text:
\section{Introduction}

Over the last decades, large systems of interacting particles (or agents)  are widely used to investigate self-organization and collective behavior. They frequently appear in modeling phenomena such as biological swarms \cite{carrillo2010asymptotic,cucker2007emergent}, crowd  dynamics \cite{albiandco2019survey,bellomo2011modeling,bellomo2012modeling}, self-assembly of nanoparticles \cite{holm2006formation} and opinion formation \cite{albiandco2017opinion,motsch2014heterophilious,helbing2010quantitative}. Similar particle models are also used in \textit{metaheuristics}
\cite{Aarts:1989:SAB:61990,Back:1997:HEC:548530,Blum:2003:MCO:937503.937505,Gendreau:2010:HM:1941310}, which provide empirically robust solutions to tackle hard optimization problems with fast algorithms. Metaheuristics are  methods that orchestrate an interaction between local improvement procedures and global/high level strategies, and combine random and deterministic decisions,
to create a process capable of escaping from local optima and performing a robust search of a solution space.
Starting with the groundbreaking work Ref. \cite{rastrigin63} of Rastrigin on Random Search in 1963,
numerous mechanisms for multi-agent global optimization have been  considered, among the most prominent instances we recall the Simplex Heuristics \cite{NeldMead65}, Evolutionary Programming \cite{Fogel:2006:ECT:1202305},  the  Metropolis-Hastings sampling algorithm \cite{hastings70}, Genetic Algorithms \cite{Holland:1992:ANA:531075}, Particle Swarm Optimization (PSO) \cite{kennedy2010particle,poli2007particle}, Ant Colony Optimization (ACO) \cite{dorigo2005ant} and Simulated Annealing (SA) \cite{holley1988simulated,kirkpatrick1983optimization}.
Despite the tremendous empirical success of these techniques, most  metaheuristical  methods  still lack  proper mathematical proof of robust convergence to global minimizers.  In fact, due to the random component of metaheuristics, their asymptotic analysis would require to discern the stochastic dependencies, which is a daunting task, especially for those methods that combine instantaneous decisions with memory mechanisms.

Recent work  Refs. \cite{pinnau2017consensus,carrillo2018analytical} on Consensus-Based Optimization (CBO) focuses on instantaneous stochastic and deterministic decisions in order to establish a consensus among particles on the location of the global minimizers within a domain.
In view of the instantaneous nature of the dynamics, the evolution of the system can be interpreted as a system of  first order stochastic differential equations, whose large particle limit is approximated by a deterministic partial differential equation of mean-field type. The large time behavior of such a deterministic PDE can be analyzed by classical techniques of large deviation bounds and the global convergence of the mean-field model can be mathematically proven in a rigorous way for a large class of optimization problems. Certainly CBO is a significantly simpler mechanism with respect to more sophisticated metaheuristics, which can include different features including memory of past exploration. Nevertheless, it seems to be powerful and robust enough to tackle many interesting nonconvex optimizations \cite{carrillo2019consensus,fhps20-2,ha2019convergence}, which would be harder to solve by gradient descent methods that have been dominating the field of optimization, especially in relevant applications such as machine learning. In fact in many of these problems the objective function is not differentiable and CBO do not use any gradient information for their exploration. Moreover, in certain models, such as training of deep neural networks, the gradient tends to explode or vanish \cite{bengio1994learning}. Finally, although there exist situations where global optimization by gradient descent methods can be ensured under {\it ad hoc} conditions, see, e.g. Refs. \cite{Chen2019,liu2019bad,recht19}, they do not offer in general guarantees of global convergence. Instead, CBO has potential of a rather general and rigorous global asymptotic/convergence analysis.
Some theoretical gaps remain open in the analysis of CBO though, in particular the rigorous derivation of the mean-field limit, due to the difficulty in establishing bounds on the moments of the probability distribution of the particles \cite{carrillo2018analytical}.  Because of this lack of compactness, a direct proof of convergence of the stochastic particle method has been recently derived in Ref. \cite{ha2019convergence}, which does not require the mean-field limit, but at the same time  it is not  able to provide a quantitative error estimate with respect to the number of particles.

Motivated by these theoretical gaps and several potential applications in machine learning, the main task of the present work is to develop a CBO approach to solve the following constrained optimization problem
\begin{equation}
v^\ast \in {\rm arg}\!\min\limits_{v\in \Gamma}\mathcal E(v)\,,
\end{equation}
where $0\leq \mathcal E:\mathbb R^{d} \to \mathbb R$ is a given continuous cost function, which we wish to minimize over a hypersurface $\Gamma$. Here we assume that $\Gamma$ is a connected smooth compact hypersurface embedded in $\mathbb R^d$, which is represented as the $0$-level set of a signed distance function $\gamma$ with $|\gamma(v)|=\mbox{dist}(v,\Gamma)$. This means that
\begin{equation*}
\Gamma=\left\{v\in \mathbb R^d|~\gamma(v)=0\right\}\,.
\end{equation*}
If $\partial \Gamma=\emptyset$, we also assume for simplicity that $\gamma<0$ on the interior of $\Gamma$ and $\gamma>0$ on the exterior. The gradient $\nabla\gamma$ is then the outward unit normal on $\Gamma$ wherever $\gamma$ is defined. Moreover, we assume that there exists an open neighborhood $\widehat\Gamma$ of $\Gamma$ such that $\gamma\in \mathcal{C}^3(\widehat\Gamma)$.  Such setting of hypersurface $\Gamma$ has been used in Ref. \cite{demlow2007adaptive}.
\begin{example}
	Examples of hypersurfaces $\Gamma$ in this setting are
	\begin{itemize}
		\item the unit sphere $\mathbb S^{d-1}$, in which case $\gamma(v)=|v|-1$, $\nabla \gamma(v)=\frac{v}{|v|}$ and $\Delta\gamma(v)=\frac{d-1}{|v|}$;
		\item a  torus radially symmetric about the $v^d$-axis  and of inner radius $r>0$ and  external radius $R>0$ that is expressed in Cartesian coordinates as the $0$-level set of the signed distance function $\gamma(v)= \sqrt{(\sqrt{|v|^2-(v^d)^2}-R)^2+(v^d)^2}-r$, where $v=(v^1,\dots,v^d)$.
	\end{itemize}
\end{example}

In particular, we consider a system of $N$ interacting particles  $((V_t^i)_{t\geq 0 })_{i=1,\dots,N}$ satisfying  the following stochastic  differential  dynamics expressed in It\^{o}'s form
\begin{align} \label{stochastic Kuramoto-Vicsek}
dV_t^i &= -\lambda P(V_t^i)(V_t^i-v_{\alpha,\mathcal E}(\rho_t^N))dt + \sigma |V_t^i - v_{\alpha,\mathcal E}(\rho_t^N)| P(V_t^i)dB_t^i-\frac{\sigma^2}{2}(V_t^i-v_{\alpha,\mathcal E}(\rho_t^N))^2\Delta\gamma(V_t^i)\nabla\gamma(V_t^i)dt\,,
\end{align}
where $\lambda>0$ is a suitable drift parameter, $\sigma>0$ a diffusion parameter,
\begin{equation}
\rho_t^N=\frac{1}{N}\sum_{i=1}^{N}\delta_{V_t^i}\,,
\end{equation}
is the empirical measure of the particles ($\delta_v$ is the Dirac measure at $v\in\mathbb R^d$), and
\begin{equation}\label{ValphaE}
v_{\alpha,\mathcal E}(\rho_t^N)= \frac{\sum_{j=1}^{N} V^j_t e^{-\alpha \mathcal E( V^j_t)}}{\sum_{j=1}^{N}e^{-\alpha \mathcal E( V^j_t )}}=\frac{\int_{\mathbb R^{d}}v\omega_\alpha^\mathcal E(v)d\rho_t^N}{\int_{\mathbb R^{d}}\omega_\alpha^\mathcal E(v)d\rho_t^N} \quad \mbox{ with }\quad  \omega_\alpha^\mathcal E(v):=e^{-\alpha\mathcal E(v)}\,.
\end{equation}
For simplicity,  we write $v^2$ to mean $|v|^2$ for any vector $v \in \mathbb R^d$,  and $|v|$ is the standard Euclidean norm.
This stochastic system is considered complemented with independent and identical distributed (i.i.d.) initial data $V_0^i\in\Gamma$ with $i=1,\cdots,N$, and the common law is denoted by $\rho_0\in \mc{P}(\Gamma)$. The trajectories $((B_t^i)_{t\geq0})_{i=1,\dots N}$ denote $N$ independent standard Brownian motions in $\mathbb R^d$.
In \eqref{stochastic Kuramoto-Vicsek} the projection operator $P(\cdot)$ is defined by
\begin{equation}\label{defP}
P(v)=I-\nabla\gamma(v)\nabla\gamma(v)^T\,.
\end{equation}
In the case of sphere we know $\nabla\gamma(v)=\frac{v}{|v|}$, so one has $P(v)=I-\frac{vv^T}{|v|^2}$,  and \eqref{stochastic Kuramoto-Vicsek} corresponds to a stochastic Kuramoto-Vicsek type model \cite{kuramoto1975self,Vicsek1995NovelTO,bolley2012mean}.

The choice of the weight function $\omega_\alpha^\mathcal E$ in \eqref{ValphaE} comes from  the  well-known Laplace's principle \cite{miller2006applied,Dembo2010,pinnau2017consensus}, a classical asymptotic method for integrals, which states that for any probability measure $\rho\in\mc{P}( \Gamma )$, it holds\footnote{
	For the sake of the reader we provide a simple argument for the validity of this well-known formula: let us take the exponential on the right-hand-side limit to obtain $\lim\limits_{\alpha\to\infty} \frac{1}{\left(\int_{ \Gamma }e^{-\alpha\mathcal E(v)}d\rho(v)\right)^{1/\alpha}} =({\rm ess}\sup_{v \in  {\rm supp }(\rho)} e^{-\mathcal E(v)})^{-1} = {\rm ess}\inf_{v \in  {\rm supp }(\rho)} e^{\mathcal E(v)} $. The first equality holds by the well-known limit of $L^\alpha$-norms to the $L^\infty$-norm for which $\lim_{\alpha\to \infty} \| e^{-\mathcal E(\cdot)}\|_\alpha =  \| e^{-\mathcal E(\cdot)}\|_\infty$. By applying now the logarithm and using its continuity, one obtains \eqref{lap_princ}.}
\begin{equation}\label{lap_princ}
\lim\limits_{\alpha\to\infty}\left(-\frac{1}{\alpha}\log\left(\int_{ \Gamma }e^{-\alpha\mathcal E(v)}d\rho(v)\right)\right)=\inf\limits_{v \in \rm{supp }(\rho)} \mathcal E(v)\,.
\end{equation}

Let us discuss the mechanism of the dynamics. The right-hand-side of the equation \eqref{stochastic Kuramoto-Vicsek} is made of three terms. The first deterministic term $-\lambda P(V_t^i)(V_t^i-v_{\alpha,\mathcal E}(\rho_t^N))dt $ imposes a drift to the dynamics towards $v_{\alpha,\mathcal E}$, which is the current consensus point at time $t$ as approximation to the global minimizer.
This drift term is the projection onto the tangent space to the hypersurfaces at $V_t^i$ of the vector $v_{\alpha,\mathcal{E}}-V_t^i$. Hence $V_t^i$ gets drifted in the direction of $v_{\alpha,\mathcal{E}}$, which is not normal to the hypersurface, and the term disappears when $V_t^i=v_{\alpha,\mathcal{E}}$. In fact, the consensus point $v_{\alpha,\mathcal E}$ is explicitly computed as in \eqref{ValphaE} and it may lay in general outside $\Gamma$. This choice of an embedded weighted barycenter is  very simple, compatible with fast computations, and, for compact manifolds $\Gamma$, it is a good proxy for a minimizer $v^*$. One could alternatively consider the computation of a weighted barycenter on the manifold
$$
v_{\alpha,\mathcal E}^\Gamma(\rho_t) = \arg \min_{v \in \Gamma} \int_\Gamma d_\Gamma(v,w)^p e^{-\alpha \mathcal E(w)} d\rho_t(w),
$$
where $d_\Gamma$ is a (Riemannian) distance on $\Gamma$, $p\geq 1$, and $\rho_t$ is the particle distribution. However, the computation of $v_{\alpha,\mathcal E}^\Gamma$ is in general not explicit (because the distance function might not be simple or explicit) and one may have to solve at each time $t$ an optimization problem over the manifold in order to compute $v_{\alpha,\mathcal E}^\Gamma$. In view of the potentially variable curvature of $\Gamma$, the distance $v \to d_\Gamma(v,w)$ could locally behave in general as a geodesically convex or nonconvex function, rendering such a global optimization rather difficult in general. These are reasons for choosing the embedded alternative \eqref{ValphaE}.
The second stochastic term $\sigma |V_t^i - v_{\alpha,\mathcal E}(\rho_t^N)| P(V_t^i)dB_t^i$ introduces a random decision to favor exploration, whose variance is a function of the distance of the particles to the current consensus points. In particular, as soon as consensus is reached, then the stochastic component vanishes. The last term  $-\frac{\sigma^2}{2}(V_t^i-v_{\alpha,\mathcal E}(\rho_t^N))^2\Delta\gamma(V_t^i)\nabla\gamma(V_t^i)dt$  combined with $P(\cdot)$ is needed to ensure that the dynamics stays on the hypersurface despite the Brownian motion component.   Intuitively, it simply compensates the normal direction of the noise produced by the isotropic Brownian motion and leaves the tangential component active.  In fact, we will initially analyze the well-posedness of the system \eqref{stochastic Kuramoto-Vicsek} as defined in the whole space $\mathbb R^d$ instead of being immediately considered constrained on the hypersurface $\Gamma$, and we will check afterwards that, if the initial data $(V_0^i)_{i=1,\dots,N}$ is set on the hypersurface, then the particles $((V_t^i)_{t\geq 0})_{i=1,\dots,N}$ remain there at all times.
We further notice that the dynamics does not make use of any derivative of $\mathcal E$, but only of its pointwise evaluations, which appear integrated in \eqref{ValphaE}.
Hence, the equation can be in principle numerically implemented at discrete times also for cost functions $\mathcal E$ which are just continuous and with no further smoothness. We require below more regularity to $\mathcal E$ exclusively to ensure formal well-posedness of the evolution.

