---
id: "20261001-0213-does-kelley-brysons-discrete"
title: "Does Kelley-Bryson's discrete costate recursion reduce to Pontryagin's continuous adjoint equations in the small-step limit?"
type: "capture"
status: "promoted"
promoted_to: ["30-notes/claim-dto-and-otd-neural-ode-gradients-can-diverge.md","30-notes/claim-no-unconditional-discrete-pontryagin-maximum-principle.md","30-notes/claim-ober-blobaum-junge-marsden-prove-discrete-adjoint-converges-to-continuous.md","30-notes/claim-kelley-bryson-pontryagin-costate-same-object-only-under-hypotheses.md","40-entities/entity-hubert-halkin.md (new hub)","40-entities/entity-william-hager.md (new hub)","40-entities/entity-jerrold-marsden.md (new hub)","40-entities/entity-discretize-then-optimize-vs-optimize-then-discretize.md (new concept hub)","40-entities/entity-lev-pontryagin.md (existing hub, updated — the capture's claim that no such page existed was wrong; see journal 2026-10-01)","40-mocs/moc-backpropagation-origins.md (updated, new bullet + bridge to entity-discretize-then-optimize-vs-optimize-then-discretize)"]
not_promoted: ["Halkin (1966) primary text — not independently read; referenced only via Kipka & Gupta's secondary citation. Carried as [unverified-mechanism — needs primary] inside claim-no-unconditional-discrete-pontryagin-maximum-principle rather than routed to 50-questions, since the attribution is not load-bearing for that note's Tier-1-grounded core claim (question-intake discipline: no new question for a non-load-bearing detail).","Boltyanskii's and Butkovskiy's specific contributions to the discrete-MP impossibility result — same reasoning as Halkin; secondary-sourced only. No entity hub created for either (thinner signal than Halkin's, not independently load-bearing enough yet).","Hager (2000/2001) primary papers — paywalled, not read directly. Carried as supporting context in claim-ober-blobaum-junge-marsden-prove-discrete-adjoint-converges-to-continuous and a thin entity-william-hager hub; no separate claim-note since his own text was never read.","Onken & Ruthotto, arXiv:2005.13420 (a second, independent ML paper on the DTO/OTD distinction) — not read this session; left as a further-research lead, not promoted.","Holtzman's generalization of Halkin's result (relaxing convexity to 'directional convexity') — secondary web-sourced only, not independently verified; not promoted.","Amir Gholami, Kurt Keutzer, George Biros (ANODE authors) — entity candidates judged UNSURE under the entity-promotion test: single-paper first appearance in the vault, no recurring presence yet. Not promoted to individual entity hubs; the paper itself is fully captured via claim-dto-and-otd-neural-ode-gradients-can-diverge.","Sina Ober-Blöbaum, Oliver Junge (DMOC co-authors) — same reasoning as the ANODE authors above. Jerrold Marsden alone was promoted to a hub, as the figure who bridges this paper to the vault's broader geometric-mechanics/optimal-control context; a flood of one-paper co-author stubs was judged not worth it.","'Discrete-time Pontryagin maximum principle' as a standalone concept hub — folded into the entity-lev-pontryagin.md update instead of a separate page, to avoid a redundant concept hub sitting beside Pontryagin's own entity page."]
origin: "batch"
writer_model: "claude-sonnet-5"
date_created: "2026-10-01T00:00:00.000Z"
provenance: "Batch research run, 2026-10-01, queue hook: unlinked pair cosine 0.878, claim-backprop-special-case-kelley-bryson-formula x claim-neural-odes-cite-pontryagin-adjoint-method-as-backprop"
derived_from: []
tags: ["kelley-bryson","pontryagin","optimal-control","adjoint-method","costate","discretization","neural-odes","backpropagation","history-of-ml","numerical-analysis"]
source_url: "https://arxiv.org/pdf/1902.10298"
source_title: "ANODE: Unconditionally Accurate Memory-Efficient Gradients for Neural ODEs"
source_author: "Amir Gholami, Kurt Keutzer, George Biros"
source_date: "2019-02-27"
source_venue: "arXiv preprint (also published: Proceedings of IJCAI-19)"
source_tier: 1
source_sha: "23cb651e73afe6937c9649f98446d6d97e3a1e7a78168fba28e8a2abbacba11d"
source_quote: "Except for rare cases [14], the error in OTD and DTO's gradient scales as O(dt). Therefore, this error can become quite large for small time step sizes."
source_url_2: "https://arxiv.org/pdf/1707.03873"
source_title_2: "The Discrete-Time Geometric Maximum Principle"
source_author_2: "Robert Kipka, Rohit Gupta"
source_date_2: "2017-07-12"
source_venue_2: "arXiv preprint (also published: SIAM Journal on Control and Optimization 57(4):2939-2961, 2019)"
source_tier_2: 1
source_sha_2: "c6253031f69262dbd946736f71f570bf27ede1f548cd561111f3ad0f2027ab54"
source_quote_2: "the formulation of the maximum principle for discrete-time optimal control problems becomes quite tricky since the maximization condition cannot be expected to hold in general and in fact some of the early literature on this topic was mathematically incorrect (see [6, 27] for a number of interesting counterexamples). However, the maximization condition does hold under appropriate convexity assumptions on the dynamics"
source_url_3: "https://arxiv.org/pdf/0810.1386"
source_title_3: "Discrete Mechanics and Optimal Control: an Analysis"
source_author_3: "Sina Ober-Bloebaum, Oliver Junge, Jerrold E. Marsden"
source_date_3: "2008-10-08"
source_venue_3: "arXiv preprint (also published: ESAIM: Control, Optimisation and Calculus of Variations 17(2):322-352, 2011, doi:10.1051/cocv/2010012)"
source_tier_3: 1
source_sha_3: "d54ad3a9199ff90a6ab2d7d57e6211577c30347ed910e5b7ea0f22f184db328e"
source_quote_3: "The purpose of this section is to establish the convergence of solutions of the discrete optimal control problem (3.15) to a solution of the Lagrangian optimal control problem (3.3) as the step size h goes to 0."
source_quote_3b: "If the symplectic partitioned Runge-Kutta discretization of the state system is of order kappa and bi > 0 for each i, then the scheme for the adjoint system is again a symplectic partitioned Runge-Kutta scheme of the same order"
seek_code_commit: "c7427a5"
---


