Classifier-free guidance: Worked examples — AI-ML deep dive

This is the "roll up your sleeves" companion to Classifier-free guidance. The parent note built the formula; here we hit every case the guidance scale $w$ can produce — including the weird ones nobody warns you about. Prefer to read this in Hinglish? Try 4.5.14 Classifier-free guidance (Hinglish).

Everything below rests on one single equation. Let us re-state it in plain words so no symbol is a stranger:

The scenario matrix

Every question this topic can throw at you is one of these cells. We will hit all of them.

Cell	Case class	What makes it tricky	Example
A	$w = 0$ (degenerate)	prompt ignored entirely	Ex 1
B	$w = 1$ (identity)	guidance does nothing	Ex 2
C	$0 < w < 1$ (interpolation)	answer sits between the two guesses	Ex 3
D	$w > 1$ (extrapolation)	unconditional weight goes negative	Ex 4
E	Degenerate arrow $Δ ϵ = 0$	prompt made no difference	Ex 5
F	Sign flip / over-shoot	large $w$ flips a component's sign	Ex 6
G	Real-world word problem	Stable-Diffusion prompt with $w = 7.5$	Ex 7
H	Negative prompting twist	subtracting an unwanted prompt	Ex 8
I	Limiting behaviour $w \to \infty$	where does $\tilde{ϵ}$ head?	Ex 9

To keep arithmetic honest, most examples use tiny 2-number noise vectors $[a, b]$ . Real models use millions of numbers, but every number obeys the same line rule independently, so 2 is enough to see everything.

Worked examples

Worked example Example 4 — Cell D:

$w > 1$ , the negative-weight extrapolation Statement. Same vectors, $w = 3.0$ . Find $\tilde{ϵ}$ and note the sign of each coefficient.

Forecast: with $w = 3$ , does the answer stay between the points or shoot past the conditional one?

Coefficients: $(1 - w) = 1 - 3 = - 2$ on the unconditional term, $+ 3$ on the conditional. Why this step? This is the famous negative coefficient: for $w > 1$ we push away from the generic distribution.
$\tilde{ϵ} = - 2 [0.2, 0.1] + 3 [0.8, - 0.4] = [- 0.4, - 0.2] + [2.4, - 1.2]$ .
Sum: $[2.0, - 1.4]$ .

Answer: $\tilde{ϵ} = [2.0, - 1.4]$ .

Verify (extrapolation check): The additive form must agree: $ϵ_{uncond} + w Δ ϵ$ with $Δ ϵ = [0.8, - 0.4] - [0.2, 0.1] = [0.6, - 0.5]$ gives $[0.2, 0.1] + 3 [0.6, - 0.5] = [0.2 + 1.8, 0.1 - 1.5] = [2.0, - 1.4]$ . ✓ Notice the point is now beyond the conditional — that is what "stronger prompt adherence" means geometrically.

Worked example Example 7 — Cell G: real Stable-Diffusion word problem

Statement. Prompt $c$ = "a cat wearing a wizard hat", $w = 7.5$ , at timestep $t = 500$ . Model outputs (simplified 3-vectors): $ϵ_{cond} = [0.2, - 0.5, 0.3]$ , $ϵ_{uncond} = [0.1, - 0.3, 0.1]$ . Find $\tilde{ϵ}$ to feed into the next DDPM sampling step.

Forecast: each component gets pushed in the direction the prompt already leans. Guess whether the first component ends above or below 0.2.

Arrow: $Δ ϵ = [0.2 - 0.1, - 0.5 + 0.3, 0.3 - 0.1] = [0.1, - 0.2, 0.2]$ . Why this step? This vector is exactly "what the words cat wizard add."
Scale: $7.5 Δ ϵ = [0.75, - 1.5, 1.5]$ . Why this step? $w = 7.5$ is Stable Diffusion's typical dial — strong adherence.
Add to baseline: $\tilde{ϵ} = [0.1, - 0.3, 0.1] + [0.75, - 1.5, 1.5] = [0.85, - 1.8, 1.6]$ .

Answer: $\tilde{ϵ} = [0.85, - 1.8, 1.6]$ .

Verify (line form): $(1 - 7.5) ϵ_{uncond} + 7.5 ϵ_{cond} = - 6.5 [0.1, - 0.3, 0.1] + 7.5 [0.2, - 0.5, 0.3] = [- 0.65, 1.95, - 0.65] + [1.5, - 3.75, 2.25] = [0.85, - 1.8, 1.6]$ . ✓ First component rose from 0.2 to 0.85 — pushed further along its own direction, as forecast.

