SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Mohammad Reza Karimi
Ya-Ping Hsieh
Interacting particle systems have proven highly successful in various machine learning tasks, including approximate Bayesian inference and neural network optimization. However, the analysis of these systems often relies on the simplifying assumption of the mean-field limit, where particle numbers approach infinity and infinitesimal step sizes are used. In practice, discrete time steps, finite particle numbers, and complex integration schemes are employed, creating a theoretical gap between continuous-time and discrete-time processes. In this paper, we present a novel framework that establishes a precise connection between these discrete-time schemes and their corresponding mean-field limits in terms of convergence properties and asymptotic behavior. By adopting a dynamical system perspective, our framework seamlessly integrates various numerical schemes that are typically analyzed independently. For example, our framework provides a unified treatment of optimizing an infinite-width two-layer neural network and sampling via Stein Variational Gradient descent, which were previously studied in isolation.
Dynamics of interacting particle systems are central to many challenges in modern machine learning. These range from algorithm design for approximate Bayesian inference, to the study of equilibria in games [26, 31, 34]. Moreover, researchers have gained valuable insights by interpreting the training process of over-parametrized two-layer neural networks as a system of interacting particles, thereby advancing our intuition in this domain [11, 12, 36-38, 46, 51].
Analyzing the behavior of such systems poses a significant challenge. To simplify the analysis, researchers often turn to the concept of the mean-field limit. In this approach, the number of particles is increased to infinity, the step-size of the algorithm shrinks to zero, and results are deduced from this continuous limit. Specifically, an initial probability distribution is assigned to the infinitely many particles, and a continuous-time evolution is derived for this distribution. The resulting evolutionary equation serves as a powerful tool for gaining insights into the system's asymptotic behavior, significantly aiding in the comprehension of systems with a large number of interacting particles [1, 5, 22, 26, 47].
Although the mean-field limit has offered valuable insights into the aforementioned problems, its rigorous justification necessitates examining systems with both a finite number of particles and the discrete-time algorithms commonly used in practice. Unfortunately, to the best of our knowledge, while a substantial body of work exists on finite-particle systems (see, e.g., the review papers [8, 9] and references therein), these studies are confined to analyzing continuous-time dynamics. As a result, a gap between theory and practice has emerged, and a precise link between continuous and discrete-time schemes is still largely lacking.
$\S$  Contributions. Our paper aims to bridge this gap by rigorously establishing the convergence of discrete-time algorithms to their continuous-time counterparts in terms of long-term behavior.
To accomplish this, we draw inspiration from the field of dynamical system theory, which traces its origins back to early developments in statistics [42] and has recently found success in various domains of machine learning, including optimization theory, games, and sampling [21, 23, 24].
This dynamical system framework offers two significant advantages. Firstly, it provides a flexible framework capable of accommodating a wide range of practical schemes commonly employed in real-world scenarios. Secondly, this framework enables a unified treatment of various machine learning tasks that were previously studied in isolation. Notably, it allows us to address tasks such as training neural networks and approximate Bayesian inference using techniques like Stein variational gradient descent in a coherent and unified manner.
In summary, our paper makes the following contributions:
1. We introduce a comprehensive framework for analyzing a broad class of algorithms called stochastic approximation schemes. These schemes are widely utilized in simulating systems of interacting particles, and our framework provides a unified approach to analyze their behavior.
2. Under mild assumptions on the discrete-time schemes and the mean-field dynamics, we prove the convergence of these schemes towards their respective mean-field limits. The convergence is established in terms of the 2-Wasserstein distance, a metric commonly used to measure the dissimilarity between probability distributions.
3. Since our framework is specifically tailored to address a wide range of fields, we instantiate our main theorem and provide novel guarantees to a diverse array of interacting particle systems across domains such as the Stein variational gradient descent, the training of wide two-layer neural networks, and the examination of game equilibria.
