SlowGuess/NeurIPS_2023_OCR huggingface dataset:
§ Step 1: The Picard Process. For a fixed  $t > 0$ , recall the construction of the interpolation (3) and the Langevin flow. Central to our analysis is the Picard process, defined as
$$
Y _ {s} ^ {(t)} = X _ {t} + \int_ {0} ^ {s} v \left(X _ {t + u}\right) d u + \int_ {0} ^ {s} \sigma \left(X _ {t + u}\right) d B _ {u} ^ {(t)}. \tag {6}
$$
4One can augment the filtration of the Brownian motion by the  $\sigma$  -algebra of  $Z_{k}$  at times  $\{\tau_k\}_{k\in \mathbb{N}}$
The Picard process is adapted and is (synchronously) coupled with the Langevin flow and the interpolation. We think of the Picard process as one step of the Picard iteration for successive approximations to solve ODEs. This means, intuitively, that its trajectory should be close to the original interpolation, as well as to that of the Langevin flow, playing the role of a "bridge".
Fix  $T > 0$ . For  $s \in [0, T]$ , we decompose the distance between the interpolation  $X_{t}$  in (3) and the Langevin flow as
$$
\frac {1}{2} \left\| X _ {t + s} - \Phi_ {s} ^ {(t)} \right\| ^ {2} \leq \left\| Y _ {s} ^ {(t)} - \Phi_ {s} ^ {(t)} \right\| ^ {2} + \left\| X _ {t + s} - Y _ {s} ^ {(t)} \right\| ^ {2}. \tag {7}
$$
We now bound each term of the decomposition. By synchronous coupling of the processes, Lipschitzness of  $v$ , and Itô isometry, Lemma 3 bounds the first term as
$$
\left\| Y _ {s} ^ {(t)} - \Phi_ {s} ^ {(t)} \right\| ^ {2} \leq 2 (T + 1) L ^ {2} \int_ {0} ^ {s} \left\| \Phi_ {u} ^ {(t)} - X _ {t + u} \right\| ^ {2} d u. \tag {8}
$$
This will be suitable for later use of Gronwall's lemma.
§ Step 2: Accumulated Noise and Bias. For the rest of the proof, we need some extra notation. Define  $m(t) \coloneqq \sup \{k \geq 0 : \tau_k \leq t\}$  and the piecewise-constant process  $\overline{X}_t \coloneqq x_{m(t)}$ . Going back to the second term of (7), observe that
$$
X _ {t + s} - Y _ {s} ^ {(t)} = \int_ {t} ^ {t + s} v \left(\bar {X} _ {u}\right) - v \left(X _ {u}\right) d u + \int_ {0} ^ {s} \sigma \left(\bar {X} _ {t + u}\right) - \sigma \left(X _ {t + u}\right) d B _ {u} ^ {(t)} - \Delta_ {Z} (t, s), \tag {9}
$$
where  $\Delta_Z(t,s)$  is the accumulated noise and bias from time  $t$  to time  $t + s$ . It is expected that  $\| \Delta_Z(t,s)\|$  eventually becomes negligible, since the step size becomes small. The next lemma confirms this intuition.
Lemma 1. Suppose Assumptions 1-3 hold. Then, for any fixed  $T > 0$  we have
$$
\lim_{t\to \infty}\sup_{0\leq s\leq T}\mathbb{E}\| \Delta_{Z}(t,s)\|^{2} = 0.
$$
§ Step 3: Gradient Moment Bounds. Based on (9) and Lemma 1, bounding the distance between the Picard process and the interpolation essentially reduces to bounding how much the discrete algorithm "moves" during one iteration in expectation. This, in turn, depends on how large the moments of  $\| v(x_k) \|$  grow per iteration, which is controlled by the following lemma:
Lemma 2. Let  $\{x_{k}\}_{k\in \mathbb{N}}$  be the iterates of (LRM) and suppose Assumptions 1-3 hold. Then,  $\mathbb{E}\| x_k\|^2 = O(1 / \gamma_{k + 1})$ . This in turn implies  $\mathbb{E}\| v(x_k)\|^2 = O(1 / \gamma_{k + 1})$  and  $\mathbb{E}\| b_{k + 1}\|^2 = O(\gamma_{k + 1})$ .
