TOBYQA：针对时变函数的无导数优化的信赖域方法
TOBYQA: A Time-Augmented Model-Based Method for Derivative-Free Optimization under Noise and Temporal Drift
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中文总结 AI 辅助
针对时变函数的无导数优化难题，提出 TOBYQA 框架，通过联合建模时空变化等技术提升漂移环境下的鲁棒性，在 Moré--Wild 88 函数套件上验证了其性能。
中文摘要 AI 辅助
当函数评估受时间漂移和观测噪声影响时，无导数优化（DFO）会变得格外困难。传统信赖域方法通常假设优化 landscape 是静态的，这可能导致过时数据偏差和误导性的信赖域更新。我们提出 TOBYQA（Time-augmented Optimization BY Quadratic Approximation，基于二次近似的时间增强优化），这是一个正则化的 DFO 框架，在统一的二次插值系统中联合建模空间几何和时间变化。具体而言，TOBYQA 在经典 KKT 插值系统的 (1,1)-块上添加了一个岭项，该岭项可适应观测噪声，并在约束块满足温和的满列秩条件时保证适定性，从而放宽了经典方法严格的几何 poisedness 要求。为处理非平稳性，我们在插值模型中嵌入了显式的线性时间漂移项，实现了经漂移补偿的比率测试，该测试会从观测到的缩减量中减去估计的时间分量。此外，对角仿射缩放可使搜索几何适应曲率的局部变化。在 Moré--Wild 88 函数套件上针对各种时间漂移和维度（n ∈ {6,10,20}）的基准评估表明，TOBYQA 在漂移环境下表现出更强的鲁棒性，同时在静态环境中保持了有竞争力的性能。
英文摘要
Derivative-free optimization (DFO) is challenging when the observation channel varies over time and evaluations are noisy. Conventional model-based methods assume stationary observations; under temporal drift, historical data biases gradient estimates and acceptance tests confound latent-objective decrease with temporal variation. We propose TOBYQA (Time-augmented Optimization BY Quadratic Approximation), a regularized model-based DFO framework that jointly incorporates spatial geometry and temporal variations within a single saddle-point interpolation system. TOBYQA augments the classical least-Frobenius-norm quadratic interpolation system with a linear-in-time drift term and a ridge regularization on the residual kernel, which accommodates noise and ensures well-posedness when the constraint block has full column rank, relaxing the geometric poisedness requirements of classical interpolation. We prove that under affine temporal drift, the recovered gradient is algebraically invariant to the drift rate for any noise scale and sample radius. This property leads to a drift-compensated acceptance test that subtracts the estimated temporal component from the observed reduction. Driven by an adaptive cubic regularization scheme with a closed-form step and a geometry-guarded statistical stationarity stopping rule, TOBYQA achieves an expected oracle complexity of $O(\varepsilon^{-2})$. Benchmark evaluations across diverse temporal drift regimes show that, at tolerance $τ=10^{-3}$, TOBYQA solves 71.0%, 60.8%, and 41.9% of instances at $n=6$, $10$, and $20$, respectively, compared with 25.1%, 18.5%, and 14.9% for the best-performing comparison method. These results demonstrate higher solve rates under temporal drift while retaining comparable performance in static environments.
发表机构
- School of Computer Science and Technology, Xi’an Jiaotong University（西安交通大学计算机科学与技术学院）
- Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室应用数学与计算研究部）
机构由 AI 辅助整理，请以论文原文为准。
