[机器学习] 1. 梯度下降 Gradient Descent 与随机梯度下降 Stochastic Gradient Descent

ML Theory 太魔怔了!!!!!

从微积分课上我们学到

  • 对一个 \(\mathscr C^2\) 函数,其二阶泰勒展开的皮亚诺余项形式

    \[f(\bm w') = f(\bm w) + \langle \nabla f(\bm w), \bm w' - \bm w\rangle + o(\|\bm w' - \bm w\|) \]

    这说明只要 \(\bm w'\)\(\bm w\) 挨得足够接近,我们就可以用 \(f(\bm w) + \langle \nabla f(\bm w), \bm w' - \bm w \rangle\) 来逼近 \(f(\bm w')\)

现在我们想定量描述这个逼近过程,来说明梯度下降 (gredient descent, GD) 的收敛性及其速率。因此考虑其拉格朗日余项

\[f(\bm w') = f(\bm w) + \langle \nabla f(\bm w), \bm w' - \bm w\rangle + \frac{g(\bm \xi)}2 \|\bm w' - \bm w\|^2 \]

我们来定量描述 \(g\) 的性质。

由于梯度下降要执行多轮,因此会有不同的 \(\bm w, \bm w'\),所以性质需要适用于所有位置。

定义 平滑性假设 (Smoothness assumption) \(\exists L, \text{ s.t. } \forall \bm w, \bm w', |g(\bm \xi)| \leq L\)。换句话说,

\[|f(\bm w') - f(\bm w) - \langle \nabla f(\bm w), \bm w' - \bm w\rangle| \leq \frac{L}2 \|\bm w' - \bm w\|^2 \]

这个假设是非常自然的,其略强于 \(\mathscr C^2\)。在有界闭集上两者等价。

平滑性是说一个函数在每个点被一个二次函数 bound 住,在梯度的视角下,这等价于其 Lipschitz 连续,在 Hessian 矩阵的视角下,这等价于矩阵的 norm 被 bound 住。

命题 梯度 \(L\)-Lipschitz 连续等价于 \(\|\nabla^2 f(x)\| \leq L\),其中 \(|\nabla^2 f(x)|\) 表示 Hessian 矩阵的 Euclidean norm,即 \(\max_{\|x\| = 1} \|Hx\| = |\lambda|_{\max}\)。梯度 \(L\)-Lipschitz 连续表示

\[\|\nabla f(\bm w) - \nabla f(\bm w')\| \leq L \|\bm w - \bm w'\| \]

证明

  • \(\Leftarrow\)

    \[\begin{aligned} \|\nabla f(\bm w') - \nabla f(\bm w)\| &= \left\|\int^1_0 \nabla^2 f(\bm w + \tau(\bm w' - \bm w))(\bm w' - \bm w) \mathrm d \tau\right\| \\ &= \left\|\int^1_0 \nabla^2 f(\bm w + \tau(\bm w' - \bm w)) \mathrm d \tau \cdot (\bm w' - \bm w)\right\| \\ &\leq \left\|\int^1_0 \nabla^2 f(\bm w + \tau(\bm w' - \bm w))\mathrm d \tau\right\| \cdot \|\bm w' - \bm w\| \\ &\leq \int^1_0 \|\nabla^2 f(\bm w + \tau(\bm w' - \bm w))\|\mathrm d \tau \cdot \|\bm w' - \bm w\| \\ &\leq L\|\bm w' - \bm w\| \end{aligned}\]

  • \(\Rightarrow\)

    \[\|\nabla^2 f(\bm w)\| = \max_{\|\bm x\| = 1} \|H\bm x\| \leq \lim_{\alpha \to 0^+}\frac{\|\nabla f(\bm w + \alpha \bm v) - \nabla f(\bm w)\|}{\alpha} \leq L \]

    其中 \(\|\bm v\| = 1\)

命题 \(L\)-平滑等价于梯度 \(L\)-Lipschitz 连续。

证明

  • \(\Leftarrow\)

