Continuous Limit of the DDPM Chain

Why does the discrete chain $x_i=\sqrt{1-{\beta }_i}{x}_{i-1}+\sqrt{{\beta }_i}{\epsilon }_i$ converge to the Variance-Preserving SDE?

Scaling. Place $N$ steps on $[0,1]$ with $\Delta t=1/N$ and set ${\beta }_i=\beta (t_i)\Delta t$ for a fixed continuous function $\beta$. The per-step noise variance must be of order $\Delta t$: only then does the accumulated noise converge to Brownian motion. Order $1$ would blow up, and order $\Delta t^2$ would vanish.

Euler–Maruyama form. Since $\sqrt{1-\beta \Delta t}=1-\frac{1}{2}\beta \Delta t+\mathcal{O}(\Delta t^2)$,

$$x(t+\Delta t)-x(t)=-\frac{1}{2}\beta (t)x(t)\Delta t+\sqrt{\beta (t)}\sqrt{\Delta t}\epsilon +\mathcal{O}(\Delta t^2),$$

which is one Euler–Maruyama step of $\mathrm{d}x=-\frac{1}{2}\beta x\mathrm{d}t+\sqrt{\beta }\mathrm{d}W$.

Convergence. The first two conditional moments per unit time converge, $\frac{1}{\Delta t}\mathbb{E}[\Delta x\mid x]\to -\frac{1}{2}\beta x$ and $\frac{1}{\Delta t}\mathbb{E}[\Delta x\Delta x^{\top }\mid x]\to \beta I$, and the fourth moment is $\mathcal{O}(\Delta t^2)$, which excludes jumps. The drift is Lipschitz, so the limiting martingale problem is well posed. The classical diffusion-limit theorem for Markov chains (Stroock–Varadhan; Ethier–Kurtz) then gives weak convergence of the interpolated chain to the SDE.

Cumulative factor. Taking logarithms turns the product into a sum:

$$\log {\bar{\alpha }}_N(t)=\sum _{t_j\le t}\log (1-\beta (t_j)\Delta t)=-\sum _{t_j\le t}\beta (t_j)\Delta t+\mathcal{O}(\Delta t)\stackrel{N\to \infty }{\to }-{\int }_0^t\beta (s)\mathrm{d}s,$$

so $\bar{\alpha }(t)=\exp (-{\int }_0^t\beta )$. This is the formula used in the Noise Schedule.

Why not differentiate the marginal formula? Differentiating $x(t)=\sqrt{\bar{\alpha }}{x}_0+\sqrt{1-\bar{\alpha }}\epsilon$ in $t$ with $\epsilon$ fixed gives a deterministic ODE. After replacing $\epsilon$ by its conditional expectation it becomes the Probability Flow ODE, which has the same marginals as the SDE but different paths. Marginals alone do not determine a diffusion; the Markov-chain limit selects the SDE.

References

1. Y. Song et al. (2021). Score-Based Generative Modeling through Stochastic Differential Equations. ICLR 2021, App. B. arXiv:2011.13456
2. S. N. Ethier, T. G. Kurtz (1986). Markov Processes: Characterization and Convergence. Wiley. doi:10.1002/9780470316658
3. D. W. Stroock, S. R. S. Varadhan (1979). Multidimensional Diffusion Processes. Springer (reprinted in Classics in Mathematics). doi:10.1007/3-540-28999-2