arXiv:2508.07928v2 Announce Type: replace 
Abstract: In this paper, we establish non-asymptotic bounds for accuracy of normal approximation for linear two-timescale stochastic approximation (TTSA) algorithms driven by martingale difference or Markov noise. Focusing on both the last iterate and Polyak-Ruppert averaging regimes, we derive bounds for normal approximation in terms of the convex distance between probability distributions. Our analysis reveals a non-trivial interaction between the fast and slow timescales: the normal approximation rate for the last iterate improves as the timescale separation increases, while it decreases in the Polyak-Ruppert averaged setting. We also provide the high-order moment bounds for the error of linear TTSA algorithm, which may be of independent interest.

أظهرت دراسة حديثة حدودًا غير أسيمptوتية لدقة التقريب الطبيعي في خوارزميات التقريب العشوائي الخطي ذات المستويين الزمني (TTSA)، والتي تتأثر بفروق مارتينغالية أو ضجيج ماركوف. تركز البحث على كل من آخر تكرار وأنظمة المتوسطات لبوليك-روبورت، كاشفًا عن تفاعلات معقدة بين المستويات الزمنية السريعة والبطيئة.

Un estudio reciente ha establecido límites no asintóticos para la precisión de la aproximación normal en algoritmos de aproximación estocástica lineal de dos escalas de tiempo (TTSA), que son influenciados por diferencias de martingala o ruido de Markov. La investigación se centra tanto en el último iterado como en los regímenes de promediado de Polyak-Ruppert, revelando interacciones complejas entre las escalas de tiempo rápidas y lentas.

Une étude récente a établi des bornes non asymptotiques pour l'exactitude de l'approximation normale dans les algorithmes d'approximation stochastique linéaire à deux échelles de temps (TTSA), influencés par des différences martingales ou du bruit de Markov. La recherche se concentre à la fois sur le dernier itéré et les régimes d'averaging de Polyak-Ruppert, révélant des interactions complexes entre les échelles de temps rapides et lentes.

A recent study has established non-asymptotic bounds for the accuracy of normal approximation in linear two-timescale stochastic approximation (TTSA) algorithms, which are influenced by martingale difference or Markov noise. The research focuses on both the last iterate and Polyak-Ruppert averaging regimes, revealing complex interactions between fast and slow timescales.

Gaussian Approximation for Two-Timescale Linear Stochastic Approximation

arXiv:2512.06270v1 Announce Type: new 
Abstract: In this work, we study contextual strongly convex simulation optimization and adopt an "optimize then predict" (OTP) approach for real-time decision making. In the offline stage, simulation optimization is conducted across a set of covariates to approximate the optimal-solution function; in the online stage, decisions are obtained by evaluating this approximation at the observed covariate. The central theoretical challenge is to understand how the inexactness of solutions generated by simulation-optimization algorithms affects the optimality gap, which is overlooked in existing studies. To address this, we develop a unified analysis framework that explicitly accounts for both solution bias and variance. Using Polyak-Ruppert averaging SGD as an illustrative simulation-optimization algorithm, we analyze the optimality gap of OTP under four representative smoothing techniques: $k$ nearest neighbor, kernel smoothing, linear regression, and kernel ridge regression. We establish convergence rates, derive the optimal allocation of the computational budget $\Gamma$ between the number of design covariates and the per-covariate simulation effort, and demonstrate the convergence rate can approximately achieve $\Gamma^{-1}$ under appropriate smoothing technique and sample-allocation rule. Finally, through a numerical study, we validate the theoretical findings and demonstrate the effectiveness and practical value of the proposed approach.

دراسة حديثة حول تحسين المحاكاة المقعرة السياقية تقدم نهج 'تحسين ثم توقع'، مما يعزز اتخاذ القرارات في الوقت الحقيقي من خلال تقريب الحلول المثلى عبر تحسين المحاكاة. تؤكد الأبحاث على أهمية فهم كيفية تأثير الحلول غير الدقيقة على فجوة الأمثلية، وهو عامل غالبًا ما يتم تجاهله في الدراسات السابقة.

Un estudio reciente sobre la optimización de simulación convexa contextual introduce un enfoque de 'optimizar y luego predecir', que mejora la toma de decisiones en tiempo real al aproximar soluciones óptimas a través de la optimización por simulación. La investigación enfatiza la importancia de comprender cómo las soluciones inexactas afectan la brecha de optimalidad, un factor a menudo pasado por alto en estudios anteriores.

Une étude récente sur l'optimisation de simulation fortement convexe contextuelle introduit une approche 'optimiser puis prédire', qui améliore la prise de décision en temps réel en approximant des solutions optimales par le biais de l'optimisation de simulation. La recherche souligne l'importance de comprendre comment les solutions inexactes impactent l'écart d'optimalité, un facteur souvent négligé dans les études précédentes.

A recent study on contextual strongly convex simulation optimization introduces an 'optimize then predict' approach, which enhances real-time decision-making by approximating optimal solutions through simulation optimization. The research emphasizes the importance of understanding how inexact solutions impact the optimality gap, a factor often overlooked in previous studies.

Gaussian Approximation for Two-Timescale Linear Stochastic Approximation

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