The main results of this paper are about the  well-posedness of \eqref{stochastic Kuramoto-Vicsek} and its rigorous mean-field limit - which is an open issue for unconstrained CBO \cite{carrillo2018analytical} -  to the following nonlocal, nonlinear Fokker-Planck equation
\begin{equation}\label{PDE}
\partial_t \rho_t= \lambda \nabla_{\Gamma} \cdot (P(v)(v-v_{\alpha,\mathcal E}(\rho_t)) \rho_t)+\frac{\sigma^2}{2}\Delta_{\Gamma} (|v-v_{\alpha,\mathcal E}(\rho_t) |^2\rho_t),\quad t>0,~v\in\Gamma\,,
\end{equation}
with the initial data $\rho_0\in\mc{P}(\Gamma)$. Here $\rho=\rho(t,v)\in \mc{P}(\Gamma)$ is a Borel probability measure on $\Gamma$ and
\[
v_{\alpha,\mathcal E}(\rho_t)  = \frac{\int_{\Gamma} v \omega_\alpha^\mathcal E(v)\,d\rho_t}{\int_{\Gamma} \omega_\alpha^\mathcal E(v)\,d\rho_t}.
\]
The operators $\nabla_{\Gamma} \cdot$ and $\Delta_{\Gamma} $ denote the divergence and Laplace-Beltrami operator on the hypersurface $\Gamma$ respectively. The mean-field limit will be achieved through the coupling method \cite{sznitman1991topics,fetecau2019propagation,huang2017error,huang2019learning,bolley2011stochastic} by introducing  an auxiliary mono-particle process, satisfying the self-consistent nonlinear SDE
\begin{align} \label{selfprocess}
d\overline V_t=-\lambda  P(\overline V_t)(\overline V_t-v_{\alpha,\mathcal E} (\rho_t)) dt + \sigma |\overline V_t - v_{\alpha,\mathcal E}(\rho_t)  | P(\overline V_t)dB_t-\frac{\sigma^2}{2}(\overline V_t-v_{\alpha,\mathcal E} (\rho_t) )^2\Delta \gamma(\overline V_t)\nabla\gamma(\overline V_t)dt\,,
\end{align}
with the initial data $\overline V_0$ distributed according to $\rho_0\in\mc{P}(\Gamma)$. Here we  require $\rho_t=\rm{law}(\overline V_t)$, and we will show that $\rho_t$ as a measure concentrated on $\Gamma$ solves the PDE \eqref{PDE}.
We call the SDE \eqref{selfprocess} the mean-field dynamic, and the PDE \eqref{PDE} is called the  mean-field PDE. In our follow up paper Ref. \cite{fhps20-2}, for more regular datum $\rho_0$, we also prove existence and uniqueness of distributional solutions $\rho \in L^2([0,T],H^1(\Gamma))$ at any finite time $T>0$.
This model and the analysis we provide in this paper are very much inspired by the homogeneous version of the kinetic Kolmogorov-Kuramoto-Vicsek model  on the sphere $\Gamma=\mathbb S^{d-1}$, which was formally derived by Degond and Motsch \cite{demo07}  as a mean-field limit of the discrete Vicsek model \cite{alhu03,coetal02,Vicsek1995NovelTO}. Recently, the stochastic Vicsek model has received extensive attention in the mathematical community to establish its mean-field limit, hydrodynamic limit, and phase transitions. Bolley, Ca\~{n}izo and Carrillo \cite{bolley2012mean} have rigorously justified the mean-field limit in case of smoothed force field, and Gamba and Kang \cite{gaka15} and Figalli, Kang, and Morales \cite{fikamo18} extended the global existence and uniqueness of weak solutions of the kinetic Kolmogorov-Kuramoto-Vicsek  model to the case of singular mean-field force field. Recently, Degond, Frouvelle and Liu \cite{defrli13} provided also a complete and rigorous description of phase transitions such as the number and nature of equilibria, stability, convergence rate, phase diagram and hysteresis. Our results build very much upon these developments.

Before starting our analysis of \eqref{stochastic Kuramoto-Vicsek}, \eqref{selfprocess}, and \eqref{PDE}, let us illustrate
numerically the behavior of the dynamics in the case of the minimum solution of the Ackley function for $d=3$  constrained over an hypersurface $\Gamma$
\begin{equation}
\mathcal E(v)= -A \exp\left(-a\sqrt{\frac{b^2}{d}\sum_{k=1}^{d} (v_k-v^*_k)^2}\right)-\exp\left(\frac1{d}\sum_{k=1}^{d} \cos(2\pi b(v^k-v^k_*))\right)+e+B,
\end{equation}
with $A=20$, $a=0.2$, $b=3$, $B=20$ and $v=(v^1,\ldots,v^d)^T$  with $\gamma(v)=0$.

The global minimum is at $v=v_*$.  In Figure \ref{fg:ackley} we report in the case $d=3$ the Ackley function constrained over the half sphere $v^3 \geq 0$ and over the torus with external radius $R=1$ and inner radius $r=0.5$. The colors over the surface represent the values attained by the function.

\begin{figure}[tb]

	\includegraphics[scale=0.39]{ackleyS3D.pdf}\,
	\includegraphics[scale=0.39]{torus0-eps-converted-to.pdf}\,
	\caption{The Ackley function for $d=3$ in the constrained case on two example hypersurfaces. The half sphere $\mathbb S^2$ (left) and the torus with $R=1$ and $r=0.5$ (right). The global minimum on $\mathbb S^2$ corresponds to the point $v_*=(0,0,1)^T$, over the torus to the point $v_*=(0,1,0.5)^T$.}
	\label{fg:ackley}
\end{figure}

\begin{figure}[htb]
	\hskip .2cm
	\includegraphics[scale=0.2]{rhot20.pdf}\hskip -.25cm
	\includegraphics[scale=0.2]{rhot50.pdf}\hskip .4cm
	\includegraphics[scale=0.2]{rhot20b.pdf}\hskip -.25cm
	\includegraphics[scale=0.2]{rhot50b.pdf}\\[-.2cm]
	\includegraphics[scale=0.39]{ackleyTraj3D.pdf}\,\hskip .2cm
	\includegraphics[scale=0.39]{ackleyTraj3Db.pdf}\,

	\caption{Particles trajectories along the simulation for the Ackley function on the half-sphere in the case $d=3$, $N=20$ with minimum at $v_*=(0,0,1)^T$ (left) and $v_*=(-1/\sqrt{2},-1/2,1/2)^T$ (right). On the top corresponding time evolution of the particle distribution $\rho(t,v)$ in angular coordinates at $t=1$ and $t=2.5$ for $N=10^6$. The simulation parameters are $\Delta t=0.05$, $\sigma=0.25$ and $\alpha=50$.  Although we run the simulation on the entire sphere, we display the result  on the half-sphere because, eventually, all particles do converge to one point, hence, all particles do eventually belong to a half sphere (mind that the stochastic process is a time continuous function). Displaying half-sphere is simply aesthetically more pleasing and it allows a better reading of the figures. On the other hand, as soon as one generates initial particles on the half-sphere containing a minimizer, then they - empirically - tend generically to remain on that half sphere, hence there is no need of displaying the rest of the sphere and we do not use any confinement mechanism rather than the given dynamics.  Also boundary conditions are actually not imposed. }
	\label{fg:ackley_traj}
\end{figure}

\begin{figure}[tb]
	\includegraphics[scale=0.39]{torus1-eps-converted-to.pdf}\,
	\includegraphics[scale=0.39]{torus3-eps-converted-to.pdf}\,
	\caption{Particles trajectories along the simulation for the Ackley function constrained on the torus with $R=1$ and $r=0.5$ in the case $d=3$, $N=20$ with minimum at $v_*=(0,1,0.5)^T$ (left) and $v_*=(0.5,0,0)^T$ (right). The simulation parameters are $\Delta t=0.05$, $\sigma=0.25$ and $\alpha=50$. }
	\label{fg:ackleyt_traj}
\end{figure}

In the reported simulations we initialize the particles with a uniform distribution over the hypersurface (this typically requires the use of a suitable algorithm, see Ref. \cite{fhps20-2})
and employ a simple Euler-Maruyama scheme with projection and time step $\Delta t>0$. We report in Figure \ref{fg:ackley_traj} the case of the half-sphere together with the particle trajectories for $t\in [0,5]$ with $N=20$, $\Delta t=0.05$, $\sigma=0.25$ and $\alpha=50$. On the left we consider the case with minimum at $v_*=(0,0,1)^T$, on the right the case with minimum at $v_*=(1/\sqrt{2},-1/2,1/2)^T$. The time evolution of the particle distribution $\rho(t,v)$ in the numerical mean field limit for $N=10^6$ is also reported in the upper part of the same figure.

In Figure \ref{fg:ackleyt_traj} we report an analogous simulation in the constrained case on the torus. The numerical parameters are the same as in the previous case. On the left the minimum is located at $v_*=(0,1,0.5)^T$, whereas on the right is located at $v_*=(0.5,0,0)^T$.

As these figures show it, the dynamics consistently converges to the global minimizers totally regardless of the many local minimizers and the large particle system does converges numerically to its mean-field approximation.  In the related paper Ref. \cite{fhps20-2}  we rigorously analyze the large time behavior of the system for  $\Gamma = \mathbb S^{d-1}$, we prove its convergence to global minimizers in a suitable sense, and we provide several applications in machine learning. A proof of convergence to global minimizers for more general hypersurfaces is the subject of current investigation.
\\

Let us conclude this introduction by mentioning that the optimization on the hypersurface offers further numerous advantages, besides allowing a rigorous proof of mean-field limit  thanks to the compactness of the hypersurface.  First of all a vast class of optimization problems can be reduced to constrained optimizations over  hypersurfaces: in a related paper Ref. \cite{fhps20-2}, where we focus on the numerical implementation of the method on the sphere and its asymptotic/convergence analysis  to global minimizers, we present also a few relevant and challenging applications in signal processing and machine learning.
Due to  compactness of the hypersurface, local smoothness and boundedness requirements on $\mathcal E$ are necessarily a uniform and global property. However, against these properties that greatly simplify the analysis of the well-posedness of the system and its mean-field limit, the specific topology of the hypersurface may make
it  harder to prove asymptotic convergence of the dynamics to global miminizers, requiring major technical variations with respect to the approach of unconstrained CBO \cite{carrillo2018analytical}. We refer to Ref. \cite{fhps20-2} for the details of the large time analysis of solutions $\rho(t,v)$ of \eqref{PDE} on the sphere and examples of applications in high-dimension. There we needed to develop {\it ad hoc} arguments to prove the asymptotic convergence.  A similar consensus dynamics could be in principle defined also on more general compact manifolds \cite{emery1989stochastic}, but we still lack a general approach to deal with its (global) asymptotic analysis and this research direction needs certainly further investigations.
\\

The rest of the paper is organized as follows: in Section \ref{sec:wellposedness} we  show that the equations \eqref{stochastic Kuramoto-Vicsek}, \eqref{PDE} and \eqref{selfprocess} are well-posed, that is, there is a unique solution for any initial distribution $\rho_0$, which depends continuously on the initial datum, and in Section \ref{sec:MFlimit} we address the mean-field limit.

\section{Well-posedness}\label{sec:wellposedness}
This section focuses on proving the well-posedness for the particle system \eqref{stochastic Kuramoto-Vicsek}, the mean-field dynamic \eqref{selfprocess} and the mean-field PDE \eqref{PDE}. To ensure this, we make the following smoothness assumption on the objective function $\mathcal E$.
\begin{assu}\label{asum1}
	The objective function $0\leq \mathcal E:\mathbb R^{d}\rightarrow \mathbb R$ is locally Lipschitz continuous.
\end{assu}

\subsection{Well-posedness for the interacting particle system \eqref{stochastic Kuramoto-Vicsek}}
Recall that the particle system \eqref{stochastic Kuramoto-Vicsek} is assumed to evolve in the whole space $\mathbb R^d$ instead of on the hypersurface $\Gamma$ directly. We choose this embedding because it provides an explicit and computable representation of the system and it allows for a global description.
The difficulty in showing first  the well-posedness of \eqref{stochastic Kuramoto-Vicsek} in the ambient space $\mathbb R^d$  is that the projection $P(V_t^{i})$, $\Delta \gamma (V_t^{i})$ and $\nabla\gamma (V_t^{i})$ may not be well defined when $V_t^{i}\notin \widehat\Gamma$.  For example, in the case of the sphere $\mathbb S^{d-1}$, the projection $P(V_t^{i})=I-\frac{V_t^{i}(V_t^{i})^T}{|V_t^{i}|^2}$ and $\Delta\gamma (V_t^{i})\nabla\gamma (V_t^{i})=(d-1)\frac{V_t^{i}}{|V_t^{i}|^2}$  are not defined for $V_t^{i}=0$.
In order to overcome this problem, we regularize the diffusion and drift coefficients, that is, we replace them with appropriate functions $P_1$, $P_2$ and $P_3$ respectively: let $P_1$ be a $d\times d$ matrix valued map on $\mathbb R^d$ with bounded derivatives such that $P_1(v)= P(v)$ for all $v\in\widehat\Gamma$, $P_2$ be a $\mathbb R^d$ valued map on $\mathbb R$ with bounded derivatives such that
$P_2(v)=\Delta\gamma(v)$ if $v\in \widehat\Gamma$, and $P_3$ be a $\mathbb R^d$ valued map on $\mathbb R^d$, again with bounded derivatives such that
$P_3(v)=\nabla\gamma(v)$ if $v\in \widehat\Gamma$.   It is also useful to mention that
\begin{equation}
P(v)\nabla\gamma(v)=0
\end{equation}
and
\begin{equation}\label{P2}
\nabla\gamma(v)\cdot P(v)y=0
\end{equation}
hold for any $y\in \mathbb R^d$. For later use, let us denote $[N] = \{1,\dots,N\}$.
Additionally, we regularize the locally Lipschitz function $\mathcal E$: Let us introduce $\tilde{\mathcal E}(v)$ satisfying the following assumptions
\begin{assu}\label{asum}
	The regularized extension function $\tilde{\mathcal E}:\mathbb R^d\rightarrow \mathbb R$ is globally Lipschitz continuous and satisfies the properties
	\begin{itemize}
		\item[1.]  $\tilde{\mathcal E}(v)=\mathcal E(v)$ when $v\in \widehat\Gamma$;
		\item[2.]  $\tilde{\mathcal E}(v)-\tilde{\mathcal E}(u)\leq L|v-u|$ for all $u,v\in\mathbb R^d$ for a suitable global Lipschitz constant $L>0$;
		\item[3.]	$-\infty < \underline{\tilde{\mathcal E}}:=\inf \tilde{\mathcal E} \leq \tilde{\mathcal E}\leq \sup \tilde{\mathcal E}=:\overline \tilde{\mathcal E} < + \infty$\,.
	\end{itemize}
\end{assu}
\begin{remark}
	Here $\tilde{\mathcal E}$ is introduced as an auxiliary function for the proof of well-posedness and mean-field limit only, and it does not play any role in actual optimization problem, which is defined on $\Gamma$. Indeed,
	as we can see in Theorem \ref{thmwellposednessofparticle} below, particles stay on the hypersurface $\Gamma$ all the time, which means that certainly $v\in \widehat\Gamma$, so one has $\tilde{\mathcal E}(v)\equiv\mathcal E(v)$.
\end{remark}

