# Hochreiter LSTM (p. 4): Maximal values of logistic sigmoid derivative times weight

My questions follow the below page 4 excerpt from Hochreiter's LSTM paper:

If $f_{l_{m}}$ is the logistic sigmoid function, then the maximal value of $f^\prime_{l_{m}}$ is 0.25. If $y^{l_{m-1}}$ is constant and not equal to zero, then $|f^\prime_{l_{m}}(net_{l_{m}})w_{l_{m}l_{m-1}}|$ takes on maximal values where

$w_{l_{m}l_{m-1}} = {1 \over y^{l_{m-1}}} \coth \left( {1 \over 2}net_{l_{m}} \right)$,

goes to zero for $|w_{l_{m}l_{m-1}}| \rightarrow \infty$, and is less than 1.0 for $|w_{l_{m}l_{m-1}}| < 4.0$.

The derivative of the sigmoid $f_{l_{m}} = f^\prime_{l_{m}} = \sigma$, is $\sigma(1-\sigma)$, so of course its maximum value is 0.25.

But I don't understand the following:

1. Where does the $\coth$ come from?
2. Why would $y^{l_{m-1}}$ be static? My understanding is that it is the activation of the unit, which changes each timestep.
3. Multiplying a non-zero positive number, which $f^\prime_{l_{m}}$ could be, by $|w_{l_{m}l_{m-1}}|$ as it approaches infinity would make it approach infinity, not zero. What am I missing here?

This paper assume $$f_i$$ is sigmoid function $$f_i(x) = \sigma(x) = \frac{1}{1 + e^{-x}}$$.

Note that

$$\frac{\partial \sigma(x)}{\partial x} = \sigma(x) (1 - \sigma(x))$$

Since

\begin{align*} & f'_{l_m}\big(\text{net}_{l_m}(t - m)\big) w_{l_m l_{m - 1}} \\ & = \sigma\big(\text{net}_{l_m}(t - m)\big) \cdot \Big(1 - \sigma\big(\text{net}_{l_m}(t - m)\big)\Big) \cdot w_{l_m l_{m - l}}, \tag{1} \end{align*}

to find the maximum value of (1) with respect to $$w_{l_m l_{m - 1}}$$, we can calculate the derivative of (1) and find the point $$w_{l_m l_{m - 1}}^*$$ where the derivative of (1) evaluated at $$w_{l_m l_{m - 1}}^*$$ equals to 0, i.e.,

$$\frac{\partial \Big[\sigma\big(\text{net}_{l_m}(t - m)\big) \cdot \Big(1 - \sigma\big(\text{net}_{l_m}(t - m)\big)\Big) \cdot w_{l_m l_{m - l}}\Big]}{\partial w_{l_m l_{m - 1}}} = 0 \tag{2}$$

Now we calculate the derivative

\begin{align*} & \frac{\partial \Big[\sigma\big(\text{net}_{l_m}(t - m)\big) \cdot \Big(1 - \sigma\big(\text{net}_{l_m}(t - m)\big)\Big) \cdot w_{l_m l_{m - l}}\Big]}{\partial w_{l_m l_{m - 1}}} \\ & = \frac{\partial \sigma\big(\text{net}_{l_m}(t - m)\big)}{\partial \text{net}_{l_m}(t - m)} \cdot \frac{\partial \text{net}_{l_m}(t - m)}{\partial w_{l_m l_{m - 1}}} \cdot \Big(1 - \sigma\big(\text{net}_{l_m}(t - m)\big)\Big) \cdot w_{l_m l_{m - l}} \\ & \quad + \sigma\big(\text{net}_{l_m}(t - m)\big) \cdot \frac{\partial \Big(1 - \sigma\big(\text{net}_{l_m}(t - m)\big)\Big)}{\partial \text{net}_{l_m}(t - m)} \cdot \frac{\partial \text{net}_{l_m}(t - m)}{\partial w_{l_m l_{m - 1}}} \cdot w_{l_m l_{m - l}} \\ & \quad + \sigma\big(\text{net}_{l_m}(t - m)\big) \cdot \Big(1 - \sigma\big(\text{net}_{l_m}(t - m)\big)\Big) \cdot \frac{\partial w_{l_m l_{m - 1}}}{\partial w_{l_m l_{m - 1}}} \\ & = \sigma\big(\text{net}_{l_m}(t - m)\big) \cdot \Big(1 - \sigma\big(\text{net}_{l_m}(t - m)\big)\Big)^2 \cdot y_{l_{m - 1}}(t - m - 1) \cdot w_{l_m l_{m - 1}} \\ & \quad - \Big(\sigma\big(\text{net}_{l_m}(t - m)\big)\Big)^2 \cdot \Big(1 - \sigma\big(\text{net}_{l_m}(t - m)\big)\Big) \cdot y_{l_{m - 1}}(t - m - 1) \cdot w_{l_m l_{m - 1}} \\ & \quad + \sigma\big(\text{net}_{l_m}(t - m)\big) \cdot \Big(1 - \sigma\big(\text{net}_{l_m}(t - m)\big)\Big) \\ & = \Big[2 \Big(\sigma\big(\text{net}_{l_m}(t - m)\big)\Big)^3 - 3 \Big(\sigma\big(\text{net}_{l_m}(t - m)\big)\Big)^2 + \sigma\big(\text{net}_{l_m}(t - m)\big)\Big] \cdot \\ & \quad \quad y_{l_{m - 1}}(t - m - 1) \cdot w_{l_m l_{m - 1}} \\ & \quad + \sigma\big(\text{net}_{l_m}(t - m)\big) \cdot \Big(1 - \sigma\big(\text{net}_{l_m}(t - m)\big)\Big) \\ & = \sigma\big(\text{net}_{l_m}(t - m)\big) \cdot \Big(2 \sigma\big(\text{net}_{l_m}(t - m)\big) - 1\Big) \Big(\sigma\big(\text{net}_{l_m}(t - m)\big) - 1\Big) \cdot \\ & \quad \quad y_{l_{m - 1}}(t - m - 1) \cdot w_{l_m l_{m - 1}} \\ & \quad + \sigma\big(\text{net}_{l_m}(t - m)\big) \cdot \Big(1 - \sigma\big(\text{net}_{l_m}(t - m)\big)\Big) \\ & = 0. \end{align*} \tag{3}

