Particle-based energetic variational inference

Abstract

We introduce a new variational inference (VI) framework, called energetic variational inference (EVI). It minimizes the VI objective function based on a prescribed energy-dissipation law. Using the EVI framework, we can derive many existing particle-based variational inference (ParVI) methods, including the popular Stein variational gradient descent (SVGD). More importantly, many new ParVI schemes can be created under this framework. For illustration, we propose a new particle-based EVI scheme, which performs the particle-based approximation of the density first and then uses the approximated density in the variational procedure, or “Approximation-then-Variation” for short. Thanks to this order of approximation and variation, the new scheme can maintain the variational structure at the particle level, and can significantly decrease the KL-divergence in each iteration. Numerical experiments show the proposed method outperforms some existing ParVI methods in terms of fidelity to the target distribution.

Appendices

Energetic variational approach

In this appendix, we gives a brief introduction to the energetic variational approach. We refer interested readers to Liu (2009) and Giga et al. (2017) for a more comprehensive description.

As mentioned previously, the energetic variational approach provides a paradigm to determine the dynamics of a dissipative system from a prescribed energy-dissipation law, which shifts the main task in the modeling of a dynamic system to the construction of the energy-dissipation law. In physics, an energy-dissipation law, yielded by the first and second law of thermodynamics (Giga et al. 2017), is often given by

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} ({\mathcal {K}} + {\mathcal {F}}) [{\varvec{\phi }}] = - 2 {\mathcal {D}}[{\varvec{\phi }}, \dot{\varvec{\phi }}], \end{aligned}$$

(A.1)

where ${\varvec{\phi }}$ is the state variable, ${\mathcal {K}}$ is the kinetic energy, ${\mathcal {F}}$ is the Helmholtz free energy, and $2 {\mathcal {D}}$ is the rate of energy dissipation. If ${\mathcal {K}} = 0$, one can view (A.1) as a generalization of gradient flow (Hohenberg and Halperin 1977).

The Least Action Principle states that the equation of motion for a Hamiltonian system can be derived from the variation of the action functional ${\mathcal {A}} = \int _{0}^T ({\mathcal {K}} - {\mathcal {F}})\mathrm {d}{\varvec{x}}$ with respect to ${\varvec{\phi }}({\varvec{z}}, t)$ (the trajectory), i.e.,

$$\begin{aligned} \begin{aligned} \delta {\mathcal {A}}&= \lim _{\epsilon \rightarrow 0} \frac{{\mathcal {A}}[{\varvec{\phi }} + \epsilon \delta {\varvec{\psi }}] - {\mathcal {A}}[{\varvec{\phi }}] }{\epsilon } \\&= \int _{0}^T \int _{{\mathcal {X}}^t} (f_{\mathrm{inertial}} - f_{\mathrm{conv}}) \cdot \delta {\varvec{\psi }} \mathrm {d}{\varvec{x}}\mathrm {d}t. \\ \end{aligned} \end{aligned}$$

This procedure yields the conservative forces of the system, that is $(f_{\mathrm{inertial}} - f_{\mathrm{conv}}) = \frac{\delta {\mathcal {A}}}{\delta {\varvec{\phi }}}$. Meanwhile, according to the MDP, the dissipative force can be obtained by minimizing the dissipation functional with respect to the “rate” $\dot{\varvec{\phi }}$, i.e.,

$$\begin{aligned} \delta {\mathcal {D}} = \lim _{\epsilon \rightarrow 0} \frac{{\mathcal {D}}[\dot{\varvec{\phi }} + \epsilon \delta {\varvec{\psi }}] - {\mathcal {D}}[\dot{\varvec{\phi }}] }{\epsilon } = \int _{{\mathcal {X}}^t} f_{\mathrm{diss}} \cdot \delta \psi \mathrm {d}{\varvec{x}}, \end{aligned}$$

or $f_{\mathrm{diss}} = \frac{\delta {\mathcal {D}}}{\delta \dot{\varvec{\phi }}}$. According to the Newton’s second law ($F = ma$), we have the force balance condition $f_{\mathrm{inertial}} = f_{\mathrm{conv}} + f_{\mathrm{diss}}$ ($f_{\mathrm{inertial}}$ plays role of ma), which defines the dynamics of the system

$$\begin{aligned} \frac{\delta {\mathcal {D}}}{\delta \dot{\varvec{\phi }}} = \frac{\delta {\mathcal {A}}}{\delta {\varvec{\phi }}}. \end{aligned}$$

