SlowGuess/NeurIPS_2023_OCR huggingface dataset:
6 Corrector: Starting from  $\widehat{x}_{T - \delta}^{\prime}$ , run underdamped Langevin Monte Carlo for total time  $\Theta (1 / \sqrt{L})$  with step size  $h_{\mathrm{corr}}$  and score estimate  $s_{T - \delta}$  to obtain  $\widehat{x}_{T - \delta}$ .
3.3 Convergence guarantees
Our main results are the following convergence guarantees for the two predictor-corrector schemes described in §3.2:
Theorem 2 (DPOM). Suppose that Assumptions 1-4 hold. If  $\widehat{q}$  denotes the output of DPOM (Algorithm 1) with  $\delta \asymp \frac{\varepsilon^2}{L^2(d\vee\mathfrak{m}_2^2)}$ , then
$$
\mathrm {T V} (\widehat {q}, q _ {\star}) \lesssim (\sqrt {d} \vee \mathfrak {m} _ {2}) \exp (- T) + L ^ {2} T d ^ {1 / 2} h _ {\text {p r e d}} + L ^ {3 / 2} T d ^ {1 / 2} h _ {\text {c o r r}} ^ {1 / 2} + L ^ {1 / 2} T \varepsilon_ {\mathfrak {s c}} + \varepsilon . \tag {7}
$$
In particular, if we set  $T = \Theta\left(\ln \left(\frac{d \vee \mathfrak{m}_2^2}{\varepsilon^2}\right)\right)$ ,  $h_{\mathrm{pred}} = \widetilde{\Theta}\left(\frac{\varepsilon}{L^2 d^{1/2}}\right)$ ,  $h_{\mathrm{corr}} = \widetilde{\Theta}\left(\frac{\varepsilon^2}{L^3 d}\right)$ , and if the score estimation error satisfies  $\varepsilon_{\mathrm{sc}} \leq \widetilde{O}\left(\frac{\varepsilon}{\sqrt{L}}\right)$ , then we can obtain TV error  $\varepsilon$  with a total iteration complexity of  $\widetilde{\Theta}\left(\frac{L^3 d}{\varepsilon^2}\right)$  steps.
The five terms in the bound (7) correspond, respectively, to: the convergence of the forward (OU) process; the discretization error from the predictor steps; the discretization error from the corrector steps; the score estimation error; and the early stopping error.
Theorem 2 recovers nearly the same guarantees as the one in [Che+23a; CLL23; LLT23], but for the probability flow ODE with overdamped Langevin corrector instead of the reverse SDE without corrector. Recall also from Remark 1 that our results can easily be extended to compactly supported data distributions without smooth score functions. This covers essentially all distributions encountered in practice. Therefore, our result provides compelling theoretical justification complementing the empirical efficacy of the probability flow ODE, which was hitherto absent from the literature.
However, in Theorem 2, the iteration complexity is dominated by the corrector steps. Next, we show that by replacing the overdamped LMC with underdamped LMC, we can achieve a quadratic improvement in the number of steps, considering the dependence on  $d$ . As discussed in the Introduction, this highlights the potential benefits of the ODE framework over the SDE.
Theorem 3 (DPUM). Suppose that Assumptions 1-4 hold. If  $\widehat{q}$  denotes the output of DPUM (Algorithm 2) with  $\delta \asymp \frac{\varepsilon^2}{L^2(d\vee\mathfrak{m}_2^2)}$ , then
$$
\mathrm {T V} (\widehat {q}, q _ {\star}) \lesssim (\sqrt {d} \vee \mathfrak {m} _ {2}) \exp (- T) + L ^ {2} T d ^ {1 / 2} h _ {\text {p r e d}} + L ^ {3 / 2} T d ^ {1 / 2} h _ {\text {c o r r}} + L ^ {1 / 2} T \varepsilon_ {\text {s c}} + \varepsilon .
$$
In particular, if we set  $T = \Theta\left(\ln\left(\frac{d \vee \mathfrak{m}_2^2}{\varepsilon^2}\right)\right)$ ,  $h_{\mathrm{pred}} = \widetilde{\Theta}\left(\frac{\varepsilon}{L^2 d^{1/2}}\right)$ ,  $h_{\mathrm{corr}} = \widetilde{\Theta}\left(\frac{\varepsilon}{L^{3/2} d^{1/2}}\right)$ , and if the score estimation error satisfies  $\varepsilon_{\mathrm{sc}} \leq \widetilde{O}\left(\frac{\varepsilon}{\sqrt{L}}\right)$ , then we can obtain TV error  $\varepsilon$  with a total iteration complexity of  $\widetilde{\Theta}\left(\frac{L^2 d^{1/2}}{\varepsilon}\right)$  steps.
