变分推断

目录

• 变分推断的基本形式
• Mean field
• 指数函数变分推断例子
• 随机梯度变分推断
• 参考

变分推断的基本形式

变分推断是使$q(z)$逼近$p(z|x)$来求得隐变量$z$的后验分布$p(z|x)$。根据贝叶斯公式，有

$$\begin{array}{rl}\underset{\text{evidence}}{\underbrace{\log (p(x))}}& =\log \left(p(x,z)\right)-\log (p(z|x))\\ & =\underset{\text{evidence low bound}}{\underbrace{{\int }_{z}q(z)\log \left(\frac{p(x,z)}{q(z)}\right)}}-\underset{\text{KL divergence}}{\underbrace{{\int }_{z}q(z)\log \left(\frac{p(z|x)}{q(z)}\right)}}\end{array}$$

$\log (p(x))$被称为Evidence的原因是因为它是来自我们观察到的，又因为KL-divergence不为负，为了使得$q(z)$逼近$p(z|x)$，优化的目标就是上面的ELOB。

Mean field

中场理论(Mean field)一般假设

$$\begin{array}{}\text{(1)}& q(z)=\prod _{i=1}^{M}q(z_i)\end{array}$$

代入$\text{1}$得到

$$\begin{gathered} {\int }_z\prod _{i=1}^{M}q(z_i)\log (p(x,z))dz- \\ {\int }_{z_1}q(z_i){\int }_{z_2}q(z_2)\cdots {\int }_{z_M}q(z_M)\sum _{i=1}^M\log (q(z_i))d{z}_M\cdots d{z}_1 \end{gathered}$$

即

$$\begin{array}{r}{\int }_z\prod _{i=1}^{M}q(z_i)\log (p(x,z))dz-\sum _{i=1}^M{\int }_{z_1}q(z_1)\log (q(z_1))\end{array}$$

令

$$\log ({\tilde{p}}_j(x,z))=E_{i\ne j}[\log (p(x,z_j))]$$

针对第$z_j$，ELOW为

$${\int }_{z_j}q(z_j)\log ({\tilde{p}}_j(x,))-{\int }_{z_j}q(z_j)\log (q(z_j))$$

因此当$q(z_j)={\tilde{q}}_j(x,z)$时上式取得最小值$0$。因此通过迭代$z_j$可以求得逼近$p(z|x)$的$q(z)$

指数函数变分推断例子

假设$p(x),p(x|z)$都来自某指数族分布，指数族分布形式如下

$$p(x|\eta )=h(x)\exp ({\eta }^{T}T(x)-A(\eta ))$$

且满足

$$\begin{array}{rl}{A}^{\prime }({\eta }_{MLE})& =\frac{1}{n}\sum _{i=1}^{n}T(x_i)\\ A^{\prime }(\eta )& =E_{p(x|\eta )}[T(x)]\\ A^{″}(\eta )& =Var[T(x)]\end{array}$$

假设隐变量$z$可以分为两部分$Z$和$\beta$，那么ELOB可以写为

$${\int }_{Z,\beta }q(Z,\beta )\log (p(x,Z,\beta ))-{\int }_{Z,\beta }q(Z,\beta )\log (q(Z,\beta ))$$

根据指数族的性质后验分布$p(\beta |Z,x)$和$p(Z|\beta ,z)$都属于指数族

$$\begin{array}{rl}p(\beta |Z,x)& =h(\beta )\exp (T(\beta {)}^T\eta (Z,x)-A(\eta (Z,x)))\\ p(Z|\beta ,x)& =h(Z)\exp (T(Z{)}^T\eta (\beta ,x)-A(\eta (\beta ,x)))\end{array}$$

这里只展示$p(\beta |Z,x)$的近似分布$q(\beta |\lambda )$求解，对于$p(Z|\beta ,x)$的近似分布$q(Z|\varphi )$也类似

$$q(\beta |\lambda )=h(\lambda )\exp (T(\beta {)}^T\lambda -A(\lambda ))$$

根据上一节的结果，ELOB是关于$\lambda ,\varphi$的函数

$$\begin{gathered} E_{q(Z,\beta )}[\log (p(\beta |Z,x))\log (p(Z|x))\log (p(x))]- \\ {E}_{q(Z,\beta )}[\log (q(Z))\log (q(\beta ))] \end{gathered}$$

固定$\varphi$，上式中与$\lambda$有关的项为

$$E_{q(Z,\beta )}[\log (q(\beta |Z,x))]-E_{q(Z,\beta )}[\log (q(\beta ))]$$

将$\log (q(\beta |Z,x))$和$\log (q(\beta ))$定义带入，得到与$\lambda$有关的项为

$$E_{q(\beta )}{[T(\beta )]}^T{E}_{q(Z)}[\eta (Z,x)]-{\lambda }^T{E}_{q(\beta )}[T(\beta )]+A(\lambda )$$

利用$A^{\prime }(\eta )=E_{p(x|\eta )}[T(x)]$得到

$$\begin{array}{rl}L(\lambda ,\varphi )& =A^{\prime }(\lambda {)}^T{E}_{q(Z)}[\eta (Z,x)]-{\lambda }^T{A}^{\prime }(\lambda )+A(\lambda )\\ \frac{\partial L(\lambda ,\varphi )}{\partial \lambda }& =A^{″}(\lambda {)}^T{E}_{q(Z)}[\eta (Z,x)]-A^{\prime }(\lambda )-{\lambda }^T{A}^{″}(\lambda )+A^{\prime }(\lambda )\end{array}$$

因为$A^{″}(\lambda )\ne 0$，因此

$$\lambda =E_{q(Z|\varphi )}[\eta (Z,x)]$$

同理

$$\varphi =E_{q(\beta |\lambda )}[\eta (\beta ,x)]$$

随机梯度变分推断

不同于mean field，随机梯度变分推断将分布$q(z|\varphi )$看为关于$\varphi$的分布，通过对$\varphi$进行优化得到最优的分布。

$$\begin{array}{rl}{\mathrm{\nabla }}_{\varphi }L& ={\mathrm{\nabla }}_{\varphi }{E}_{q(z|\varphi )}[\log (p(x,z))-\log (q(z|\varphi ))]\\ & =E_{q(z|\varphi )}\left[{\mathrm{\nabla }}_{\varphi }[\log q(z|\varphi ](\log p(x,z)-\log q(z|\varphi ))\right]\end{array}$$

随后用蒙塔卡罗就可以近似出梯度，虽然直接使用蒙特卡洛会造成方差较大，可以通过重参数技巧进行减小方差（在VAE中也有用到）。重参数后的计算参见SGVI。

参考

1. WallE-Chang SGVI repository

2. ws13685555932 machine learning derivative repository

3. shuhuai008 SGVI