Retrieval error propagates into generation

ProofGraduate⏱ 1 hourRAGerror propagationtotal variationtightnessrag

Theory: Retrieval as a Latent-Variable Model →

Problem

Prove that the total variation distance between the RAG output distribution and the oracle distribution is at most $1-\gamma$ , where $\gamma$ is the probability that a sufficient document is retrieved, and show the bound is attained. Then compute the end-to-end $\gamma$ for a realistic pipeline and interpret.

Write a complete argument. Every inequality you use should be one you could state precisely.

Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

Related problems

All problems

Problem	Field	Type	Difficulty	Time
Markov and Chebyshev are attained	High-Dim Probability	Proof	Foundation	15 min
When is the union bound tight?	Probability	Proof	Foundation	15 min
End-to-end recall arithmetic for a RAG system	RAG	Numerical	Beginner	15 min
Tightness of the descent lemma	Foundations	Proof	Beginner	15 min
RAG-sequence and RAG-token are different models	RAG	Proof	Intermediate	30 min