2 Illustration: Training Two-Layer Neural Networks via Noisy SGD
To illustrate the particle system under study and its corresponding mean-field limit, let us consider the example of training a two-layer neural network with  $N$  neurons and squared loss using the noisy SGD algorithm. The network takes an input  $z \in \mathbb{R}^{d-1}$  and computes function:
$$
h _ {\boldsymbol {\theta}} (z) := \frac {1}{N} \sum_ {i = 1} ^ {N} \varphi (\theta^ {i}, z), \quad \text {w i t h} \quad \theta^ {i} = (a ^ {i}, b ^ {i}) \in \mathbb {R} ^ {d - 1} \times \mathbb {R} \quad \text {a n d} \quad \varphi (\theta^ {i}, z) = b ^ {i} \kappa (\langle a ^ {i}, z \rangle).
$$
Here,  $\pmb{\theta} = (\theta^{1},\dots,\theta^{N})$  denotes the collection of neurons' parameters, and  $\kappa$  is an activation function (such as sigmoid or tanh). To train this network, one ideally minimizes the regularized risk  $L(\pmb{\theta}) \coloneqq \mathbb{E}_{(y,z) \sim \mathcal{D}} \frac{1}{2}(y - h_{\pmb{\theta}}(z))^{2} + \lambda U(\pmb{\theta})$ , where  $\mathcal{D}$  is the data distribution, and  $U(\pmb{\theta}) = \frac{1}{N} \sum_{i} U(\theta^{i})$  is a regularizer, such as  $U(\theta) = \frac{1}{2} |\theta|^{2}$ . Applying the noisy gradient descent algorithm to the regularized risk then results in the update rule:
$$
\theta_ {k + 1} ^ {i} = \theta_ {k} ^ {i} - \gamma_ {k + 1} \nabla_ {\theta^ {i}} L (\theta_ {k}) + \sigma \sqrt {2 \gamma_ {k + 1}} \xi_ {k + 1} ^ {i}, \quad i = 1, \dots , N, \quad \left(\operatorname {S G D} _ {\sigma}\right)
$$
where  $\gamma_{k + 1}$  is the step-size,  $\sigma \geq 0$  is the noise level, and  $\{\xi_{k + 1}^i\}$  is a collection of i.i.d. standard Gaussians. To cast this algorithm as a system of interacting particles, we define the interaction kernel  $W$  and the potential  $V$  as
$$
W (\theta , \theta^ {\prime}) = \mathbb {E} _ {z \sim \mathcal {D}} [ \varphi (\theta , z) \varphi (\theta^ {\prime}, z) ] \quad \text {a n d} \quad V (\theta) = \mathbb {E} _ {(y, z) \sim \mathcal {D}} [ y \varphi (\theta , z) ] + \lambda U (\theta). \tag {1}
$$
It is then easily seen (cf. [36, Eqn. 4]) that the update rule  $(\mathrm{SGD}_{\sigma})$  can be rewritten as
$$
\theta_ {k + 1} ^ {i} = \theta_ {k} ^ {i} - \gamma_ {k + 1} \left(\frac {1}{N} \sum_ {j = 1} ^ {N} \nabla_ {\theta} W \left(\theta_ {k} ^ {i}, \theta_ {k} ^ {j}\right) + \nabla V \left(\theta_ {k} ^ {i}\right)\right) + \sigma \sqrt {2 \gamma_ {k + 1}} \xi_ {k + 1} ^ {i}, \quad i = 1, \dots , N. \tag {2}
$$
By considering neurons as particles, the equation (2) reveals that the training dynamics of each particle is influenced by the potential  $V$  and the interaction energy  $W$  with other particles. This interpretation highlights that  $(\mathrm{SGD}_{\sigma})$  represents the evolution of a system of interacting particles. In a broader context, one may consider more sophisticated algorithms, several of which (see Section 4 for examples) can be formulated as
$$
\theta_ {k + 1} ^ {i} = \theta_ {k} ^ {i} - \gamma_ {k + 1} \left(\nabla_ {\theta^ {i}} L \left(\theta_ {k}\right) + P _ {k + 1} ^ {i}\right) + \sigma \sqrt {2 \gamma_ {k + 1}} \xi_ {k + 1} ^ {i}, \quad i = 1, \dots , N \tag {3}
$$
where  $P_{k + 1}^{i}$  is the (random or deterministic) perturbation in evaluating the gradient. These algorithms are usually called stochastic approximation algorithms. For example, Noisy SGD corresponds to setting  $P_{k + 1}^{i} = \nabla_{\theta^{i}}\widetilde{L} (\pmb{\theta}_{k}) - \nabla_{\theta^{i}}L(\pmb{\theta}_{k})$ , where  $\widetilde{L}$  is the loss of a random batch of data.