We note here that $\mathcal E$ and $\tilde{\mathcal E}$ appearing the sequel always satisfy Assumptions \ref{asum1} and \ref{asum}.
Given such $P_1$, $P_2$, $P_3$ and $\tilde{\mathcal E}$, we introduce the following regularized particle system
\begin{align} \label{RSKV}
dV_t^i &= -\lambda P_1(V_t^i)(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))dt + \sigma |V_t^i - v_{\alpha,\tilde{\mathcal E}}(\rho_t^N)| P_1(V_t^i)dB_t^i-\frac{\sigma^2}{2}(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))^2P_2(V_t^i)P_3(V_t^i)dt\,,
\end{align}
for $i \in [N]$,  where
\begin{equation}\label{ValphaERd}
v_{\alpha,\tilde{\mathcal E}}(\rho_t^N)=\frac{\int_{\mathbb R^{d}}v\omega_\alpha^{\tilde{\mathcal E}} (v)d\rho_t^N}{\int_{\mathbb R^{d}}\omega_\alpha^{\tilde{\mathcal E}}(v)d\rho_t^N}\,, \qquad  \omega_\alpha^{\tilde{\mathcal E}} (v)=e^{-\alpha{\tilde{\mathcal E}} (v)}\,.
\end{equation}

Next we can deduce that the coefficients in \eqref{RSKV}  are locally Lipschitz continuous and have linear growth. More precisely, we have the following result.

\begin{lemma}\label{lemlocal}
	Let $N\in \mathbb{N}$, $\alpha>0$ be arbitrary and $\tilde{\mathcal E}$ satisfy Assumption \ref{asum}. Then for any $\mb{V}^N,\widehat{\mb{V}}^N\in \mathbb R^{Nd}$,
	and corresponding empirical measures $\rho^N=\frac{1}{N}\sum_{i=1}^{N}\delta_{V^i}\,,$ and $\widehat \rho^N=\frac{1}{N}\sum_{i=1}^{N}\delta_{\widehat V^i}\,,$
	it holds
	\begin{equation}
	|v_{\alpha,\tilde{\mathcal E}}(\rho^N)|\leq  \frac{1}{N}C_{\alpha,\tilde{\mathcal E}}\|\mb{V}^N\|_1
	\end{equation}
	and
	\begin{equation}
	|v_{\alpha,\tilde{\mathcal E}}(\widehat \rho^N)-v_{\alpha,\tilde{\mathcal E}}(\rho^N)|\leq \left(\frac{C_{\alpha,\tilde{\mathcal E}}}{N}+\frac{2\alpha LC_{\alpha,\tilde{\mathcal E}}}{N}\|\mb{\widehat V}^N\|_\infty\right)\|\mb{V}^N- \mb{\widehat V}^N\|_1\,,
	\end{equation}
	where $C_{\alpha,\tilde{\mathcal E}}=e^{\alpha(\overline \tilde{\mathcal E}-\underline \tilde{\mathcal E})}$. Here we used the notations for norms of vectors $\|\mb{V}\|_\infty=\sup_{i \in [N]}|V^i|$ and $\|\mb{V}\|_1=\sum_{i=1}^N|V^i|$.
\end{lemma}

\begin{proof}
	First, one notes that
	\begin{equation}
	e^{-\alpha \overline \tilde{\mathcal E}}\leq \omega_{\alpha}^{\tilde{\mathcal E}}(V^j)=e^{-\alpha \tilde{\mathcal E}( V^j )}\leq  e^{-\alpha \underline \tilde{\mathcal E}}\,.
	\end{equation}
	Then we have
	\begin{align*}
	|v_{\alpha,\tilde{\mathcal E}}(\rho^N)|=\left| \frac{\sum_{j=1}^{N} V^j e^{-\alpha \tilde{\mathcal E}( V^j )}}{\sum_{j=1}^{N}e^{-\alpha \tilde{\mathcal E}( V^j )}}\right|\leq \frac{1}{N}e^{\alpha( \overline \tilde{\mathcal E}-\underline \tilde{\mathcal E})} \sum_{j=1}^N | V^j | = \frac{1}{N}C_{\alpha,\tilde{\mathcal E}} \|\mb{V}^N\|_1\,,\end{align*}
	where  $C_{\alpha,\tilde{\mathcal E}} :=e^{\alpha( \overline \tilde{\mathcal E}-\underline \tilde{\mathcal E})}$. Next we split the error
	\begin{align}
	v_{\alpha,\tilde{\mathcal E}}(\widehat \rho^N)-v_{\alpha,\tilde{\mathcal E}}(\rho^N)&=\frac{\sum_{j=1}^{N}( V^j - \widehat V^j )\omega_{\alpha}^{\tilde{\mathcal E}}(V^j)}{\sum_{j=1}^{N}\omega_{\alpha}^{\tilde{\mathcal E}}(V^j)}+\frac{\sum_{j=1}^{N}\widehat{V}^j(\omega_{\alpha}^{\tilde{\mathcal E}}(V^j)-\omega_{\alpha}^{\tilde{\mathcal E}}(\widehat V^j))}{\sum_{j=1}^{N}\omega_{\alpha}^{\tilde{\mathcal E}}(V^j)}\notag\\
	&\quad +\frac{\sum_{j=1}^{N} \widehat V^j \omega_{\alpha}^{\tilde{\mathcal E}}(\widehat V^j)\sum_{j=1}^{N}(\omega_{\alpha}^{\tilde{\mathcal E}}(\widehat V^j)-\omega_{\alpha}^{\tilde{\mathcal E}}( V^j))}{(\sum_{j=1}^{N}\omega_{\alpha}^{\tilde{\mathcal E}}(V^j))(\sum_{j=1}^{N}\omega_{\alpha}^{\tilde{\mathcal E}}(\widehat V^j))}\notag\\
	&=:I_1+I_2+I_3\,.
	\end{align}
	For $I_1$, the previous computation yields
	\begin{equation}\label{I1}
	|I_1|\leq \frac{1}{N}C_{\alpha,\tilde{\mathcal E}}\|\mb{V}^N- \mb{\widehat V}^N\|_1\,,
	\end{equation}
	Notice that it holds
	\begin{align*}
	|\omega_{\alpha}^{\tilde{\mathcal E}}(V^j)-\omega_{\alpha}^{\tilde{\mathcal E}}(\widehat V^j)|=|e^{-\alpha \tilde{\mathcal E}( V^j )}-e^{-\alpha \tilde{\mathcal E}( \widehat V^j )}|\leq  \alpha e^{-\alpha \underline \tilde{\mathcal E}}   |\tilde{\mathcal E}( V^j )-\tilde{\mathcal E}( \widehat V^j )|\leq \alpha Le^{-\alpha \underline \tilde{\mathcal E}}|V^j-\widehat V^j|\,.
	\end{align*}
	Thus we conclude that
	\begin{equation}\label{I2}
	\max \{|I_2|\,,|I_3|\}\leq  \frac{\alpha LC_{\alpha,\tilde{\mathcal E}}}{N}\|\mb{\widehat V}^N\|_\infty\|\mb{V}^N- \mb{\widehat V}^N\|_1\,.
	\end{equation}
	Collecting estimates \eqref{I1} and \eqref{I2} completes the proof.
\end{proof}

\begin{theorem}\label{thmwellposednessofparticle}
	Under Assumptions \ref{asum1} and \ref{asum}  let $\rho_0$ be a probability measure on $\Gamma$ and, for every $N\in\mathbb{N}$, $(V_0^i)_{i \in [N]}$ be $N$ i.i.d. random variables with the common law $\rho_0$.
	For every $N\in\mathbb{N}$, there exists a pathwise unique strong solution $((V_t^i)_{t\geq 0})_{i \in [N]}$ to the particle system \eqref{stochastic Kuramoto-Vicsek} with the initial data $(V_0^i)_{i \in [N]}$. Moreover it holds that $V_t^i\in \Gamma$ for all $i \in [N]$ and any $t>0$.
\end{theorem}
\begin{proof}
	Given $P_1$, $P_2$, $P_3$ and $\tilde{\mathcal E}$, the SDE \eqref{RSKV} has locally Lipschitz coefficients, so it admits a pathwise unique local strong solution by standard SDE well-posedness result \cite[Chap. 5, Theorem 3.1]{durrett2018stochastic}.  Moreover, it follows from It\^{o}'s formula \eqref{Mulito} (choosing $\varphi(x)=\gamma(x)$), as long as $V_t^i\in \widehat\Gamma$ (the process is continuous and we can nevertheless consider a smooth extension of $\gamma$ out of $\widehat\Gamma$), that
	\begin{align}\label{dVti2=0}
	d\gamma(V_t^i)&= -\lambda \nabla\gamma(V_t^i)\cdot P(V_t^i)(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))dt + \sigma |V_t^i - v_{\alpha,\tilde{\mathcal E}}(\rho_t^N)| \nabla\gamma(V_t^i)\cdot P(V_t^i)dB_t^i \notag\\
	&\quad -\frac{1}{2}\sigma^2(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))^2\Delta \gamma(V_t^i)dt\notag\\
	&\quad+\frac{1}{2}\sigma^2(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))^2\sum_{k,\ell=1}^d\partial_{k,\ell}^2\gamma(V_t^i)\left(e_k^T-\partial_k\gamma(V_t^i)\nabla\gamma(V_t^i)^T\right)\left(e_\ell-\partial_{\ell}\gamma(V_t^i)\nabla\gamma(V_t^i)\right)dt\notag\\
	&=-\frac{1}{2}\sigma^2(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))^2\Delta \gamma(V_t^i)dt+\frac{1}{2}\sigma^2(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))^2\sum_{k}^d\partial_{k}^2\gamma(V_t^i) dt\notag\\
	&\quad +\frac{1}{2}\sigma^2(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))^2\left(-\sum_{k,\ell=1}^d\partial_{k,\ell}^2\gamma(V_t^i)\partial_{k}\gamma(V_t^i)\partial_{\ell}\gamma(V_t^i)\right)dt \notag\\
	&= -\frac{1}{2}\sigma^2(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))^2\Delta \gamma(V_t^i)dt
	+\frac{1}{2}\sigma^2(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))^2\Delta \gamma(V_t^i)dt\notag\\
	&\quad +\frac{1}{2}\sigma^2(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))^2\left(-\frac{1}{2}\sum_{\ell}^d\partial_{\ell}|\nabla\gamma(V_t^i)|^2\partial_{\ell}\gamma(V_t^i)\right)dt\quad (\mbox{ since }|\nabla\gamma|=1)\notag\\
	&=0
	\,,
	\end{align}
	where $\partial_k\gamma(V_t^i)$ is the $k$-th component of $\nabla\gamma(V_t^i)$, $e_k$ is the unit vector along the $k$th dimension, and  we have used the property of $P(V_t^i)$ as in \eqref{P2}. Hence $\gamma(V_t^i)=\gamma(V_0^i)=0$ for all $t>0$, which ensures that the solution keeps bounded at finite time, hence we have a global solution. Since all $V_t^i\in\Gamma$, the solution to the regularized system \eqref{RSKV} is a solution to \eqref{stochastic Kuramoto-Vicsek}, which provides the global existence of solutions to \eqref{stochastic Kuramoto-Vicsek}.

	To show pathwise uniqueness let us consider two solutions to \eqref{stochastic Kuramoto-Vicsek} for the same initial distribution and Brownian motion. According to the above computation these two solutions stay on the hypersurface for any $t\geq 0$, hence they are solutions to the regularized system \eqref{RSKV}, whose solutions are pathwise unique due to the locally Lipschitz continuous coefficients.  Hence we have uniqueness for solutions to \eqref{stochastic Kuramoto-Vicsek}.
\end{proof}

\subsection{Well-posedness for the mean-field dynamic \eqref{selfprocess}}
For readers' convenience, we give a brief introduction of the Wasserstein metric in the following definition, we refer to Ref. \cite{ambrosio2008gradient} for more details.
\begin{definition}[Wasserstein Metric]
	Let $1\leq p < \infty$ and $\mc{P}_p(\mathbb R^{d})$ be the space of Borel probability measures on $\mathbb R^{d}$ with finite $p$-moment. We equip this space with the Wasserstein distance
	\begin{equation}
	W_p^{p}(\mu, \nu):=\inf\left\{\int_{\mathbb R^d\times \mathbb R^d} |z-\hat{z}|^{p}\ d\pi(\mu, \nu)\ \big| \ \pi \in \Pi(\mu, \nu)\right\}
	\end{equation}
	where $\Pi(\mu, \nu)$ denotes the collection of all Borel probability measures on $\mathbb R^d\times \mathbb R^d$ with marginals $\mu$ and $\nu$ in the first and second component respectively. The Wasserstein distance can also be expressed as
	\begin{equation}
	W_p^{p}(\mu, \nu) = \inf \left\{\mathbb{E}[|Z-\overline {Z}|^{p}]\right\}
	\end{equation}
	where the infimum is taken over all joint distributions of the random variables $Z$, $\overline {Z}$ with marginals $\mu$, $\nu$ respectively.
\end{definition}
Notice now that, for any $\rho \in \mathcal P_1(\mathbb R^d)$
\begin{align}
\frac{e^{-\alpha \underline \tilde{\mathcal E}}}{\|{e^{-\alpha \tilde{\mathcal E}}}\|_{L^1(\rho)}}\leq e^{\alpha(\overline \tilde{\mathcal E}-\underline \tilde{\mathcal E})}=:C_{\alpha,\tilde{\mathcal E}}\,.
\end{align}
A direct application of above leads to
\begin{equation}\label{esv}
|v_{\alpha,\tilde{\mathcal E}}(\rho)|:= \left|\frac{\int_{\mathbb R^{d}}  v  \omega_{\alpha}^{\tilde{\mathcal E}}(v)\,d\rho}{\int_{\mathbb R^{d}}\omega_{\alpha}^{\tilde{\mathcal E}}(v)\,d\rho}\right|= \left|\frac{\int_{\mathbb R^{d}}  v  e^{-\alpha\tilde{\mathcal E}( v )}\,d\rho}{\|{e^{-\alpha \tilde{\mathcal E}}}\|_{L^1(\rho)}}\right|\leq  C_{\alpha,\tilde{\mathcal E}}\int_{\mathbb R^d}|v|d\rho\,.
\end{equation}