(A.2)

In the case that ${\mathcal {K}} = 0$, we have

$$\begin{aligned} \frac{\delta {\mathcal {D}}}{\delta \dot{\varvec{\phi }}} = - \frac{\delta {\mathcal {F}}}{\delta {\varvec{\phi }}}, \end{aligned}$$

(A.3)

Notice that the free energy is decreasing with respect to the time when ${\mathcal {K}} = 0$. As an analogy, if we consider VI objective functional as ${\mathcal {F}}$, (A.1) gives a continuous mechanism to decrease the free energy, and (A.2) or (A.3) gives the equation ${\varvec{\phi }}({\varvec{z}}, t)$.

Derivation of the transport equation

The transport equation can be derived from the conservation of probability mass directly. Let

$$\begin{aligned} {\mathsf{F}}({\varvec{z}}, t) = \nabla _{{\varvec{z}}} \varvec{\phi }({\varvec{z}}, t) \end{aligned}$$

(B.1)

be the deformation tensor associate with the flow map $\phi ({\varvec{z}}, t)$, i.e., the Jacobian matrix of ${\varvec{\phi }}$, then due to the conservation of probability mass, we have

$$\begin{aligned} \begin{aligned} 0&= \frac{\mathrm {d}}{\mathrm {d}t} \int _{{\mathcal {X}}^t} \rho ({\varvec{x}}, t) \mathrm {d}{\varvec{x}}= \frac{\mathrm {d}}{\mathrm {d}t} \int _{{\mathcal {X}}^0} \rho (\phi ({\varvec{z}}, t), t) \det ({\mathsf{F}}({\varvec{z}}, t)) \mathrm {d}{\varvec{z}}\\&\quad = \int _{{\mathcal {X}}^0} \left( {\dot{\rho }} + \nabla \rho \cdot {\mathbf {u}}+ \rho ({\mathsf{F}}^{-\mathrm T} : \frac{\mathrm {d}{\mathsf{F}}}{\mathrm {d}t}) \right) \det {\mathsf{F}}\mathrm {d}{\varvec{z}}\\&\quad = \int _{{\mathcal {X}}^t} \left( {\dot{\rho }} + \nabla \rho \cdot {\mathbf {u}}+ \rho (\nabla \cdot {\mathbf {u}}) \right) \mathrm {d}{\varvec{x}}= 0, \\ \end{aligned} \end{aligned}$$

which implies that

$$\begin{aligned} {\dot{\rho }} + \nabla \cdot (\rho {\mathbf {u}}) = 0. \end{aligned}$$

Here the operation “ : ” between two matrix, $\mathsf{A} : \mathsf{B} = \sum _{i} \sum _{j} A_{ij} B_{ij}$, is the Frobenius inner product between two matrices $\mathsf{A}, \mathsf{B} \in {\mathbb {R}}^{n \times m}$.

Computation of Equation (2.10)

In this part, we give a detailed derivation of the variation of $\mathrm {KL}(\rho _{[\varvec{\phi }]} | \rho ^{*})$ with respect to the flow map $\varvec{\phi }({\varvec{z}}): {\mathcal {X}}^0 \rightarrow {\mathcal {X}}^t$. Consider a small perturbation of $\varvec{\phi }$

$$\begin{aligned} \varvec{\phi }^{\epsilon }({\varvec{z}}):= \varvec{\phi }({\varvec{z}}) + \epsilon \varvec{\psi }({\varvec{z}}), \end{aligned}$$

where $\varvec{\psi }({\varvec{z}}) = \widetilde{\varvec{\psi }}(\varvec{\phi }({\varvec{z}}))$ is a smooth map satisfying

$$\begin{aligned} \widetilde{\varvec{\psi }} \cdot \varvec{\nu } = 0, \quad \text {on} ~~ \partial {\mathcal {X}}^t \end{aligned}$$