$\S$  Continuous-time limit and the mean-field approximation. We now go from the discrete-time algorithm (3) to a continuous-time process and then derive the mean-field approximation. First, observe that the risk  $L(\theta)$  depends on  $(\theta^1,\dots ,\theta^N)$  only through their empirical measure  $\widehat{\mu} = \frac{1}{N}\sum_{i}\delta_{\theta^{i}}$ . For example,
$$
\frac {1}{N} \sum_ {j = 1} ^ {N} \nabla_ {\theta} W (\theta , \theta^ {j}) + \nabla V (\theta) = \int \nabla_ {\theta} W (\theta , \theta^ {\prime}) \widehat {\mu} (d \theta^ {\prime}) + \nabla V (\theta) =: b (\theta , \widehat {\mu}), \tag {4}
$$
where we have introduced the drift function  $b$ . To simplify the analysis, previous studies then consider the idealized setting where an infinitesimal step-size is employed, thereby further reducing (3) to the following system of stochastic differential equations (SDEs):
$$
d \theta_ {t} ^ {i} = b \left(\theta_ {t} ^ {i}, \widehat {\mu} _ {t}\right) d t + \sigma \sqrt {2} d W _ {t} ^ {i}, \quad i = 1, \dots , N, \tag {5}
$$
where  $\widehat{\mu}_t = \frac{1}{N}\sum_i\delta_{\theta_t^i}$  is the empirical measure of the particles at time  $t$ , and  $\{W_i^i\}$  is a collection of i.i.d. standard Brownian motions. In the over-parametrized regime where the number of particles  $N$  becomes very large, one can approximate the initial setting of the particles  $\widehat{\mu}_0$  with a probability density  $\rho_0$ , and consider the following mean-field approximation defined over  $\mathbb{R}^d$  as
$$
d \theta_ {t} = b \left(\theta_ {t}, \rho_ {t}\right) d t + \sigma \sqrt {2} d W _ {t}, \quad \rho_ {t} = \text {d e n s i t y} \left(\theta_ {t}\right). \tag {6}
$$
This dynamics captures the behavior of an individual particle  $\theta$  within a density of particles distributed according to  $\rho$ . The analysis of (6) turns out to be considerably simpler compared to the system of SDEs in (5), e.g., via studying the evolutionary PDE  $\partial_t\rho_t = \nabla \cdot (\rho_t b(\cdot ,\rho_t)) + \sigma^2\Delta \rho_t$ . As a result, it has attracted significant interest in the field of deep learning theory [11, 12, 36-38].
In conclusion, the mean-field dynamics (6) offers a powerful and elegant framework, but its validity rests on two simplifying assumptions: an infinite number of particles and a step-size approaching zero. A rigorous justification of these two steps is by no means trivial. While the existing literature has made progress in addressing the infinite particle issue, the second assumption has received comparatively less attention. Bridging this gap is one of the primary objectives of our paper. Specifically, we aim to establish the Wasserstein convergence of the discrete-time dynamics (3) to the same limit sets<sup>1</sup> as the continuous-time particle dynamics (5), under mild conditions on the drift  $b$  and perturbations  $\{P_{k+1}^i\}$ , as well as the step-size rule  $\gamma_{k+1}$ .
3 Dynamics of Systems of Interacting Particles
This section presents the fundamental master theorem that forms the basis for all the applications discussed in Section 4. Our objectives are two-fold. Firstly, we aim to provide a set of assumptions for the discrete-time scheme that can be easily verified by practical algorithms. Secondly, we establish a set of assumptions on the mean-field dynamics, which, as demonstrated in Section 4, are readily implied by the standard assumptions in the finite particle regime. The key result of our paper asserts that, under these assumptions, any discrete-time scheme converges to its continuous counterpart, thus closing the existing theoretical gap.