    \[\begin{aligned} f(\bm w') &= f(\bm w) + \int^1_0 \langle \nabla f(\bm w + \tau(\bm w' - \bm w)), \bm w' - \bm w \rangle \mathrm d \tau \\ &= f(\bm w) + \langle \nabla f(\bm w), \bm w' - \bm w \rangle + \int^1_0 \langle \nabla f(\bm w + \tau(\bm w' - \bm w)) - \nabla f(\bm w), \bm w' - \bm w \rangle \mathrm d \tau \\ &\leq f(\bm w) + \langle \nabla f(\bm w), \bm w' - \bm w \rangle + \int^1_0 \|\nabla f(\bm w + \tau(\bm w' - \bm w)) - \nabla f(\bm w)\| \cdot \|\bm w' - \bm w \| \mathrm d \tau && \text{Cauchy-Schwarz} \\ &\leq f(\bm w) + \langle \nabla f(\bm w), \bm w' - \bm w \rangle + \int^1_0 L\tau\|\bm w' - \bm w\| \cdot \|\bm w' - \bm w \| \mathrm d \tau \\ &= f(\bm w) + \langle \nabla f(\bm w), \bm w' - \bm w \rangle + \frac L2\|\bm w' - \bm w\|^2 \\ \end{aligned}\]

  • \(\Rightarrow\):考虑 \(f\) 的 Lagrange 余项的 Taylor 展开

    \[f(\bm w') = f(\bm w) + \langle f(\bm w), \bm w' - \bm w \rangle + \frac 12 \langle\nabla^2 f(\bm \xi)(\bm w' - \bm w), \bm w' - \bm w \rangle \]

    \[|f(\bm w')- f(\bm w) - \langle \nabla f(\bm w), \bm w' - \bm w\rangle| = \frac 12 |\langle \nabla^2 f(\bm \xi)(\bm w' - \bm w), \bm w' - \bm w \rangle|\leq \frac L2\|\bm w' - \bm w\|^2 \]

    \(\bm w' = \bm w + t \bm v, \|\bm v\| = 1\),有

    \[|\langle \nabla^2 f(\bm w + t \bm v) \bm v, \bm v\rangle| \leq L \]

    \(t \to 0^+\),由于 \(f\)\(\mathscr C^2\) 函数,可得

    \[|\langle \nabla^2 f(\bm w) \bm v, \bm v\rangle| \leq L \]

    注意到 \(\nabla^2 f(\bm w)\) 是一个 self-adjoint 的矩阵,因此

    \[\max_{\bm v} \|\nabla^2 f(\bm w)\bm v\|_2 = \max_{\bm v} \langle \nabla^2 f(\bm w) \bm v, \bm v\rangle = |\lambda|_{\max} \]

    根据上一条命题,该命题得证。

回到梯度下降中。对平滑的 \(f\),有

\[\begin{cases} f(\bm w') \leq f(\bm w) + \langle \nabla f(\bm w), \bm w' - \bm w \rangle + \frac L2 \| \bm w' - \bm w \|^2 \\ f(\bm w') \geq f(\bm w) + \langle \nabla f(\bm w), \bm w' - \bm w \rangle - \frac L2 \| \bm w' - \bm w \|^2 \\ \end{cases}\]

这给出了一个从 \(\bm w\) 出发,走到某个 \(\bm w'\)\(f\) 的上下界,就像这样(灵魂画手 yy)

下界并不重要,我们关心的是上界。在 \(\bm w, \bm w'\) 足够接近时,\(f\) 总是下降的,定量地,假设在梯度下降中采取学习速率 \(\eta\)\(\bm w' = \bm w - \eta \nabla f(\bm w)\)

\[\begin{aligned} f(\bm w') - f(\bm w) &\leq \langle \nabla f(\bm w), \bm w' - \bm w \rangle + \frac L2 \|\bm w' - \bm w\|^2 \\ &= \langle \nabla f(\bm w), - \eta \nabla f(\bm w)\rangle + \frac{L\eta^2}2 \|\nabla f(\bm w)\|^2 \\ &= -\eta\left(1 - \frac{L\eta}2\right) \|\nabla f(\bm w)\|^2 \end{aligned}\]

因此当 \(\eta < \frac 2L\) 时,式子总是 \(< 0\) 的,这保证我们每次梯度下降都会有进步。

但是这个假设还是不够。首先它可能会落入局部最优,其次虽然每次都有进步,但是全局的收敛速度没有保证。考虑 \(f(x) = \mathrm{sigmoid}(x)\),从 \(x\) 很大的开始向中间靠拢,速度是负指数级的。这要求我们给函数更多的整体性质。

定义 一个函数 \(f\) 是凸的,如果 \(f(t\bm x_1 + (1 - t)\bm x_2) \leq tf(\bm x_1) + (1 - t)f(\bm x_2),\ t \in [0, 1]\)

