What CFG actually does

Classifier-free guidance (CFG)^[1] is simple to state. At every sampling step, run the model twice — once with the condition, once without — and combine the two outputs with a coefficient γ. A larger γ gives outputs that follow the condition more closely and, in practice, look better. This is reliable enough that almost nobody samples at γ=1.

The earliest explanation of why is that CFG lowers the sampling temperature: the distribution gets sharper and concentrates on high-probability regions. That account is everywhere, including most tutorials.

It is probably wrong. Over the past two years several papers have each rejected the others’ picture of CFG, and the disagreement is not about wording — it is about what CFG does to the output distribution.

Above all of that sits a contradiction that everyone assumes away:

If CFG pushes the distribution away from the real data, why is γ=5 better than γ=1?

This article answers in three layers.

First, what CFG is trying to do (Section 2). Under an idealised model — CFG applied once, and the sampler returning exactly the distribution that one application defines — what happens as γ grows: which modes become more common, where each mode’s centre moves, and how much variation is left within a mode. All three have closed forms.

Second, where reality departs from the ideal (Sections 3–4). The sampled distribution is not the idealised one, for two reasons: the tilt and the noising happen in the wrong order, and CFG is applied not once but at every step. The first reason has nothing to do with step count or with whether the sampler is deterministic — infinitely many steps would not remove it.

Third, why quality improves anyway (Sections 5–6). Because CFG is not correcting the distribution. It is correcting the model’s error.

Section 1 fixes the formula and notation. The exposition uses rectified flow / flow matching; DDPM and DDIM appear only for comparison.

01The formula, and the distribution behind it

Start with the sampler. Rectified flow begins at pure noise and follows a trajectory to clean data: at each step the model takes the current x and the noise level t and returns a velocity — which way to move, and how far. Integrating those steps to t=0 gives a sample.

Without CFG the model runs once per step: it sees the condition, returns v_c, and the sampler steps along v_c.

CFG changes exactly this step. The model runs twice — once with the condition, giving v_c, once with the condition removed, giving v_u — and the sampler steps along a linear combination of the two:

step taken = v_u + γ · (v_c − v_u)

γ is the guidance_scale of every pipeline. At γ=1 the right-hand side collapses to v_c and CFG is off; at γ>1 the sampler continues a little further along the difference the condition made.

DDPM and DDIM predict noise ε rather than velocity, but the CFG formula is identical with ε in place of v. Everything below holds in both parameterisations; the reason is in the collapsible box.

What this step does to the distribution

Velocity and score (∇log p) determine each other — given one you can compute the other. So a linear combination of two velocities is a linear combination of two scores:

(1−γ) · ∇log p(x)  +  γ · ∇log p(x|c)

and that is the gradient of log[ p(x|c)^γ · p(x)^(1−γ) ]. Each CFG step follows the score of that distribution. Rearranged:

p_γ(x|c)  ∝  p(x|c) · r(x)^(γ−1) where  r(x) = p(x|c) / p(x)

So at the level of distributions, γ reweights p(x|c) by r: regions where r is large are amplified, regions where r is small are suppressed, and γ sets how hard.

Multiplying a distribution by a weight function and renormalising has a name: a tilt. Throughout, “tilt” means this operation — CFG is a tilt of p(x|c) with weight r^(γ−1).

Three terms recur below; definitions first:

r is the most important quantity in this article and deserves a moment. It measures how much an output owes its existence to the condition:

• r large — this kind of output essentially only appears when the condition is given
• r near 1 — it appears just as readily without the condition, so it says little about the condition

Equivalently, r is proportional to the probability that the condition was c given this output — the classifier implicit in the model. CFG never trains a classifier, but it steers by this ratio at every step.

Notation

02The single-step ideal: what CFG is trying to do

This section works with an idealised model that assumes two things:

1. CFG is applied once, to the distribution of clean data
2. The sample is drawn from the p_γ that one application defines

Neither holds in real sampling; the next two sections take them in turn. The ideal is still worth working through: the three conclusions below survive real sampling in direction, though not in magnitude — and without the ideal there would be nothing specific to measure.

