基于零阶残差跟踪的部分信息随机优化
Partially Informed Stochastic Optimization via Zeroth-Order Residual Tracking
AI总结：
针对仅能获取部分梯度的优化问题，提出PISO方法，结合可用梯度与残差跟踪，在非凸优化中获领先复杂度，在多任务中优于多数基线方法。
摘要：
许多优化问题仅能直接获取部分梯度，其余分量需通过函数评估推断，该结构出现在表演式预测、表演式强化学习以及带有黑箱下层的双层优化中。我们提出部分信息随机优化（PISO），将可用梯度分量与残差的运行估计相结合，通过将每个零阶观测以该估计为中心，使随机探测导致的方差依赖于跟踪误差而非残差的大小，从而得到具有显式收缩机制和动量原理性解释的随机草图迭代。对于光滑非凸目标，我们建立了依赖于残差变化和初始大小的收敛保证，无需对不可用梯度分量施加统一边界。在 Lipschitz 梯度和额外 Lipschitz-Hessian 假设下，当使用独立采样的函数评估及高斯或球形探测时，PISO 分别达到领先的函数评估复杂度 O(d²ε⁻⁶) 和 O(d²ε⁻⁵)，这些速率每次更新仅需一个探测方向、两个函数样本和一个已知分量样本。分析还量化了批处理如何用观测换取更少部署。在五个设置中，PISO 变体在三个设置中优于具有相当神谕访问权限的测试基线，在双层路由中更快达到相当的解，且在无响应模型导数的表演式强化学习方法中表现最佳。
英文摘要：
Many optimization problems provide direct access to only part of the gradient, while the remaining component must be inferred from function evaluations. This structure arises in performative prediction, performative reinforcement learning, and bilevel optimization with a black-box lower level. We introduce Partially Informed Stochastic Optimization (PISO), which combines the available gradient component with a running estimate of the residual. By centring each zeroth-order observation on this estimate, PISO makes the variance due to random probing depend on the tracking error rather than the residual's magnitude. This yields a randomized sketch iteration with an explicit contraction mechanism and a principled interpretation of momentum. For smooth nonconvex objectives, we establish convergence guarantees that depend on the residual's variation and initial magnitude, without requiring a uniform bound on the unavailable gradient component. With independently sampled function evaluations and Gaussian or spherical probes, PISO attains leading function-evaluation complexities of \(O(d^{2}\varepsilon^{-6})\) under Lipschitz gradients and \(O(d^{2}\varepsilon^{-5})\) under an additional Lipschitz-Hessian assumption. These rates require only one probe direction, two function samples, and one known-component sample per update. The analysis also quantifies how batching trades observations for fewer deployments. Across five settings, PISO variants improve on the tested baselines with comparable oracle access in three, reach a comparable solution faster in bilevel routing, and perform best among methods without response-model derivatives in performative reinforcement learning.
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