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<?xml version = "1.0" encoding = "UTF-8"?>
<xsl:stylesheet version = "1.0" xmlns:xsl = "http://www.w3.org/1999/XSL/Transform">
<xsl:template match = "/">
<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<title>Memristec Hands-On Tutorial</title>
<meta name="description" content="NNML">
<meta name="author" content="Emre Neftci">
<meta name="apple-mobile-web-app-capable" content="yes">
<meta name="apple-mobile-web-app-status-bar-style" content="black-translucent">
<link rel="stylesheet" href="dist/reset.css">
<link rel="stylesheet" href="dist/reveal.css">
<link rel="stylesheet" href="nmilab.css">
<!-- Theme used for syntax highlighted code -->
<link rel="stylesheet" href="plugin/highlight/monokai.css">
<script src="jquery.js"></script>
<script>
$(function(){
$("#sdlides").load("slides_1_inner.html");
});
</script>
</head>
<body>
<div class="reveal">
<div class="slides">
<section data-markdown data-vertical-align-top data-background-color=#B2BA67><textarea data-template>
<h1> Neuromorphic Computing and Algorithms<br/> </h1>
</textarea></section>
<section data-markdown data-vertical-align-top><textarea data-template>
<h2> Focus of this Lecture:<br/> Machine Learning and Neural Networks for Neuromorphic Engineering </h2>
<img src= img/pgi15_triangle.png class=large />
<p> What you'll learn: </p>
<ul>
<li/> Foundations of Neural Networks (Perceptrons, gradient descent)
<li/> Neural networks and their implementation in Python/PyTorch
<li/> Basic hardware-aware training using Pytorch and IBM AI hardware kit
</ul>
</textarea>
</section>
<script>
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document.getElementById('bval').innerHTML =eval(b).toFixed(2);
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<section>
<h2>The First Artificial Neuron</h2><ul>
<li><p>In 1943, McCulloch and Walter Pitts propose the first artificial neuron, the Linear Threshold Unit. </p>
<img src="images/artificial_neuron.png" class="large"/>
</li>
<li>In the Linear Threshold Unit, $f$ is a step function: $f(x) = 1$ if $x>0$
</li>
<li>"Modern" artificial neurons are similar, but $f$ is typically a sigmoid or rectified linear function</li>
</ul>
</section>
<section>
<h2>Basic Mathematical Model of the Artificial Neuron</h2>
<div class=row>
<div class=column>
<img src="images/artificial_neuron.png"/>
</div>
<div class=column>
<ul>
<li>$x_i$ is the state of the input neurons</li>
<li>$w_i$ is the weight of the connection</li>
<li>$b$ is a bias</li>
<li>The total input to the neuron is: $ a = \sum_i w_i x_i +b $</li>
<li>The output of the neuron is: $ y = f(a) $</li>
<li>where $f$ is the activation function</li>
</ul>
</div>
</div>
</section>
<section data-markdown><textarea data-template>
<h2>The Perceptron</h2>
<img src="images/rosenblatt57_title.png" />
<blockquote>
<img src="images/rosenblatt57_quote1.png" class=small />
</blockquote>
<ul>
<li/> Further reading: <a href="https://news.cornell.edu/stories/2019/09/professors-perceptron-paved-way-ai-60-years-too-soon">Professor’s perceptron paved the way for AI – 60 years too soon </a>
</ul>
</textarea></section>
<section>
<h2>The Perceptron</h2>
<ul>
<li> The Perceptron is a special case of the artificial neuron where:
$$
\begin{eqnarray}
\mbox{y} & = & \begin{cases}
-1 & \mbox{if } a = \sum_j w_j x_j + b \leq 0 \\\\
1 & \mbox{if } a = \sum_j w_j x_j + b > 0
\end{cases}
\end{eqnarray}
$$</li>
<img src=images/single_perceptron.svg />
<li> Three inputs $x_1$, $x_2$, $x_3$ with weights $w_1$, $w_2$, $w_3$, and bias $b$</li>
</ul>
</section>
<section>
<h2> Perceptron Example</h2>
<ul>
<li/> Like McCulloch and Pitts neurons, Perceptrons can be hand-constructed to solve simple logical tasks
<li/> Let's build a "sprinkler" that activates only if it is dry and sunny.
<li/> Let's assume we have a dryness detector $x_0$ and a light detector $x_1$ (two inputs)
<li/> Find $w_0$, $w_1$ and $b$ such that output $y$ matches target $t$
</ul>
<div class=row>
<div class=column>
<img src="images/twonode_perceptron_template.svg" style="height:200px" />
</div>
<div class=column>
<table>
<thead>
<tr>
<th>Sunny</th>
<th>Dry</th>
<th>$a$</th>
<th>$y$</th>
<th>$t$</th>
</tr>
</thead>
<tbody>
<tr>
<td>1 (yes)</td>
<td>1 (yes)</td>
<td> <div id="a11"></div></td>
<td> <div id="y11"></div></td>
<td>1</td>
</tr>
<tr>
<td>1 (yes)</td>
<td>0 (no)</td>
<td> <div id="a10"></div></td>
<td> <div id="y10"></div></td>
<td>0</td>
</tr>
<tr>
<td>0 (no)</td>
<td>1 (yes)</td>
<td> <div id="a01"></div></td>
<td> <div id="y01"></div></td>
<td>0</td>
</tr>
<tr>
<td>0 (no)</td>
<td>0 (no)</td>
<td> <div id="a00"/></div></td>
<td> <div id="y00"/></div></td>
<td>0</td>
</tr>
</tbody>
</table>
</div>
</div>
<table border="1">
<tr>
<td>$w_0 =$ <span id=w0val>0</span></td>
<td>$w_1 =$ <span id=w1val>0</span></td>
<td>$b =$ <span id=bval>0</span></td>
</tr>
<tr>
<td colspan="1"><input type="range" min="-50." max="100." step=0.01 onchange="solve();" id="w0" /></td>
<td colspan="1"><input type="range" min="-50." max="100." step=0.01 onchange="solve();" id="w1" /></td>
<td colspan="1"><input type="range" min="-50." max="100." step=0.01 onchange="solve();" id="b" /></td>
</tr>
</table>
</section>
<section data-markdown><textarea data-template>
## The Perceptron Learning Rule
<img src=images/perceptron_convergence.png class=large />
<p class=ref>(Bishop, 2006 Pattern Recognition and Machine Learning)</p>
- Perceptron convergence theorem: if the training dataset is linearly separable, then the perceptron learning rule is guaranteed to find an exact solution
<p class=ref>(Rosenblatt, 1962, Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms)</p>
</textarea></section>
<section data-markdown><textarea data-template>
## Cost Functions
<ul>
<li/> The Cost (Error) function returns a number representing how well a model performed.