Framing note: the vault already holds the "same algorithm or distant cousin" tension
at the historical/mathematical level —
[[claim-backprop-special-case-kelley-bryson-formula]] (Mizutani–Dreyfus–Nishio: backprop
is a "simplified version"/"equivalent" to the Kelley-Bryson discrete gradient formula) versus
[[claim-kelley-bryson-optimal-control-precursor]] (Schmidhuber: a precursor *lacking*
backprop's efficiency) versus [[claim-neural-odes-cite-pontryagin-adjoint-method-as-backprop]]
(Chen et al. 2018 name Pontryagin's *continuous* adjoint method, not Kelley-Bryson's discrete
one, as the thing equal to backprop). This capture asks a narrower, checkable mathematical
question underneath all three: does the discrete costate recursion (Kelley 1960, Bryson 1961,
formalized by [[claim-recht-adjoints-equivalence-not-transmission|Recht's Lagrangian-duality
argument]]) actually converge to Pontryagin's continuous-time adjoint ODE as the step size
shrinks to zero — making them the same object under two discretizations — or is that
convergence a vault assumption the control-theory and numerical-analysis literature does not
actually support unconditionally?

Direct primaries for Kelley (1960) and Bryson (1961) were not re-read for this capture; it
instead searches the control-theory and numerical-optimal-control literature that treats the
discrete-vs-continuous maximum-principle question as its own research object, independent of
the neural-network history.

## Claim: The discretize-then-optimize recursion and the optimize-then-discretize continuous adjoint ODE are provably different computations, and their disagreement does not have to vanish as the step size shrinks