with $\varvec{\nu }$ be the outward pointing unit normal on the boundary, $\partial {\mathcal {X}}^t$. Thus, $\widetilde{\varvec{\psi }}=\varvec{\psi }(\varvec{\phi }^{-1}({\varvec{x}}))$ and the above condition indicates that $\widetilde{\varvec{\psi }}$ is diffused to zero at the boundary of ${\mathcal {X}}^t$. For ${\mathcal {X}}^0 = {\mathcal {X}}^d = {\mathbb {R}}^d$, $\widetilde{\varvec{\psi }} \in C_0^{\infty } ({\mathbb {R}}^d)$. We denote ${\mathsf{F}}$ as the Jacobian matrix of $\varvec{\phi }$, i.e., $F_{ij}=\frac{\partial \phi _i}{\partial z_j}$, and ${\mathsf{F}}^{\epsilon }$ is the Jacobian matrix of $\varvec{\phi }^{\epsilon }$, i.e.,

$$\begin{aligned} {\mathsf{F}}^{\epsilon }:= \nabla _{{\varvec{z}}} \varvec{\phi } + \epsilon \nabla _{z} \varvec{\psi }. \end{aligned}$$

Then we have

$$\begin{aligned} \begin{aligned}&\frac{\mathrm {d}}{\mathrm {d}\epsilon } \Big |_{\epsilon = 0} \mathrm {KL}(\rho _{[\varvec{\phi }^{\epsilon }]} || \rho ^{*}) \\&\quad = \frac{\mathrm {d}}{\mathrm {d}\epsilon } \Big |_{\epsilon =0 }\left( \int _{{\mathcal {X}}^0} \frac{\rho _0}{\det ({\mathsf{F}}^{\epsilon })} \ln \left( \frac{\rho _0}{\det ({\mathsf{F}}^{\epsilon })} \right) \det ({\mathsf{F}}^{\epsilon })\mathrm {d}{\varvec{z}}\right. \\&\qquad \left. \quad \quad \quad \quad + \int _{{\mathcal {X}}^0} \frac{\rho _0}{\det ({\mathsf{F}}^{\epsilon })} V( \varvec{\phi }^{\epsilon }({\varvec{z}})) ~\det ({\mathsf{F}}^{\epsilon }) \mathrm {d}{\varvec{z}}\right) \\&\quad = \frac{\mathrm {d}}{\mathrm {d}\epsilon } \Big |_{\epsilon =0 }\left( \int _{{\mathcal {X}}^0} \rho _0\ln \rho _0-\rho _0\ln \det {\mathsf{F}}^{\epsilon } + \rho _0 V( \varvec{\phi }^{\epsilon }({\varvec{z}})) \mathrm {d}{\varvec{z}}\right) \\&\quad =\int _{{\mathcal {X}}^0} -\rho _0\frac{\mathrm {d}}{\mathrm {d}\epsilon } \Big |_{\epsilon =0 }\left( \ln \det ({\mathsf{F}}^{\epsilon }) + V( \varvec{\phi }^{\epsilon }({\varvec{z}}))\right) \mathrm {d}{\varvec{z}}\\&\quad = \int _{{\mathcal {X}}^0} - \rho _0 ({\mathsf{F}}^{-\top }:\nabla _{{\varvec{z}}} \varvec{\psi }) + (\nabla _{{\varvec{x}}} V \cdot \varvec{\psi }) \rho _0 \mathrm {d}{\varvec{z}}. \end{aligned} \end{aligned}$$

(C.1)

For two matrices of the same size, define $\mathsf{A}:\mathsf{B}=\sum _i\sum _j A_{ij}B_{ij}=\mathrm {tr}(\mathsf{A}^\top \mathsf{B})$. Since

$$\begin{aligned} \frac{\mathrm {d}\det ({\mathsf{F}}^{\epsilon })}{\mathrm {d}\epsilon }=\det ({\mathsf{F}}^{\epsilon })\mathrm {tr}\left[ ({\mathsf{F}}^{\epsilon })^{-1}\frac{\mathrm {d}{\mathsf{F}}^{\epsilon }}{\mathrm {d}\epsilon }\right] , \end{aligned}$$

we have

$$\begin{aligned} \frac{\mathrm {d}\det ({\mathsf{F}}^{\epsilon })}{\mathrm {d}\epsilon }\Big |_{\epsilon =0 }&=\det ({\mathsf{F}})\mathrm {tr}\left[ {\mathsf{F}}^{-1}\nabla _{{\varvec{z}}}\varvec{\psi }\right] \\&=\det ({\mathsf{F}})({\mathsf{F}}^{-\top }:\nabla _{{\varvec{z}}}\varvec{\psi }). \end{aligned}$$

Hence, we have the last result in (C.1).