Under the ideal, raising γ does three things:

Effect 1Modes are reweighted by r

Why this holds: p_γ is p(x|c) times r^(γ−1). A single mode is narrow, so r is approximately a constant r_j across it, and the mode’s share is simply its original share times r_j^(γ−1):

share_j  ≈  w_j · r_j^(γ−1)        w_j = share of mode j under p(x|c) share_A / share_B  =  (w_A / w_B) · (r_A / r_B)^(γ−1)

γ sits in the exponent, so the gap opens exponentially. But the point is not the exponent. It is that the quantity being amplified is the ratio of r, not the ratio of densities. In the figure, mode A starts at share 0.6 and B at 0.4 — but A sits at the centre of the unconditional distribution (dashed) and B in its tail, so r_B > r_A. Raise γ to 4 and the taller peak sinks while the shorter one rises, crossing at γ≈1.8.

CFG does not pick the most probable mode. It picks the mode that is best explained only by the condition.

MeasuredSame seed: the bird becomes a crane as γ grows

γ = 1γ = 2γ = 6γ = 25

Why a popular account fails: CFG is not low-temperature sampling

Low-temperature sampling draws from p(x|c)^γ — the numerator alone — and its share ratio is (w_A/w_B)^γ, which can only make the larger mode larger. CFG carries the denominator p(x)^(1−γ), which penalises precisely the outputs that are common without the condition. On the figure’s example:

low temperature:  A keeps winning     (0.6/0.4)^γ grows monotonically CFG:              B overtakes         reversal beyond γ≈1.8

Same distribution, same γ, and the two operations push samples toward opposite modes. “CFG is roughly a temperature” is therefore not imprecise but wrong: on the question of which mode wins, the sign is reversed.

Effect 2Each mode’s centre moves, then stops

Within a mode, the centre moves away from the unconditional distribution — but not indefinitely. Give the difference inside the CFG bracket a name:

Δ  =  v_c − v_u        CFG is: start from v_u, then go γ times further along Δ

x* is where Δ = 0 — the point at which both branches make the same prediction.

Δ is proportional to ∇log r (the conversion is in the box above), and x* is the maximum of r, where the gradient vanishes — so Δ vanishes there too, CFG switches itself off, and samples are attracted to the point. The second small panel plots mode B’s centre against γ: a fast rise, then a plateau on x*, with almost no movement between γ=5 and γ=12.

Note the counter-intuitive consequence: the larger γ, the closer samples sit to the point where CFG itself disappears. Section 4 will show that no such point exists in the model we measured.

Effect 3Each mode narrows, at rate 1/(γ−1)

The variance within a mode falls as 1/(γ−1). The standard deviation therefore falls as 1/√(γ−1), more slowly than the centre settles — so beyond a certain γ, the samples stop changing and only become more alike.

MeasuredThe shell where seeds disagree thins as γ grows

γ = 1γ = 2γ = 6γ = 25

Drag the second slider, σu, until the unconditional distribution is as wide as a single mode (0.5): the narrowing vanishes. In other words:

Whether a mode narrows depends on how narrow the mode is relative to how wide the unconditional distribution is — the larger the gap, the stronger the narrowing. γ is only an amplifier: when the two are equally wide, any γ merely moves the mode without shrinking it.

That premise does not always hold in a real model: half of the directions violate it. Take the latent distribution under one condition, run PCA, keep the twelve leading directions, and compare conditional and unconditional widths direction by direction:

And the expansion peaks at γ=2–5 — the range people actually use. So:

CFG is not an isotropic contraction. It compresses the directions in which the conditional was already narrower and expands the ones in which it was wider. A scalar variance sums the two, shows a net contraction, and hides the fact that the two families of directions move in opposite senses.

Where samples concentrate as γ → ∞

r^(γ−1) is an exponential amplifier: in the limit, all of the mass collapses onto the point where r is largest — not the point where p(x|c) is largest.

The limit is “the c that is most distinguishable from unconditional”, not “the most typical c”. At moderate γ the two barely differ; at large γ they separate quickly.

03Where the gap between ideal and real comes from

The sampled distribution does not match the ideal p_γ, and the reason is specific: the tilt and the noising happen in the wrong order.

In the ideal model, the tilt acts on the clean-data distribution and produces p_γ. For the sampler to actually end at p_γ, what it sees at every step must be p_γ after noising — tilt first, then noise.