Using this lemma and Lemma 1 we can obtain  $A_{t} \coloneqq \sup_{0 \leq s \leq T} \mathbb{E}\|X_{t+s} - Y_{s}^{(t)}\|^{2} \to 0$  as  $t \to \infty$ , which shows that the Picard process gets arbitrarily close to the interpolation as  $t \to \infty$ .
§ Step 4: Concluding the Proof. Let us go back to the decomposition (7). Taking expectation and using (8) and Gronwall's lemma, we obtain  $\mathbb{E}\left[\| X_{t + s} - \Phi_s^{(t)}\|^2\right] \leq 4A_t\exp (T^2 L^2)$ , Thus,
$$
\lim  _ {t \to \infty} \sup  _ {s \in [ 0, T ]} \mathbb {E} \left[ \| X _ {t + s} - \Phi_ {s} ^ {(t)} \| ^ {2} \right] = 0.
$$
As we coupled  $X_{t+s}$  and  $\Phi_s^{(t)}$  in a specific way (via synchronizing the Brownian motions), we directly get an upper bound on the Wasserstein distance.
5 Last-Iterate Convergence of Sampling Schemes
In this section we focus on last-iterate convergence of LRM schemes in Wasserstein space. We first explore the interplay between the convergence of WAPTs and stability. We then show that the existing stability results for simple Euler-Maruyama discretization of the Langevin diffusion can be extended, with little to no extra assumptions, to the class of LRM schemes in Section 2. This in turn readily implies the last-iterate convergence of a wide class of LRM schemes.
5.1 From WAPTs to Convergence in  $\mathbb{W}_2$
Since convergence of the distribution of  $x_{k}$  to  $\pi$  in Wasserstein distance implies convergence of the second moments of  $x_{k}$  to that of  $\pi$  [3], convergence in the Wasserstein space should at least require:
$$
\sup  _ {k \in \mathbb {N}} \mathbb {E} \| x _ {k} \| ^ {2} <   \infty . \tag {10}
$$
It turns out that, for WAPTs, the exceedingly weak necessary condition (10) is also sufficient:
Theorem 2. Let  $(X_{t})$  be a Wasserstein asymptotic pseudotrajectory of the Langevin diffusion (LD) generated by an LRM scheme  $\{x_{k}\}$  via (3). Then  $W_{2}(x_{k},\pi)\to 0$  if and only if (10) holds.
Proof. The proof relies on the structure of compact sets in the Wasserstein space and limit-set theorems for dynamical systems [7]. Specifically, the closure of bounded subsets of  $\mathbb{W}_2$  is compact [3], so condition (10) implies that  $(\mathrm{law}(X_t))_{t\geq 0}$  is pre-compact in  $\mathbb{W}_2$ . Moreover, Assumption 1 implies that the Langevin flow is globally integrable. Thus,  $(\mathrm{law}(X_t))_{t\geq 0}$  is a pre-compact WAPT of a globally integrable flow, and we can apply the limit-set theorem for metric spaces [7, Theorem 0.1] to conclude that the limit-set of  $(\mathrm{law}(X_t))_t$  is an internally chain transitive (ICT) set.
Next, we show that for the case of the Langevin flow, the only ICT set is  $\{\pi\}$ , implying the desired convergence of our theorem. To see this, define  $V(\cdot) = D_{\mathrm{KL}}(\cdot \mid \pi)$ . It can be observed that  $V$  is a Lyapunov function for (LD), whose value is strictly decreasing along the flow (as the time derivative of  $V$  along the flow is negative of the relative Fisher information, which is strictly positive for all measures other than  $\pi$ ). Thus, all requirements of [6, Prop. 6.4] are satisfied, showing that the only point in the ICT set is  $\pi$ . This also shows the uniqueness of the stationary distribution of (LD).
Remark. From the proof of Theorem 1, we observe that the supremum of the Wasserstein distance between  $(X.)_{[t,t + T]}$  and  $(\Phi_t^{(t)})_{[0,T]}$  typically scales exponentially with  $T$ , which is common for weak approximation error in the literature, see [40]. Despite the exponential dependence on  $T$ , the convergence of the last iterate is assured by Theorem 2 without a need of a uniform control in  $T$ . This is primarily attributed to the adoption of a dynamical system viewpoint and the application of corresponding tools, effectively harnessing the paradigm established by Benaim and Hirsch.
