---
title: "Recht's \"Mates of Costate\" argues mathematical equivalence (Lagrangian duality), not historical transmission — and never mentions Kelley"
type: "claim"
status: "budding"
audit_status: "verified-verbatim (quotes reproduced identically across four independent fetch passes at capture level + matched against the sibling 0208 capture's independent fetch; Kelley absence checked against full text)"
source_url: "https://archives.argmin.net/2016/05/18/mates-of-costate/"
source_title: "Mates of Costate – arg min blog"
source_author: "Ben Recht (arg min blog, 2016-05-18)"
source_date: "2016-05-18T00:00:00.000Z"
source_tier: 1
source_quote: "A few years ago, Steve Wright introduced me to an older method from optimal control, called the method of adjoints, which is equivalent to backpropagation."
provenance: "Promotion from 10-inbox/raw/20260707-0229-what-duality-argument-does.md, 2026-07-07, queen cycle 21"
origin: "batch"
derived_from: "10-inbox/raw/20260707-0229-what-duality-argument-does.md"
date_created: "2026-07-07T00:00:00.000Z"
tags: ["recht","optimal-control","lagrangian-duality","adjoint-method","backpropagation","history-of-ml"]
---


**The duality argument, precisely.** Attach Lagrange multipliers to a
discrete-time LQ problem's dynamics constraint and the multipliers "become
a trajectory of a related linear system called the *adjoint* or *dual*
system," whose "dynamics are linear in the costate p_t, with time running
in reverse and the state transition matrix being the transpose (also known
as the adjoint) of A." Stationarity w.r.t. the multipliers recovers the
forward pass; stationarity w.r.t. the states yields the backward costate
recursion. Same transpose structure the AD literature states as
[[claim-jvp-vjp-transpose-duality]] — control theory's clothing for it. He
also derives Kalman filtering as a special case of the same machinery,
placing Kalman (1960) and Bryson (1961) "very much at the birth of modern
control theory."

**What the post is and isn't evidence for.** It is an explicit,
Tier-1-worked *equivalence* claim ("the method of adjoints, which is
equivalent to [[entity-backpropagation|backpropagation]]"). It is *not* a transmission claim — Recht's
own narrative frame is late rediscovery (a colleague showed him the method
"a few years ago"), and his Bryson-1961 priority line is expressly relayed
("According to Dreyfus…" — he couldn't locate the 1961 proceedings; his
"1968" book date is off by one, preserved not corrected). Checked against
the full text: **"Kelley" never appears** — the post's chain is
Kalman → Bryson → Bryson & Ho only, so it cannot carry the joint
Kelley–Bryson priority story ([[claim-kelley-bryson-optimal-control-precursor]],
[[claim-bryson-ho-1969-curriculum-vector]]).

Equivalence-vs-transmission is the same distinction the cluster's spine
keeps enforcing ([[claim-rhw-1986-demonstration-not-invention]],
[[claim-reverse-mode-multiple-independent-discovery]]): "these are the same
mathematics" and "one caused the other" are different claims with different
evidence, and this post — read carefully — only makes the first.