4 Proof overview
Here we give a detailed technical overview for the proof of our main results, Theorems 2 and 3. As in [Che+23a; CLL23; LLT23], the three sources of error that we need to keep track of are (1) estimation of the score function; (2) discretization of time when implementing the probability flow ODE and corrector steps; and (3) initialization of the algorithm at  $\gamma^d$  instead of the true law of the end of the forward process,  $q_0 = q_T^\rightarrow$ . It turns out that (1) is not so difficult to manage as soon as we can control (2) and (3). Furthermore, as in prior work, we can easily control (3) via the data-processing inequality: the total variation distance between the output of the algorithm initialized at  $q_0$  versus at  $\gamma^d$  is at most  $\mathsf{TV}(q_T^\rightarrow, \gamma^d)$ , which is exponentially small in  $T$  by rapid mixing of the OU process. So henceforth in this overview, let us assume that both the algorithm and the true process are initialized at  $q_0$ . It remains to control (2).
Failure of existing approaches. In the SDE implementation of diffusion models, prior works handled (2) by directly bounding a strictly larger quantity, namely the KL divergence between the laws of the trajectories of the algorithm and the true process; by Girsanov's theorem, this has a clean formulation as an integrated difference of drifts. Unfortunately, in the ODE implementation, this KL divergence is infinite: in the absence of stochasticity in the reverse process, these laws over trajectories are not even absolutely continuous with respect to each other.
In search of an alternative approach, one might try a Wasserstein analysis. As a first attempt, we could couple the initialization of both processes and look at how the distance between them changes over time. If  $(\widehat{x}_t)_{0\leq t\leq T}$  and  $(x_{t})_{0\leq t\leq T}$  denote the algorithm and true process, then smoothness of the score function allows us to naively bound  $\partial_t\mathbb{E}[\| \widehat{x}_t - x_t\| ^2 ]$  by  $O(L)\mathbb{E}[\| \widehat{x}_t - x_t\| ^2 ]$ . While this ensures that the processes are close if run for time  $\ll 1 / L$ , it does not rule out the possibility that they drift apart exponentially quickly after time  $1 / L$ .
Restarting the coupling—first attempt. What we would like is some way of "restarting" this coupling before the processes drift too far apart, to avoid this exponential compounding. We now motivate how to achieve this by giving an argument that is incorrect but nevertheless captures the intuition for our approach. Namely, let  $p_t \coloneqq \mathrm{law}(\widehat{x}_t)$  denote the law of the algorithm, let  $P_{\mathrm{ODE}}^{t_0,h}$  denote the result of running the ideal probability flow ODE for time  $h$  starting from time  $t_0$ , and let  $\widehat{P}_{\mathrm{ODE}}^{t_0,h}$  denote the same but for the discretized probability flow ODE with estimated score. For  $h \lesssim 1 / L$ , consider the law of the two processes at time  $2h$ , i.e.,
$$
p _ {2 h} = q _ {0} \widehat {P} _ {\mathrm {O D E}} ^ {0, 2 h} \quad \text {a n d} \quad q _ {2 h} = q _ {0} P _ {\mathrm {O D E}} ^ {0, 2 h}. \tag {8}
$$
The discussion above implies that  $q_0P_{\mathrm{ODE}}^{0,h}$  and  $q_0\widehat{P}_{\mathrm{ODE}}^{0,h}$  are close in 2-Wasserstein distance, so by the data-processing inequality, this implies that  $q_0P_{\mathrm{ODE}}^{0,h}\widehat{P}_{\mathrm{ODE}}^{h,h}$  and  $q_0\widehat{P}_{\mathrm{ODE}}^{0,h}\widehat{P}_{\mathrm{ODE}}^{h,h}$  are also close. To show that  $p_{2h}$  and  $q_{2h}$  in Eq. (8) are close, it thus suffices to show that  $q_0P_{\mathrm{ODE}}^{0,2h}$  and  $q_0P_{\mathrm{ODE}}^{0,h}\widehat{P}_{\mathrm{ODE}}^{h,h}$  are close. But these two distributions are given by running the algorithm and the true process for time  $h$ , both starting from  $q_0P_{\mathrm{ODE}}^{0,h}$ . So if we "restart" the coupling by coupling the processes based on their locations at time  $h$ , rather than time 0, of the reverse process, we can again apply the naive Wasserstein analysis.