Theorems 1-2 in tandem thus show that, as long as an LRM scheme satisfies Assumptions 1-3 and the moment condition (10), the desirable last-iterate convergence in  $\mathbb{W}_2$  is immediately attained. Therefore, in the rest of this section, we turn our focus to establishing (10) for LRM schemes.
5.2 Bounded Moments of LRM Schemes
There is a long history of study on conditions that ensure (10) for iterative algorithms, which has culminated in the so-called dissipativity properties. We consider two such examples below.
Assumption 4 (Dissipativity). There exist constants  $\alpha >0$  and  $\beta \geq 0$  such that
$$
\langle x, v (x) \rangle \leq - \alpha \| x \| ^ {2} + \beta , \quad \forall x \in \mathbb {R} ^ {d}.
$$
Under Assumption 4, it is classical that (10) holds for the simple Euler-Maruyama discretization of (LD) with deterministic or stochastic gradient oracles [23, 29, 30, 38, 48, 51, 54]. These studies, however, cannot handle non-zero bias, which, as seen in Examples 1-3, is crucial for incorporating more advanced sampling schemes.
To this end, our next result shows that for a wide class of LRM schemes, the stability (10) essentially comes for free under Assumption 4. The proof is provided in Appendix D.
Theorem 3. Let  $v$  be a vector field satisfying Assumptions 1 and 4 and  $\sigma$  be a diffusion coefficient satisfying Assumption 1, and let  $\{x_{k}\}$  be an LRM scheme. Assume that  $\lim_{k\to \infty}\gamma_k = 0$ ,  $\sup_k\mathbb{E}\| U_k\|^2 < \infty$ , and the bias satisfies (5). Then, the stability condition (10) holds for  $\{x_{k}\}$ .
A weaker notion of dissipativity that has been studied in the literature is:
Assumption 5 (Weak dissipativity). There exist constants  $\alpha >0$ ,  $\kappa \in (0,1]$ , and  $\beta \geq 0$  such that
$$
\langle x, v (x) \rangle \leq - \alpha \| x \| ^ {1 + \kappa} + \beta , \quad \forall x \in \mathbb {R} ^ {d}.
$$
NOISE	BIAS	LAST-ITERATE
LAMBERTON AND PAGES [29], LEMAIRE [30]	X	X	X
TEH ET AL. [54]	✓	X	X
BENAÇM ET AL. [8]	X	X	✓
DURMUS AND MOULINES [20]	X	X	✓
BALASUBRAMANIAN ET AL. [4]	X	X	X
THIS WORK	✓	✓	✓
Table 1: Comparison to existing works on convergence of LRM schemes. All methods, except for [4], require bounded second moments of the iterates.
When  $\kappa = 1$ , Assumption 5 is simply Assumption 4. As opposed to Assumption 4, which requires quadratic growth of  $f$  outside a compact set (when  $v = -\nabla f$ ), Assumption 5 only entails superlinear growth and therefore is considerably weaker.
For Euler-Maruyama discretization of (LD) with deterministic gradients, [20] prove that Assumption 5 is sufficient to guarantee bounded moments of the iterates. As for a generic LRM scheme, we consider the following general condition on the bias terms, which will suffice to cover all our examples in Section 2: For some constant  $c$ ,
$$
\left\| b _ {k + 1} \right\| ^ {2} \leq c \left(\gamma_ {k + 1} ^ {2} \| v (x _ {k}) \| ^ {2} + \gamma_ {k + 1} ^ {2} \| U _ {k + 1} ^ {\prime} \| ^ {2} + \gamma_ {k + 1} \| \xi_ {k + 1} ^ {\prime} \| ^ {2} + \gamma_ {k + 1} \| \xi_ {k + 1} \| ^ {2}\right), \tag {11}
$$
where  $U_{k+1}^{\prime}$  is an extra noise term, and  $\xi_{k+1}^{\prime}$  is a standard Gaussian independent of the noises and  $\xi_k$ . The price to pay with the weaker Assumption 5, however, is that we need to assume sub-Gaussianity of the noise. For a proof, see Appendix D.