Based on the definition of $\varvec{\phi }$, we have the following.

$$\begin{aligned}&{\varvec{x}}=\varvec{\phi }({\varvec{z}}), \quad {\varvec{z}}=\varvec{\phi }^{-1}({\varvec{x}})\\&\varvec{\psi }({\varvec{z}})=\widetilde{\varvec{\psi }}(\varvec{\phi }({\varvec{z}}))=\widetilde{\varvec{\psi }}({\varvec{x}})\\&\rho ({\varvec{x}})=\rho _0(\varvec{\phi }^{-1}({\varvec{x}}))\det (\nabla _{{\varvec{x}}}\varvec{\phi }^{-1}({\varvec{x}}))\\&\rho _0({\varvec{z}})=\rho _0(\varvec{\phi }^{-1}({\varvec{x}}))=\frac{\rho ({\varvec{x}})}{\det (\nabla _{{\varvec{x}}}\varvec{\phi }^{-1}({\varvec{x}}))}. \end{aligned}$$

The second summand of (C.1) becomes

$$\begin{aligned}&\int _{{\mathcal {X}}_0}\rho _0\left[ (\nabla _{{\varvec{x}}}V)^\top \varvec{\psi }\right] \mathrm {d}{\varvec{z}}\\&\quad =\int _{{\mathcal {X}}_t}\frac{\rho ({\varvec{x}})}{\det (\nabla _{{\varvec{x}}}\varvec{\phi }^{-1}({\varvec{x}}))} \left[ (\nabla _{{\varvec{x}}}V)^\top \varvec{\psi }\right] \mathrm {d}\varvec{\phi }^{-1}({\varvec{x}})\\&\quad =\int _{{\mathcal {X}}_t}\frac{\rho ({\varvec{x}})}{\det (\nabla _{{\varvec{x}}}\varvec{\phi }^{-1}({\varvec{x}}))} \left[ (\nabla _{{\varvec{x}}}V)^\top \varvec{\psi }\right] \det (\nabla _{{\varvec{x}}}\varvec{\phi }^{-1}({\varvec{x}})) \mathrm {d}{\varvec{x}}\\&\quad =\int _{{\mathcal {X}}_t}\rho ({\varvec{x}})\left[ (\nabla _{{\varvec{x}}}V)\cdot \varvec{\psi }\right] \mathrm {d}{\varvec{x}}. \end{aligned}$$

Now, we investigate the first summand in (C.1). Based on the definition of $\widetilde{\varvec{\psi }}$, we can see that

$$\begin{aligned} (\nabla _{{\varvec{x}}} \widetilde{\varvec{\psi }})_{i,j}&=\sum _{k=1}^d \frac{\partial \psi _i}{\partial z_k}\cdot \frac{\partial z_k}{\partial x_j}=\sum _{k=1}^d (\nabla _{{\varvec{z}}}\varvec{\psi })_{i,k}(\nabla _{{\varvec{x}}}\varvec{\phi }^{-1})_{k,j}\\ \nabla _{{\varvec{x}}} \widetilde{\varvec{\psi }}&=\nabla _{{\varvec{z}}}\varvec{\psi }\left[ \nabla _{{\varvec{x}}}\varvec{\phi }^{-1} \right] ^\top =(\nabla _{{\varvec{z}}}\varvec{\psi }) {\mathsf{F}}^{-\top }, \end{aligned}$$

because ${\mathsf{F}}=\nabla _{{\varvec{z}}}\varvec{\phi }(z)=\left( \frac{\partial x_i}{\partial z_j}\right) _{i,j}$, ${\mathsf{F}}^{-1}=\left( \frac{\partial z_i}{\partial x_j}\right) _{i,j}=(\nabla _{{\varvec{x}}} \varvec{\phi }^{-1}({\varvec{x}}))^{-1}$. Divergence of $\widetilde{\varvec{\psi }}(\varvec{x})$ is

$$\begin{aligned}&\nabla _{{\varvec{x}}} \cdot \widetilde{\varvec{\psi }}(\varvec{x})=\sum _{i=1}^d \frac{\partial {\tilde{\psi }}_i}{\partial x_i}=\mathrm {tr}(\text {Jacobian of }\widetilde{\varvec{\psi }})\\&\quad =\mathrm {tr}(\nabla _{{\varvec{x}}} \widetilde{\varvec{\psi }})=\mathrm {tr}((\nabla _{{\varvec{z}}}\varvec{\psi }) {\mathsf{F}}^{-\top })=\mathrm {tr}({\mathsf{F}}^{-1}\nabla _{{\varvec{z}}}\varvec{\psi }). \end{aligned}$$