At this juncture, it would seem that we have miraculously sidestepped the exponential blowup and shown that the expected distance between the processes only increases linearly over time! The issue of course is in the application of the "data-processing inequality," which simply does not hold for the Wasserstein distance.
Restarting the coupling with a corrector step. This is where the corrector comes in. The idea is to use short-time regularization: if we apply a small amount of noise to two distributions which are already close in Wasserstein, then they become close in KL divergence, for which a data-processing inequality holds. The upshot is that if the noise doesn't change the distributions too much, then we can legitimately restart the coupling as above and prove that the distance between the processes, now defined by interleaving the probability flow ODE and its discretization with periodic injections of noise, increases only linearly in time.
It turns out that naive injection of noise, e.g., convolution with a Gaussian of small variance, is somewhat wasteful as it fails to preserve the true process and leads to poor polynomial dependence in the dimension. On the other hand, if we instead run the overdamped Langevin diffusion with potential chosen so that the law of the true process is stationary, then we can recover the linear in  $d$  dependence of Theorem 2. Then by replacing overdamped Langevin diffusion with its underdamped counterpart, which has the advantage of much smoother trajectories, we can obtain the desired quadratic speedup in dimension dependence in Theorem 3.
Score perturbation lemma. In addition to the switch from SDE to ODE and the use of the underdamped corrector, a third ingredient is essential to our improved dimension dependence. The former two ensure that the trajectory of our algorithm is smoother than that of DDPMs, so that even over time windows that scale with  $1 / \sqrt{d}$ , the process does not change too much. By extension, as the score functions are Lipschitz, this means that any fixed score function evaluated over iterates in such a window does not change much. This amounts to controlling discretization error in space.
It is also necessary to control discretization error in time, i.e., proving what some prior works referred to as a score perturbation lemma [LLT22]. That is, for any fixed iterate  $x$ , we want to show that the score function  $\nabla \ln q_t(x)$  does not change too much as  $t$  varies over a small window. Unfortunately, prior works were only able to establish this over windows of length  $1 / d$ . In this work, we improve this to windows of length  $1 / \sqrt{d}$  (see Lemma 3 and Corollary 1).
In our proof, we bound the squared  $L^2$  norm of the derivative of the score along the trajectory of the ODE. The score function evaluated at  $y$  can be expressed as  $\mathbb{E}_{P_{0|t}(\cdot | y)}[\nabla U]$ ; here, the posterior distribution  $P_{0|t}(\cdot | y)$  is essentially the prior  $q_{\star}$  tilted by a Gaussian of variance  $O(t)$ . Hence we need to bound the change in the expectation when we change the distribution from  $P_{0|t}$  to  $P_{0|t + \Delta t}$ ; because  $\nabla U$  is  $L$ -Lipschitz, we can bound this by the Wasserstein distance between the distributions. For small enough  $t$ ,  $P_{0|t}$  is strongly log-concave, and a transport cost inequality bounds this in terms of KL divergence, which is more easily bounded. Indeed, we can bound it with the KL divergence between the joint distributions  $P_{0,t}$  and  $P_{0,t + \Delta t}$ , which reduces to bounding the KL divergence between Gaussians of unequal variance.
However, since our score perturbation lemma degrades near the beginning of the forward process, we require better control of the discretization error during this part of the algorithm, hence leading to our choice of geometrically decreasing step sizes. Alternatively, we could use a two-stage step size schedule, see Remark 4.
In this work, we have provided the first polynomial-time guarantees for the probability flow ODE implementation of SGMs with corrector steps and exhibited improved dimension dependence of the ODE framework over prior results for DDPMs (i.e., the SDE framework). Our analysis raises questions relevant for practice, of which we list a few.
- Although we need the corrector steps for our proof, are they in fact necessary for the algorithm to work efficiently in practice?
- Is it possible to obtain even better dimension dependence, perhaps using higher-order solvers and stronger smoothness assumptions?
- Can we obtain improved dimension dependence even in the non-smooth setting, compared to the result of [CLL23]?
We also list several limitations of our work, namely:
- Our analysis only covers the probability flow ODE corresponding to the OU forward process. We leave the study of more general dynamics for future study.
- Our guarantees require the score function to be learned to  $L^2$  accuracy  $\widetilde{O}(\varepsilon/\sqrt{L})$ , which is more stringent than the prior works [Che+23a; CLL23; LLT23] and may be an technical artefact of our proof.
- We have not validated our theoretical findings with large-scale experiments. In particular, it is still unclear whether flow-based methods can outperform the standard DDPM algorithm in practical, high dimensional settings.
