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<!DOCTYPE html>
<html>
<head>
<title>14.2 - The problems with depth [Andrei Bursuc]</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/>
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<textarea id="source">
layout: true
.center.footer[Marc LELARGE and Andrei BURSUC | Deep Learning Do It Yourself | 14.2 The problems with depth]
---
class: center, middle, title-slide
count: false
## Going Deeper
# 14.2 The problems with depth
<br/>
<br/>
.bold[Andrei Bursuc ]
<br/>
url: https://dataflowr.github.io/website/
.citation[
With slides from A. Karpathy, F. Fleuret, G. Louppe, C. Ollion, O. Grisel, Y. Avrithis ...]
---
class: middle
.bigger[Although it was known that _deeper is better_, for decades training deep neural networks was highly challenging and unstable. Besides limited hardware and data there were a few algorithmic flaws that have been fixed/softened in the last decade.]
---
# Vanishing gradients
Training deep MLPs with many layers has for long (pre-2011) been very difficult due to the **vanishing gradient** problem.
- Small gradients slow down, and eventually block, stochastic gradient descent.
- This results in a limited capacity of learning.
.center.width-60[<img src="images/part14/vanishing-gradient.png">]
.caption[Backpropagated gradients normalized histograms (Glorot and Bengio, 2010).<br> Gradients for layers far from the output vanish to zero. ]
.citation[X. Glorot and Y. Bengio, Understanding the difficulty of training deep feedforward neural networks; AISTAT 2010]
---
Consider a simplified 3-layer MLP, with $x, w\_1, w\_2, w\_3 \in\mathbb{R}$, such that
$$f(x; w\_1, w\_2, w\_3) = \sigma\left(w\_3\sigma\left( w\_2 \sigma\left( w\_1 x \right)\right)\right). $$
Under the hood, this would be evaluated as
$$\begin{aligned}
u\_1 &= w\_1 x \\\\
u\_2 &= \sigma(u\_1) \\\\
u\_3 &= w\_2 u\_2 \\\\
u\_4 &= \sigma(u\_3) \\\\
u\_5 &= w\_3 u\_4 \\\\
\hat{y} &= \sigma(u\_5)
\end{aligned}$$
.credit[Credits: G. Louppe, [INFO8010 Deep Learning](https://github.com/glouppe/info8010-deep-learning), ULiège]
--
count:false
and its derivative $\frac{\text{d}\hat{y}}{\text{d}w\_1}$ as
$$\begin{aligned}\frac{\text{d}\hat{y}}{\text{d}w\_1} &= \frac{\partial \hat{y}}{\partial u\_5} \frac{\partial u\_5}{\partial u\_4} \frac{\partial u\_4}{\partial u\_3} \frac{\partial u\_3}{\partial u\_2}\frac{\partial u\_2}{\partial u\_1}\frac{\partial u\_1}{\partial w\_1}\\\\
&= \frac{\partial \sigma(u\_5)}{\partial u\_5} w\_3 \frac{\partial \sigma(u\_3)}{\partial u\_3} w\_2 \frac{\partial \sigma(u\_1)}{\partial u\_1} x
\end{aligned}$$
---
class: middle
The derivative of the sigmoid activation function $\sigma$ is:
.center[<img src="images/part14/activation-grad-sigmoid.png">]
$$\frac{\text{d} \sigma}{\text{d} x}(x) = \sigma(x)(1-\sigma(x))$$
Notice that $0 \leq \frac{\text{d} \sigma}{\text{d} x}(x) \leq \frac{1}{4}$ for all $x$.
.credit[Credits: G. Louppe, [INFO8010 Deep Learning](https://github.com/glouppe/info8010-deep-learning), ULiège]
---
class: middle
Assume that weights $w\_1, w\_2, w\_3$ are initialized randomly from a Gaussian with zero-mean and small variance, such that with high probability $-1 \leq w\_i \leq 1$.
Then,
$$\frac{\text{d}\hat{y}}{\text{d}w\_1} = \underbrace{\frac{\partial \sigma(u\_5)}{\partial u\_5}}\_{\leq \frac{1}{4}} \underbrace{w\_3}\_{\leq 1} \underbrace{\frac{\partial \sigma(u\_3)}{\partial u\_3}}\_{\leq \frac{1}{4}} \underbrace{w\_2}\_{\leq 1} \underbrace{\frac{\sigma(u\_1)}{\partial u\_1}}\_{\leq \frac{1}{4}} x$$
This implies that the gradient $\frac{\text{d}\hat{y}}{\text{d}w\_1}$ **exponentially** shrinks to zero as the number of layers in the network increases.
Hence the vanishing gradient problem.
- In general, bounded activation functions (sigmoid, tanh, etc) are prone to the vanishing gradient problem.
- Note the importance of a proper initialization scheme.