\begin{lemma}\label{lemsta} Assume that $\rho,\widehat \rho\in\mc{P}_c(\mathbb R^d)$ (with compact support), then the following stability estimate holds
	\begin{equation}\label{lemstaeq}
	|v_{\alpha,\tilde{\mathcal E}}(\rho)-v_{\alpha,\tilde{\mathcal E}}(\widehat \rho)|\leq CW_p(\rho,\widehat \rho)\,,
	\end{equation}
	for any $1\leq p<\infty$, where $C= C( C_{\alpha,\tilde{\mathcal E}},\alpha,L)>0$. \end{lemma}
\begin{proof}
	Let us compute the difference
	\begin{align}
	v_{\alpha,\tilde{\mathcal E}}(\rho)-v_{\alpha,\tilde{\mathcal E}}(\widehat \rho)&=\frac{\int_{\mathbb R^{d}}  v  e^{-\alpha \tilde{\mathcal E}( v )}\,d\rho(v)}{\|{e^{-\alpha \tilde{\mathcal E}}}\|_{L^1(\rho)}}-\frac{\int_{\mathbb R^{d}}  \widehat v  e^{-\alpha \tilde{\mathcal E}( \widehat v )}\,d\widehat \rho(\widehat v)}{\|{e^{-\alpha \tilde{\mathcal E} }}\|_{L^1(\widehat \rho)}} \notag\\
	&=\iint_{\mathbb R^{d}\times \mathbb R^{d}} \underbrace{\frac{ v e^{-\alpha \tilde{\mathcal E}( v )}}{\|{e^{-\alpha \tilde{\mathcal E}}}\|_{L^1(\rho)}} -\frac{ \widehat v e^{-\alpha \tilde{\mathcal E}( \widehat v )}}{\|{e^{-\alpha \tilde{\mathcal E} }}\|_{L^1(\widehat \rho)}}}_{:=h(v)-h(\widehat{v})} d\pi(v,\widehat v) \notag
	\end{align}
	where $\pi \in \Pi(\rho, \widehat{\rho})$ is an arbitrary coupling of $\rho$ and $\widehat \rho$. We can write the integrand as follows
	\begin{align*}
	h(v)-h(\widehat v)&= \underbrace{\frac{( v - \widehat v )e^{-\alpha \tilde{\mathcal E}( \widehat v )}}{\|{e^{-\alpha \tilde{\mathcal E} }}\|_{L^1(\rho)}}}_{=:I_1} +\underbrace{\frac{ \widehat v (e^{-\alpha \tilde{\mathcal E}( v )}-e^{-\alpha \tilde{\mathcal E}( \widehat v )})}{\|{e^{-\alpha \tilde{\mathcal E}}}\|_{L^1(\rho)}}}_{=:I_2} +  \underbrace{\frac{\widehat{v}e^{-\alpha \tilde{\mathcal E}( \widehat v )}}{\|e^{-\alpha \tilde{\mathcal E} }\|_{L^{1}(\rho)}} - \frac{ \widehat v e^{-\alpha \tilde{\mathcal E}( \widehat v )}}{\|e^{-\alpha \tilde{\mathcal E}}\|_{L^{1}(\widehat \rho)}}}_{:=I_3}
	\end{align*}
	where the last two terms on the right hand side can be rewritten as
	\begin{align*}
	&\frac{ \widehat v e^{-\alpha \tilde{\mathcal E}( \widehat v )}}{\|e^{-\alpha \tilde{\mathcal E}}\|_{L^{1}(\rho)}} - \frac{ \widehat v e^{-\alpha \tilde{\mathcal E}( \widehat v )}}{\|e^{-\alpha \tilde{\mathcal E} }\|_{L^{1}(\widehat \rho)}}\notag\\
	&=  \frac{ \widehat v e^{-\alpha \tilde{\mathcal E}( \widehat v )}\int e^{-\alpha \tilde{\mathcal E}( \widehat v )}\ d\widehat{\rho}(\widehat{v})}{\|e^{-\alpha \tilde{\mathcal E} }\|_{L^{1}(\rho)}\|e^{-\alpha \tilde{\mathcal E} }\|_{L^1(\widehat{\rho})}} - \frac{ \widehat v e^{-\alpha \tilde{\mathcal E}( \widehat v )}\int e^{-\alpha \tilde{\mathcal E}( v )}\ d\rho(v)}{\|e^{-\alpha \tilde{\mathcal E} }\|_{L^1(\widehat{\rho})}\|e^{-\alpha \tilde{\mathcal E} }\|_{L^{1}(\rho)}} \\&=\frac{ \widehat v e^{-\alpha\tilde{\mathcal E}( \widehat v )}\iint_{\mathbb R^{d}\times \mathbb R^{d}}(e^{-\alpha\tilde{\mathcal E}( \widehat v )}-e^{-\alpha\tilde{\mathcal E}( v )})d\pi(v,\widehat v)}{\|{e^{-\alpha \tilde{\mathcal E}}}\|_{L^1(\rho)}\|{e^{-\alpha \tilde{\mathcal E} }}\|_{L^1(\widehat \rho)}} =: I_3\,.
	\end{align*}
	Under  Assumption \ref{asum}, one can obtain the bounds
	\begin{equation}
	|I_1|\leq \frac{e^{-\alpha \underline \tilde{\mathcal E}}}{\|{e^{-\alpha \tilde{\mathcal E}}}\|_{L^1(\rho)}}| v - \widehat v |\leq  C_{\alpha,\tilde{\mathcal E}} |v-\widehat v|\,,
	\end{equation}
	and
	\begin{equation}
	|I_2|\leq \frac{| \widehat v |\alpha e^{-\alpha \underline \tilde{\mathcal E}}L| v - \widehat v |}{\|{e^{-\alpha \tilde{\mathcal E}}}\|_{L^1(\rho)}}\leq \alpha  C_{\alpha,\tilde{\mathcal E}}L|\widehat v||v-\widehat v|\,.
	\end{equation}
	Similar to the estimate of $|I_2|$, we estimate $I_3$
	\begin{align}
	|I_3|\leq  \alpha  C_{\alpha,\tilde{\mathcal E}}^2L|\widehat v| \iint_{\mathbb R^{d}\times \mathbb R^{d}} |v-\widehat v|d\pi(v,\widehat v)\,.
	\end{align}

	Notice that $\rho,\widehat \rho\in\mc{P}_c(\mathbb R^d)$, so for any $1\leq p<\infty$ one has
	\begin{equation}
	\max\left\{\int_{\mathbb R^d}  |v|^p d\rho(v),\int_{\mathbb R^d}  |\widehat v|^p d\widehat \rho(\widehat v)\right\}<\infty\,.
	\end{equation}
	Collecting the above estimates, we obtain
	\begin{align}
	| v_{\alpha,\tilde{\mathcal E}}(\rho)-v_{\alpha,\tilde{\mathcal E}}(\widehat \rho)|&\leq  C_{\alpha,\tilde{\mathcal E}} \iint_{\mathbb R^d\times \mathbb R^d}|v-\widehat v|d\pi(v,\widehat v)
	+\alpha C_{\alpha,\tilde{\mathcal E}}L \iint_{\mathbb R^d\times \mathbb R^d}|\widehat v||v-\widehat v|d\pi(v,\widehat v)\notag\\
	&\quad+\alpha C_{\alpha,\tilde{\mathcal E}}^2L\int_{\mathbb R^{d}}|\widehat v| d\widehat \rho(\widehat v) \iint_{\mathbb R^{d}\times \mathbb R^{d}} |v-\widehat v|d\pi(v,\widehat v)  \notag\\
	&\leq C\left( \iint_{\mathbb R^{d}\times \mathbb R^{d}}|v-\widehat v|^pd\pi(v,\widehat v)\right)^{\frac{1}{p}}\,,
	\end{align}
	where $C$ depends only on $ C_{\alpha,\tilde{\mathcal E}}$ and $\alpha,L$. Lastly, optimizing over all couplings $\pi$ yields  \eqref{lemstaeq}.
\end{proof}

The following theorem states the well-posedness for the mean-field dynamic \eqref{selfprocess}.
\begin{theorem}
	Let $\mathcal E$ and $\tilde{\mathcal E}$ satisfy Assumptions \ref{asum1} and \ref{asum}. Then there exists a unique process $\overline V\in \mc{C}([0,T],\mathbb R^{d})$, $T>0$, satisfying the nonlinear SDE \eqref{selfprocess}
	\begin{equation}
	d\overline V_t=- \lambda P(\overline V_t)(\overline V_t-v_{\alpha,\mathcal E} (\rho_t)) dt + \sigma |\overline V_t - v_{\alpha,\mathcal E}(\rho_t)  | P(\overline V_t)dB_t-\frac{\sigma^2}{2}(\overline V_t-v_{\alpha,\mathcal E} (\rho_t) )^2\Delta \gamma (\overline V_t)\nabla\gamma (\overline V_t)dt\,, \notag
	\end{equation}
	in strong sense for any initial data $\overline V_0\in \Gamma$ distributed according to $\rho_0\in \mc{P}(\Gamma)$, where
	\begin{equation}
	v_{\alpha,\mathcal E}(\rho_t)  = \frac{\int_{\mathbb R^{d}}  v  e^{-\alpha \mathcal E( v )}\,d\rho_t}{\int_{\mathbb R^{d}}e^{-\alpha \mathcal E( v )}\,d\rho_t},  \notag
	\end{equation}
	and $\rho_t=\rm{law}(\overline V_t)$ for all $t\in[0,T]$.
	Moreover $\overline V_t\in\Gamma$ for all $t\in[0,T]$.
\end{theorem}
\begin{proof}
	The proof in the following is based on the Leray-Schauder fixed point theorem, see, e.g., \cite[Chapter 10]{gilbarg2015elliptic}, and it will be carried out through 5 steps.

	$\bullet$ \textit{Step 1:} For some given $\xi\in \mc{C}([0,T],\mathbb R^d)$,  a distribution $\rho_0$ on $\Gamma$ and $\overline{V}_0$ with law $\rho_0$, we can uniquely solve the SDE
	\begin{equation}\label{Rself}
	d\overline V_t=-\lambda  P_1(\overline V_t)(\overline V_t-\xi_t ) dt + \sigma |\overline V_t - \xi_t | P_1(\overline V_t)dB_t-\frac{\sigma^2}{2}(\overline V_t-\xi_t )^2P_2(\overline V_t)P_3(\overline V_t)dt\,
	\end{equation}
	because the coefficients are locally Lipschitz continuous and $\xi$ is independent of $\overline{V}$. In addition, following the same argument as in \eqref{dVti2=0}  we have $d\gamma(\overline V_t)=0$, so that $\overline V_t\in \Gamma$ for all time.  This introduces $\rho_t=\rm{law}(\overline V_t)$ and $\rho\in \mc{C}([0,T],\mc{P}_c(\mathbb R^d))$. Setting $\mc{T}\xi:=v_{\alpha,\tilde{\mathcal E}}(\rho)\in \mc{C}([0,T],\mathbb R^d)$ we define the map
	\begin{equation}
	\mc{T}: \mc{C}([0,T],\mathbb R^d)\rightarrow \mc{C}([0,T],\mathbb R^d),\quad  \xi\mapsto \mc{T}(\xi):=v_{\alpha,\tilde{\mathcal E}}(\rho)\,
	\end{equation}
	which we will show to be compact in the next step.\\

	$\bullet$ \textit{Step 2:} For any $0<s<t\leq T$, one has
	\begin{align}
	\overline V_t-\overline V_s=-\lambda \int_s^tP_1(\overline V_\tau)(\overline V_\tau-\xi_\tau)  d\tau+\int_s^t\sigma |\overline V_\tau - \xi_\tau | P_1(\overline V_\tau)dB_\tau-\int_s^t\frac{\sigma^2}{2}(\overline V_\tau-\xi_\tau )^2P_2(\overline V_\tau)P_3(\overline V_\tau)d\tau\,.
	\end{align}
	Then applying It\^{o}'s isometry yields
	\begin{align}
	\mathbb{E}\left[|\overline V_t-\overline V_s|^2\right]&\leq C_1|t-s|^2+3\sigma^2\int_s^t\mathbb{E}\left[|\overline V_\tau - \xi_\tau | ^2| P_1(\overline V_\tau)|^2\right]d\tau +C_2|t-s|^2\\
	&\leq C|t-s|\,.
	\end{align}
	where the constants $C_1,C_2,C$ only depend on $\sigma,d,T,\lambda,\|P_1\|_\infty,\|P_2\|_\infty,\|P_3\|_\infty\|\xi\|_\infty$. Therefore, by definition of the Wasserstein distance we have
	\begin{equation}
	W_2(\rho_t,\rho_s)\leq C|t-s|^{\frac{1}{2}}.
	\end{equation}
	By an application of Lemma \ref{lemsta} one obtains
	\begin{equation}
	|v_{\alpha,\tilde{\mathcal E}}(\rho_t)-v_{\alpha,\tilde{\mathcal E}}(\rho_s)|\leq C|t-s|^{\frac{1}{2}}\,.
	\end{equation}
	This provides the H\"{o}lder continuity of $t\to v_{\alpha,\tilde{\mathcal E}}(\rho_t)$.
	Thus one has $\mc{T}(\mc{C}([0,T],\mathbb R^d))\subset \mc{C}^{0,\frac{1}{2}}([0,T],\mathbb R^d)\hookrightarrow \mc{C}([0,T],\mathbb R^d)$, which implies the compactness of the map $\mc{T}$.\\

	$\bullet$ \textit{Step 3:} Let us define the set
	\begin{equation}
	\mathcal A:=\left\{\xi \in \mc{C}([0,T],\mathbb R^d): \xi=\vartheta \mc{T}\xi\mbox{ for some }0\leq \vartheta\leq 1 \right\}\,.
	\end{equation}
	For $\xi\in \mathcal A$, there exists some $\overline V_t$ satisfying \eqref{Rself} with its law  $\rho\in \mc{C}([0,T],\mc{P}_c(\mathbb R^d))$ such that $\xi=\vartheta v_{\alpha,\tilde{\mathcal E}}(\rho)$. Due to \eqref{esv}, we have that for any $t\in[0,T]$
	\begin{equation}
	|\xi_t|^2=\vartheta^2|v_{\alpha,\tilde{\mathcal E}}(\rho_t)|^2\leq \vartheta^2\left( C_{\alpha,\tilde{\mathcal E}}\int_{\mathbb R^d}|v|d\rho\right)^2<\infty\,,
	\end{equation}
	since $\rho$ has a compact support, which verifies the boundedness of the set $\mathcal A$. Applying the Leray-Schauder fixed point theorem,  there exists a fixed point $\xi$ for the  mapping $\mc{T}$ and thereby a solution of
	\begin{equation}\label{Rnonlinear}
	d\overline V_t= -\lambda P_1(\overline V_t)(\overline V_t-v_{\alpha,\tilde{\mathcal E}}(\rho_t) ) dt + \sigma |\overline V_t - v_{\alpha,\tilde{\mathcal E}}(\rho_t) | P_1(\overline V_t)dB_t-\frac{\sigma^2}{2}(\overline V_t-v_{\alpha,\tilde{\mathcal E}}(\rho_t) )^2P_2(\overline V_t)P_3(\overline V_t)dt
	\end{equation}
	with $\mbox{law}(\overline V_t)=\rho_t$. \\

