Pipeline Amplification

How Lipschitz constants multiply through an agent tool-chain, and why this makes deeper pipelines worse to defend.

See Pipeline Degradation for the formal statement.

A five-stage pipeline

Effective Lipschitz constant:

$$K_{\mathrm{eff}}=K_D\cdot 1\cdot 2\cdot 2\cdot 2\cdot 2=16K_D$$

and the ε-robust band width grows in proportion.

Band width vs. depth

The band width scales linearly in $K_{\mathrm{eff}}$, so it grows exponentially in the number of tools. A single-stage defense with a 1-unit band becomes a 32-unit band at depth 5 and a ~1024-unit band at depth 10.

Why naive "defense in depth" fails

The difference between "good" defense in depth and "bad" defense in depth is whether the stages compose by rejection (discontinuous) or by continuous pass-through. The impossibility theorems only cover the latter.

How the pipeline breaks the dilemma optimum

Recall the defense dilemma: a single-stage defense must pick $K$ near $K^{\star }=G/\ell -1$ to balance the band and the persistent region.

Once the pipeline fixes $K_{\mathrm{eff}}\gg K^{\star }$, the persistent unsafe region might become empty, but the $\epsilon$-robust band becomes so wide that the defense can no longer push any point far below $\tau$. The problem just shifts from "unsafe" to "barely safe".

Related

• Pipeline Degradation theorem
• Defense dilemma
• Lean file: MoF_15_NonlinearAgents.