GANs as Two Factors

model

Can the GAN diagram be achieved with two factors?

The wiring: yes — two parameter sets, three factor nodes, and the third is the second one again under weight tying. The objective: no, and the obstruction is one sign.

Implemented in lib/Adversarial.jl; see Adversarial.

Sources: original to this vault (design and analysis; no single paper).

Theory (CT-ML wiki): Open Game · Statistical Game · Parametric Lens · Para Construction · Variational Free Energy · Lens · Symmetric Monoidal Category

1. The diagram

Circles are variables, grey squares factors, the dark square a data clamp. The dashed line is not an edge — it is a shared parameter set.

2. Counting

	count	what
learnable parameter sets	2	$\theta$ (generator), $\phi$ (discriminator)
factor nodes	3	$G$, $D$ at the fake branch, $D$ at the real branch
structural factors	2	the latent prior $q(z)$, the data clamp
variables	3	$z$, $x_{\text{fake}}$, $x_{\text{real}}$

So the answer to “two factors?” is: two in the sense that matters — two things with parameters — and three nodes, because the discriminator is evaluated twice:

$${\mathbb{E}}_{x\sim p_{\text{data}}}[\log D_{\phi }(x)]+{\mathbb{E}}_{z\sim q}\left[\log \right(1-D_{\phi }(G_{\theta }(z))\left)\right]$$

Two evaluation sites, one parameter set. A factor node in Mycelium has fixed channels, so one node cannot attach to two different variables; you need two nodes that share $\phi$.

Weight tying is already in the vault's vocabulary

Lux as a Parametric Lens’s dictionary lists reparametrisation $\alpha :Q\to P$ as “Optimisers.jl rules, weight tying, LoRA”. Two factor nodes sharing one parameter set is exactly a $\mathbf{Para}$ reparametrisation, and it is the mechanism the GAN diagram needs. Mycelium has no implementation of it — ps is a flat NamedTuple keyed by factor name, so two nodes cannot name the same entry.

That is the one genuinely missing piece of graph machinery, and it is small.

You can collapse to literally two nodes by making the discriminator’s input a mixture variable carrying both real and fake particles with a label. That is what the implementation does in effect, and it trades a node for a bookkeeping obligation.

3. And the whole thing is a DAG

All three factors in lib/Adversarial.jl are unidirectional — isunidirectional is true for each, asserted in the test suite. So:

A GAN uses none of Mycelium’s bidirectionality. The diagram is a DAG; it is Lux-shaped.

Which is worth sitting with, given that Implicit Learners files diffusion, equilibrium and algebraic models as the three implicit families. A GAN is implicit in a different sense — Three Senses of Implicit — and that sense does not buy you a polarity. The factor-graph formulation gains parameter management, channel names and free-energy accounting, and gains no directions.

4. The obstruction: one sign

Here is what the graph cannot hold.

The Bethe free energy (Bethe Free Energy) is a single scalar:

$F={\sum }_c{F}_c+{\sum }_v(1-d_v)H_v$

and every learnable factor descends $\partial F/\partial {\theta }_c$. A GAN is a minimax:

${\min }_{\theta }{\max }_{\phi }V(\theta ,\phi )$

$\theta$ descends, $\phi$ ascends, on the same quantity. There is no assignment of per-factor free energies whose common descent reproduces that, because descent has one direction and the game has two.

lib/Adversarial.jl therefore computes the number and cannot act on it: local_free_energy(::RatioFactor, …) returns ${\mathbb{E}}_b[-\log r]$, an estimate of $\mathrm{KL}(q\Vert p)$ — the quantity $G$ descends and $D$ ascends. The graph holds the number; it cannot hold the two signs.

The minimal patch, and why it is not enough

A per-factor sign — objective_sign(factor) ∈ {+1,-1} — would make simultaneous gradient descent-ascent expressible. That is genuinely all GAN training does in practice.

It is not enough as theory, for a reason the vault should care about: simultaneous gradient descent-ascent is not guaranteed to converge to anything, and the fixed points it does find are not characterised by minimising any function. A sign flag would let you run the algorithm while telling you nothing about what it computes. The framework’s whole selling point is that the free energy means something (Variational Free Energy).

The obstruction generalises — and so does the escape

The Two-Part Diagram places this finding in a taxonomy: GANs are the $f=-g$ row of a bilevel problem, and the sign obstruction is specific to competitive architectures. The $f=g$ row — EM, VAE, active inference, LQG control — is not obstructed, because coordinate descent on one objective is exactly what a single-signed free energy expresses.

So the right summary is not “the framework cannot do two-part architectures”. It is “the framework does the cooperative ones and not the arguing ones”.

5. What the missing structure actually is

A GAN is a two-player zero-sum game. The categorical treatment of games that composes like a lens exists: open games, Ghani, Hedges, Winschel & Zahn (arXiv:1603.04641), with the Bayesian generalisation in Bolt, Hedges & Zahn (arXiv:1910.03656).

An open game is a morphism in a symmetric monoidal category with a forward play map and a backward pass carrying coutility and a best-response condition, drawn with string diagrams, composing sequentially and in parallel — and faithful in the sense of preserving Nash equilibria and off-equilibrium best responses.

That completes a three-way pattern the vault is halfway through documenting:

framework	the backward pass carries	vault note
parametric lens (Cruttwell et al.)	a gradient	Parametric Lens
statistical game (AutoBayes)	a posterior	Factors are Parameterized Statistical Games
open game (Ghani et al.)	a best response	—

All three are lens-shaped, all three have a forward and a backward pass, and they differ only in what flows backwards. Lenticulum implements the second. A GAN needs the third.

A terminological collision worth flagging

AutoBayes calls its central object a statistical game (Definition 20), and Lenticulum’s factors are parameterized statistical games (Definition 27). That is not a game in the GAN or game-theoretic sense — there is one player and one loss. It is “game” as in “a lens with an objective attached”.

So “Lenticulum factors are games, and a GAN is a game, therefore…” is a pun, not an argument. The two notions meet only in the open-game framework, where a statistical game would be the one-player degenerate case.

6. What would follow from doing it properly

Not a to-do list — a note of what the open-game reading would buy, if anyone took it up:

• Equilibria instead of minima. The graph’s solution concept becomes Nash rather than stationary, which is what a GAN actually converges to when it converges.
• Adversarial factors compose. Open games compose sequentially and monoidally, so a graph containing several adversarial pairs would have a meaning rather than a training script.
• The discriminator’s inexactness gets a home. ratio §5 records that a RatioFactor’s energy is estimated and nothing accounts for it. In a game, “the other player has not best responded yet” is a first-class off-equilibrium condition rather than an unmodelled error.

Related: Implicit Generative Models, Three Senses of Implicit, Adversarial, The Two-Part Diagram, The Inferencer and the Optimizer, Factors are Parameterized Statistical Games, Parametric Lens, Bethe Free Energy, Lux as a Parametric Lens, Variational Free Energy