The Two-Part Diagram

design

GANs, actor–critic reinforcement learning, and a controller with an observer all draw the same picture: the architecture splits in two and the halves close a loop.

This note says what is actually shared, and — more usefully — what is not. The shared part is the wiring, and the vault already has the structure for it. The unshared part is the objective, and that is what decides whether an architecture fits in this framework at all.

Sources: Willems, The Behavioral Approach to Open and Interconnected Systems, IEEE CSM 2007; Ghani, Hedges, Winschel & Zahn, Compositional Game Theory, arXiv:1603.04641; Friston, The free-energy principle: a unified brain theory?, Nat. Rev. Neurosci. 2010; Friston et al., Active Inference: A Process Theory, Neural Computation 2017; RxInfer.jl and ForneyLab.jl; Feldbaum, Dual control theory I–IV, 1960–61

Theory (CT-ML wiki): Hypergraph Category · Compact Closed Category · Open Game · Open Model · Parametric Lens · Variational Free Energy · Lens

1. The picture

One half emits a configuration; the other half judges it; the judgement returns. Filling in the boxes:

	acts / produces	evaluates / estimates	what returns
GAN	generator $G_{\theta }$	discriminator $D_{\phi }$	$\log (1-D(G(z)))$
actor–critic	policy ${\pi }_{\theta }$	value $V_{\phi }$	the advantage
control	controller	plant + observer	the measurement / error
VAE	decoder	encoder	the ELBO
EM	M-step	E-step	the posterior
active inference	action	perception	the free energy

The convergence is real. What it is not is a single theory — the last three rows behave completely differently from the first, and §3 is about why.

2. The wiring is not the problem

A loop in a diagram is a trace, and the vault has already dealt with traces.

Lux as a Parametric Lens §“Constraint 1” records why a Lux Chain cannot express this: lens composition is function composition, and function composition around a cycle does not terminate. Open Models and Latent Channels is the fix — compact closure lets you bend an output wire into an input wire, so a cycle becomes a straight line with a bent end, and every compact closed category has a canonical trace. Acausal Composition is a Hypergraph Category goes one rung further.

So:

Lenticulum can already draw every diagram in §1

A factor graph has no direction, so it has no cycles to worry about. Two factors sharing two variables is a “loop” only if you insist on reading the edges as arrows, and a factor graph does not.

The obstruction is never the wiring. It is always the objective.

3. What actually differs: one objective or two

Every architecture in §1 is an instance of bilevel optimisation:

$$\underset{\theta }{\min }f(\theta ,{\phi }^{\ast }(\theta ))\text{subject to}{\phi }^{\ast }(\theta )=\arg \underset{\phi }{\min }g(\theta ,\phi )$$

and the whole taxonomy is the relationship between $f$ and $g$:

case	is	examples	fits Lenticulum?
$f=g$	coordinate descent on one objective	EM, VAE, active inference, LQG	yes
$f=-g$	minimax, zero-sum	GAN, $H_{\infty }$ / robust control	no
$f\ne \pm g$	general bilevel / Stackelberg	actor–critic, meta-learning	no

The Bethe free energy (Bethe Free Energy) is a single scalar that every learnable factor descends. That is exactly the top row and nothing else.

GANs as Two Factors §4 established this for GANs — two parameter sets, three factor nodes, and one sign the graph cannot hold. The point of this note is that the finding generalises, and generalises favourably: the sign obstruction is specific to the competitive rows. The cooperative row is not obstructed at all, and it contains most of control theory and all of variational inference.

Why $f=g$ is not really a loop

Coordinate descent alternates, but it descends one function. There is no equilibrium to seek, no oscillation to damp, no best-response to compute — just a sequence of partial minimisations of a single objective, each of which decreases it.

That is the precise sense in which the cooperative architectures are not feedback loops even though they are drawn as one. They are alternating minimisation wearing a loop’s clothes, and it matters because alternating minimisation converges under conditions you can state, while simultaneous gradient descent–ascent does not.

4. Willems: control is interconnection

Control theory has its own version of “the loop is an artefact of insisting on arrows”, and the vault already imported it.