其有若干个等价定义,这是微积分课上讲过的。

命题\(f\)\(\mathscr C^2\) 函数,则凸等价于 \(\nabla^2 f(\bm w)\) 半正定。

也就是说,凸性和平滑性一个保证的是 \(|\lambda|_{\max}\) 的界,一个保证的是 \(\lambda_{\min}\) 的符号。

凸性能够保证收敛速度。

命题 \(\bm w^* = \operatorname{argmin}_{\bm w} f(\bm w)\),采用学习速率 \(\eta \leq \frac 1L\) 进行 \(t\) 轮梯度下降时,有

\[f(\bm w_t) \leq f(\bm w^*) + \frac 1{2\eta t}\|\bm w_0 - \bm w^*\|^2 \]

证明 考虑裂项法

\[\begin{aligned} f(\bm w_{i+1}) &\leq f(\bm w_i) - \eta\left(1 - \frac{L\eta}2\right) \|\nabla f(\bm w_i)\|^2 && \text{Smoothness}\\ &\leq f(\bm w_i) - \frac \eta 2\|\nabla f(\bm w_i)\|^2 \\ &\leq f(\bm w^*) + \langle \nabla f(\bm w_i), \bm w_i - \bm w^*\rangle - \frac \eta 2 \|\nabla f(\bm w_i)\|^2 && \text{Convexity} \\ &= f(\bm w^*) - \frac 1{\eta} \langle \bm w_{i+1} - \bm w_i, \bm w_i - \bm w^*\rangle - \frac 1{2\eta} \|\bm w_i - \bm w_{i+1}\|^2 && \text{梯度下降} \\ &= f(\bm w^*) + \frac 1{2 \eta} \|\bm w_i - \bm w^*\|^2 - \frac 1{2\eta}(\|\bm w_i - \bm w^*\|^2 - 2 \langle \bm w_i - \bm w_{i+1}, \bm w_i - \bm w^* \rangle + \|\bm w_i - \bm w_{i+1}\|^2) && 配方 \\ &= f(\bm w^*) + \frac 1{2 \eta} \|\bm w_i - \bm w^*\|^2 - \frac 1{2\eta} \|(\bm w_i - \bm w_{i+1}) - (\bm w_i - \bm w^*)\|^2 \\ &= f(\bm w^*) + \frac 1{2 \eta} (\|\bm w_i - \bm w^*\|^2 - \|\bm w_{i+1} - \bm w^*\|^2) \end{aligned}\]

\[\sum_{i=0}^{t - 1} (f(\bm w_{i+1}) - f(\bm w^*)) \leq \frac 1{2\eta} (\|\bm w_0 - \bm w^*\|^2 - \|\bm w_t - \bm w^*\|^2) \leq \frac 1{2\eta} \|\bm w_0 - \bm w^*\|^2 \]

由于 \(f(\bm w_i)\) 不升,

\[f(\bm w_t) \leq f(\bm w^*) + \frac 1{2\eta t}\|\bm w_0 - \bm w^*\|^2 \]

令总训练轮数

\[T = \frac{L\|\bm w_0 - \bm w^*\|^2}{2 \epsilon} \]

即可得到 \(f(\bm w_t) \leq f(\bm w^*) + \epsilon\)

接下来考虑一个很常用的技巧,随机梯度下降 (stochastic gradient descent, SGD)。如果我们每次都仅选取小批量数据计算梯度,那么便要考虑收敛性的问题。

\[\bm w_{t+1} = \bm w_t - \eta \bm G_t \]

\[\mathbb E[\bm G_t] = \nabla f(\bm w_t) \]

其中

\[\nabla f(\bm w, \bm X, \bm Y) = \frac 1N\sum_i \nabla l(\bm w, x_i, y_i) \]

\[\bm G_t = \frac 1{|S|} \sum_{i \in S} \nabla l(\bm w, x_i, y_i) \]

如果采取随机选取 \(S\) 的策略,我们可以不再考虑 \(\bm G_t\) 的由来,而是仅把其当作一个随机变量对待。

命题 \(f\) 是一个凸的 \(L\)-平滑函数,\(\bm w^* = \operatorname{argmin}_{\bm w} f(\bm w)\),采用学习速率 \(\eta \leq \frac 1L\) 且使得 \(\mathrm{Var}(\bm G_t) \leq \sigma^2\) 进行 \(t\) 轮梯度下降时,有

\[\mathbb E[f(\overline{\bm w_t})] \leq f(\bm w^*) + \frac{\|\bm w_0 - \bm w^*\|^2}{2t\eta} + \eta \sigma^2 \]

其中 \(\overline {\bm w_i} = \frac 1t \sum_{i=1}^t \bm w_i\)

