RBM

RBM

---

目录

• RBM
  • 基础知识
    • 马尔可夫链
      • 马尔可夫性质
      • 马尔可夫链
      • n次转移矩阵
      • 细致平稳条件
    • 分布抽样
      • Box-Muller变换
      • CDF变换
      • Metropolis 采样算法
      • MCMC采样算法
      • Metropolis-Hasting 采样算法
      • Gibbs 采样算法
      • 与Boilzmann 机的联系
  • Boilzmann 机学习规则
    • 基础函数
      • 能量函数
      • 状态概率分布
      • sigmoid激活函数
    • 概率分布推导
    • 激活函数反向推导验证
    • 似然函数求取
    • 学习目标
      • 偏导求解
  • 结构
    • 计算图
    • 迭代过程
  • CD学习算法
    • 更新法则
    • 学习步骤
  • 优化设置
    • 训练方式
    • 学习率
      • 动量学习率
    • 权重和偏置的初始值
    • 权衰减
    • 隐单元个数

基础知识

马尔可夫链

随机过程即状态变化路径，可以表示为 (x₀，x₁，...)。还有：高斯过程，泊松过程，自回归模型，移动平均模型...

马尔可夫性质

在给定现在状态时，它与过去状态（即该过程的历史路径）是条件独立的。可以用一个自反有向完全图表示，图中所有路径皆有转移概率。每一个状态转移的下一点位置由概率决定，每一个状态转移概率之和为1。

马尔可夫链

具有马尔可夫性质的随机过程为马尔可夫链。

n次转移矩阵

n次转移矩阵 = \(P^n\)，意义在于表示此随机过程中从某一状态出发走n步到其他状态的概率。

• 遍历性：只要转移矩阵每个元素大于0，\(\lim_{n\rightarrow \infty} P^n\)收敛，且结果为一次转移矩阵的特征向量转置集

因为走n步后分布平稳，所以可以使用采样的方式求n次转移矩阵。采样一般分为3个阶段：初始化、burn-in（随机产生不记录）、sampling（随机产生并记录）[可以间隔采样]

细致平稳条件

即马尔可夫链收敛条件。P为转移矩阵，\(\pi(x)\)为分布

\[\pi(i)P_{ij}=\pi(j)P_{ji} \]

一般马尔科夫链未收敛，要使其收敛，则可以改变其转移矩阵为Q，有

\[Q_{ij}=P_{ij}·\alpha_{ij}\qquad \alpha_{ij}=\pi(j)P_{ji}\\ Q_{ji}=P_{ji}·\alpha_{ji}\qquad \alpha_{ji}=\pi(i)P_{ij} \]

其中\(\alpha\)为接受概率，于是

\[\pi(i)Q_{ij}=\pi(j)Q_{ji} \]

分布抽样

设均匀分布随机数为U，则

Box-Muller变换

可以通过U生成标准正态分布

\[Z_0=\sqrt{-2\ln U_1}\cos(2\pi U_2)\\ Z_1=\sqrt{-2\ln U_1}\sin(2\pi U_2) \]

CDF变换

生成符合高斯分布或者其他任意分布的随机数

\[p(x)=\frac{\tilde{p}(x)}{\int \tilde{p}(x)dx} \]

Metropolis 采样算法

从一个复杂的目标分布获取近似的样本的采样算法

MCMC采样算法

目的：以任意状态为始点采样转移序列，p和q已知

步骤：

• 初始化状态 x₀
• 随着时间递增，循环采样：
  • y ~ q(x|x_t) —— 以分布q找下一个状态y
  • u ~ Uniform[0,1] —— 均匀分布采样u
  • u < \(\alpha\)(x_t , y) = p(y) q(x_t|y) —— 考虑是否接受转移
    • TRUE：x_t+1 = y
    • FALSE：x_t+1 = x_t

存在的问题：\(\alpha\)值可能会非常小，会导致很难跳转

Metropolis-Hasting 采样算法

MCMC的改进版，使

\[\alpha(i,j)=min\left\{ \frac{p(j)q(j,i)}{p(i)q(i,j)},\ 1\right\} \]

