\section{Notation and overview}


In this section, we collect together the notation used throughout the proofs and provide a road map for the end-to-end analysis in \S\ref{sec:end_to_end}.

Throughout the analysis, Assumptions~\ref{ass:second_moment}--\ref{ass:score_error} are in full force.

We will reserve $q$ for the law of the reverse process (and denote the forward process by $q^\rightarrow$ when needed). In \S\ref{sec:end_to_end}, the law of the algorithm is denoted by $p$.

We use the following Markov kernels:
\begin{enumerate}
    \item $\Podeth th$ is the output of running the ODE for time $h$, starting at (reverse) time $t$.
    \item $\Plan$ (resp.\ $\Puld$) is the output of running the continuous-time overdamped (resp.\ underdamped) Langevin diffusion for time $h$.
    In this notation, we have suppressed mention of the stationary distributions of the diffusion, which will be provided by context.
    \item $\Podesth th$ and $\Plans$ (resp.\ $\Pulds$) are the corresponding processes once discretized and using the estimated score.
\end{enumerate}

For the ODE\@, we are more precise with the notation because even within a single epoch of predictor steps, the kernel for the probability flow ODE depends on time (as opposed to the kernels for the diffusions, which are constant within any epoch of corrector steps); moreover, for our analysis in \S\ref{sec:end_to_end}, we also need to take time-varying step sizes for the predictor steps.
We will omit the dependencies on $t$ and $h$ when clear from context.
When $P=\Pode$ or $\Podes$, we use $P^{t,h_1,\ldots, h_N}$ to denote $P^{t,h_1}P^{t+h_1,h_2}\dotsm P^{t+h_1+\cdots +h_{N-1}, h_N}$ (we compose kernels on the right).

We refer to \S\ref{sec:pf_overview} for a high-level description of the proof strategy. We begin in \S\ref{sec:score_perturb} with our improved score perturbation lemma (Corollary~\ref{c:sp}); this is the only section of the analysis which is indexed by \emph{forward} time (instead of reverse time).
In Lemma~\ref{lem:pred} in \S\ref{sec:predictor}, we establish our main result for the predictor steps, which combines together standard ODE discretization analysis with the score perturbation lemma of \S\ref{sec:score_perturb}.
Since Corollary~\ref{c:sp} degrades near the end of the reverse process (or equivalently, near the start of the forward process, when the regularization has not yet kicked in), our analysis requires a geometrically decreasing step size schedule, which leads to the two-stage Algorithms~\ref{alg:over} and~\ref{alg:under}.

In \S\ref{sec:corrector}, we prove our main regularization results for the overdamped corrector (Theorem~\ref{thm:main_overdamped}) and the underdamped corrector (Theorem~\ref{thm:main_underdamped}).
Finally, we put together the various constituent results in the end-to-end analysis in \S\ref{sec:end_to_end}.