Therefore, the first summand in (C.1) becomes,

$$\begin{aligned} -\int _{{\mathcal {X}}_0}\rho _0 ({\mathsf{F}}^{-\top } : \nabla _{{\varvec{z}}} \varvec{\psi })\mathrm {d}z =-\int _{{\mathcal {X}}_t}\rho ({\varvec{x}}) (\nabla _{{\varvec{x}}}\cdot \widetilde{\varvec{\psi }})\mathrm {d}{\varvec{x}}. \end{aligned}$$

Following the corollary of divergence theorem,

$$\begin{aligned}&\int _{{\mathcal {X}}_t}\rho ({\varvec{x}}) (\nabla _{{\varvec{x}}}\cdot \widetilde{\varvec{\psi }})\mathrm {d}{\varvec{x}}+ \int _{{\mathcal {X}}_t}\widetilde{\varvec{\psi }}^\top (\nabla _{{\varvec{x}}}\rho ({\varvec{x}}))\mathrm {d}{\varvec{x}}\\&\quad =\oint _{\partial {\mathcal {X}}_t} \rho ({\varvec{x}})(\widetilde{\varvec{\psi }}\cdot \varvec{\nu })\mathrm {d}S=0, \end{aligned}$$

because the boundary condition $\widetilde{\varvec{\psi }}\cdot \varvec{\nu }=0$ on $\partial {\mathcal {X}}_t$. So

$$\begin{aligned}&-\int _{{\mathcal {X}}_t}\rho ({\varvec{x}}) (\nabla _{{\varvec{x}}}\cdot \widetilde{\varvec{\psi }})\mathrm {d}{\varvec{x}}\\&\quad =\int _{{\mathcal {X}}_t}\widetilde{\varvec{\psi }}^\top (\nabla _{{\varvec{x}}}\rho ({\varvec{x}}))\mathrm {d}{\varvec{x}}=\int _{{\mathcal {X}}_t}\nabla _{{\varvec{x}}}\rho ({\varvec{x}})\cdot \widetilde{\varvec{\psi }}\mathrm {d}{\varvec{x}}. \end{aligned}$$

Therefore, in ${\mathcal {X}}^t$, by performing integration by parts, we have

$$\begin{aligned} \begin{aligned}&\frac{\mathrm {d}}{\mathrm {d}\epsilon } \Big |_{\epsilon = 0} \mathrm {KL}(\rho _{[\varvec{\phi }^{\epsilon }]} || \rho ^{*}) \\&\quad = \int _{{\mathcal {X}}^t} - \rho _{[\varvec{\phi }]} (\nabla _{{\varvec{x}}} \cdot \widetilde{ \varvec{\psi }}) + \rho \nabla V \cdot \widetilde{\varvec{\psi }} \mathrm {d}{\varvec{x}}\\&\quad = \int _{{\mathcal {X}}^t} (\nabla \rho + \rho \nabla V) \cdot \widetilde{\varvec{\psi }} \mathrm {d}{\varvec{x}}, \\ \end{aligned} \end{aligned}$$

which implies that

$$\begin{aligned} \frac{\delta \mathrm {KL}(\rho _{[\varvec{\phi }]} || \rho ^{*})}{\delta \varvec{\phi }} = \nabla \rho + \rho \nabla V \end{aligned}$$

(C.2)

Recall $V = - \ln \rho ^{*}$. One can notice that if F is an identity matrix, the result in (C.1) can be written as

$$\begin{aligned} - {\mathbb {E}}_{{\varvec{z}}\sim \rho _0} [\mathrm {trace} (\nabla _{{\varvec{z}}} \varvec{\psi } + \nabla \ln \rho ^{*} \varvec{\psi }^{\mathrm{T}})], \end{aligned}$$

which is exactly the form given by the Stein operator in Liu and Wang (2016).