	$\bullet$ \textit{Step 4:}  In this step, we shall prove the uniqueness. Suppose we have two fixed points $\xi^1$ and $\xi^2$, and their corresponding process $\overline V_t^1$, $\overline V_t^2$ satisfying \eqref{Rself} respectively, denote by $\rho_t^1$ and $\rho_t^2$ their laws. We compute the difference $\overline Z_t:=\overline V_t^1-\overline V_t^2$ and applying It\^{o}'s isometry yields
	\begin{align}
	\mathbb{E}[|\overline Z_t|^2]\leq C \left ( \mathbb{E}[|\overline Z_0|^2]+\int_0^t\mathbb{E}[|\overline Z_s|^2]ds+\int_0^t|\xi^1_s-\xi^2_s|^2ds \right )\,,
	\end{align}
	where $C$ depends on $\lambda, \sigma,d,t,\|\nabla P_3\|_\infty,\| P_3\|_\infty,\|\nabla P_2\|_\infty,\| P_2\|_\infty,\|\nabla P_1\|_\infty,\| P_1\|_\infty,\|\xi^1\|_\infty,\|\xi^2\|_\infty$.
	According to Lemma \ref{lemsta}, one has
	\begin{equation}\label{xi-xi}
	|\xi^1_s-\xi^2_s|^2= |v_{\alpha,\tilde{\mathcal E}}(\rho^1_s)-v_{\alpha,\tilde{\mathcal E}}(\rho^2_s)|^2\leq CW_2^2(\rho_s^1,\rho_s^2)\leq C\mathbb{E}[|\overline Z_s|^2]\,,
	\end{equation}
	which implies that
	\begin{align}
	\mathbb{E}[|\overline Z_t|^2]\leq C\mathbb{E}[|\overline Z_0|^2]+C\int_0^t\mathbb{E}[|\overline Z_s|^2]ds\,.
	\end{align}
	Therefore, applying Gronwall's inequality with $\mathbb{E}[|\overline Z_0|^2]=0$, one obtain $\mathbb{E}[|\overline Z_t|^2]=0$ for all $t\in[0,T]$.
	This leads to $\xi^1\equiv \xi^2$ by \eqref{xi-xi}. Hence, we obtain the uniqueness for solutions to \eqref{Rnonlinear}.\\

	$\bullet$ \textit{Step 5:}
	Similar to the argument in Theorem \ref{thmwellposednessofparticle}, the unique solution to the regularized SDE \eqref{Rnonlinear} is also the unique solution to the nonlinear SDE  \eqref{selfprocess} due to the fact that   $\gamma(\overline V_t)=0$  for all $t\in[0,T].$
\end{proof}

\subsection{Well-posedness for the PDE \eqref{PDE}}

Let $\rho_0 \in \Gamma$, and let $\{\overline V_t: t\geq 0 \}$ be the solution to \eqref{selfprocess} obtained in the last section with the initial data $\overline V_0$ distributed according to $\rho_0$.  For any $\varphi\in C_c^\infty(\mathbb R^d)$, it follows from It\^{o}'s formula \eqref{Mulito} that
\begin{align}\label{pippoeq}
d \varphi(\overline V_t)&=\nabla\varphi(\overline V_t)\cdot \left(-\lambda P(\overline V_t)(\overline V_t-v_{\alpha,\mathcal E}(\rho_t))  -\frac{\sigma^2}{2}(\overline V_t-v_{\alpha,\mathcal E}(\rho_t)  )^2\Delta \gamma(\overline V_t)\nabla\gamma(\overline V_t)\right)dt \notag\\
&\quad+\sigma |\overline V_t - v_{\alpha,\mathcal E}(\rho_t)  | \nabla\varphi(\overline V_t)\cdot P(\overline V_t)dB_t\notag\\
&\quad+\frac{\sigma^2}{2}\left(\overline V_t - v_{\alpha,\mathcal E}(\rho_t)  \right)^2\nabla^2\varphi(\overline V_t):\left (I-\nabla\gamma(\overline V_t)\nabla\gamma(\overline V_t)^T\right)dt\,,
\end{align}
where $\nabla^{2}\varphi$ is the Hessian and $A:B := \operatorname{Tr}(A^{T} B)$.  Taking expectation on both sides of \eqref{pippoeq}, the law $\rho_t$ of $\overline V_t$ as a measure on $\mathbb R^d$ satisfies
\begin{align}\label{wholeweak}
\frac{d}{dt}\int_{\mathbb R^d}\varphi(v)d\rho_t(v)&=\int_{\mathbb R^d}\nabla\varphi(v)\cdot \left(-\lambda(I-\nabla\gamma(v)\nabla\gamma(v)^T)(v-v_{\alpha,\mathcal E}(\rho_t)) - \frac{\sigma^2}{2}(v-v_{\alpha,\mathcal E}(\rho_t)  )^2\Delta \gamma(v)\nabla\gamma(v)\right)d\rho_t(v)
\notag\\
&\quad+\int_{\mathbb R^d}\frac{\sigma^2}{2}\left(v- v_{\alpha,\mathcal E}(\rho_t)  \right)^2\nabla^2\varphi(v):\left (I-\nabla\gamma(v) \nabla\gamma(v)^T\right)d\rho_t(v)\,.
\end{align}
As we have proved that $\overline V_t\in \Gamma$, almost surely, that is, the density $\rho_t$ is concentrated on $\Gamma$ for any $t$, we have $\operatorname{supp}(\rho_t)\subset \Gamma$. Let us now define the restriction $\mu_t$ of $\rho_t$ on $\Gamma$ by
\begin{equation}
\int_{\Gamma}\Phi(v)d \mu_t(v)=\int_{\mathbb R^d}\varphi(v)d\rho_t(v)
\end{equation}
for all continuous maps $\Phi\in \mc{C}(\Gamma)$, where $\varphi\in \mc{C}_b(\mathbb R^d)$ equals $\Phi$ on $\Gamma$.

Next we define the projection
\begin{equation*}
\Pi_\Gamma (v)=v-\gamma(v)\nabla\gamma(v)\in \Gamma,\quad \mbox{for }v\in \mathbb R^d\,.
\end{equation*}
We then let $\Gamma_\delta \subset\mathbb R^d$ be a strip of width $\delta>0$ about $\Gamma$, where $\delta>0$ is sufficiently small to ensure that the decomposition
\begin{equation}
v=\Pi_\Gamma (v)+\gamma(v)\nabla\gamma(v)
\end{equation}
is unique for $v\in \Gamma_\delta$. We know such $\delta$ exists since $\gamma \in \mathcal{C}^2(\widehat\Gamma)$, see for example \cite[Section 2.1]{demlow2007adaptive}.
Let now $\Phi\in \mc{C}^\infty(\Gamma)$ and define
a function $\varphi\in \mc{C}_c^\infty(\mathbb R^d)$ such that
\begin{equation}
\varphi(v)= \Phi\left(\Pi_\Gamma (v)\right)\quad  \mbox{ for all } v\in\Gamma_\delta\,.
\end{equation}
Then $\varphi$ defined above  is $0$-homogeneous in $v$ in the strip $\Gamma_\delta$, so that $\nabla\varphi(v)\cdot \nabla\gamma(v)=0$ for all $v$ in the support $\Gamma$ of $\rho_t$, which leads to $\nabla^2\varphi(v):\nabla\gamma(v)\nabla\gamma(v)^T=0$.  Hence,
\begin{align}
&\frac{d}{dt}\int_{\Gamma}\Phi(v)d\mu_t(v)=\frac{d}{dt}\int_{\mathbb R^d}\varphi(v)d\rho_t(v)\notag\\
=&-\lambda \int_{\mathbb R^{d}}\nabla\varphi(v)\cdot \left((I-\nabla\gamma(v)\nabla\gamma(v)^T)(v-v_{\alpha,\mathcal E}(\rho_t)) \right)d\rho_t(v)+\int_{\mathbb R^{d}}\frac{\sigma^2}{2}\left(v- v_{\alpha,\mathcal E}(\rho_t)  \right)^2\Delta\varphi(v)d\rho_t(v)\,.\notag\,
\end{align}
\noindent
Let us now relate the Euclidean differential operators to corresponding operators on $\Gamma$, so that for $v\in\Gamma$  it holds
$\nabla_{\Gamma} \Phi(v) = \nabla \varphi(v)$ and $\Delta_{\Gamma}\Phi(v)=\Delta \varphi(v)$. Therefore, we obtain
\begin{align}
\frac{d}{dt}\int_{\Gamma}\Phi(v)d\mu_t(v)&=-\lambda \int_{\Gamma}\nabla_{\Gamma}\Phi(v)\cdot \left((I-\nabla\gamma(v)\nabla\gamma(v)^T)(v-v_{\alpha,\mathcal E}(\mu_t))\right)d\mu_t(v)\notag\\
&\quad +\int_{\Gamma}\frac{\sigma^2}{2}\left(v- v_{\alpha,\mathcal E}(\mu_t) \right)^2\Delta_{\Gamma}\Phi(v)d\mu_t(v)\,,
\end{align}
where
\begin{equation}
v_{\alpha,\mathcal E}(\mu_t) = \frac{\int_{\Gamma} v e^{-\alpha \mathcal E(v)}\,d \mu_t}{\int_{\Gamma}e^{-\alpha \mathcal E(v)}\,d \mu_t}.
\end{equation}
Thus by this construction we obtain a weak solution $\mu_t$ to the PDE \eqref{PDE}.\\

Next we prove the uniqueness of solutions to \eqref{PDE}. Assume that $\rho_t^1$ and $\rho_t^2$ are two solutions to \eqref{PDE} with the same initial data $\rho_0$, and at each time $t$ we treat them as measures on $\mathbb R^d$ concentrated on the hypersurface $\Gamma$. Then we construct two linear process $(\overline V_t^i)_{t\geq 0}$ $(i=1,2)$ satisfying
\begin{equation}
d\overline V_t^i=-\lambda P_1(\overline V_t^i)(\overline V_t^i-v_{\alpha,\mathcal E} (\rho_t^i)) dt + \sigma |\overline V_t^i - v_{\alpha,\mathcal E}(\rho_t^i)  | P_1(\overline V_t^i)dB_t-\frac{\sigma^2}{2}(\overline V_t^i-v_{\alpha,\mathcal E} (\rho_t^i) )^2P_2(\overline V_t^i)P_3(\overline V_t^i)dt\,, \label{Eqn for Vbar}
\end{equation}
with the common initial data $\overline V_0$ distributed  according to $\rho_0$. Let us denote $\rm{law}(\overline V_t^i)=\bar{\rho}_t^i$ $(i=1,2)$ as measures  on $\mathbb R^{d}$, which are solutions to the following linear PDE
\begin{align*}
\partial_t \bar \rho_t^i&= \nabla\cdot \left(\bar \rho_t^i\left(\lambda P_1(v)(v-v_{\alpha,\mathcal E} (\rho_t^i))+\frac{\sigma^2}{2}(v-v_{\alpha,\mathcal E} (\rho_t^i) )^2P_2(v)P_3(v)\right)\right)\\
&\quad +\frac{\sigma^2}{2}\sum_{k,\ell=1}^{d}\frac{\partial^2}{\partial v_k\partial v_\ell}\left(|v-v_{\alpha,\mathcal E}(\rho_t^i) |^2(P_1P_1^T)_{k\ell}\bar \rho_t^i\right)\,.
\end{align*}
Since the uniqueness for the above linear PDE holds and $\rho_t^i$ is also a solution to the above PDE on $\mathbb R^d$ (see \eqref{wholeweak}), it follows that
$\bar \rho_t^i=\rho_t^i$ $(i=1,2)$. Consequently, the process $(\overline V_t^i)_{(t\geq 0)}$ are solutions to the nonlinear SDE \eqref{selfprocess}, for which the uniqueness has been obtained. Hence $(\overline V_t^1)_{(t\geq 0)}$ and $(\overline V_t^2)_{(t\geq 0)}$ are equal, which implies $\rho_t^1=\bar \rho_t^1=\bar \rho_t^2=\rho_t^2$. Thus the uniqueness is obtained.

\section{Mean-field limit}\label{sec:MFlimit}
The well-posedness of  \eqref{stochastic Kuramoto-Vicsek}, \eqref{PDE} and \eqref{selfprocess} obtained in the last section provides all the ingredients we need for the mean-field limit. Let $((\overline V_t^i)_{t\geq 0})_{i \in [N]}$ be $N$ independent copies of solutions to \eqref{selfprocess}. They are i.i.d. with the same distribution $\rho_t$. Assume that $((V_t^i)_{t\geq 0})_{i \in [N]}$ is the solution to the particle system \eqref{stochastic Kuramoto-Vicsek}. Since  $\overline V_t^i,V_t^i\in \Gamma$ for all $i$ and $t$, $((\overline V_t^i)_{t\geq 0})_{i \in [N]}$ and $((V_t^i)_{t\geq 0})_{i \in [N]}$ are solutions to the corresponding regularized systems \eqref{Rnonlinear} and \eqref{RSKV} respectively. We denote below $\overline \rho_t^N = \frac{1}{N} \sum_{j=1}^N \delta_{\overline V_t^j}$ and $\rho_t = \rm{law}(\overline V_t)$.