In neural-ODE training there are two ways to get a gradient through a numerically-discretized
ODE layer: solve the continuous adjoint (costate) ODE backward in time and then discretize
that ODE ("optimize-then-discretize," OTD — the move Chen et al. 2018 make, see
[[claim-neural-odes-cite-pontryagin-adjoint-method-as-backprop]]); or differentiate the
already-discretized forward recursion directly, which reproduces the Kelley-Bryson/backprop
chain-rule recursion exactly ("discretize-then-optimize," DTO). Gholami, Keutzer & Biros (2019)
prove these give literally different gradient expressions for an explicit Euler step
(z1 = z0 + f(z0,θ)): DTO backpropagates ∂L/∂z1·(I + ∂f(z0,θ)/∂z0), while OTD's adjoint
recursion computes α1·(I + ∂f(z1,θ)/∂z1) — "the results from DTO approach (Eq. 9), and OTD
(Eq. 10) can be completely different. This is because in general
∂f(z1,θ)/∂z1 ≠ ∂f(z0,θ)/∂z0." They state the per-step discrepancy is formally small
("Except for rare cases [14], the error in OTD and DTO's gradient scales as O(dt)") but add
immediately — in the paper's own words, kept verbatim rather than smoothed into the more
intuitive-sounding "large step" reading — "Therefore, this error can become quite large for
small time step sizes," and demonstrate empirically that for ReLU/non-ReLU activations and
general convolution operators the backward-in-time (OTD-style) integration is numerically
unstable, producing training that diverges outright rather than converging as resolution
increases (Fig. 1, Fig. 3–5: RK45 integration of the OTD adjoint "leads to divergent training
in the first epoch"). Their proposed fix, ANODE, is explicitly a DTO (discretize-then-optimize)
method — i.e., it recovers the Kelley-Bryson-style discrete recursion rather than the
continuous Pontryagin costate ODE — specifically *because* DTO "provides unconditionally
accurate gradients" where OTD does not.

**What kind of claim, and why this source clears the floor:** specific technical-mechanism
claim (how two gradient-computation procedures differ and under what conditions). Tier 1 —
author's own arXiv preprint (Gholami, Keutzer, Biros; UC Berkeley / UT Austin), also published
at IJCAI-19; read directly via extract_pdf, quote verified verbatim against the extracted text
(quote_check: grounded).

> [!note] Seek's commentary:
> This is the sharpest available evidence that "same object under two
> time-discretizations" is the wrong frame to default to. It's not that the discrete
> Kelley-Bryson recursion and the continuous Pontryagin adjoint are unrelated — they visibly
> rhyme, and under the right conditions (see the convergence claim below) they do provably
> coincide in the limit. But "coincide in the limit, generically" is false: for the exact
> nonlinearities neural networks actually use (ReLU), integrating the continuous adjoint and
> then discretizing it is not a harmless approximation to the discrete recursion — it can
> diverge. The family resemblance the vault flagged is closer to correct than the single-object
> claim.
> — Seek

## Claim: A direct, unconditional discrete-time analogue of Pontryagin's full maximum principle is mathematically false in general — this was established by counterexample in classical control theory, independent of neural networks entirely

Before neural-ODE training ever raised the question, control theory already had this exact
dispute and resolved it negatively in the general case. Kipka & Gupta (2017/2019, SIAM J.
Control Optim.) summarize the state of the field: "the formulation of the maximum principle
for discrete-time optimal control problems becomes quite tricky since the maximization
condition cannot be expected to hold in general and in fact some of the early literature on
this topic was mathematically incorrect... However, the maximization condition does hold under
appropriate convexity assumptions on the dynamics and moreover an approximate maximization
condition can be derived (under suitable assumptions) in the absence of such assumptions." The
counterexamples this passage cites trace to Halkin's 1966 SIAM J. Control paper ("A Maximum
Principle of the Pontryagin Type for Systems Described by Nonlinear Difference Equations," SIAM
J. Control 4(1):90–111) and later work by Boltyanskii — this capture did not read Halkin or
Boltyanskii directly, so the counterexamples themselves rest on Kipka & Gupta's secondary
account, not a primary read. The point this establishes: it is specifically the *maximization*
(Hamiltonian-optimality) condition of Pontryagin's apparatus — not the costate/adjoint equation
alone — that fails to carry over to discrete time without extra convexity assumptions the
continuous theory does not need. Butkovskiy is credited (per web secondary sources, not
independently read) with first showing a direct extension of Pontryagin's principle to discrete
systems is in general impossible.

**What kind of claim, and why this source clears the floor:** specific technical-mechanism
claim about where exactly the discrete/continuous correspondence breaks (the maximization
condition, under which assumptions). Tier 1 — Kipka & Gupta's own arXiv preprint (also
published SIAM J. Control Optim. 57(4):2939–2961, 2019), read directly via extract_pdf, quote
verified verbatim (quote_check: grounded). The attribution of the counterexamples specifically
to Halkin (1966) and Boltyanskii is carried from this Tier-1 source's own citations, not
independently verified against Halkin's or Boltyanskii's primary texts — flagged
`[unverified-mechanism — needs primary]` for the Halkin/Boltyanskii attribution specifically,
though the Tier-1-sourced general claim (no unconditional direct analogue exists) stands on its
own.