证明 考虑转化为和 GD 类似的形式。一次项用期望的线性性,二次项用方差 \(\mathrm{Var}(\bm G_t) = \mathbb E\|\bm G_t\|^2 - (\mathbb E \bm G_t)^2 = \mathbb E\|\bm G_t\|^2 - \|\nabla f(\bm w_i)\|^2\)。由此不断转化 \(\bm G_i\)\(\nabla f(\bm w_i)\),分离固定部分和随机部分。

\[\begin{aligned} E[f(\bm w_{i+1})] &\leq f(\bm w_i) + \mathbb E\langle \nabla f(\bm w_i), \bm w_{i+1} - \bm w_i \rangle + \frac L2\mathbb E\|\bm w_{i+1} - \bm w_i\|^2 && \text{Smoothness} \\ &= f(\bm w_i) + \langle \nabla f(\bm w_i), \mathbb E(\bm w_{i+1} - \bm w_i) \rangle + \frac L2\mathbb E\|\bm w_{i+1} - \bm w_i\|^2 && 期望的线性性 \\ &= f(\bm w_i) - \eta \langle \nabla f(\bm w_i), \nabla f(\bm w_i) \rangle + \frac{L\eta^2}2 \mathbb E\|\bm G_i\|^2 \\ &= f(\bm w_i) - \eta \langle \nabla f(\bm w_i), \nabla f(\bm w_i) \rangle + \frac{L\eta^2}2(\|\nabla f(\bm w_i) \|^2 + \mathrm{Var}(\bm G_i)) && \mathrm{Var}(\bm G_t) = \mathbb E\|\bm G_t\|^2 - \|\nabla f(\bm w_i)\|^2 \\ &\leq f(\bm w_i) - \eta \left(1 - \frac{L \eta}2\right) \|\nabla f(\bm w_i)\|^2 + \frac{L\eta^2}2 \sigma^2 \\ &\leq f(\bm w_i) - \frac \eta 2 \|\nabla f(\bm w_i)\|^2 + \frac \eta 2 \sigma^2 && \eta < \frac 1L \\ &\leq f(\bm w^*) + \langle \nabla f(\bm w_i), \bm w_i - \bm w^* \rangle - \frac \eta 2\|\nabla f(\bm w_i)\|^2 + \frac \eta 2\sigma^2 \\ &\leq f(\bm w^*) - \frac 1 \eta\mathbb E\langle \bm w_{i+1} - \bm w_i, \bm w_i - \bm w^* \rangle - \frac \eta 2\|\bm G_i\|^2 + \eta\sigma^2 && \|\nabla f(\bm w_i)\|^2 = \mathbb E\|\bm G_i\|^2 - \mathrm{Var}(\bm G_i) \\ &= \frac 1{2\eta} \mathbb E(\|\bm w_i - \bm w^*\|^2 - \|\bm w_{i+1} - \bm w^*\|^2) + \eta \sigma^2 && 同 \text{ GD} \\ \end{aligned}\]

\[\begin{aligned} \mathbb E[f(\overline{\bm w_t})] - f(\bm w^*) &= \mathbb Ef\left(\frac 1t\sum_{i=1}^t \bm w_t\right) - f(\bm w^*) \\ &\leq \frac 1t\mathbb E\left(\sum_{i=1}^t f(\bm w_i)\right) - f(\bm w^*) && \text{Jensen's Ineq} \\ &\leq \frac 1t\sum_{i=0}^{t - 1} (\mathbb Ef(\bm w_{i+1}) - f(\bm w^*)) \\ &\leq \frac 1{2\eta t}(\|\bm w_0 - \bm w^*\|^2 - \mathbb E\|\bm w_t - \bm w^*\|^2) + \eta \sigma^2 \\ &\leq \frac 1{2\eta t}\|\bm w_0 - \bm w^*\|^2 + \eta \sigma^2 \end{aligned}\]

\[T = \frac{2 \|\bm w_0 - \bm w^*\|^2 \sigma^2}{\epsilon^2}, \eta = \frac \epsilon {2\sigma^2} \]

即可得到 \(\mathbb Ef(\overline{\bm w_t}) \leq f(\bm w^*) + \epsilon\)

也就是说,误差项是不随 \(t\) 改变的,因此只能通过缩小学习速率降低误差。这导致 GD 有 \(\frac 1T\) 的收敛速率时 SGD 只有 \(\frac 1{\sqrt T}\) 的收敛速率。

posted @ 2023-10-05 17:41  shiys22  阅读(393)  评论(0编辑  收藏  举报