Before stating our theorem on the mean-field limit, let us introduce the following lemma on a large deviation bound.
\begin{lemma}\label{lemLLN}  Let $\mathcal E$ and $\tilde{\mathcal E}$ satisfy Assumptions \ref{asum1} and \ref{asum}.  Let $((\overline V_t^i)_{t\geq 0})_{i \in [N]}$ be the solution to the mean-field dynamics \eqref{Rnonlinear}, which are i.i.d. with common distribution $\rho\in \mc C([0,T],\mc{P}_c(\mathbb R^d))$. Then there exists a constant $C$ depending  only on  $\Gamma$ and $C_{\alpha,\tilde{\mathcal E}}$ such that
	\begin{equation} \label{estimazza}
	\sup_{t\in[0,T]}\mathbb{E}\left[|v_{\alpha,\tilde{\mathcal E}}(\overline \rho_t^N)-v_{\alpha,\tilde{\mathcal E}}(\rho_t)|^2\right]\leq C N^{-1}\,.
	\end{equation}
\end{lemma}
\begin{proof}
	We start by estimating
	\begin{align}\label{splites}
	&|v_{\alpha,\tilde{\mathcal E}}(\overline \rho_t^N)-v_{\alpha,\tilde{\mathcal E}}(\rho_t)|= \left|\frac{\sum_{j=1}^{N} \overline V_t^j e^{-\alpha \tilde{\mathcal E}( \overline V_t^j )}}{\sum_{i=1}^{N}e^{-\alpha \tilde{\mathcal E}( \overline V_t^i)  }}-\frac{1}{\int_{\mathbb R^d}e^{-\alpha\tilde{\mathcal E}(v)}d\rho_t}\int_{\mathbb R^d} v e^{-\alpha\tilde{\mathcal E}(v)}d\rho_t\right| \notag\\
	\leq & \left|\frac{1}{\frac{1}{N}\sum_{i=1}^{N} e^{-\alpha \tilde{\mathcal E}( \overline V_t^i )}}\left(\frac{1}{N}\sum_{j=1}^{N} \overline V_t^j e^{-\alpha \tilde{\mathcal E}( \overline V_t^j )}-\int_{\mathbb R^d} v e^{-\alpha\tilde{\mathcal E}(v)}d\rho_t\right)\right| \notag\\
	&+\left|\int_{\mathbb R^d} v e^{-\alpha\tilde{\mathcal E}(v)}d\rho_t\left(\frac{1}{\frac{1}{N}\sum_{i=1}^{N} e^{-\alpha \tilde{\mathcal E}( \overline V_t^i )}}-\frac{1}{\int_{\mathbb R^d} e^{-\alpha\tilde{\mathcal E}(v)}d\rho_t}\right)\right| \notag\\
	\leq& e^{ \alpha\overline\tilde{\mathcal E}}\left|\frac{1}{N}\sum_{j=1}^{N} \overline V_t^j e^{-\alpha \tilde{\mathcal E}( \overline V_t^j )}-\int_{\mathbb R^d} v e^{-\alpha\tilde{\mathcal E}(v)}d\rho_t\right|+e^{\alpha(2\overline\tilde{\mathcal E}-\underline\tilde{\mathcal E})}\left|\frac{1}{N}\sum_{j=1}^{N} e^{-\alpha \tilde{\mathcal E}( \overline V_t^j )}-\int_{\mathbb R^d} e^{-\alpha\tilde{\mathcal E}(v)}d\rho_t\right|\,.
	\end{align}
	Here we used explicitly the upper and lower bounds $\overline\tilde{\mathcal E}, \underline\tilde{\mathcal E}$.
	Denote
	\begin{equation}
	\overline Z_t^j:=\overline V_t^j e^{-\alpha \tilde{\mathcal E}( \overline V_t^j )}-\int_{\mathbb R^d} v e^{-\alpha\tilde{\mathcal E}(v)}d\rho_t\,,
	\end{equation}
	then one can rewrite
	\begin{equation}
	\frac{1}{N}\sum_{j=1}^{N} \overline V_t^j e^{-\alpha \tilde{\mathcal E}( \overline V_t^j )}-\int_{\mathbb R^d} v e^{-\alpha\tilde{\mathcal E}(v)}d\rho_t=\frac{1}{N}\sum_{j=1}^{N}\overline Z_t^j\,.
	\end{equation}
	It is obvious that $\mathbb E[\overline Z_t^j]=0$, since $\{(\overline V_t^j)_{t\geq 0}\}_{j=1}^N$ are i.i.d. with common distribution $\rho$. Next we compute
	\begin{align*}
	&\mathbb{E}\left[\left|\frac{1}{N}\sum_{j=1}^{N} \overline V_t^j e^{-\alpha \tilde{\mathcal E}( \overline V_t^j )}-\int_{\mathbb R^d} v e^{-\alpha\tilde{\mathcal E}(v)}d\rho_t\right|^2\right]=\frac{1}{N^2}\mathbb E\left[(\sum_{j=1}^N\overline Z_t^j)(\sum_{k=1}^N\overline Z_t^k)\right]\\
	=&\frac{1}{N^2}\mathbb E\left[\sum_{j=1}^N(\overline Z_t^j)^2\right]=\frac{1}{N}\mathbb E[(\overline Z_t^1)^2]\,,
	\end{align*}
	where we have used the fact that $\mathbb E[Z_t^jZ_t^k]=0$ for $j\neq k$.  It is also easy to check that
	\begin{equation}
	\mathbb E[(\overline Z_t^1)^2]\leq 2\mathbb E\left[|\overline V_t^1 e^{-\alpha \tilde{\mathcal E}( \overline V_t^1 )}|^2\right]+2\left(\int_{\mathbb R^d} v e^{-\alpha\tilde{\mathcal E}(v)}d\rho_t\right)^2\leq 4C_{\Gamma}e^{-2\alpha\underline\tilde{\mathcal E}}\,.
	\end{equation}
	Here we have used the fact that $\overline V_t^1\in \Gamma$, so it holds $\mathbb E\left[|\overline V_t^1|^2\right]=\int_{\mathbb R^d}|v|^2d\rho_t\leq C_{\Gamma}$ for some $C_{\Gamma}>0$ depending on $\Gamma$.
	Thus we conclude
	\begin{equation}
	\mathbb{E}\left[\left|\frac{1}{N}\sum_{j=1}^{N} \overline V_t^j e^{-\alpha \tilde{\mathcal E}( \overline V_t^j )}-\int_{\mathbb R^d} v e^{-\alpha\tilde{\mathcal E}(v)}d\rho_t\right|^2\right]\leq \frac{1}{N}4C_{\Gamma}e^{-2\alpha\underline\tilde{\mathcal E}}\,.
	\end{equation}
	By following the same argument it also holds that
	\begin{equation}
	\mathbb{E}\left[\left|\frac{1}{N}\sum_{j=1}^{N} e^{-\alpha \tilde{\mathcal E}( \overline V_t^j )}-\int_{\mathbb R^d}e^{-\alpha\tilde{\mathcal E}(v)}d\rho_t\right|^2\right]\leq \frac{1}{N}4C_{\Gamma}e^{-2\alpha\underline\tilde{\mathcal E}}\,.
	\end{equation}
	Combining the above two estimates and \eqref{splites}, one has
	\begin{align}
	\mathbb{E}\left[|v_{\alpha,\tilde{\mathcal E}}(\overline \rho_t^N)-v_{\alpha,\tilde{\mathcal E}}(\rho_t)|^2\right]\leq\frac{1}{2}C_{\Gamma}e^{2\alpha(\overline{\tilde{\mathcal E}}-\underline{\tilde{\mathcal E}})}\frac{1}{N}+\frac{1}{2}C_{\Gamma}e^{\alpha(2\overline{\tilde{\mathcal E}}-3\underline \tilde{\mathcal E})}\frac{1}{N}\leq C_{\Gamma}e^{3\alpha(\overline{\tilde{\mathcal E}}-\underline \tilde{\mathcal E}))}\frac{1}{N}=C_{\Gamma}C_{\alpha,\tilde{\mathcal E}}^3\frac{1}{N}\,.
	\end{align}
	Thus we have completed the proof.
\end{proof}

Then we get the following mean-field limit result by the classical Sznitman's theory.
\begin{theorem}\label{thmmean}
	Under the  Assumptions \ref{asum1} and \ref{asum}, for any $T>0$, let $((V_t^i)_{t\in [0,T]})_{i \in [N]}$  and $((\overline V_t^i)_{t\in [0,T]})_{i \in [N]}$  be respective solutions to  \eqref{stochastic Kuramoto-Vicsek} and \eqref{selfprocess} up to time $T$ with the same initial data  $V_0^i=\overline V_0^i$  and the same Brownian motions $((B_t^i)_{t\in [0,T]})_{i \in [N]}$. Then there exists a constant $C>0$ depending only on $\Gamma$, $\lambda$, $\alpha$,  $\sigma$, $\|\nabla P_1\|_\infty$, $\|P_1\|_\infty$, $\|\nabla P_2\|_\infty$, $\|P_2\|_\infty$, $\|\nabla P_3\|_\infty$, $\|P_3\|_\infty$, $L$ and $C_{\alpha,\tilde{\mathcal E}}$, such that
	\begin{equation}\label{MFlimit}
	\sup_{i=1,\cdots,N}\mathbb{E}[|V_t^i-\overline V_t^i|^2]\leq CT\left(1+CTe^{CT}\right)\frac{1}{N}\,,
	\end{equation}
	holds for all $0\leq t\leq T$.
\end{theorem}
\begin{remark}
	The estimate above guarantees the weak convergence of the empirical measure $\rho_t^N$ towards $\rho_t$, in the following sense
	\begin{equation}
	\sup_{t\in[0,T]}\mathbb E\left[|\langle\rho_t^N,\phi\rangle-\langle \rho_t,\phi\rangle|^2\right]\to 0\mbox{ as }N\to \infty\quad \mbox{ for any test funtion }\phi\in \mc{C}_b^1(\mathbb R^{d})\,.
	\end{equation}
	Indeed, one has
	\begin{align*}
	&\mathbb E\left[|\langle\rho_t^N,\phi\rangle-\langle \rho_t,\phi\rangle|^2\right]=\mathbb E\left[\left|\frac{1}{N}\sum_{i=1}^{N}\phi(V_t^i)-\int_{\mathbb R^{d}}\phi(v)d\rho_t(v)\right|^2\right]\\
	\leq& 2\mathbb E\left[|\phi(V_t^1)-\phi(\overline V_t^1)|^2\right]+2\mathbb E\left[\left|\frac{1}{N}\sum_{i=1}^{N}\phi(\overline V_t^i)-\int_{\mathbb R^{d}}\phi(v)d\rho_t(v)\right|^2\right]\leq \frac{C}{N}\|\phi\|_{\mc{C}^1}^2
	\,.
	\end{align*} We refer to Refs. \cite{sznitman1991topics,bolley2011stochastic} for more details.
\end{remark}
\begin{proof}
	Notice that $((\overline V_t^i)_{t\geq 0})_{i \in [N]}$  and $((V_t^i)_{t\geq 0})_{i \in [N]}$  are also solutions to the corresponding regularized systems \eqref{Rnonlinear} and \eqref{RSKV} respectively, and
	\begin{align}
	d(V_t^i-\overline V_t^i)&=-\lambda\left(  P_1(V_t^i)(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))-P_1(\overline V_t^i)(\overline V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t)) \right)dt \notag\\
	&+\sigma \left(|V_t^i - v_{\alpha,\tilde{\mathcal E}}(\rho_t^N)| P_1(V_t^i)- |\overline V_t^i- v_{\alpha,\tilde{\mathcal E}}(\rho_t) | P_1(\overline V_t^i)\right)dB_t^i\notag\\
	&-\frac{\sigma^2}{2}\left((V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))^2P_2(V_t^i)P_3(V_t^i)-(\overline V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t) )^2P_2(\overline V_t^i)P_3(\overline V_t^i)\right)dt
	\,.
	\end{align}
	By applying It\^{o}'s formula one has
	\begin{align}\label{expec}
	d(V_t^i-\overline V_t^i)^2&=-2 \lambda(V_t^i-\overline V_t^i) \cdot\left(  P_1(V_t^i)(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))-P_1(\overline V_t^i)(\overline V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t)) \right)dt \notag\\
	&+2 \sigma  (V_t^i-\overline V_t^i) \cdot \left(|V_t^i - v_{\alpha,\tilde{\mathcal E}}(\rho_t^N)| P_1(V_t^i)- |\overline V_t^i- v_{\alpha,\tilde{\mathcal E}}(\rho_t) | P_1(\overline V_t^i)\right)dB_t^i\notag\\
	&-\sigma^2 (V_t^i-\overline V_t^i)\cdot\left((V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))^2P_2(V_t^i)P_3(V_t^i)-(\overline V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t) )^2P_2(\overline V_t^i)P_3(\overline V_t^i)\right)dt \notag\\
	&+\sigma^2\sum_{k=1}^d\left(|V_t^i - v_{\alpha,\tilde{\mathcal E}}(\rho_t^N)| P_1^k(V_t^i)- |\overline V_t^i- v_{\alpha,\tilde{\mathcal E}}(\rho_t) | P_1^k(\overline V_t^i)\right)\notag\\
	&\cdot \left(|V_t^i - v_{\alpha,\tilde{\mathcal E}}(\rho_t^N)| P_1^k(V_t^i)- |\overline V_t^i- v_{\alpha,\tilde{\mathcal E}}(\rho_t) | P_1^k(\overline V_t^i)\right)dt
	\,,
	\end{align}
	where $P_1^k(\cdot)$ represents the $k$-th row of the matrix $P_1(\cdot)$. Using Lemma \ref{lemlocal} it is easy to compute that
	\begin{align*}
	&\left|P_1(V_t^i)(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))-P_1(\overline V_t^i)(\overline V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t)) \right|\\
	\leq&\left | P_1(V_t^i)v_{\alpha,\tilde{\mathcal E}}(\rho_t^N)-P_1(\overline V_t^i)v_{\alpha,\tilde{\mathcal E}}(\rho_t) \right |+(\|P_1\|_\infty+\|\nabla P_1\|_\infty)|V_t^i-\overline V_t^i|\\
	\leq &\left |P_1(V_t^i)\left(v_{\alpha,\tilde{\mathcal E}}(\rho_t^N)-v_{\alpha,\tilde{\mathcal E}}(\overline \rho_t^N)\right)+v_{\alpha,\tilde{\mathcal E}}(\overline \rho_t^N)\left(P_1(V_t^i)-P_1(\overline V_t^i)\right)+P_1(\overline V_t^i)\left(v_{\alpha,\tilde{\mathcal E}}(\overline \rho_t^N)-v_{\alpha,\tilde{\mathcal E}}(\rho_t)\right) \right |\\
	&+(\|P_1\|_\infty+\|\nabla P_1\|_\infty)|V_t^i-\overline V_t^i|\\
	\leq&\|P_1\|_\infty \left(\frac{C_{\alpha,\tilde{\mathcal E}}}{N}+\frac{2\alpha LC_{\alpha,\tilde{\mathcal E}}}{N}\|\mb{\overline V}_t^N\|_\infty\right)\|\mb{V}_t^N- \mb{\overline V}_t^N\|_1+\|\nabla P_1\|_\infty\frac{C_{\alpha,\tilde{\mathcal E}}\|\mb{\overline V}_t^N\|_1}{N}|V_t^i-\overline V_t^i|\\
	&+\|P_1\|_\infty |v_{\alpha,\tilde{\mathcal E}}(\overline \rho_t^N)-v_{\alpha,\tilde{\mathcal E}}(\rho_t)|+(\|P_1\|_\infty+\|\nabla P_1\|_\infty)|V_t^i-\overline V_t^i| \\
	\leq & C(\alpha,\|P_1\|_\infty,L,C_{\alpha,\tilde{\mathcal E}},C_{\Gamma})\frac{\sum_{i=1}^N|V_t^i-\overline V_t^i|}{N}+C(C_{\alpha,\tilde{\mathcal E}},C_{\Gamma},\|\nabla P_1\|_\infty,\|P_1\|_\infty)|V_t^i-\overline V_t^i|+\|P_1\|_\infty |v_{\alpha,\tilde{\mathcal E}}(\overline \rho_t^N)-v_{\alpha,\tilde{\mathcal E}}(\rho_t)|\,,
	\end{align*}
	where we have used the fact that  $V_t^i,\overline V_t^i\in\Gamma$ with $\Gamma$ being compact, so there exists some $C_\Gamma>0$ such that
	\begin{equation*}
	|V_t^i|,|\overline V_t^i|\leq C_{\Gamma},\quad \frac{\|\mb{\overline V}_t^N\|_1}{N}=\frac{\sum_{i=1}^N|\overline V_t^i|}{N}\leq C_{\Gamma},\quad \|\mb{\overline V}_t^N\|_\infty=\sup_{i=1,\cdots,N}|\overline V_t^i|\leq C_{\Gamma}\,.
	\end{equation*}
	Hence it yields
	\begin{align}\label{ter1}
	&-2\lambda(V_t^i-\overline V_t^i) \cdot \left(  P_1(V_t^i)(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))-P_1(\overline V_t^i)(\overline V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t)) \right)\notag\\
	\leq & C \left ( \frac{\sum_{i=1}^N|V_t^i-\overline V_t^i|^2}{N}+ |V_t^i-\overline V_t^i|^2+  |v_{\alpha,\tilde{\mathcal E}}(\overline \rho_t^N)-v_{\alpha,\tilde{\mathcal E}}(\rho_t)|^2 \right)\,,
	\end{align}
	where we have used Cauchy's inequality and $C$ depends only on $\alpha,\|\nabla P_1\|_\infty, \|P_1\|_\infty,L,C_{\Gamma}$ and $C_{\alpha,\tilde{\mathcal E}}$, and $\|P_1\|_\infty$ is the spectral norm of the matrix $P_1$.

