional gain in the controlled low-rank regime, beats isotropic smoothing there, and gives validation-selected deployment gains on both real datasets when the Air Quality drift subspace is estimated from target-orthogonal sensor motion. Moderate drift-subspace misspecification is tolerable while orthogonal misspecification largely removes the benefit.

This study addresses the estimation of the time-homogeneous drift function in multivariate stochastic differential equations (SDEs) with known diffusion coefficients, based on high-frequency observations from multiple trajectories. To this end, the authors propose formulating drift estimation as a conditional denoising problem conditioned on historical observations and introduce a conditional diffusion model that dynamically generates new trajectories from which the drift estimator is extracted. This approach represents the first application of conditional denoising diffusion models to drift estimation in SDEs. It significantly outperforms classical methods in high-dimensional settings without relying on any specific neural network architecture, while achieving comparable performance to existing approaches in low-dimensional cases, thereby demonstrating both its effectiveness and scalability.

This paper addresses drift optimization for stochastic processes under Lipschitz-continuous controller constraints—i.e., optimizing the drift term subject to path-dependent regulatory constraints to enhance system performance. To tackle this infinite-dimensional, nonconvex, path-constrained optimization problem, we first formulate a novel drift control framework incorporating regulated path constraints. We then propose a sample-average approximation (SAA) method integrating path discretization, function-space discretization, and Monte Carlo sampling, and derive a computationally tractable path-guided directional derivative. A recursive mirror-descent-based optimization algorithm is further designed. Theoretically, we establish consistency guarantees for the SAA estimator and quantify its convergence complexity, explicitly characterizing the computational trade-offs among discretization accuracy, sample size, and iteration count. This work provides a new implementable paradigm for high-dimensional controlled stochastic systems.

This work addresses temporal drift in existing finite-dimensional diffusion policies—caused by discretization artifacts—that hinders their performance in long-horizon physical tasks. The authors elevate policy modeling to the Cameron–Martin space, leveraging Gaussian measure theory to construct a colored-noise covariance operator that enhances trajectory regularity. They reformulate stochastic score matching as a deterministic boundary-value PDE problem grounded in the Kolmogorov backward equation. The proposed framework introduces a precision-weighted Cameron–Martin loss and PDE residual diagnostics, yielding dimension-independent convergence guarantees and enabling reward-free anomaly detection. Experiments demonstrate a 17% increase in maximum episode reward and a 67.6% reduction in inter-step drift on the PushT task; in manufacturing line scheduling, it achieves a 28.4% lower RMSE, perfect bottleneck identification accuracy (1.0), and a 96% reduction in deadlock events.

This work addresses the limitation of classical stochastic optimization theory, which relies on the uniform escape (UE) assumption to avoid strict saddle points—a condition often violated in over-parameterized, interpolation, or finite-sum settings. The authors establish a stochastic recursive almost sure saddle avoidance theorem without requiring the UE assumption. By introducing a path-dependent variable transformation and a pathwise Lyapunov–Perron method, they extend the center-stable manifold framework to sequences of random mappings lacking common fixed points, under assumptions of local smoothness, finite moment conditions, and without-replacement sampling structure. This unified framework applies to stochastic mirror descent (including SGD), stochastic reshuffling, and proximal stochastic gradient methods for nonsmooth composite objectives, proving their almost sure convergence to local minima by first avoiding strict saddle points and then ensuring iterative convergence.

This study investigates the high-variance issue in temporal-difference (TD) learning within reinforcement learning and its impact on estimation stability. Under tabular representations and episodic settings, the authors theoretically derive and numerically validate, for the first time, an asymptotic upper bound on the variance of TD estimates relative to Monte Carlo (MC) methods, demonstrating that shorter update horizons effectively reduce variance. They further interpret Direct Advantage Estimation (DAE) as a regression-adjusted control variate and establish a tighter asymptotic variance bound than TD under large-sample conditions. The findings indicate that aggregating multiple trajectories, reducing update horizons, and incorporating DAE all substantially improve variance control, with empirical validation of estimator variance behavior conducted in tailored environments.

This study addresses the loss incurred by Bayesian investors when asset drifts are unknown and the observation model may be misspecified. The authors develop a robust investment framework in path space by integrating Kalman–Bucy filtering with mean–variance optimization, incorporating adversarial perturbations penalized by relative entropy to account for potential misspecification of the price distribution. They innovatively reveal the joint perturbation effect on wealth and belief processes driven by a common Brownian motion and prove that value scaling exactly preserves the affine structure of the optimal strategy. The resulting closed-form robust policy incurs, at leading order, only half the loss variance of its non-robust counterpart and automatically trims large positions through third-order corrections, thereby guaranteeing finite losses and controlled risk exposure.

This study addresses the unresolved non-asymptotic sample complexity of single-trajectory least squares estimation for linear systems under heavy-tailed noise. For exponentially stable systems, this work proposes a unified analytical framework that accommodates both sub-Gaussian and sub-exponential noise regimes, subject to persistence of excitation, bounded noise covariance, and finite moment conditions. The primary contribution lies in establishing a non-asymptotic error bound of $\widetilde{O}(r^{1/2}T^{-1/2+1/p})$, which is shown to be independent of the model order. This result significantly improves system identification performance in the presence of heavy-tailed noise, offering tighter theoretical guarantees than existing approaches.