Compositional Generalization (CG) embodies the ability to comprehend novel
combinations of familiar concepts, representing a significant cognitive leap in
human intellectual advancement. Despite its critical importance, the deep
neural network (DNN) faces challenges in addressing the compositional
generalization problem, prompting considerable research interest. However,
existing theories often rely on task-specific assumptions, constraining the
comprehensive understanding of CG. This study aims to explore compositional
generalization from a task-agnostic perspective, offering a complementary
viewpoint to task-specific analyses. The primary challenge is to define CG
without overly restricting its scope, a feat achieved by identifying its
fundamental characteristics and basing the definition on them. Using this
definition, we seek to answer the question "what does the ultimate solution to
CG look like?" through the following theoretical findings: 1) the first No Free
Lunch theorem in CG, indicating the absence of general solutions; 2) a novel
generalization bound applicable to any CG problem, specifying the conditions
for an effective CG solution; and 3) the introduction of the generative effect
to enhance understanding of CG problems and their solutions. This paper's
significance lies in providing a general theory for CG problems, which, when
combined with prior theorems under task-specific scenarios, can lead to a
comprehensive understanding of CG.

本研究从无任务偏见的视角探索了合成泛化问题，通过理论发现：1）在合成泛化中没有通用解决方案；2）提出了适用于任何合成泛化问题的新型泛化界限，指定了有效合成泛化解决方案的条件；和 3）引入生成效应以增强对合成泛化问题及其解决方案的理解，为合成泛化问题提供了一个普适理论，可以与特定任务场景下的先前定理相结合以全面理解合成泛化。