	Similarly we compute
	{\small 	\begin{align*}
		&\left |(V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))^2P_2(V_t^i)P_3(V_t^i)-(\overline V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t) )^2P_2(\overline V_t^i)P_3(\overline V_t^i)\right |\\
		=&\left | (V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))^2\left(P_2(V_t^i)P_3(V_t^i)-P_2(\overline V_t^i)P_3(\overline V_t^i)\right)
		+P_2(\overline V_t^i)P_3(\overline V_t^i)\left((V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))^2-(V_t^i-v_{\alpha, \tilde{\mathcal E}}(\overline\rho_t^N))^2\right)\right . \\
		&+ \left . P_2(\overline V_t^i)P_3(\overline V_t^i)\left((V_t^i-v_{\alpha, \tilde{\mathcal E}}(\overline\rho_t^N))^2-(\overline V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t) )^2\right) \right |\\
		\leq &C|V_t^i-\overline V_t^i|+C\frac{\sum_{i=1}^N|V_t^i-\overline V_t^i|}{N}+C|v_{\alpha,\tilde{\mathcal E}}(\overline \rho_t^N)-v_{\alpha,\tilde{\mathcal E}}(\rho_t)|\,,
		\end{align*} }
	where we have used Lemma \ref{lemlocal} and the fact that $|V_t^i|,|\overline V_t^i|\leq C_\Gamma$.
	Here $C$ depends only on $\alpha,\|\nabla P_2\|_\infty, \|P_2\|_\infty$, $\|\nabla P_3\|_\infty, \|P_3\|_\infty,L,C_{\Gamma}$ and $C_{\alpha,\tilde{\mathcal E}}$.  This leads to
	\begin{align}\label{ter2}
	&-\sigma^2 (V_t^i-\overline V_t^i)\cdot\left((V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t^N))^2P_2(V_t^i)P_3(V_t^i)-(\overline V_t^i-v_{\alpha,\tilde{\mathcal E}}(\rho_t) )^2P_2(\overline V_t^i)P_3(\overline V_t^i)\right)\notag\\
	\leq &C \left (\frac{\sum_{i=1}^N|V_t^i-\overline V_t^i|^2}{N}+|V_t^i-\overline V_t^i|^2+ |v_{\alpha,\tilde{\mathcal E}}(\overline \rho_t^N)-v_{\alpha,\tilde{\mathcal E}}(\rho_t)|^2 \right)\,,
	\end{align}
	where $C$ depends only on $\alpha, \lambda,\sigma,\|\nabla P_2\|_\infty, \|P_2\|_\infty,\|\nabla P_3\|_\infty, \|P_3\|_\infty,L,C_{\Gamma}$ and $C_{\alpha,\tilde{\mathcal E}}$.

	Now let us compute
	\begin{align*}
	&\left | |V_t^i - v_{\alpha,\tilde{\mathcal E}}(\rho_t^N)| P_1^k(V_t^i)- |\overline V_t^i- v_{\alpha,\tilde{\mathcal E}}(\rho_t) | P_1^k(\overline V_t^i) \right |\\
	=&\left | |V_t^i - v_{\alpha,\tilde{\mathcal E}}(\rho_t^N)|\left(P_1^k(V_t^i)-P_1^k(\overline V_t^i)\right)+P_1^k(\overline V_t^i)\left(|V_t^i - v_{\alpha,\tilde{\mathcal E}}(\rho_t^N)|-|V_t^i - v_{\alpha,\tilde{\mathcal E}}(\overline \rho_t^N)|\right) \right .\\
	&\left . + P_1^k(\overline V_t^i)\left(|V_t^i - v_{\alpha,\tilde{\mathcal E}}(\overline \rho_t^N)|-|\overline V_t^i- v_{\alpha,\tilde{\mathcal E}}(\rho_t) |\right)\right |\\
	\leq&C \left (|V_t^i-\overline V_t^i|+\frac{\sum_{i=1}^N|V_t^i-\overline V_t^i|}{N}+|v_{\alpha,\tilde{\mathcal E}}(\overline \rho_t^N)-v_{\alpha,\tilde{\mathcal E}}(\rho_t)| \right)\,,
	\end{align*}
	where $C$ depends only on $\alpha,\|\nabla P_1\|_\infty, \|P_1\|_\infty,L,C_{\Gamma}$ and $C_{\alpha,\tilde{\mathcal E}}$. This implies that
	{\small \begin{align}\label{ter3}
		&\sigma^2\sum_{k=1}^d\left(|V_t^i - v_{\alpha,\tilde{\mathcal E}}(\rho_t^N)| P_1^k(V_t^i)- |\overline V_t^i- v_{\alpha,\tilde{\mathcal E}}(\rho_t) | P_1^k(\overline V_t^i)\right)\cdot \left(|V_t^i - v_{\alpha,\tilde{\mathcal E}}(\rho_t^N)| P_1^k(V_t^i)- |\overline V_t^i- v_{\alpha,\tilde{\mathcal E}}(\rho_t) | P_1^k(\overline V_t^i)\right)\notag\\
		\leq &C \left (\frac{\sum_{i=1}^N|V_t^i-\overline V_t^i|^2}{N}+|V_t^i-\overline V_t^i|^2+|v_{\alpha,\tilde{\mathcal E}}(\overline \rho_t^N)-v_{\alpha,\tilde{\mathcal E}}(\rho_t)|^2 \right)\,,
		\end{align} }
	where $C$ depends only on $\alpha,\sigma,\|\nabla P_1\|_\infty, \|P_1\|_\infty,L,C_{\Gamma}$ and $C_{\alpha,\tilde{\mathcal E}}$.

	Taking expectation of both sides of \eqref{expec}, and collecting estimates \eqref{ter1}, \eqref{ter2} and \eqref{ter3}, one concludes
	\begin{align*}
	\mathbb{E}[|V_t^i-\overline V_t^i|^2]&\leq\mathbb{E}[|V_0^i-\overline V_0^i|^2]+C\int_0^t\frac{\sum_{i=1}^N\mathbb{E}[|V_s^i-\overline V_s^i|^2]}{N}ds+C\int_0^t\mathbb{E}[|V_s^i-\overline V_s^i|^2]ds\\
	&+C\int_0^t\mathbb{E}\left[|v_{\alpha,\tilde{\mathcal E}}(\overline \rho_s^N)-v_{\alpha,\tilde{\mathcal E}}(\rho_s)|^2\right] ds\\
	&\leq \mathbb{E}[|V_0^i-\overline V_0^i|^2]+C\int_0^t\sup_{i=1,\cdots,N}\mathbb{E}[|V_s^i-\overline V_s^i|^2]ds+CT\frac{1}{N}\,,
	\end{align*}
	where we have used  the fact that
	\begin{align*}
	\mathbb E\left[(V_t^i-\overline V_t^i) \cdot \left(|V_t^i - v_{\alpha,\tilde{\mathcal E}}(\rho_t^N)| P_1(V_t^i)- |\overline V_t^i- v_{\alpha,\tilde{\mathcal E}}(\rho_t) | P_1(\overline V_t^i)\right)dB_t^i\right]=0\,,
	\end{align*}
	and Lemma \ref{lemLLN} in the last inequality.
	Applying Gronwall's inequality with $\mathbb{E}[|V_0^i-\overline V_0^i|^2]=0$, one concludes
	\begin{equation}
	\sup_{i=1,\cdots,N}\mathbb{E}[|V_t^i-\overline V_t^i|^2]\leq C T\left(1+CTe^{CT}\right)\frac{1}{N}\,,
	\end{equation}
	for all $t\in[0,T]$, which completes the proof.
\end{proof}

\begin{remark}
	The constant $C>0$ appearing in the estimate \eqref{MFlimit} may depend on the dimension through the norm of $P_2$ or $\nabla P_2$, see \eqref{ter2}. Nevertheless we expect this dependency to scale
	at most linearly as $d-1$. In fact, for the case of  sphere $\Gamma=\mathbb S^{d-1}$ it is $P_2(v)=\Delta\gamma(v)= \frac{d-1}{|v|}$. We conclude that in general there is no curse of dimensionality involved in  the estimate \eqref{MFlimit}.
\end{remark}

\section{Conclusions}

We presented a new consensus-based model for global optimization on hypersurfaces, which is inspired by the kinetic Kolmogorov-Kuramoto-Vicsek equation on the sphere. The main results of this paper are the well-posedness of the stochastic particle system and its mean-field limit, which is obtained by the coupling method through introducing an auxiliary self-consistent nonlinear SDE. The presented mean-field limit is not affected by curse of dimension, i.e., the rate of convergence is of order $N^{-1}$ in the particle number $N$. This favorable theoretical rate is confirmed by numerical experiments in very high dimension ($d \approx 3000$) \cite{fhps20-2}.  In the related paper Ref. \cite{fhps20-2}  we analyze the large time behavior of the system for  $\Gamma = \mathbb S^{d-1}$, we prove its convergence to global minimizers in a suitable sense, and we provide several applications in machine learning.  We consider the results of this paper as a preliminary step towards the formulation of consensus-based optimization on compact manifolds \cite{emery1989stochastic}, which requires  a more general approach for the analysis of the global large time asymptotics.

\section{Appendix}
\begin{theorem}[Multidimensional It\^{o}'s formula]
	Let
	$$dX_t=\textbf{u}(t)dt+\textbf{v}(t)dB_t$$
	be an $d$-dimensional It\^{o} process, where $X_t\in \mathbb R^{d}$, $\textbf{u}(t)\in\mathbb R^d$, $\textbf{v}(t)\in\mathbb R^{d\times d'}$, and $B_t$ is $d'$-dimensional Brownian motion. Assume $\varphi(x)$ be a $\mc{C}^2$ map from $\mathbb R^d$ to $\mathbb R$, then it holds that
	\begin{align}\label{Mulito}
	\varphi(X_t)&=\varphi(X_0)+\int_0^t\left(\nabla \varphi(X_s)\cdot \textbf{u}(s) +\frac{1}{2}\sum_{i,j=1}^{d}\frac{\partial^2\varphi}{\partial x_i\partial x_j}(X_s)v_i(s)v_j(s)^T\right)ds\notag\\
	&\quad+\int_0^t\nabla\varphi(X_s)\cdot \textbf{v}(s)dB_s
	\end{align}
	with $v_i(s)$ being the $i$-th row of the matrix $\textbf{v}(s)$.
\end{theorem}

\section*{Acknowledgment}

Massimo Fornasier and Hui Huang acknowledge the support of the DFG Project "Identification of Energies from Observation of Evolutions" and the DFG SPP 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization".     The present project and Philippe  S\"{u}nnen  are supported by the National Research Fund, Luxembourg (AFR PhD Project Idea ``Mathematical Analysis of Training Neural Networks'' 12434809).
Lorenzo Pareschi acknowledges the support of the John Von Neumann guest Professorship program of the Technical University of Munich and the PRIN Project 2017KKJP4X "Innovative numerical methods for evolutionary partial differential equations and applications".

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
Nelly van Doesburg (née Petronella Johanna van Moorsel; The Hague, 27 July 1899 – Meudon, 1 October 1975) was a Dutch avant-garde musician, dancer, artist and art collector. She performed under her dadaïst alias Pétro van Doesburg and used the pseudonym Cupera for her work as a painter.

Biography
She was born on 27 July 1899 as the daughter of shopkeeper Petrus (Piet) Bartholomeus van Moorsel and his wife Helena (Leentje) Maria Büch as the second youngest of the six van Moorsel children.

She started classical piano studies at the age of 14, graduating 4 years later in 1918. After which she continued her musical studies in Rotterdam for a time, while also working as a piano teacher.

She was a life long champion of De Stijl movement and advisor to notable American art collector, bohemian and socialite Peggy Guggenheim.

In 1920 she met Theo van Doesburg through her brother, who had a subscription to the magazine De Stijl, published by Van Doesburg. During the opening of the La Section d'Or exhibition in The Hague on 11 July 1920, she was completely captivated by this man, and fell in love in what she later called a "coup de foudre". Her Catholic parents did not appreciate the association with Van Doesburg, who was not only Protestant, but also already married. Because of this, her parents came up with the idea to send her to a boarding-school for nuns in the UK, hastily sewing the wardrobe needed for this.  She ran away from home around the turn of the year from 1920-1921 and was received in Leiden by J.J.P. Oud and Van Doesburg's wife Lena Milius. During her relationship with Van Doesburg she was introduced to the world of modern art. Van Doesburg took her with him on a lecture tour through Europe in March 1921, during which they also visited Piet Mondriaan in Paris.

