Volume III · Fewer Steps

Skipping Steps Without Cheating

Climb a ladder rung by rung and you touch every step on the way up. But if the ladder was only ever tracing a smooth curve — bolted to a wall in an arc rather than straight up — then a handful of well-chosen footholds get you to the top just as surely as the whole ladder does. Nobody had to add or remove any rungs from the original ladder to see this; they only had to notice the curve was there all along.

A choice nobody questioned

Song, Meng and Ermon's 2020 paper, "Denoising Diffusion Implicit Models" (DDIM), starts from an observation about DDPM that is easy to miss precisely because it looks so natural: DDPM's reverse process is Markovian by construction — each step depends only on the step immediately before it. That Markovian structure was a modelling choice made when the forward process was defined. It was never a requirement imposed by the forward process's marginal distributions — the actual quantities, q(x_t | x_0) at each t, that training depends on.

DDIM's contribution is to define an entire family of forward processes that are non-Markovian — each step can depend on more than just the immediately preceding one — while sharing exactly the same marginal q(x_t | x_0) at every t that DDPM's Markovian process has. Because the training objective from Lecture 1 depends only on those marginals, and not on the Markovian structure itself, a network trained under ordinary DDPM can be reused, unmodified, under any member of this non-Markovian family.

Picking the deterministic member

Within that family, one particular choice sets the injected randomness at every reverse step to exactly zero. Sampling then behaves like solving Lecture 4's probability flow ODE with a simple numerical integrator, rather than like running a stochastic process. And because a deterministic trajectory tolerates larger integration steps with controlled error — the same reason any ODE solver can take bigger steps than a stochastic one — DDIM sampling can skip most of the intermediate timesteps that DDPM's original stochastic sampler had to visit one by one.

The practical payoff

Reported speedups run roughly 10 to 50 times fewer sampling steps for comparable sample quality, applied to a model that was never retrained — the entire gain comes from changing how an already-trained network is queried at inference time, not from any change to the training loop.

The teaching image is the ladder from the opening paragraph: DDPM takes every rung, because a Markovian process, by its own construction, has no way to look further ahead than the immediately preceding step. DDIM notices the ladder was tracing a smooth curve all along, and a smooth curve can be approximated well by a few well-chosen points instead of every rung.

That last point is worth underlining because it is easy to read "10 to 50 times fewer steps" and assume some new training trick is responsible. It is not. The entire saving sits in a change to which member of the reverse-process family does the consuming at sampling time — the clearest demonstration in this series so far that inference-time choices can be as consequential as anything decided during training.

Where this volume goes next

DDIM is the first entry in a broader family of fast samplers, and it is worth being honest about the edges of what can be asserted here: no single comprehensive, settled survey of the current 2025–26 sampler landscape is confirmed for this lecture, so none is claimed. What is safe to name is that solvers in the DPM-Solver style exist as a further, higher-order development of the same probability-flow-ODE idea — worth knowing the name of, without this lecture asserting details about them that have not been verified. The next two lectures take the deterministic trajectory this one establishes and push on it from two different directions: teaching a model to skip itself entirely, and straightening the path so there is less of it to walk in the first place.