3.1 The algorithmic template
In this paper, we study the stochastic approximation algorithms for simulating systems of interacting particles of the form:
$$
x _ {k + 1} ^ {i} = x _ {k} ^ {i} + \gamma_ {k + 1} \left\{b \left(x _ {k} ^ {i}, \widehat {\mu} _ {k}\right) + P _ {k + 1} ^ {i} \right\} + \sqrt {\gamma_ {k + 1}} \sigma \left(x _ {k} ^ {i}, \widehat {\mu} _ {k}\right) \xi_ {k + 1} ^ {i}, \quad i = 1, \dots , N, \tag {SAA}
$$
where  $b:\mathbb{R}^d\times \mathcal{P}_2(\mathbb{R}^d)\to \mathbb{R}^d$  is the (non-local) drift,  $\sigma :\mathbb{R}^d\times \mathcal{P}_2(\mathbb{R}^d)\to \mathbb{R}^{d\times d}$  is the (state-dependent and non-local) diffusion coefficient,  $P_{k + 1}^{i}$  is the noise and bias in evaluating the drift, and  $\widehat{\mu}_k$  is the empirical measure of the particles at iteration  $k$ . The system (SAA) can be written in a more succinct way by stacking all the variables in a larger vector. That is, define  $x\coloneqq (x^{i})_{i\in [N]}\in (\mathbb{R}^{d})^{\otimes N}$ , and let  $\mu_{x}$  be the empirical distribution corresponding to the  $N$  vectors in  $x$ , and define the aggregated drift and diffusion terms as
$$
\boldsymbol {b} (x) := \left(b \left(x ^ {i}, \mu_ {x}\right)\right) _ {i \in [ N ]} \in \left(\mathbb {R} ^ {d}\right) ^ {\otimes N}, \quad \sigma (x) := \operatorname {d i a g} \left(\left(\sigma \left(x ^ {i}, \mu_ {x}\right)\right) _ {i \in [ N ]}\right). \tag {7}
$$
Define  $\{P_{k + 1}\}$  and  $\{\xi_{k + 1}\}$  analogously. We can then rewrite (SAA) as:
$$
x _ {k + 1} = x _ {k} + \gamma_ {k + 1} \left\{\boldsymbol {b} \left(x _ {k}\right) + P _ {k + 1} \right\} + \sqrt {\gamma_ {k + 1}} \sigma \left(x _ {k}\right) \xi_ {k + 1}. \tag {PSAA}
$$
1The limit set of a curve  $(c(t))_{t\geq 0}$  in a metric space is  $\cap_{t\geq 0}\operatorname {cl}(c([t,\infty)))$  , that is, the set of all limits of convergent sequences  $\{c(t_k),t_k\to \infty \}$
$^{2}\mathcal{P}_{2}(\mathbb{R}^{d})$  is the space of probability measures on  $\mathbb{R}^d$  with bounded second moments.
3.2 Dynamical system theory
By considering infinitesimal step-sizes and neglecting the perturbations  $P_{k + 1}^{i}$ , we can derive a system of corresponding SDEs as follows:
$$
d X _ {t} ^ {i} = b \left(X _ {t} ^ {i}, \widehat {\mu} _ {t}\right) d t + \sigma \left(X _ {t} ^ {i}, \widehat {\mu} _ {t}\right) d W _ {t} ^ {i}, \quad i = 1, \dots , N, \quad \widehat {\mu} _ {t} = \frac {1}{N} \sum_ {i} \delta_ {X _ {t} ^ {i}}, \tag {Sys-SDE}
$$
with the corresponding aggregated version
$$
d X _ {t} = \boldsymbol {b} \left(X _ {t}\right) d t + \sigma \left(X _ {t}\right) d W _ {t}. \tag {PSDE}
$$
It is important to emphasize that even though we have rearranged the particles into a unified vector and introduced the concepts of aggregated drift and diffusion, the original process retains its exchangeability. This property implies that the distribution of  $X_{t}^{i}$  in (Sys-SDE) remains invariant under permutations of the particles (for further details, refer to [8, Def. 2.1]).
The primary objective of our paper is to rigorously establish the convergence of the stochastic approximation scheme (PSAA) to its continuous-time counterpart (PSDE). To accomplish this, we employ the dynamical system theory introduced by Benaim and Hirsch [4]. First, we construct a continuous-time interpolated process associated with the discrete-time algorithm (PSAA):
$$
X _ {t} = x _ {k} + (t - \tau_ {k}) (\boldsymbol {b} (x _ {k}) + \mathbb {E} [ P _ {k + 1} \mid \mathcal {F} _ {t} ]) + \sigma (x _ {k}) (W _ {t} - W _ {\tau_ {k}}), \quad \tau_ {k} \leq t <   \tau_ {k + 1}. \tag {Int}
$$
Here,  $\tau_{k} = \sum_{j=1}^{k} \gamma_{j}$  represents the cumulative time until step  $k$ . It is worth noting that we construct (Int) in such a way that it is adapted to the same filtration  $(\mathcal{F}_{t})$  as the Brownian motion.