We want to thank Yin Tat Lee for his valuable comments, shaping the direction of this research project in its early stages. SC was supported by NSF Award 2103300 for part of this work.
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A 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 §E.
Throughout the analysis, Assumptions 1-4 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 §E, the law of the algorithm is denoted by  $p$ .
We use the following Markov kernels:
1.  $P_{\mathrm{ODE}}^{t,h}$  is the output of running the ODE for time  $h$ , starting at (reverse) time  $t$ .
2.  $P_{\mathrm{LD}}$  (resp.  $P_{\mathrm{ULD}}$ ) 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.
3.  $\widehat{P}_{\mathrm{ODE}}^{t,h}$  and  $\widehat{P}_{\mathrm{LMC}}$  (resp.  $\widehat{P}_{\mathrm{ULMC}}$ ) are the corresponding processes once discretized and using the estimated score.
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 §E, 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  $P = P_{\mathrm{ODE}}$  or  $\widehat{P}_{\mathrm{ODE}}$ , we use  $P^{t,h_1,\dots,h_N}$  to denote  $P^{t,h_1}P^{t + h_1,h_2}\dots P^{t + h_1 + \dots +h_{N - 1},h_N}$  (we compose kernels on the right).
We refer to §4 for a high-level description of the proof strategy. We begin in §B with our improved score perturbation lemma (Corollary 1); this is the only section of the analysis which is indexed by forward time (instead of reverse time). In Lemma 5 in §C, we establish our main result for the predictor steps, which combines together standard ODE discretization analysis with the score perturbation lemma of §B. Since Corollary 1 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 1 and 2.
In §D, we prove our main regularization results for the overdamped corrector (Theorem 4) and the underdamped corrector (Theorem 5). Finally, we put together the various constituent results in the end-to-end analysis in §E.
B Score perturbation
In this section, we prove a score perturbation lemma which refines that of [LLT22]. This improved lemma is necessary in order to obtain  $O(\sqrt{d})$  dependence for the probability flow ODE.
Lemma 1 (Score perturbation). Suppose  $p_t = p_0 * \mathcal{N}(0, tI)$  and  $x_0 \sim p_0$ ,  $\dot{x}_t = -\frac{1}{2} \nabla \ln p_t(x_t)$ . Suppose that  $\| \nabla^2 \ln p_{(t - \frac{1}{2L}) \vee 0}(x) \|_{\mathrm{op}} \leq L$  for all  $x$ . Then
$$
\mathbb {E} \left[ \| \partial_ {t} \nabla \ln p _ {t} (x _ {t}) \| ^ {2} \right] \leq L ^ {2} d \left(L + \frac {1}{t}\right).
$$
Proof. Without loss of generality, we may assume  $t \leq \frac{1}{2L}$ , as otherwise, noting that  $p_t = p_{t - \frac{1}{2L}} * N(0, \frac{1}{2L} I)$ , we may replace  $p_0$  with  $p_{t - \frac{1}{2L}}$  and  $t$  with  $\frac{1}{2L}$ . Suppose  $p_0(x) = e^{-V(x)}$ . Let  $P_{0,t}$  denote the joint distribution of  $(X_0, X_t)$  where  $X_t = X_0 + \sqrt{t} Z$  with  $Z \sim \mathcal{N}(0, I)$  independent of  $X_0$ , and let  $P_{0|t}(\cdot | x_t)$  denote the conditional distribution of  $X_0$  given  $X_t = x_t$ . We first note that since
$$
\ln p _ {t} (y) = \ln \int \exp \left(- V (x) - \frac {1}{2 t} \| y - x \| ^ {2}\right) \mathrm {d} x,
$$
we have the following calculations:
$$
\nabla \ln p _ {t} (y) = - \frac {1}{t} \mathbb {E} _ {P _ {0 | t} (\cdot | y)} (y - \cdot) = - \mathbb {E} _ {P _ {0 | t} (\cdot | y)} (\nabla V),
$$
$$
\nabla^ {2} \ln p _ {t} (y) = \operatorname {C o v} _ {P _ {0 | t} (\cdot | y)} (\nabla V) - \mathbb {E} _ {P _ {0 | t} (\cdot | y)} (\nabla^ {2} V).