Worked example Example 8 — Cell H: the negative-prompting twist

Statement. We want "a photo of a dog" but not blurry. Negative prompting replaces the unconditional branch with a negative prompt $c_{neg}$ = "blurry". So $\tilde{ϵ} = ϵ_{neg} + w (ϵ_{cond} - ϵ_{neg})$ . With $ϵ_{cond} = [0.6, 0.2]$ (dog), $ϵ_{neg} = [0.4, 0.5]$ (blurry), $w = 6.0$ , find $\tilde{ϵ}$ .

Forecast: the arrow now points from blurry to dog — so the answer should flee the "blurry" point.

Arrow: $Δ ϵ = [0.6 - 0.4, 0.2 - 0.5] = [0.2, - 0.3]$ . Why this step? Same machinery as CFG, but the "baseline" is now the thing we want to move away from.
Guided: $\tilde{ϵ} = [0.4, 0.5] + 6.0 [0.2, - 0.3] = [0.4, 0.5] + [1.2, - 1.8] = [1.6, - 1.3]$ . Why this step? Walking 6 arrow-lengths from blurry toward dog strongly suppresses blur features.

Answer: $\tilde{ϵ} = [1.6, - 1.3]$ .

Verify: Line form with $ϵ_{neg}$ as baseline: $(1 - 6) [0.4, 0.5] + 6 [0.6, 0.2] = - 5 [0.4, 0.5] + [3.6, 1.2] = [- 2, - 2.5] + [3.6, 1.2] = [1.6, - 1.3]$ . ✓ Identical algebra to plain CFG — negative prompting is just CFG with $\emptyset$ swapped for an unwanted prompt.

Worked example Example 9 — Cell I: limiting behaviour

$w \to \infty$ Statement. Using Example 4's vectors ( $ϵ_{cond} = [0.8, - 0.4]$ , $ϵ_{uncond} = [0.2, 0.1]$ ), describe the direction $\tilde{ϵ}$ heads as $w \to \infty$ , and compute $w = 100$ .

Forecast: as the dial goes to infinity, does $\tilde{ϵ}$ settle down or run off — and in which direction?

Factor the formula: $\tilde{ϵ} = ϵ_{uncond} + w Δ ϵ$ with $Δ ϵ = [0.6, - 0.5]$ . Why this step? Only the $w Δ ϵ$ term grows; the constant $ϵ_{uncond}$ becomes negligible.
As $w \to \infty$ , $\tilde{ϵ} \approx w Δ ϵ$ , i.e. it runs off to infinity along the arrow direction $[0.6, - 0.5]$ (up to positive scaling).
At $w = 100$ : $\tilde{ϵ} = [0.2, 0.1] + 100 [0.6, - 0.5] = [0.2 + 60, 0.1 - 50] = [60.2, - 49.9]$ .

Answer: direction $\to [0.6, - 0.5]$ ; at $w = 100$ , $\tilde{ϵ} = [60.2, - 49.9]$ .

Verify: Line form: $(1 - 100) [0.2, 0.1] + 100 [0.8, - 0.4] = - 99 [0.2, 0.1] + [80, - 40] = [- 19.8, - 9.9] + [80, - 40] = [60.2, - 49.9]$ . ✓ The magnitude explodes — this is exactly why unbounded $w$ destroys images: the noise prediction leaves any sensible range. This connects to the score-scaling view in Score-based diffusion models and the classifier-gradient origin in Classifier guidance.

Recall Quick self-test

With $w = 0$ , which prediction do you get? ::: The unconditional one, $ϵ_{uncond}$ (prompt ignored). With $w = 1$ , which prediction do you get? ::: Exactly the conditional one, $ϵ_{cond}$ . What is the sign of the unconditional coefficient when $w = 7.5$ ? ::: Negative ( $1 - 7.5 = - 6.5$ ). If $ϵ_{cond} = ϵ_{uncond}$ , what does guidance do? ::: Nothing — the arrow is zero for any $w$ . As $w \to \infty$ , which direction does $\tilde{ϵ}$ run off along? ::: The conditioning arrow $Δ ϵ = ϵ_{cond} - ϵ_{uncond}$ .