To compare (PSAA) with (PSDE), we integrate (PSDE) using the following approach: For a fixed  $t \geq 0$ , we define  $W_{s}^{(t)} = W_{t + s} - W_{t}$  and denote the solution of (PSDE) as the flow:
$$
\Phi_ {s} ^ {(t)} = X _ {t} + \int_ {0} ^ {s} \boldsymbol {b} \left(\Phi_ {u} ^ {(t)}\right) d u + \int_ {0} ^ {s} \sigma \left(\Phi_ {u} ^ {(t)}\right) d W _ {u} ^ {(t)}. \tag {Flow}
$$
It is important to observe that the flow starts at  $X_{t}$  and continues according to the true SDE.
We now introduce the central concept in our paper, which is the asymptotic pseudotrajectory theory of Benaim and Hirsch [4].
Definition 1 (Wasserstein asymptotic pseudotrajectory). We say the stochastic process  $(X_{t})_{t\geq 0}$  is a Wasserstein asymptotic pseudotrajectory (WAPT) of the flow  $\Phi$  if for any fixed  $T > 0$
$$
\lim  _ {t \rightarrow \infty} \sup  _ {0 \leq s \leq T} \mathcal {W} _ {2} \left(X _ {t + s}, \Phi_ {s} ^ {(t)}\right) = 0, \tag {8}
$$
where  $\mathcal{W}_2(\cdot, \cdot)$  denotes the 2-Wasserstein distance between two distributions.
The notion of WAPT provides a measure of "asymptotic closeness" between two stochastic processes. In particular, (8) requires that  $(X_{t})_{t\geq 0}$  closely tracks the flow  $\Phi_s^{(t)}$  over arbitrarily long time intervals  $T$  with arbitrary precision. The key aspect of the WAPT is that it serves as a tool specifically designed to establish the convergence of a stochastic approximation scheme to its continuous-time counterparts. In particular, it is known that proving the convergence of a stochastic approximation algorithm (PSAA) to its continuous-time counterparts (PSDE) can be accomplished by demonstrating the following two conditions [3, 4]:
- The interpolation (Int) satisfies the WAPT condition with respect to the corresponding flow.
- The iterates  $\{x_{k}\}_{k}$ 's in (PSAA) have bounded second moments.
Below, we present a set of general conditions that are straightforward to verify and ensure the satisfaction of the above two conditions.
Remark. There are two major reasons for choosing  $\mathcal{W}_2$  as the distance in (8): Firstly, the 2-Wasserstein space is a metric space on which McKean-Vlasov equations can be seen as a flow, both aspects indispensable for invoking the dynamical system theory of Benaim and Hirsch [3, 4]. Secondly, it is a popular metric in the propagation of chaos literature. This allows a seamless transition from convergence guarantees for stochastic approximation schemes to their mean-field limit counterparts via combining our results with the propagation of chaos results in the literature, see (13).
3.3 Technical Assumptions
We proceed to present our technical assumptions and discuss their generality.
$\S$  On the mean-field dynamics. We begin by introducing three assumptions that pertain to the drift and diffusion coefficients of the continuous-time dynamics (Sys-SDE):
Assumption 1 (Lipschitzness of drift and diffusion). There is some  $L > 0$  such that for all  $x, y \in \mathbb{R}^d$  and all  $\mu, \nu \in \mathcal{P}_2(\mathbb{R}^d)$ ,
$$
\left| b (x, \mu) - b (y, \nu) \right| + \left\| \sigma (x, \mu) - \sigma (y, \nu) \right\| _ {\mathrm {F}} \leq L \left(\left| x - y \right| + \mathcal {W} _ {2} (\mu , \nu)\right).
$$
Assumption 2 (Drift growth condition). For all  $\mu \in \mathcal{P}_2(\mathbb{R}^d)$ , there is some  $C_v > 0$  such that
$$
\int \langle x, b (x, \mu) \rangle \mu (d x) \leq C _ {v} \int (| x | + 1) \mu (d x).
$$
Assumption 3 (Boundedness of the diffusion). There is some  $K > 0$  such that for all  $x \in \mathbb{R}^d$  and all  $\mu \in \mathcal{P}_2(\mathbb{R}^d)$ ,  $\| \sigma(x, \mu) \|_F \leq K$ .
We note that the first assumption is standard and is commonly used to prove existence of strong solutions for the mean-field equation (see [26, Thm. 3.3]). The other two assumptions are exceedingly weak and are satisfied by all the applications we consider.
