AI 中文总结
针对非光滑估计器的最优实验设计，提出双层优化框架，通过凹稀疏代价自动决定测量数量与选择，并开发单循环近端梯度算法，实验验证其在稀疏恢复和图像重建中的有效性。
AI 中文摘要
最优实验设计（OED）决定为下游估计获取哪些测量。经典上，这通过优化从线性-高斯模型导出的信息准则来完成，该模型的估计器具有闭式解。然而，许多现代估计器（如Lasso、弹性网或基于全变差的估计器）是非光滑的，缺乏闭式解。为了将OED扩展到这些场景，我们重新将问题表述为其隐含的形式：一个双层程序，其下层计算所部署的非光滑估计器，上层评估其验证预测风险。我们不预先固定测量数量，而是为每个连续获取权重收取凹稀疏代价。因此，保留多少和哪些测量都是优化的结果。一个测量仅在其估计器感知价值超过其代价时存活，并且逐步增加代价会产生一系列设计，探索测量基数和估计精度之间的权衡。利用值函数惩罚重表述，我们开发了一种单循环近端梯度算法，避免对非光滑解映射求微分，并建立其收敛到（近似）稳定点。在合成稀疏恢复和图像重建上的实验证明了所提出方法的优势。
英文摘要
Optimal experimental design (OED) decides which measurements to acquire for downstream estimation. Classically, this is done by optimizing an information criterion derived from a linear-Gaussian model, for which the estimator is available in closed form. Many modern estimators such as Lasso, elastic net, or total-variation-based estimators, however, are nonsmooth and lack a closed-form solution. To extend OED to these scenarios, we recast the problem as what it implicitly is: a bilevel program whose lower level computes the deployed nonsmooth estimator and whose upper level scores its validation prediction risk. Rather than fixing the number of measurements in advance, we charge each continuous acquisition weight a concave sparsity price. As a result, how many and which measurements to keep are both outcomes of the optimization. A measurement survives only if its estimator-aware value exceeds its price, and progressively increasing the price yields a sequence of designs that explores the trade-off between measurement cardinality and estimation accuracy. Using a value-function penalty reformulation, we develop a single-loop proximal-gradient algorithm that avoids differentiating the nonsmooth solution map and establish its convergence to an (approximate) stationary point. Experiments on synthetic sparse recovery and image reconstruction demonstrate the benefits of the proposed method.