But what the model learned is p_t(x|c) and p_t(x): clean data noised directly, with no tilt in between. CFG can only tilt those two already-noised distributions — noise first, then tilt.

needed:     clean data ──tilt──▶ p_γ ──noise──▶ what the sampler sees CFG gives:  clean data ──noise──▶ two distributions ──tilt──▶ what the sampler sees

The two operations do not commute. Only with no noise at all do the two paths agree. p_γ is defined at zero noise, so p_γ itself is not miscalculated; the sampler simply follows the other path and lands somewhere else. This is not a precision issue — with infinitely many steps, a perfectly trained model and a deterministic sampler, the path is still the other one, and the result is still not p_γ.

04Many steps: how the three effects change

The ideal applies CFG once. Real sampling applies it at every one of several dozen steps, each at a different noise level. The same tilt does different things at different noise levels.

Prediction: each effect has its own noise window

One quantity governs all three effects: how narrow a conditional mode is against how wide the unconditional distribution is. Noise widens both, so the gap changes with the noise level — and each effect gets its own active window.

Measured: one prediction out of three holds

Every measurement is on our own 3D rectified-flow model. The CFG and no-CFG arms use the same seeds and the same initial noise, and the two arms’ estimates of the clean sample are compared step by step. t′ is the noise level the model actually receives, 1 being pure noise; sampling runs from right to left over 45 steps.

Effect 1 | Reweighting between modes: finished in the high-noise band

At every step, both arms are projected onto a mode axis and the mass on each mode is read off. The fit allows the whole cloud to translate along the axis, so what is read is mass genuinely moving from one mode to the other, not an artefact of translation.

The second row is how much of the eventual difference in mode weight between the two arms has already appeared by that noise level. 61% in the first step, 98% by t′=0.832, nothing further below t′=0.71. The prediction said mid-way; the measurement says it is essentially over at the highest noise.

This contradicts the prediction head-on: at t′=0.99 the two modes overlap completely — how can mass move between them? Because noising does not change mixture weights. The forward process shrinks each component toward the origin and blurs them into one another, but the weight in front of each component is untouched at every noise level. What high noise destroys is the components’ positions — which is why a single sample cannot be classified there — not the weight that CFG acts on.

Separating the two arms shows one more thing: the CFG arm’s mode weight stops changing at t′=0.96 and moves by 0.002 over the remaining 39 steps, while the no-CFG arm keeps redistributing until t′≈0.71. The gap keeps widening because the no-CFG arm is moving — not because CFG is still acting.

Effect 2 | Centres move, then stop: the first half holds, the second does not

The unit is the no-CFG arm’s spread at the end. The displacement is largest at the highest noise and decays monotonically to one seventh — the “strongest at high noise” prediction holds, the only one of the three that does.

But the ideal also said the centre would stop at x*, where both branches agree. It does not. Fit a saturating curve to the displacement against γ, and the measured points overshoot the curve’s own limit — this model has no x*. Samples stop because the 45 steps run out, not because they have reached that point.

Effect 3 | Each mode narrows: expansion first, narrowing late

The prediction said strongest at low noise. In the high-noise band the measured direction is the reverse: CFG does not narrow the distribution there — it expands it, to 1.9× (figure above; the reason is in its caption). The expansion decays as noise falls, net narrowing appears only below t′≈0.65, and what survives to the end is 8% of the expansion (28% and 16% on two other inputs).

This explains why Section 2’s 1/(γ−1) is so far off: γ−1 spans a factor of 96 and the variance only falls to 30–61% — most of the “contraction” is undoing the earlier expansion.

Half the prediction still stands: at low noise, effect 3 is the only effect still active — not because narrowing is strongest there, but because the other two have stopped.

What survives from Section 2

Every directional conclusion holds; no numerical form matches the ideal model.

The directional statements are not merely “still looking right”; several are theorems. Wu et al.^[2] prove that CFG lowers the differential entropy of the output distribution and raises classification confidence, for DDPM and DDIM alike; Li and Jiao^[3] prove for general data distributions that CFG lowers the expected reciprocal of the classifier probability. The directional conclusions are the most reliable ones — and they are also the useful ones.

05So why does quality improve?

Here is the real problem. None of the three effects — reweighting, centre shift, narrowing — explains why generation quality goes up. Section 2’s last conclusion makes it worse: at large γ, samples concentrate on the point most distinguishable from the unconditional distribution, not on the most typical one. That should look less like real data.

One premise is missing:

Everything so far assumes the model is correct. Admit that the model itself is biased, and the contradiction disappears.