Proof of Theorem 1

Proof

Let ${\mathbf {X}}\in {\mathbb {R}}^D$ be vectorized $\{ {\varvec{x}}_i \}_{i = 1}^N$, that is

$$\begin{aligned} {\mathbf {X}}= (x_1^{(1)},\ldots x_N^{(1)}, \ldots x_1^{(d)} \ldots x_N^{(d)} ), \end{aligned}$$

where $D = N \times d$. Recall that $V({\varvec{x}}) = - \ln \rho ^{*}$. For a sufficient smooth target distribution $\rho ^{*}({\varvec{x}})$, it is easy to show that

$$\begin{aligned} {\mathcal {F}}_h ( \{ {\varvec{x}}_i \}_{i=1}^{N}) = \frac{1}{N} \sum _{i=1}^N \left( \ln \left( \frac{1}{N} \sum _{j=1}^N K({\varvec{x}}_i, {\varvec{x}}_j) \right) + V({\varvec{x}}_i) \right) \end{aligned}$$

is continuous, coercive and bounded from below as a function of ${\mathbf {X}}\in {\mathbb {R}}^D$. We denote ${\mathcal {F}}_h ( \{ {\varvec{x}}_i \}_{i=1}^{N})$ by ${\mathcal {F}}_h ({\mathbf {X}})$.

For any given $\{ {\varvec{x}}_i^n \}_{i=1}^N$, recall

$$\begin{aligned} J_n({\mathbf {X}}) = \frac{1}{2 \tau } \Vert {\mathbf {X}}- {\mathbf {X}}^n\Vert ^2 + {\mathcal {F}}_h({\mathbf {X}}), \end{aligned}$$

where $\Vert \cdot \Vert ^2_{{\mathbf {X}}}$ is a norm for ${\mathbf {X}}$, defined by

$$\begin{aligned} \Vert {\mathbf {X}}- {\mathbf {X}}^n\Vert ^2_{{\mathbf {X}}} = \frac{1}{N} \sum _{i=1}^N \Vert {\varvec{x}}_i - {\varvec{x}}_i^n \Vert ^2. \end{aligned}$$

Since

$$\begin{aligned} {\mathcal {S}} = \{ J(\varvec{{\mathbf {X}}}) \le J(\varvec{{\mathbf {X}}}^n) \} \end{aligned}$$

is a non-empty, bounded, and closed set, by the coerciveness and continuity of ${\mathcal {F}}_h({\mathbf {X}})$, $J_n({\mathbf {X}})$ admits a global minimizer ${\mathbf {X}}^{n+1}$ in ${\mathcal {S}}$. Since ${\mathbf {X}}^{n+1}$ is a global minimizer of $J({\mathbf {X}})$, we have

$$\begin{aligned} \frac{1}{2 \tau } \Vert {\mathbf {X}}^{n+1} - {\mathbf {X}}^{n} \Vert ^2_{{\mathbf {X}}} + {\mathcal {F}}_h({\mathbf {X}}^{n+1}) \le {\mathcal {F}}_h({\mathbf {X}}^{n}), \end{aligned}$$

which gives us equation (3.19).

For series $\{ {\mathbf {X}}^n \}$, since

$$\begin{aligned} \Vert {\mathbf {X}}^{k} - {\mathbf {X}}^{k-1} \Vert ^2_{{\mathbf {X}}} \le 2 \tau ({\mathcal {F}}_h({\mathbf {X}}^{k-1}) - {\mathcal {F}}_h({\mathbf {X}}^{k})), \end{aligned}$$

we have

$$\begin{aligned} \sum _{k=1}^n \Vert {\mathbf {X}}^{k} - {\mathbf {X}}^{k-1} \Vert ^2_{{\mathbf {X}}} \le 2 \tau ({\mathcal {F}}_h({\mathbf {X}}^{0}) - {\mathcal {F}}_h({\mathbf {X}}^{n})) \le C, \end{aligned}$$

for some constant C that is independent with n. Hence

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert {\mathbf {X}}^{n} - {\mathbf {X}}^{n-1} \Vert _{{\mathbf {X}}} = 0, \end{aligned}$$

which indicates the convergence of $\{ {\mathbf {X}}^n \}$. Moreover, since

$$\begin{aligned} {\mathbf {X}}^{n} = {\mathbf {X}}^{n-1} - \tau \nabla _{{\mathbf {X}}} {\mathcal {F}}_h({\mathbf {X}}^{n}), \end{aligned}$$

we have

$$\begin{aligned} \lim _{n \rightarrow \infty } \nabla _{{\mathbf {X}}} {\mathcal {F}}_h({\mathbf {X}}^{n}) = 0, \end{aligned}$$

so $\{{\mathbf {X}}^n\}$ converges to a stationary point of ${\mathcal {F}}_h({\mathbf {X}})$. $\square$