也就是两边同比例放大，只是概率不能大于1，所以要min

Gibbs 采样算法

发现

\[\begin{align} p(x_1,y_1)p(y_2|x_1)&=p(x_1)p(x_1,y_2)p(y_2|x_1)\\&=p(x_1,y_2)p(y_1|x_1) \end{align} \]

然后居然是细致平稳条件！最终步骤很完美：

• 初始化状态 x₀，y₀
• 随着时间递增，循环采样：
  • y_t+1 ~ q(y|x_t)
  • x_t+1 ~ q(x|y_t)

转移样本序列 (x₀, y₀)、(x₀, y₁)、(x₁, y₁)、(x₁, y₂) ...

然后推广到n维：

• 初始化状态 x₀，x₁，x₂，...
• 随着时间递增，循环采样：
  • x₀ ~ q(x₀|x₁, x₂, ...)
  • x₁ ~ q(x₁|x₀, x₂, ...)
  • x₂ ~ q(x₂|x₀, x₁, ...)
  • ......

与Boilzmann 机的联系

Boilzmann 机能量有

\[E(x)=-\frac{1}{2}\sum_i\sum_{j\neq i}w_{ij}x_ix_j \]

某一节点位于某一状态的概率（玻尔兹曼分布）

\[P(X=x)=\frac{1}{Z}e^{-\frac{E(x)}{T}} \]

我们有当其他节点不改变状态时单节点状态分布概率（A：j节点状态分布，B：除了此节点以外的状态分布）

\[P(A|B)=\frac{P(A,B)}{P(B)}=\frac{1}{1+exp\left( -\frac{x_i}{T}\sum_{i\neq j}w_{ji}x_i\right)} \]

也可以表示成

\[P(A|B)=\varphi\left( \frac{x_i}{T}\sum_{i\neq j}w_{ji}x_i\right)\\ \varphi(v)=\frac{1}{1+exp(-v)} \]

由此概率决定状态转化，而前面的抽样方法将用于确定\(P(A,B)\)，运行长时间后即可达到热平衡。

Boilzmann 机学习规则

基础函数

能量函数

\[E(v,h)=-a^Tv-b^Th-h^TWv \]

状态概率分布

\[P(v,h)=\frac{1}{Z}e^\frac{-E(v,h)}{T} \]

sigmoid激活函数

\[f(x)=\frac{1}{1+e^x} \]

概率分布推导

\[\begin{align} P(h|v)&=\frac{P(h,v)}{P(v)} \\&=\frac{1}{P(v)}\frac{1}{Z}\exp\{a^Tv+b^Th+h^TWv\} \\&=\frac{1}{Z'}\prod^{n_h}_{j=1}\exp\{b^T_jh_j+h^T_jW_{j,:}v\}\\ \\ \frac{1}{Z'}&=\frac{1}{P(v)}\frac{1}{Z}\exp\{a^Tv\} \end{align} \]

激活函数反向推导验证

\[\begin{align} P(h_j=1|v)&=\frac{P(h_j=1|v)}{P(h_j=1|v)+P(h_j=0|v)} \\&=\frac{\exp\{b_j+W_{j,:}v\}}{\exp\{0\}+\exp\{b_j+W_{j,:}v\}} \\&=\frac{1}{1+\exp\{b_j+W_{j,:}v\}} \\&=sigmoid(b_j+W_{j,:}v) \end{align} \]

所以只需要确定边的权重，偏置，激活函数选取Sigmoid，就符合上述能量公式。所以分隔值选取0.5。

似然函数求取

因为分布为乘积形式，所以选取对数似然函数

\[L(w,a,b)=\sum \log P(v=V,h=H) \]

然后代入求w和b的导数，利用随机梯度上升法优化。有

学习目标

\[\arg\max\mathcal{L}(\theta)=\arg\max\sum^T_{i=1}\log P(v^{(i)}|\theta)\\ \theta=\{w,a,b\} \]