$$
Using  $\dot{x}_t = -\frac{1}{2}\nabla \ln p_t(x_t)$ , we calculate
$$
\partial_ {t} \nabla \ln p _ {t} (x _ {t}) = [ \partial_ {t} \nabla \ln p _ {t} (y) ] | _ {y = x _ {t}} - \frac {1}{2} \nabla^ {2} \ln p _ {t} (x _ {t}) \nabla \ln p _ {t} (x _ {t}). \tag {9}
$$
We bound each term above separately. For the first term, a quick calculation shows that  $\partial_t\nabla \ln p_t(y) = -\operatorname{Cov}_{P_{0|t}(\cdot |y)}\left(\frac{\|y-\cdot\|^2}{2t^2},\nabla V\right)$  is finite a.s.: by Cauchy-Schwarz, it suffices to show  $\mathbb{E}_{P_{0|t}(\cdot |y)}[\|y-\cdot\|^4]$  and  $\mathbb{E}_{P_{0|t}(\cdot |y)}[\|\nabla V\|^2]$  are finite for all  $y$ , and this follows because for  $t \leq \frac{1}{2L}$  the measure  $P_{0|t}(\cdot |y)$  is strongly log-concave and  $\|\nabla V\|^2$  can be bounded by a quadratic. Because  $\nabla V$  is  $L$ -Lipschitz,
$$
\begin{array}{l} \left. \right.\left\| \right.\left[ \partial_ {t} \nabla \ln p _ {t} (y) \right]\left. \right| _ {y = x _ {t}} \left. \right\| ^ {2} = \left\| \right.\left[ \partial_ {t} \mathbb {E} _ {P _ {0 | t} (\cdot | y)} (\nabla V) \right]\left. \right| _ {y = x _ {t}} \left. \right\| ^ {2} \\ = \left\| \right. \lim  _ {\Delta t \rightarrow 0} \frac {1}{\Delta t} \left[ \mathbb {E} _ {P _ {0 | t + \Delta t} (\cdot | y)} [ \nabla V ] - \mathbb {E} _ {P _ {0 | t} (\cdot | y)} [ \nabla V ] \right]\left. \right| _ {y = x _ {t}} \left. \right\| ^ {2} \\ \leq L ^ {2} \lim  _ {\Delta t \rightarrow 0} \inf  \frac {1}{(\Delta t) ^ {2}} W _ {1} ^ {2} \left(P _ {0 | t + \Delta t} (\cdot \mid x _ {t}), P _ {0 | t} (\cdot \mid x _ {t})\right). \tag {10} \\ \end{array}
$$
Now  $P_{0|t}(\cdot \mid y)$  has density  $p_{0|t}(x \mid y) \propto p_0(x)e^{-\frac{\|x - y\|^2}{2t}}$  so if  $\| \nabla^2\ln p_0\|_{\mathrm{op}} \leq L$  and  $t \leq \frac{1}{2L}$ , then  $P_{0|t}$  is  $\frac{1}{2t}$ -strongly log-concave. By Talagrand's transport cost inequality,
$$
W _ {1} ^ {2} \left(P _ {0 | t + \Delta t} (\cdot \mid x _ {t}), P _ {0 | t} (\cdot \mid x _ {t})\right) \leq 4 t \mathrm {K L} \left(P _ {0 | t + \Delta t} (\cdot \mid x _ {t}) \| P _ {0 | t} (\cdot \mid x _ {t})\right).
$$
Plugging this back in (10) and using Fatou's lemma and the chain rule for KL,
$$
\begin{array}{l} \mathbb {E} [ \| [ \partial_ {t} \nabla \ln p _ {t} (y) ] | _ {y = x _ {t}} \| ^ {2} ] \leq L ^ {2} \mathbb {E} \lim  _ {\Delta t \rightarrow 0} \inf  _ {\Delta t ^ {2}} \frac {1}{(\Delta t) ^ {2}} 4 t \operatorname {K L} \left(P _ {0 | t + \Delta t} (\cdot \mid x _ {t}) \| P _ {0 | t} (\cdot \mid x _ {t})\right) \\ \leq L ^ {2} \lim  _ {\Delta t \rightarrow 0} \inf  _ {\Delta t) ^ {2}} \frac {1}{(\Delta t) ^ {2}} 4 t \mathbb {E} \mathsf {K L} \left(P _ {0 | t + \Delta t} (\cdot \mid x _ {t}) \| P _ {0 | t} (\cdot \mid x _ {t})\right) \\ \leq L ^ {2} \lim  _ {\Delta t \rightarrow 0} \inf  _ {\Delta t} \frac {1}{(\Delta t) ^ {2}} 4 t \mathrm {K L} \left(P _ {0, t + \Delta t} \| P _ {0, t}\right). \tag {11} \\ \end{array}
$$