ModelingToolkit as an Acausal Relation §3 records Willems’ behavioural view: a system is its set of admissible trajectories, and interconnection is variable sharing — not plugging an output into an input. In that framework a controller is simply another system you interconnect with the plant, restricting the joint behaviour to what you want.

A block diagram with a feedback arrow and a factor graph with a shared variable are the same object. The arrow is a choice of representation; the shared variable is the system.

Which means the controller/plant loop, in this project’s terms, is two factors sharing two variables — and nothing about that is hard. The vault demonstrated exactly this shape already, and found it loopy in the graph-theoretic sense: the resistive divider in ModelingToolkit as an Acausal Relation §6 is a three-factor cycle, and the honest finding there was that message passing handles it badly compared to a direct solve.

So the difficulty with control in Lenticulum is not the feedback. It is that a loopy graph gets exact means and wrong variances (Loopy Message Passing), which for a controller means your gains are right and your confidence in them is not.

5. Active inference: the case that fits

If the cooperative row is the one that fits, the natural question is what lives there. The fullest answer is active inference: perception and action both minimising one variational free energy — perception over beliefs, action over policies.

That is $f=g$ exactly, and it is the same functional the Bethe machinery already computes. It is also not speculative in Julia: ForneyLab.jl and its successor RxInfer.jl do Forney-style factor graphs with (variational) message passing and build active-inference agents that minimise free energy by message passing. That ecosystem is the closest existing neighbour to this project, and the closest thing to a demonstration that the cooperative architectures work as factor graphs.

The difference in ambition is worth stating plainly: RxInfer does inference on a specified probabilistic model, extremely well. Lenticulum is trying to do inference on a graph whose factors may be learned, implicit and non-probabilistic — an arbitrary residual, a DEQ, a diffusion prior. That is more general and much less finished.

6. What this means for the three competitive rows

They are not out of reach, but they need structure that is not here:

• Minimax ($f=-g$) needs GANs as Two Factors §5’s open games — a third lens-shaped object where the backward pass carries a best response rather than a gradient or a posterior. The solution concept becomes Nash instead of stationary.

• General bilevel ($f\ne \pm g$) — actor–critic, meta-learning — needs the hypergradient $\mathrm{d}f/\mathrm{d}\theta$ through ${\phi }^{\ast }(\theta )$, which is the implicit function theorem applied to the inner optimum. The vault has that machinery: Backpropagation by the Implicit Function Theorem, and ImplicitLayers’s ift_sensitivity computes exactly this shape of object for a fixed point.

  That is a genuine and unexploited connection: bilevel optimisation and deep equilibrium models have the same backward pass. Both differentiate through an argmin/fixed point via one linear solve. ImplicitLayers implements it for the equilibrium family and nobody has pointed it at the outer problem.

7. The summary

	wiring	objective	status
the loop	a trace; compact closure	—	solved, twice over
cooperative ($f=g$)	shared variables	one free energy	expressible today
minimax ($f=-g$)	shared variables	two signs	needs open games
general bilevel	shared variables	hypergradient	needs the IFT at the outer level

The one-line version: the two-part diagram is never a wiring problem, and whether it is a problem at all depends entirely on whether the two halves are arguing.

And the architecture in The Inferencer and the Optimizer is deliberately one where they are not.

Sources

• Willems, The Behavioral Approach to Open and Interconnected Systems, IEEE CSM 2007 — control as interconnection.
• Ghani, Hedges, Winschel & Zahn, Compositional Game Theory, arXiv:1603.04641 — open games, for the minimax row.
• Friston, The free-energy principle: a unified brain theory?, Nat. Rev. Neurosci. 2010; Friston et al., Active Inference: A Process Theory, Neural Computation 2017.
• RxInfer.jl and ForneyLab.jl — Forney-style factor graphs, message passing, and active-inference agents, in Julia.
• Feldbaum, Dual control theory I–IV, 1960–61 — why separation fails; see The Inferencer and the Optimizer §5.

Related: The Inferencer and the Optimizer, GANs as Two Factors, Open Models and Latent Channels, Acausal Composition is a Hypergraph Category, ModelingToolkit as an Acausal Relation, Bethe Free Energy, Loopy Message Passing, Lux as a Parametric Lens, Backpropagation by the Implicit Function Theorem