Source
Ho & Salimans (2022), “Classifier-Free Diffusion Guidance”, arXiv v1, LIT-839, read in full in NOTE-603. The NeurIPS 2021 workshop version was not read.
What was actually shown
Classifier guidance as tilting (§3.1). ε̃ = ε(z, c) − wσ∇ log p_θ(c|z) gives approximate samples from p(z|c)p(c|z)
Classifier-free guidance (Eq. 6). ε̃(z, c) = (1 + w)ε(z, c) − wε(z), where one network, trained with the condition dropped with probability p_uncond, supplies both scores. With exact scores ε*(z, c) − ε*(z) is −σ times the gradient of log p_i(c|z), the implicit classifier p_i(c|z) ∝ p(z|c)/p(z), so Eq. 6 is classifier guidance by that classifier (§3.2).
With learned scores (§§3.2, 5). The networks are arbitrary vector fields, so the same arithmetic “is not in general the (scaled) gradient of any classifier”. The guided field need not be the gradient of any scalar potential (C3, strong).
It works anyway (Tables 1–2). At 64×64 with p_uncond = 0.1: w = 0 gives FID 1.80, IS 53.7; w = 0.1 gives FID 1.55, IS 66.1; w = 4 gives FID 26.22, IS 260.2. The same trade-off classifier guidance gives, with no classifier.
What could have come out otherwise: without a classifier, the IS gain could have vanished. It did not.
What this does not say
- What the guided sampler samples. If the field is not conservative, the stationary law of guided sampling is uncharacterised (NOTE-603, Open questions). It is not shown to be p(z|c)p(c|z)
- That guidance’s gain is not an adversarial effect on classifiers. That follows from the form of the sampler (C4, moderate). No held-out classifier or non-Inception metric was used.
- Anything about text or compositional conditions. One task family, class labels on ImageNet.
- That FID falls monotonically with w. Section 4.1 says so; Tables 1–2 show it rising after w ≈ 0.1–0.3. The tables are right.
- That sums of several guidance terms are conservative. The paper states the point for its own Eq. 6. Extending it to sums of several terms, as Composable Diffusion uses, is NOTE-603’s reading, by the same argument.
Connections
- CLAIM-123 (adding scores composes conditions only under conditional independence at the noisy state) names this as a second, independent caveat: even one guided condition does not sample the tilted target.
- Composable Diffusion (LIT-770, NOTE-597) sums guidance terms, one per concept, and its AND reduces to Eq. 6 for one concept.
- The anthology holds the account of the guidance weight’s effect on metrics, beside Dhariwal & Nichol’s classifier guidance (ANTH-LIT-699).
- The conditional Schrödinger bridge (THEORY-515) conditions by changing the endpoints and path rather than the field.
Note of 2026-10-25: guided fields combined convexly
Two works read on 2026-10-25 apply classifier-free guidance inside each component and then combine the guided fields convexly: MultiDiffusion by region (LIT-1522, NOTE-1158, THEORY-958) and LoRA Composite over LoRA-modified models (LIT-1529, NOTE-1160). Each guided field is in general the gradient of no density, and a mean of such fields is no better. Spatial masking adds a second source. Across a region boundary the Jacobian of Σᵢ Mᵢ ⊙ sᵢ is symmetric only if Σᵢ (Mᵢ(p) − Mᵢ(q)) ∂sᵢ,p/∂x_q vanishes, which in general it does not when the models couple pixels across the boundary. That argument is the record’s (NOTE-1158), not the paper’s. LoRA Composite’s “enabling CFG to condition on LoRAs” is text guidance within each LoRA-modified model, not guidance toward a LoRA (NOTE-1160). The account is unchanged.