Theorem 4. Let  $\pi \propto e^{-f}$  be the target distribution, where  $v = -\nabla f$  satisfies Assumptions 1 and 5, and let  $\{x_{k}\}$  be an LRM scheme. Assume that  $\lim_{n\to \infty}\gamma_k = 0$ , the noises  $U_{k}$  and  $U_{k}^{\prime}$  are sub-Gaussian, and the bias term of  $\{x_{k}\}$  satisfies (11). Then, (10) holds for  $\{x_{k}\}$  in when (i)  $\sigma \equiv 1$ , or (ii)  $f$  is Lipschitz and the LRM follows the Mirror Langevin algorithm (Example 4).
5.3 Examples of Convergent LRM Schemes
We now illustrate the use of Theorems 1-4 on our examples in Section 2.
Proposition 1. Under Assumption 1 and noise with uniformly bounded second moments, the following holds for Examples 1-6: (i) The bias has the form (11) and satisfies (5), (ii) As a result, under Assumptions 2 and 3, Examples 1-6 produce iterates that generate a WAPT of (SDE). (iii) Under the additional conditions of Theorem 3 or Theorem 4, Examples 1-6 enjoy last-iterate convergence to the target distribution in Wasserstein distance.
5.4 Comparison to Existing Work
We now give a more detailed comparison of our results to existing literature; a summary is given in Table 1, and additional comparison with prior works can be found in Appendix B.
$\S$  Guarantees for LRM Schemes. Lamberton and Pages [29] and Lemaire [30] study the simple Euler-Maruyama discretization of (LD) with deterministic gradients (i.e.,  $U_{k} = b_{k} = 0$ ) and establish the weak convergence of the average iterates under a moment condition that is slightly weaker than (10). Their analysis is further extended by [54] to incorporate stochastic gradients. Later, the last-iterate convergence of the simple Euler-Maruyama discretization of (LD) is studied by [20], who prove the convergence in the total variation distance under Assumption 5. Another work on a similar setting as [20] is [8], where the convergence criterion is given in an integral probability metric (IPM) [42] of the form  $d_{\mathcal{B}}(\mu ,\nu)\coloneqq \sup_{\varphi \in \mathcal{B}}|\mathbb{E}_{\mu}\varphi -\mathbb{E}_{\nu}\varphi |$  for a certain class of test functions  $\mathcal{B}$  that is known to imply weak convergence, but not convergence in total variation or Wasserstein distances.
Compared to these results, our guarantees possess the following desirable features:
Although the condition in [29, 30] is stated in a weaker form than (10), it is typically only verified on a special case that is equivalent to our Assumption 4, and thus implies (10). See e.g., [29, Remark 3].
- The convergence is always on the last iterates instead of the average iterates.
- As we tolerate biased algorithms, the class of LRM schemes we consider is significantly more general than the ones in existing work.
Finally, we note that our results are incomparable to the recent work of Balasubramanian et al. [4], who derive the same result as in [29, 30], i.e., average-iterate, weak convergence, deterministic Euler-Maruyama discretization. A remarkable feature of the analysis in [4] is that it does not require any bounded moments, and, in particular, their bounds can be applied to target distributions with unbounded variance. However, the downside of [4] is that, in the presence of  $U_{k}$  and  $b_{k}$ , their analysis produces a bound that does not vanish as  $k\to \infty$ ; see [4, Theorem 15]. In contrast, our framework can tolerate quite general  $U_{k}$  and  $b_{k}$ , gives stronger guarantees (W2 vs. weak convergence; last-iterate vs. average-iterate).
$\S$  On Analysis Techniques. While, to our knowledge, our framework is significantly different from previous works on sampling, we acknowledge that similar ideas of creating an auxiliary process in-between the iterates and the continuous-time flow is not entirely new and has been touched upon in the literature, e.g., [10, 12]. That being said, our specific approach in building the Picard process and its development into a wider array of algorithms, i.e., Langevin-Robbins-Monro schemes, undoubtedly plays a pivotal role in our analysis. Moreover, the integration of the Picard process with the theory of asymptotic pseudo-trajectories offers dual benefits to our study, and we view these as our unique contributions to this area of research.
Furthermore, the novel Picard process gives a significant advantage in all of our results. The work of [8] also hinges on dynamical system theory-related ideas. Yet, missing the critical step of the Picard process has seemingly resulted in much weaker findings compared to our work. This observation is not meant as a critique; rather, it merely highlights the potency of the unique method we have integrated into our study.
