用于Nesterov加速的统一连续-离散框架：凸与强凸状态间的过渡
A unified continuous-discrete framework for Nesterov acceleration: transitions between convex and strongly convex regimes
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中文总结 AI 辅助
本研究提出统一连续-离散框架，解决Nesterov加速在强凸参数较小时早期收敛慢的问题，可实现凸与强凸状态过渡，恢复经典速率，生成多种过渡族，数值实验验证其有效性。
中文摘要 AI 辅助
经典Nesterov加速在凸与强凸场景下，无论是连续时间动力学还是离散算法，都采用不同的阻尼参数和惯性参数选择。当强凸参数较小时，直接使用强凸阻尼或惯性系数，尽管其具有良好的渐近指数或线性速率，但可能会导致早期阶段的收敛速度慢于对应的凸选择。我们开发了一种统一的连续-离散框架，涵盖两种经典状态并提供它们之间的系统过渡。所得的系数族在早期阶段保留了加速凸行为，同时达到强凸渐近速率。连续时间动力学源于两状态耦合，并在统一的Lyapunov框架内进行分析，该框架同时产生$\u2134(1/t^2)$和指数收敛估计，从而恢复经典的凸和强凸速率。我们进一步通过离散化所提出的动力学推导出两类加速前向-后向算法，并建立了涵盖凸、强凸和中间状态的收敛估计。该框架恢复了经典Nesterov惯性系数，并生成双曲、指数、代数和多项式过渡族。数值实验证明，当强凸参数较小时，所提出方法的有效性。
英文摘要
Classical Nesterov acceleration employs different choices of damping and inertial parameters in the convex and strongly convex settings, both for continuous-time dynamics and for discrete algorithms. When the strong convexity parameter is small, directly using the strongly convex damping or inertial coefficient may lead to slower early-stage convergence than the corresponding convex choice, despite its favorable asymptotic exponential or linear rate. We develop a unified continuous-discrete framework that encompasses both classical regimes and provides systematic transitions between them. The resulting coefficient families retain the accelerated convex behavior at early stages while attaining the strongly convex asymptotic rate. The continuous-time dynamics arise from a two-state coupling and are analyzed within a unified Lyapunov framework that yields simultaneous $\mathcal{O}(1/t^2)$ and exponential convergence estimates, thereby recovering the classical convex and strongly convex rates. We further derive two classes of accelerated forward-backward algorithms by discretizing the proposed dynamics and establish convergence estimates covering the convex, strongly convex, and intermediate regimes. The framework recovers the classical Nesterov inertial coefficients and generates hyperbolic, exponential, algebraic, and polynomial transition families. Numerical experiments demonstrate the effectiveness of the proposed methods when the strong convexity parameter is small.