偏导求解

中括号表示在此分布下的期望

\[\frac{\partial\log P(v|\theta)}{\partial W_{ij}}=\langle v_ih_j\rangle_{data}-\langle v_ih_j\rangle_{model}\\ \frac{\partial\log P(v|\theta)}{\partial a_i}=\langle v_i\rangle_{data}-\langle v_i\rangle_{model}\\ \frac{\partial\log P(v|\theta)}{\partial h_j}=\langle h_j\rangle_{data}-\langle h_j\rangle_{model} \]

其中

\[data\equiv P(h|v^{(t)},\theta)\\ model\equiv P(v,h|\theta) \]

结构

计算图

![][fig1]

迭代过程

CD学习算法

• 中文名：基于对比散度的快速学习算法
• 优点：不需要经由Gibbs采样计算期望，一步计算完成

更新法则

其中\(\langle *\rangle_{recon}\)表示一次重构可见层后的分布

\[\Delta W_{ij}=\eta(\langle v_ih_j\rangle_{data}-\langle v_ih_j\rangle_{recon})\\ \Delta a_i=\eta(\langle v_i\rangle_{data}-\langle v_i\rangle_{recon})\\ \Delta b_j=\eta(\langle h_j\rangle_{data}-\langle h_j\rangle_{recon}) \]

学习步骤

（注：以下直接使用概率运算是因为其属于0-1分布，期望等于概率）

• 输入：一个训练样本\(x_0\)，隐层单元个数m，学习率\(\epsilon\)，最大训练周期T
• 输出：连接权重矩阵W，可见层的偏置向量a，隐层的偏置向量b
• 训练：
  • 初始化
    • \(v_1=x_0\)
    • \(w,i,j\) 随机取较小数值
  • \(1:T\rightarrow t\)，循环
    • \(1:m\rightarrow j\)，循环遍历隐层单元
      • \(P(h_{1j}|v_1)=\sigma(b_j+\sum_iv_{1i}W_{ij})\)
      • 抽取 \(h_{1j}\in {0,1}\)
    • \(1:n\rightarrow i\)，循环遍历隐层单元
      • \(P(v_{2j}|h_1)=\sigma(a_i+\sum_jW_{ij}h_{1j})\)
      • 抽取 \(v_{2i}\in {0,1}\)
    • \(1:m\rightarrow j\)，循环遍历隐层单元
      • \(P(h_{2j}|v_2)=\sigma(b_j+\sum_iv_{2i}W_{ij})\)
    • 更新：
      • \(W\leftarrow W+\epsilon\ [P(h_1=1|v_1)v^T_1-P(h_2=1|v_1)v^T_1]\)
      • \(a\leftarrow a+\epsilon\ [v_1-v_2]\)
      • \(b\leftarrow b+\epsilon\ [P(h_1=1|v_1)-P(h_2=1|v_2)]\)

优化设置

训练方式

• 小批量梯度下降（最优）
  • 容量：几十 ~ 几百
• 随机梯度下降
• 梯度下降

学习率

一般设置为权重的\(10^{-3}\)

动量学习率

k为动量项学习率，开始时昬 k可设为0.5 在重构误差处于平稳增加状态时，k可取为0.9。

\[W^{(t+1)}_{ij}=kW^{(t)}_{ij}+\epsilon \frac{\partial \mathcal{L}}{\partial W^{(t)}_{ij}} \]

权重和偏置的初始值

权重 = N(0, 0.01)

b = 0

a = log [ p_i / (1 - p_i) ]，p_i 为训练样本中第 i 个特征处于激活状态所占比率

权衰减

避免过拟合，设置罚函数如下，系数取值可取0.01~0.001。

\[\mathcal{L}_2:\frac{\lambda}{2}\sum_i\sum_j W^2_{ij} \]

隐单元个数

可以先估算一下用一个好的模型描述一个数据所需的比特数，用其乘上训练集容量（即总描述所需比特数）。基于所得的数，选择比其低一个数量级的值作为隐元个数。如果训练数据是高度冗余的（比如数据集容量非常大），则可以使用更少一些的隐元。

©️ Copyrights.RSMX.GUILIN.2020-05-17

[fig1]:data:image/jpg;base64,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