In this paper, we provided a new, unified framework for analyzing a wide range of sampling schemes, thus laying the theoretical ground for using them in practice, as well as motivating new and more efficient sampling algorithms that enjoy rigorous guarantees. We built on the ideas from dynamical system theory, and gave a rather complete picture of the asymptotic behavior of many first-order sampling algorithms. In short, our results help with the following:
- Validating existing methods: Methods like mirror Langevin and randomized mid-point currently lack even asymptotic guarantees in fully non-convex scenarios, such as sampling from neural network-defined distributions. Our work fills this gap by offering the first rigorous justification for these schemes, supporting practitioners in utilizing these methods confidently.
- Facilitating new algorithm design: Our work motivates novel sampling methods through a straightforward verification of Assumptions 1-3. An illustrative instance involves the randomized mid-point method and Runge-Kutta integrators, wherein a substantial  $50\%$  reduction in computation per iteration can be achieved without compromising convergence by simply recycling past gradients, shown in Example 2. The balance between the benefits of saving gradient oracles and potential drawbacks remains an open question, necessitating case-by-case practical evaluation. Nevertheless, our theory provides a flexible algorithmic design template that extends beyond the current literature's scope.
While our WAPT result holds under very mild conditions, a severe limitation of our current framework is that it only applies to Langevin-based algorithms, whereas there exist numerous practical sampling schemes, such as Metropolis-Hastings, that are not immediately linked to (LD). We believe that this restriction arises as an artifact of our analysis, as the WAPT framework can in principle be applied equally well to any continuous-time dynamics. Lifting such constraint is an interesting future work.
This work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program grant agreement No 815943. YPH acknowledges funding through an ETH Foundations of Data Science (ETH-FDS) postdoctoral fellowship.
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A Further Examples of LRM Schemes
Example 5. The classic Stochastic Gradient Langevin Dynamics [57] iterates as
$$
x _ {k + 1} = x _ {k} - \gamma_ {k + 1} \widetilde {\nabla} f (x _ {k}) + \sqrt {2 \gamma_ {k + 1}} \xi_ {k + 1}, \tag {SGLD}
$$
where  $\widetilde{\nabla} f$  is the gradient of the negative log-likelihood of a random batch of the data. (SGLD) fits the LRM template by setting  $U_{k + 1}\coloneqq \widetilde{\nabla} f(x_k) - \nabla f(x_k)$ , and  $b_{k + 1}\coloneqq 0$ .
Example 6. The Proximal Langevin Algorithm [9, 46, 59] is defined via
$$
x _ {k + 1} = x _ {k} - \gamma_ {k + 1} \nabla f \left(x _ {k + 1}\right) + \sqrt {2 \gamma_ {k + 1}} \xi_ {k + 1}. \tag {PLA}
$$
This algorithm is implicit, and it is assumed that one can solve (PLA) for  $x_{k + 1}$ . By setting  $b_{k + 1} \coloneqq \nabla f(x_{k + 1}) - \nabla f(x_k)$  and  $U_{k + 1} \coloneqq 0$ , we see that this algorithm also follows the LRM template.
B Additional Related Work
Our paper studies the behavior of a wide range of Langevin-based sampling algorithms proposed in the literature in the asymptotic setting under minimal assumptions. This allows us to give last-iterate guarantees in Wasserstein distance. As stressed in Section 1, our goal is not to provide non-asymptotic rates in this general setting as the problem is inherently NP-Hard. However, given more assumptions and structures on the potential  $f$ , there is a plethora of works which prove convergence rates for the last iterates in Wasserstein distance. In this appendix, we provide additional background for these works and the methods used in the literature.
A powerful framework for quantifying the global discretization error of a numerical algorithm is the mean-square analysis framework [40]. This framework furnishes a general recipe for controlling short and long-term integration errors. For sampling, this framework has been applied to prove convergence rates for Langevin Monte-Carlo (the Euler-Maruyama discretization of (LD)) in the strongly-convex setting [32, 34]. Similar to our work, the convergence obtained in these works is last-iterate and in Wasserstein distance. One of the essential ingredients in the latter work is the contraction property of the SDE, which is ensured by the strong convexity assumption. This, in turn, implies strong non-asymptotic convergence guarantees.
It is an interesting future direction to study the combination of the Mean-Squared analysis together with the Picard process and its applicability to more sophisticated algorithms (such as LRM schemes with bias and noise), as well as non-convex potentials.