.credit[Credits: G. Louppe, [INFO8010 Deep Learning](https://github.com/glouppe/info8010-deep-learning), ULiège]
---
# Rectified linear units
Instead of the sigmoid activation function, modern neural networks
are for most based on **rectified linear units** (ReLU) (Glorot et al, 2011):
$$\text{ReLU}(x) = \max(0, x)$$
.center[<img src="images/part14/activation-relu.png">]
.credit[Credits: G. Louppe, [INFO8010 Deep Learning](https://github.com/glouppe/info8010-deep-learning), ULiège]
---
class: middle
Note that the derivative of the ReLU function is
$$\frac{\text{d}}{\text{d}x} \text{ReLU}(x) = \begin{cases}
0 &\text{if } x \leq 0 \\\\
1 &\text{otherwise}
\end{cases}$$
.center[<img src="images/part14/activation-grad-relu.png">]
For $x=0$, the derivative is undefined. In practice, it is set to zero.
.credit[Credits: G. Louppe, [INFO8010 Deep Learning](https://github.com/glouppe/info8010-deep-learning), ULiège]
---
class: middle
Therefore,
$$\frac{\text{d}\hat{y}}{\text{d}w\_1} = \underbrace{\frac{\partial \sigma(u\_5)}{\partial u\_5}}\_{= 1} w\_3 \underbrace{\frac{\partial \sigma(u\_3)}{\partial u\_3}}\_{= 1} w\_2 \underbrace{\frac{\partial \sigma(u\_1)}{\partial u\_1}}\_{= 1} x$$
This **solves** the vanishing gradient problem, even for deep networks! (provided proper initialization)
Note that:
- The ReLU unit dies when its input is negative, which might block gradient descent.
- This is actually a useful property to induce *sparsity*.
- This issue can also be solved using **leaky** ReLUs, defined as $$\text{LeakyReLU}(x) = \max(\alpha x, x)$$ for a small $\alpha \in \mathbb{R}^+$ (e.g., $\alpha=0.1$).
---
class: middle
The steeper slope in the loss surface speeds up the training.
.center.width-30[]
.citation[A. Krizhevsky et al., ImageNet Classification with Deep Convolutional Neural Networks; NIPS 2012]
---
# Other activation functions
- Several $\text{ReLU}$-like alternatives have been proposed in recent years: $\text{PReLU}$, $\text{ELU}$, $\text{SeLU}$, $\text{SReLU}$
.center.width-50[]
.citation[[Visualising Activation Functions in Neural Networks](https://dashee87.github.io/data%20science/deep%20learning/visualising-activation-functions-in-neural-networks/)]
---
class: center, middle
# Regularization
---
# Under-fitting and over-fitting
What if we consider a hypothesis space $\mathcal{F}$ in which candidate functions $f$ are either too "simple" or too "complex" with respect to the true data generating process?
.center[]
---
class: middle
.center.width-40[]
Our goal is to find a function $f$ that makes good predictions on average from given data.
We can consider $f$ as a polynomial of degree $D$ defined through its parameters $\mathbf{w} \in \mathbb{R}^D$ such that
$$\hat{y} = f(x; \mathbf{w}) = \sum\_{d=0}^D w\_d x^d$$
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
Although it is difficult to define precisely, it is quite clear in practice how to increase or decrease it for a given class of models. For example:
- The degree of polynomials;
- The number of layers in a neural network;
- The number of training iterations;
- Regularization terms.
---
class: middle
# Regularization
---
.center.width-100[]
---
# Regularization
We can reformulate the previously used squared error loss
$$\ell(y, f(x;\mathbf{w})) = (y - f(x;\mathbf{w}))^2$$
to
$$\ell(y, f(x;\mathbf{w})) = (y - f(x;\mathbf{w}))^2 + \rho \sum_{d}^{D} w\_d^2$$
<br>
.Q.big.center[What will happen now?]
---
# Regularization
We can reformulate the previously used squared error loss
$$\ell(y, f(x;\mathbf{w})) = (y - f(x;\mathbf{w}))^2$$
to
$$\ell(y, f(x;\mathbf{w})) = (y - f(x;\mathbf{w}))^2 + \rho \sum_{d}^{D} w\_d^2$$
<br>
.center.big[This is called __$L\_2$ regularization__.]
---
class: middle
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
class: middle
count: false
.center.width-60[]
.credit[Credits: F. Fleuret, [EE-559 Deep Learning](https://fleuret.org/dlc/), EPFL]
---
# $L\_2$ regularization
- in Deep Learning it is ofter referred to as __weight decay__
```py
torch.optim.SGD(params, lr=<required parameter>, momentum=0, dampening=0, weight_decay=0, nesterov=False)
```
```py
torch.optim.Adam(params, lr=0.001, betas=(0.9, 0.999), eps=1e-08, weight_decay=0, amsgrad=False)
```
---
class: middle, center
.bigger[So far we have explored generic regularization methods. In the following we will learn deep learning specific regularization strategies.]
---
class: end-slide, center
count: false
The end.
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