A very short argument

Suppose the model at γ=1 returns the true conditional distribution. Then any γ>1 can only move the result away from the truth, and every quality metric should degrade monotonically from γ=1 upward.

In practice nobody samples at γ=1. The best γ is always above 1 — universal experience, and the shape of every FID-versus-γ curve in the literature. Therefore:

some γ > 1 beats γ = 1   ⟹   the model at γ = 1 was already biased

Simple reasoning, large conclusion: CFG is not correcting the distribution. It is correcting the model’s error.

What error, and why r can fix it

Karras et al.^[4] give a concrete account: score-matching training makes the model place probability where real data never occur, producing outliers, and CFG removes those outliers.

MeasuredAn outlier repaired by a small γ

γ = 1γ = 1.25γ = 1.5γ = 2

What matters is the nature of this error: it comes from the model being insufficiently accurate, not from the condition — the conditional and unconditional branches make the same kind of mistake, only more severely on the unconditional side. Because both branches have it, it cancels in the ratio r: those regions have r ≈ 1, and r^(γ−1) suppresses them.

This yields a falsifiable prediction: only the error shared by both branches is removed; error specific to the conditional branch is amplified. That is different from “CFG removes outliers”, under which it would not matter which branch the error came from.

The three effects, read again

This agrees with the line of work that decomposes Δ into components — Li et al.^[5] isolate a term that “suppresses generic features common across unconditional data”. Two descriptions of the same thing.

Where the optimum γ lies

Too small, and the error is not fully cancelled; too large, and the bias the tilt introduces exceeds the error it removes. The best γ balances the two — which is why quality metrics are U-shaped in γ.

And a part that has nothing to do with distributions

Some of the “improvement” is not distributional at all. People prefer high contrast, saturated colour and clean composition, and high γ delivers exactly those. Distribution metrics cannot see this; human evaluation can — and aesthetic preference is a different thing from “closer to real data”, sometimes the opposite.

So “looks better” is at least three things stacked together: error cancelled (a genuine improvement), bias from the tilt (a genuine degradation), and aesthetic preference (a property of the metric). Separating them is what this article set out to do.

06What this means for training

If CFG is fundamentally correcting model error, a few conclusions follow directly.

The CFG strength a model needs reflects its error

In the limit, a perfectly accurate model needs no CFG — so the more accurate the model, the smaller the γ it needs. The same fact can be exploited the other way round: Karras et al.^[4] replace the unconditional branch with a deliberately under-trained version of the same model, so that r points more purely along the error direction — FID 1.01 on ImageNet. Both routes share a premise: what CFG acts on is the error, not the condition.

Two different thresholds when tuning γ

Raising γ meets two events in turn: a value where quality peaks, and, further on, visible defects — oversaturation in images, structural collapse in geometry. These are two different thresholds, set by different factors.

So the usable range between them widens as training proceeds. We have not seen this separation reported, and it can be measured on any series of checkpoints. Tracking the optimal γ also shows how much the conditional branch has improved relative to the unconditional one — γ stops being merely an inference-time knob.

Designing a schedule

So a schedule that applies CFG only within some noise band can buy very different things on different models, depending on where that model’s modes are decided. That position is measurable — apply CFG band by band and compare mode shares — so measure it before choosing a schedule.

References

1. Jonathan Ho, Tim Salimans. Classifier-Free Diffusion Guidance. NeurIPS 2021 Workshop on Deep Generative Models. arXiv:2207.12598
2. Yuchen Wu, Minshuo Chen, Zihao Li, Mengdi Wang, Yuting Wei. Theoretical Insights for Diffusion Guidance: A Case Study for Gaussian Mixture Models. ICML 2024. arXiv:2403.01639
3. Gen Li, Yuchen Jiao. Provable Efficiency of Guidance in Diffusion Models for General Data Distribution. 2025. arXiv:2505.01382
4. Tero Karras, Miika Aittala, Tuomas Kynkäänniemi, Jaakko Lehtinen, Timo Aila, Samuli Laine. Guiding a Diffusion Model with a Bad Version of Itself. NeurIPS 2024. arXiv:2406.02507
5. Xiang Li, Rongrong Wang, Qing Qu. Towards Understanding the Mechanisms of Classifier-Free Guidance. 2025. arXiv:2505.19210

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