Adaptivity Under Realizability Constraints: Comparing In-Context and Agentic Learning
For researchers in learning theory and neural network approximation, this work clarifies how realizability constraints alter the benefits of adaptive querying, revealing nuanced interactions that were previously unknown.
The paper compares in-context learning with fixed queries and agentic learning with adaptive queries for uniform approximation of task families, finding that adaptivity never hinders performance but its advantage can change under realizability constraints imposed by ReLU neural networks. Four distinct scenarios are identified, showing that representational constraints interact with the effect of adaptivity.
We compare in-context learning with fixed queries and agentic learning with adaptive queries for uniform approximation of task families. We consider two settings: an unrestricted regime, where querying and approximation are arbitrary functions, and a realizable regime, where we require these operations to be implemented by ReLU neural networks. In both settings, adaptivity never hinders approximation performance. However, this advantage can change when one passes from the unrestricted regime to the realizable regime. We identify four distinct approximation scenarios, each witnessed by an explicit task family: (a) no advantage of adaptivity; (b) an advantage in the unrestricted regime that persists under ReLU realizability; (c) an advantage that arises only under realizability; and (d) an advantage that disappears under realizability. This demonstrates that representational constraints interact profoundly with the effect of adaptivity.