§ On the stochastic approximation schemes. The following assumptions concern the time-discretization scheme and the induced noise and bias.
Assumption 4 (Noise and bias). The perturbation  $P_{k + 1}$  decomposes into noise and bias as  $P_{k + 1} = U_{k + 1} + \varepsilon_{k + 1}$ , where the noises  $\{U_{k + 1}\}$  form a martingale difference sequence, i.e.,  $\mathbb{E}[U_{k + 1} | U_k] = 0$  and have second moments uniformly bounded by  $M_U$ . In addition, the bias terms satisfy  $\varepsilon_k \in \mathcal{F}_{\tau_k}$  and
$$
\mathbb {E} \left[ \left| \varepsilon_ {k + 1} \right| ^ {2} \mid \mathcal {F} _ {\tau_ {k}} \right] = \mathcal {O} \left(\gamma_ {k + 1} ^ {2} | \boldsymbol {b} (x _ {k}) | ^ {2} + \gamma_ {k + 1}\right). \tag {9}
$$
Assumption 5 (Step-sizes). The step-sizes are decreasing and satisfy the Robbins-Monro summability conditions
$$
\sum_ {k} \gamma_ {k + 1} = \infty \quad a n d \quad \sum_ {k} \gamma_ {k + 1} ^ {2} <   \infty . \tag {10}
$$
Moreover, we require, for some constant  $P > 0$
$$
\gamma_ {k + 1} / \gamma_ {k} + P \gamma_ {k} \gamma_ {k + 1} \leq 1 - \gamma_ {k + 1}. \tag {11}
$$
The condition specified in equation (9) is algorithm-dependent, and as we will prove in Section 4, it is satisfied by numerous practical schemes. In Assumption 5, equation (10) is a commonly used condition in the literature [42], while (11) imposes a mild growth condition on the step size, which remains satisfied even for slowly-decreasing step-sizes such as  $\gamma_{k + 1}\sim (\sqrt{k}\log k)^{-1}$ . Therefore, this condition is not overly restrictive and accommodates a wide range of scenarios.
§ On dissipativity. In the context of dynamical system theory, it is important to ensure that the iterates of stochastic approximation schemes have bounded second moments. In the literature, this requirement is often met by imposing dissipativity-type conditions. Building upon this concept, we introduce the following definition:
Definition 2 (Average Dissipativity). We call the drift  $b$  to be  $(\alpha, \beta)$ -dissipative on average for some  $\alpha > 0$  and  $\beta \in \mathbb{R}$ , if for all probability measures  $\mu \in \mathcal{P}_2(\mathbb{R}^d)$ , it holds
$$
\int \langle x, b (x, \mu) \rangle \mu (d x) \leq - \alpha \int | x | ^ {2} \mu (d x) + \beta .
$$
The concept of average dissipativity, as introduced in Definition 2, provides a novel formulation specifically designed to capture the dissipativity property of a drift function that depends on measures, as in equation (Sys-SDE). In contrast to the traditional notion of dissipativity [19], which focuses on the dissipative behavior of individual particles in isolation, this formulation allows for a more fine-grained control over the collective behavior exhibited by  $N$  particles, each running in parallel with the same stochastic approximation scheme. In Section 4, we will provide concrete examples from the applications of machine learning to demonstrate the satisfaction of average dissipativity.
We are now ready to state our main theorem, whose proof can be found in Appendix B.
Theorem 1. Consider the algorithm (SAA), where the drift  $b$  and diffusion  $\sigma$  satisfy Assumptions 1-3, and the step-sizes  $\{\gamma_{k+1}\}$  and the perturbations  $\{P_{k+1}^i\}$  satisfy Assumptions 4 and 5. Then the following holds:
- The interpolation (Int) of iterates of the algorithm is a WAPT of the flow in (Flow).
- Moreover, if the drift  $b$  is dissipative on average (see Definition 2), the iterates of (PSAA) are bounded in second moments, and their limit set is included in that of the original SDE (PSDE).
Proof sketch. The main step of the proof is the construction of the Picard process, which is inspired by [23] and defined as follows:
$$
\Pi_ {s} ^ {(t)} = X _ {t} + \int_ {0} ^ {s} \boldsymbol {b} \left(X _ {t + u}\right) d u + \int_ {0} ^ {s} \sigma \left(X _ {t + u}\right) d W _ {u} ^ {(t)}. \tag {Picard}
$$