> [!note] Seek's commentary:
> This is the oldest and cleanest version of the question, asked by control theorists with no
> stake in neural-network history at all, and it still comes out "no, not unconditionally." That
> should weigh heavily against any framing — including some the vault itself carries — that
> treats the discrete-continuous correspondence as a settled formality.
> — Seek

## Claim: Where smoothness, stability and the right discretization structure hold, convergence of the discrete costate recursion to the continuous adjoint IS a proven theorem — so the vault's intuition is right inside a narrower regime than it assumes

The positive case exists and is rigorous, not folklore. Ober-Blöbaum, Junge & Marsden
(2008 preprint / 2011 ESAIM: COCV) prove, for Lagrangian optimal-control problems meeting a
stated Smoothness assumption and a Coercivity assumption (ensuring a locally unique,
Lipschitz-continuous control-from-costate map), that the discrete adjoint/costate recursion
generated by a symplectic partitioned Runge-Kutta discretization of the state equations
converges to the continuous costate system as the step size h goes to zero, with matching
order: "If the symplectic partitioned Runge-Kutta discretization of the state system is of
order κ and bi > 0 for each i, then the scheme for the adjoint system is again a symplectic
partitioned Runge-Kutta scheme of the same order" (Theorem 3.15), building toward an explicit
statement that "the purpose of this section is to establish the convergence of solutions of the
discrete optimal control problem... to a solution of the Lagrangian optimal control problem...
as the step size h goes to 0" (§3.6), via a general consistency-plus-stability convergence
theorem credited to Hager (2001) and Dontchev, Hager & Veliov (2000).

**What kind of claim, and why this source clears the floor:** specific technical-mechanism
claim (a stated convergence theorem and its hypotheses). Tier 1 — co-authored by Jerrold
Marsden; read directly via extract_pdf from arXiv (TLS verified), quotes verified verbatim
(quote_check: grounded). A near-identical typeset copy is also hosted on co-author Marsden's
own Caltech CDS group wiki
(`cds.caltech.edu/~marsden/wiki/uploads/dmoc/papers/ObJuMa08.pdf`) — confirmed to be the same
paper (matching title/author/abstract header) but fetched over an **unverified TLS
connection**, so per the safety spec this mirror is not used as the citation of record or a
sole source for any claim here; the arXiv copy (verified TLS) is the read of record, with the
published venue (ESAIM: Control, Optimisation and Calculus of Variations 17(2):322–352, 2011,
doi:10.1051/cocv/2010012) recorded as source_venue_3.

> [!note] Seek's commentary:
> Put together with the two claims above, the shape of the real answer is: yes, this is "the
> same object" — but only inside a hypothesis set (smoothness, coercivity/control-uniqueness,
> a structure-preserving discretization scheme, bi > 0) that a generic feedforward network with
> ReLU activations and an arbitrary forward-Euler step does not automatically satisfy. The
> vault's embedding model clustered these two claim-notes at cosine 0.878 because they really
> are pointing at the same underlying mathematical fact; the thing the cosine similarity can't
> see is that the fact has a hypothesis clause, and the neural-network case lives right at the
> boundary where the clause can fail.
> — Seek

## Core question status

Partially resolved, with a precise shape rather than a flat yes/no. The vault's two notes
([[claim-backprop-special-case-kelley-bryson-formula]] and
[[claim-neural-odes-cite-pontryagin-adjoint-method-as-backprop]]) are not simply in tension —
they describe two ends of a genuine conditional: the discrete Kelley-Bryson costate recursion
*does* converge to Pontryagin's continuous adjoint ODE as the step size shrinks, provably, under
stated smoothness/coercivity/discretization-structure hypotheses (claim 3); but there is no
unconditional version of that convergence, either in the general discrete-time Pontryagin
maximum principle (claim 2, a classical-control-theory result predating neural networks) or in
the specific neural-ODE setting with realistic nonsmooth activations and naive discretizations
(claim 1, where the two computations can diverge outright rather than converge). "Same object
under two time-discretizations" is correct as a limiting statement inside a regularity regime;
"family resemblance" is the more accurate description once that regime's hypotheses are
dropped — which is exactly what happens in the ReLU/non-convex settings neural networks
actually use.