From April 1921 she lived with Van Doesburg in Weimar. In December she spent a few weeks in Vienna to reflect on her future. Here she met the Hungarian artist Lajos Kassák, publisher of the magazine Ma. Aktivista Folyóirat (Present Activists Magazine). After this a correspondence between Kassák and Theo van Doesburg arose.

On 24 September 1922 she visited with Theo, Cor van Eesteren and others, the Städtische Kunstverein by Walter Dexel in Jena. The following day she played compositions by Vittorio Rieti during a dada evening at Hotel Fürstenhof in Weimar, in which Jean Arp and Tristan Tzara also participated.

In January–February 1923 she participated with her future husband and the artists Kurt Schwitters and Vilmos Huszár, under her stage name Pétro, in the Dutch Dada tour. During these evenings she played music by Satie and Rieti. Following this tour she played compositions by Arthur Honegger, Daniël Ruyneman, Francis Poulenc, Josef Hauer and Egon Wellesz during a "Modern Soirée" on 12 March at the Lily Green dance school in The Hague. "Proeven Van Stijlkunst" (Experiments In Artistic Style) by the composer Jakob van Domselaer also featured in her repertoire. The final performance, together with Schwitters took place on the 14th of February 1925 at the avant-garde-publisher Müller & Co. in Potsdam. In addition, under the pseudonym Cupera she made a number of paintings, in which influences from De Stijl can be recognized. In Paris she enjoyed some fame as a dancer under the name Sonia Pétrowska.

From 2 October 1929 to 5 January 1930 she organized the ESAC (Expositions Sélectes d'Art Contemporain) of "contemporary young Parisian painting" at the Stedelijk Museum in Amsterdam and Pulchri Studio in The Hague. Van Doesburg also participated in this exhibition under her pseudonym P. Cupera.

After Theo van Doesburg's death in 1931, widow Nelly van Doesburg committed herself to keeping the memory of her husband alive. She wanted to help change the one-sided image of Theo being  primarily a follower of artistic movements instead of an innovator of modern art . To achieve this, she organized a series of exhibitions of his work. She also provided a good shelter for a large part of his paintings by selling them to the best collections of modern art in the world. Many paintings by Van Doesburg ended up in the United States, for example in the Museum of Modern Art and the Guggenheim collections, because Van Doesburg was friends with Peggy Guggenheim, who acted as her agent in America.

Though they shared a lot of their social circles, in the artistic avant-garde at the time, Peggy and Nelly did not meet in person until 1938 in London after which they developed a close friendship. During the Second World War Peggy bought several of the artworks in Nelly's collection, allowing Nelly to sustain herself in a time where there was practically no market for abstract art.  While a few of these works were by Theo van Doesburg, it also included paintings by El Lissitzky, Giacomo Balla and Gino Severini. Nelly's influence on Peggy Guggenheim's collection didn't end here. Peggy preferred to buy works for her collection directly from the artists themselves, through her many connections and relations within the art-world Nelly was able to introduce Peggy to an array of abstract artists and subsequently helped broker a great deal of Peggy's acquisitions in that time.

In the fifties she also made some reproductions in silkscreen of her late husband's works. Van Doesburg's remaining collection, consisting mainly of archive material, studies, photographs and drawings, left Nelly van Doesburg to her cousin Wies van Moorsel, who donated it to the Dutch Nation in 1981. The collection is managed by the Netherlands Institute for Cultural Heritage and the Netherlands Institute for Art History.

She also remained active in the artists' circuit. She had a short relationship with the German architect Ludwig Mies van der Rohe, who helped her with setting up an exhibition of her late husband's work in Chicago in 1947. She was also, as an important witness of the entire Stijl movement, a good source for, among others, the book Holland Dada by K. Schippers. At the end of her life she returned to her roots and she became Catholic again.

It is rumoured that Nelly van Doesburg had an affair with Jazz musician Thelonious Monk who she met whilst in at the Minton's Playhouse, and continued the relationship in the De Stijl house in Meudon.

Nelly van Doesburg died at the age of 76 from the consequences of breast cancer in the home and studio designed by Theo van Doesburg in Meudon, France.

References

External links
 Nelly van Doesburg in the RKD

1899 births
1975 deaths
Artists from The Hague
Dutch performance artists
Dutch women painters
20th-century Dutch painters
20th-century Dutch women artists
Women performance artists
Entertainers from The Hague
Dutch dadaists

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text:
Pubs and bars to close across England as part of second lockdown
Anna PaulMonday 2 Nov 2020 2:45 pm
Pubs and bars are to close in England (Picture: Getty/PA
Pubs, bars and restaurants will close in England from Thursday, as the country faces new lockdown restrictions.
Schools and essential shops will stay open but all other businesses will have to shut.
Announcing the changes today, Prime Minister Boris Johnson said he is under 'no illusion' how difficult the new rules will be for businesses and people, but that he 'could not ignore' bleak figures which suggest the second wave could be twice as deadly as the first.
What do the new rules mean for pubs?
When will pubs close in England?
Pubs will close up at one minute past midnight on Thursday, November 5.
Or more precisely, 10pm, which is the time hospitality venues are currently required to close under curfew rules
They'll remain closed for at least four weeks, with the restrictions being reviewed on December 2.
Mr Johnson said: 'Pubs, bars and restaurants must close except for takeaway and delivery services, workplaces should stay open where people can't work from home, for example in the construction and manufacturing sectors.'
People will be asked to leave their home only for essential reasons such as a food shop, exercise, work and education.
Anyone who is able to work from home must do so.
The news is likely to concern many hospitality workers (Picture: AFP)
There are varying lockdown rules in place across the UK: Northern Ireland is currently in a four-week circuit breaker lockdown, which sees closures across the hospitality sector, tighter social gathering restrictions, and a ban on wedding receptions, among other rules.
Meanwhile, Wales is halfway through their firebreak lockdown, which lasts for a fortnight. There is a country-wide ban on mixing with other households either inside or outside, hospitality venue closures, and a ban on buying non-essential items.
Scotland is not currently in a second national lockdown, however, the country did endure a 16-day circuit breaker lockdown in October. The central belt has extra restrictions in place until November 2, including a ban on pubs selling alcohol past 6pm.
Could beer gardens stay open?
While there was no specific reference made to outdoor areas of hospitality venues, it is understood beer gardens will also be required to close, however, takeaway orders will be permitted.
MORE : How long will the national lockdown last in England?
MORE : Furlough scheme extended as England enters second national lockdown
Follow Metro across our social channels, on Facebook, Twitter and Instagram.
Share your views in the comments below.

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text:
Feature / Software
Instant Messaging Services of the Past
by Louie Diangson · March 14, 2019
Last week, we provided you a list of social media sites of the past. Now it's time to look at another internet service that has become essential in our life online – Instant Messaging. Again, let's take a short trip down memory lane as we look back at the popular instant messaging services of the past.
AIM (AOL Instant Messenger)
Let's start with AIM or AOL Instant Messenger. It was released in May 1997 for Windows and is one of the most widely-used instant messaging services at the time before social networks like MySpace, Friendster, Facebook, and Twitter gained popularity. Although mostly used on desktops, AIM also arrived on mobile via Palm OS, Symbian OS, BlackBerry, and Windows Mobile, then eventually on Android and iOS. Usage has declined around 2011 as people moved to social networks and other messaging services. AIM completely shut down on December 15, 2017.
Also known as Google Chat, which eventually became Hangouts, was released in August 2005. It was available in Windows, Android, BlackBerry, and Chrome OS. The service was eventually integrated into Gmail so users could send IMs to other Gmail users. Google+, Google's social network, was also integrated into Google Talk. Google added new features like voice and video for users with Google Voice accounts In 2006, Google introduced offline messaging to Google Talk, allowing users to send messages to their contacts, even if they are not signed in. The messages will be received when they go online even if the user who has sent it is offline.
An Israeli company called Mirabilis developed ICQ and was released in November 1996. The name derives from the phrase "I Seek You." Like AIM, ICQ is also widely used at the time and include offline user messaging, multi-user chats, free daily-limited SMS sending, resumable file transfers, greeting cards, multiplayer video games, and a searchable user directory. It also featured emoticons for chats. In 1998, AOL acquired ICQ. Then in April 2010, AOL sold it to Digital Sky Technologies which later changed its name to Mail.Ru Group. ICQ is still available for download, but it has already stopped working on some unofficial clients since December 28, 2018.
MSN/Windows Live Messenger
MSN Messenger was a cross-platform instant messaging client developed by Microsoft. It was initially released in July 22, 1999. It was later rebranded under Windows Live Messenger in 2005. Versions of MSN/Windows Live Messenger were released for Windows, Xbox 360, Mac OS X (later under the name Microsoft Messenger for Mac), BlackBerry OS, iOS, Java ME, S60 on Symbian OS 9.x, Zune HD, Windows Phone, Windows Mobile and Windows CE. In November 2012, Microsoft announced that Messenger and Skype would merge in Q1 2013. Microsoft then officially retired Windows Live Messenger on October 31, 2014.
Perhaps the most popular instant messaging service in the world, especially in the Philippines, is Yahoo! Messenger or also known as YM. Aside from instant messaging, the service also offered VoIP, file transfers, webcam hosting, a text messaging service, and chat rooms in various categories. The chat room service shut down in 2012. In December 2015, Yahoo! announced a brand new YM for mobile, the Web and in Yahoo Mail on the desktop. The company then cut off support for the legacy app on August 5, 2016. Yahoo! Messenger completely shut down on July 17, 2018.
Chikka Text Messenger
Chikka Text Messenger was released by Chikka Philippines, Inc., in 2000 and is an Internet-based instant messaging application that supports free SMS or text messaging between online users and offline mobile subscribers of Globe, Smart, and Sun. In 2009, Smart acquired Chikka. On March 7, 2018, Chikka announced on their social media accounts that they were shutting down their service on March 31, 2018.
Have you tried any of these IM services? If we missed anything, let us know in the comments below.
Chat Apps We Never Got to Use
Dead Social Media Sites: What Happened?
Social Media Sites of the Past
The challenges and rewards of Facebook Live Selling
Let's confuse the kids: Classic 90s games we miss playing
Article Under:aim aol chikka google talk icq instant messengers msn windows live messenger yahoo messenger
Louie Diangson
This article was written by Louie Diangson, Managing Editor of YugaTech. You can follow him at @John_Louie.
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capricorn1976 says:
tried and tested, CHISMOSA Messaging. It was developed since the start of the civilization and still working up to now. No need for internet connection and electricity. This is the mother of all the FAKE news.
Michael Agustin says:
MRIC is not included.
Ronaldo Gonzales says:
IRC DALNet…using that old noisy modem to connect to pacific internet saka yung 100 pesos prepaid card!


text:
NJ Judge Sides With Morris County Municipal Utilities Authority in Public Bidding Dispute
On January 7, 2014 April 10, 2014 By Sheri Siegelbaum
The Appellate Division of the New Jersey Superior Court recently ruled in favor of the Morris County Municipal Utilities Authority in a public bidding dispute. In Waste Management of N.J. Inc. v. Morris County Municipal Utilities Authority, the court held that the trial judge mistakenly overlooked his authority to impose interlocutory injunctive relief by considering only the plaintiffs' likelihood of success.
The Facts of the Case
On July 9, 2012, defendant Morris County Municipal Utilities Authority (the Authority) issued a public notice seeking sealed bids for a five-year contract to operate the two Morris County solid waste transfer stations and to provide related transportation and disposal services. The request for bids mandated, among other things, that bidders "[s]upply . . . the certified financial statement of the Bidder and/or, if applicable, the Guarantor for each of the three (3) recent fiscal years.
Plaintiff Covanta 4Recovery, L.P. (Covanta) submitted the lowest bid, defendant Solid Waste Services, Inc. d/b/a J.P. Mascaro & Sons (Mascaro) submitted the second lowest bid, and plaintiff Waste Management of New Jersey, Inc. (Waste Management) submitted the third lowest bid. The Authority concluded that Mascaro submitted the lowest responsible bid, and the losing bidders objected. They contended that Mascaro failed to comply with the requirement for a certified financial statement.
After review of the issues, the Authority again concluded Mascaro was the lowest responsible bid, and the other bidders filed suit. The trial judge denied interlocutory injunctive relief, finding that Waste Management and Covanta failed to satisfy by clear and convincing evidence a likelihood of success on the merits. On appeal, the Appellate Division considered the narrow question of whether the trial judge mistakenly exercised his discretion in denying interlocutory injunctive relief.
The Court's Decision
As explained by the court, the issuance of an interlocutory injunction turns on several factors, including whether "the movant has demonstrated a reasonable probability of success on the merits; that a balancing of the equities and hardships favors injunctive relief; that the movant has no adequate remedy at law and that the irreparable injury to be suffered in the absence of injunctive relief is substantial and imminent; and that the public interest will not be harmed."
In this case, the judge only considered the first factor. After concluding that plaintiffs had not clearly and convincingly shown they were likely to succeed, he failed to consider any additional evidence. According to the Appellate Division, this decision was erroneous because it "overlooks a court's authority to impose interlocutory restraints regardless of doubts about the movants' likelihood of success."
With regard to the other factors, the court noted that [t]hese circumstances weighed heavily in favor of interlocutory injunctive relief, and the judge's failure to consider these other Crowe factors constitutes an abuse of discretion warranting intervention notwithstanding the deference owed to the judge's prediction of the likelihood of success." Accordingly, it reversed the decision.
For more information about this case or the legal issues involved, we encourage you to contact a member of Scarinci Hollenbeck's Government Law Group.
Categories : Public ContractsTags : New Jersey public bidding lawNew Jersey public contracts
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Rainbow of Books (Explore #86). | Flickr - Photo Sharing!
Rainbow of Books (Explore #86).
ellenator, galendara, fabiwa, and 139 other people added this photo to their favorites.
this is a thing of beauty.
who'd do that? thats a whole lot of work! please tell me its in a library where somebody got paid for doing that.
Beautiful! Have a good weekend my friend!
books are like rainbows: there are treasures hidden in them.
Hi, I'm an admin for a group called ~The Rainbow Spectrum~, and we'd love to have this added to the group!
Rainbow really is the best order :3 I only have 1 shelf like this though, it's awesome to see the whole thing!
Hi, I'm an admin for a group called Rainbow Organization, and we'd love to have this added to the group!
Great photo! Using it in my blog! Will credit!
Love this photo. Using it on my blog www.businesslifehack.de/high-speed-reading-oder-wie-du-de... Thanks a lot!
I love this! So I used it on my blog (Motherhood and Music) for a post about parents fostering their children's healthy inner narrative through promoting good literature. Check it out here if you'd like.