The last equality in (3) follows from (2). By swapping terms we can further reduce our equation:

\begin{align*} & \sigma\big(\text{net}_{l_m}(t - m)\big) \cdot \Big(2 \sigma\big(\text{net}_{l_m}(t - m)\big) - 1\Big) \Big(1 - \sigma\big(\text{net}_{l_m}(t - m)\big)\Big) \cdot \\ & \quad \quad y_{l_{m - 1}}(t - m - 1) \cdot w_{l_m l_{m - 1}} = \sigma\big(\text{net}_{l_m}(t - m)\big) \cdot \Big(1 - \sigma\big(\text{net}_{l_m}(t - m)\big)\Big) \\ \implies & \Big(2 \sigma\big(\text{net}_{l_m}(t - m)\big) - 1\Big) \cdot y_{l_{m - 1}}(t - m - 1) \cdot w_{l_m l_{m - 1}} = 1 \\ \implies & w_{l_m l_{m - 1}} = \frac{1}{y_{l_{m - 1}}(t - m - 1)} \cdot \frac{1}{2 \sigma\big(\text{net}_{l_m}(t - m)\big) - 1} \\ \implies & w_{l_m l_{m - 1}} = \frac{1}{y_{l_{m - 1}}(t - m - 1)} \cdot \coth\bigg(\frac{\text{net}_{l_m}(t - m)}{2}\bigg). \end{align*} \tag{4}

The last implication in (4) use the following equations:

\begin{align*} \tanh(x) & = 2 \sigma(2x) - 1 \\ \tanh(\frac{x}{2}) & = 2 \sigma(x) - 1 \\ \coth(\frac{x}{2}) & = \frac{1}{\tanh(\frac{x}{2})} = \frac{1}{2 \sigma(x) - 1} \end{align*}

The lengthy writing above should be enough to answer your question 1.

For 2., I believe is just an assumption to make analysis easier. But I agree it's not a good assumption (why? you can see that even if we let $$y_{l_{m - 1}}(t - m - 1)$$ non-zero, we can still make $$\text{net}_{l_m}(t - m) = 0$$, which make $$\coth$$ undefined).

For 3., since $$f_{l_m}'$$ is sigmoid function, which use exponential in it, we can see that

\begin{align*} & \lim_{w_{l_m l_{m-1}} \to \infty} \frac{w_{l_m l_{m - 1}}}{1 + e^{-\text{net}_{l_m}(t - m)}} \frac{e^{-\text{net}_{l_m}(t - m)}}{1 + e^{-\text{net}_{l_m}(t - m)}} \\ &= \lim_{w_{l_m l_{m-1}} \to \infty} \frac{w_{l_m l_{m - 1}}}{1 + e^{-\sum_{l_m^*} w_{l_m^* l_{m - 1}} y_{l_{m - 1}}(t - m - 1) }} \frac{e^{-\sum_{l_m^*} w_{l_m^* l_{m - 1}} y_{l_{m - 1}}(t - m - 1)}}{1 + e^{-\sum_{l_m^*} w_{l_m^* l_{m - 1}} y_{l_{m - 1}}(t - m - 1)}} \\ &= 0. \end{align*}

$$\frac{\partial \vartheta(t - q)}{\partial \vartheta(t)} = \sum_{l_1 = 1}^n \cdots \sum_{l_{q - 1} = 1}^n \prod_{m = 1}^q f_{l_{m}}'(\text{net}_{l_m}(t - m)) w_{l_{m - 1} l_m}$$
(he wrote $$w_{l_m l_{m - 1}}$$ instead of $$w_{l_{m - 1} l_m}$$.)