> [!note] Seek's commentary:
> I don't think this fully closes the vault's flagged contradiction so much as it relocates it
> onto solid ground: the disagreement between Mizutani-Dreyfus-Nishio's "special case" framing
> and Schmidhuber's "precursor lacking efficiency" framing was never really about whether the
> math is the same — it's about whether "the math being the same in the limit, under
> hypotheses" counts as "the same algorithm." That's a definitional choice, not a fact this
> capture can settle. What this capture adds is that the choice isn't arbitrary: there's a real,
> provable boundary (smoothness + coercivity + stable discretization) where the identification
> holds, and a real, demonstrated failure mode (ReLU nonsmoothness, numerically unstable
> backward integration) just outside it.
> — Seek

## Further leads

- Halkin, H. (1966), "A Maximum Principle of the Pontryagin Type for Systems Described by
  Nonlinear Difference Equations," SIAM J. Control 4(1):90–111 — the original counterexample
  paper; not read directly (paywalled via SIAM/DOI 10.1137/0304009), worth a primary read in a
  future session.
- Hager, W.W. (2000), "Runge-Kutta methods in optimal control and the transformed adjoint
  system," Numerische Mathematik 87:247–282 — the result Ober-Blöbaum/Junge/Marsden build on
  for order-of-consistency between discrete and continuous adjoint systems; paywalled at
  Springer, not independently read; Hager's own faculty page
  (people.clas.ufl.edu/hager/optimal-control/) lists it without a direct PDF link.
- Butkovskiy is credited (via secondary web sources, not independently verified) as the first to
  show a direct discrete analogue of Pontryagin's principle is impossible in general — worth
  chasing the primary citation.
- Onken & Ruthotto, "Discretize-Optimize vs. Optimize-Discretize for Time-Series Regression and
  Continuous Normalizing Flows" (arXiv:2005.13420) — a second, independent ML paper on the same
  OTD/DTO distinction; not read for this capture.
- Holtzman is noted (secondary web source) to have generalized Halkin's maximum principle for
  difference equations by relaxing the convexity requirement to "directional convexity" — not
  independently verified.

## Entity candidates

- Hubert Halkin — person — the foundational figure the modern discrete-time Pontryagin maximum
  principle literature measures itself against; his 1966 counterexample is the reason "does the
  discrete analogue hold" became its own research question rather than an assumed formality.
  Flagged first, per the known blind spot about under-catching the older figure a priority/
  ancestry claim rests on.
- Lev Pontryagin — person — already present via other claim-notes
  ([[claim-pontryagin-worked-blind-via-mothers-spoken-symbol-glosses]] etc.) but worth a
  dedicated entity page given how load-bearing "Pontryagin's maximum principle" is across this
  whole cluster; no entity-pontryagin page currently exists in the vault.
- William W. Hager — person — control theorist whose Runge-Kutta/adjoint consistency results
  (Hager 2000, 2001) underlie the one rigorous convergence proof this capture found.
- Jerrold E. Marsden — person — co-author of the DMOC convergence paper; major figure in
  geometric mechanics/variational integrators, a different lineage than Kelley-Bryson-Dreyfus
  that arrives at the same discrete/continuous adjoint question from the mechanics side.
- Amir Gholami, Kurt Keutzer, George Biros — people — authors of ANODE; the paper that makes
  the OTD/DTO distinction concrete and empirically demonstrates divergence in realistic
  (ReLU/CNN) neural-ODE training.
- Sina Ober-Blöbaum, Oliver Junge — people — co-authors of the DMOC convergence analysis with
  Marsden.
- discretize-then-optimize / optimize-then-discretize — concept — the general numerical-analysis
  distinction this whole capture turns on; could anchor its own note and would usefully link
  [[claim-neural-odes-cite-pontryagin-adjoint-method-as-backprop]] (an OTD example) against
  [[claim-backprop-special-case-kelley-bryson-formula]] (a DTO example) directly.
- discrete-time Pontryagin maximum principle — concept — the classical-control-theory object;
  distinct from (and narrower than) "the costate/adjoint equation" alone, a distinction this
  capture found genuinely load-bearing and the vault does not yet separate explicitly.

## Safety flags

None. All sources this session (arxiv.org PDFs, two independently fetched via extract_pdf with
TLS verified; the Caltech CDS mirror fetched with TLS unverified but containing nothing more
than a standard academic paper header, cross-checked against the verified arXiv copy rather
than used as a citation) were ordinary academic preprints and journal articles. No
addressed-to-AI language, override language, claimed authority, tier self-assignment,
file-system instructions, credential requests, or urgency framing encountered.