Data-driven algorithm design automatically adapts algorithms to specific application domains, achieving better performance. In the context of parameterized algorithms, this approach involves tuning the algorithm's hyperparameters using problem instances drawn from the problem distribution of the target application domain. This can be achieved by maximizing empirical utilities that measure the algorithms' performance as a function of their hyperparameters, using problem instances. While empirical evidence supports the effectiveness of data-driven algorithm design, providing theoretical guarantees for several parameterized families remains challenging. This is due to the intricate behaviors of their corresponding utility functions, which typically admit piecewise discontinuous structures. In this work, we present refined frameworks for providing learning guarantees for parameterized data-driven algorithm design problems in both distributional and online learning settings. For the distributional learning setting, we introduce the \textit{Pfaffian GJ framework}, an extension of the classical \textit{GJ framework}, that is capable of providing learning guarantees for function classes for which the computation involves Pfaffian functions. Unlike the GJ framework, which is limited to function classes with computation characterized by rational functions, our proposed framework can deal with function classes involving Pfaffian functions, which are much more general and widely applicable. We then show that for many parameterized algorithms of interest, their utility function possesses a \textit{refined piecewise structure}, which automatically translates to learning guarantees using our proposed framework.

翻译：数据驱动的算法设计能够自动调整算法以适应特定应用领域，从而获得更好的性能。在参数化算法的背景下，这种方法涉及使用从目标应用领域的问题分布中抽取的问题实例来调整算法的超参数。这可以通过最大化经验效用函数来实现，该函数利用问题实例来衡量算法性能作为其超参数的函数。尽管经验证据支持数据驱动算法设计的有效性，但为多个参数化家族提供理论保证仍然具有挑战性。这是由于它们对应的效用函数具有复杂的行为，通常呈现出分段不连续的结构。在本工作中，我们提出了改进的框架，为参数化数据驱动算法设计问题在分布学习和在线学习两种场景下提供学习保证。对于分布学习场景，我们引入了\textit{普法夫GJ框架}，这是经典\textit{GJ框架}的扩展，能够为计算涉及普法夫函数的函数类提供学习保证。与仅限于计算由有理函数表征的函数类的GJ框架不同，我们提出的框架可以处理涉及普法夫函数的函数类，后者更为通用且适用性更广。随后我们证明，对于许多感兴趣的参数化算法，它们的效用函数具有\textit{精细的分段结构}，这通过我们提出的框架可以自动转化为学习保证。