<li/> Perceptrons: Cost function = Number of Misclassified Samples
<li/> Other common cost functions are
<ul>
<li/> Mean Squared Error: $ C_\text{MSE} = \frac{1}{2N} \sum_{n \in \text{train}} (\mathbf{y}^n - \mathbf{t}^n) ^2 $
<li/> Cross-Entropy: $ C_{XENT} = - \frac1N \sum_{n \in \text{train}} \sum_k y_{k}^n \log t_{k}^n $
</ul>
<li /> The objective is to minimize the cost function.
<li /> Cost functions can be minimized using an optimization algorithm
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
## Optimization Algorithm Gradient Descent
Example: Find $x$ that minimizes $C(x) = x^2$
<img src=images/quadratix_function.png class=small />
- Incremental change in $\Delta x$:
$$
\begin{eqnarray}
\Delta C \approx \underbrace{\frac{\partial C}{\partial x}}_{\text{=Slope of }C(x)} \Delta x
\end{eqnarray}
$$
With $\Delta x = - \eta \frac{\partial C}{\partial x}$, $\Delta C \approx - \eta \left( \frac{\partial C}{\partial x} \right)^2$
- Gradient Descent for finding the optimal $x$:
$
\begin{eqnarray}
x \leftarrow x - \eta \frac{\partial C}{\partial x}
\end{eqnarray}
$
</textarea></section>
<section data-markdown><textarea data-template>
## Linear separability
A perceptron is equivalent to a decision boundary.
- A straight line can separate blue vs. red
<img src=images/perceptron_rain_hyperplane.png class="small">
- There is no straight line that can separate blue vs. red <!-- .element: class="fragment" -->
<img src=images/perceptron_xor_hyperplane.png class="small">
<p class="pl">Problems where a straight line can separate two classes are called <em>Linearly Separable</em></p>
</textarea></section>
<section data-markdown><textarea data-template>
## Limitations of Perceptrons
The limitation of a Perceptron to linearly separable problems caused its downfall:
<img src="images/perceptrons_minsky_papert.png" class=stretch />
<p class=ref> Minsky and Papert, 1969 </p>
</textarea></section>
<section data-markdown><textarea data-template>
## XOR can be solved with an intermediate perceptron
<div class=row>
<div class=column>
<img src="images/perceptron_xor_hyperplane.svg" class="large" />
</div>
<div class=column>
<img src="images/perceptron_xor_hyperplane_extended.svg" class=large />
</div>
</div>
- We need an intermediate unit that is on only when $x_1$ and $x_2$ are both on.
</textarea></section>
<section data-markdown><textarea data-template>
## A Neural Network Solving XOR
- Find the parameters in the following network:
<div class=row>
<div class=column>
<img src="images/xor_template.svg" class="stretch" />
</div>
<div class=column>
<table>
<tbody><tr bgcolor="#ddeeff" align="center">
<td colspan="2"><b>INPUT</b></td>
<td><b>Y</b>
</td></tr>
<tr bgcolor="#ddeeff" align="center">
<td>A</td>
<td>B</td>
<td>A XOR B
</td></tr>
<tr bgcolor="#ddffdd" align="center">
<td>0</td>
<td>0</td>
<td>0
</td></tr>
<tr bgcolor="#ddffdd" align="center">
<td>0</td>
<td>1</td>
<td>1
</td></tr>
<tr bgcolor="#ddffdd" align="center">
<td>1</td>
<td>0</td>
<td>1
</td></tr>
<tr bgcolor="#ddffdd" align="center">
<td>1</td>
<td>1</td>
<td>0
</td></tr></tbody>
</table>
</div>
<div class=fragment><p class=pl> The strategy of automatically extending networks with intermediate units is the main idea of representation learning </p></div>
</textarea></section>
<section>
<section data-markdown><textarea data-template>
## Neural Network
- We can connect Perceptrons together to form a multi-layered network.
<img src=images/mlp.png />
<ul>
<li/> If a neuron produces an input, it is called an <em>input neuron</em>
<li/> If a neuron's output is used as a prediction, we will call it an <em>output neuron</em>
<li/> If a neuron is neither and input or an output neuron, it is a <em>hidden neuron</em>
</ul>
- Increased level of abstraction from layer to layer
- Also called Multilayer Perceptrons (although units are not always Perceptrons)
</textarea></section>
<section data-markdown><textarea data-template>
## Multiple Linear Layers are Equivalent to one Single Layer
<ul>
<li/> A linear transformation is of the type:
$$
\mathbf{y} = W \mathbf{x}
$$
<li/> Linear networks are mathematically tractable, so why not build multilayer linear networks?
$$
\mathbf{y}^{(2)} = {W}^{(2)} ({W}^{(1)} \mathbf{x}^{(1)} )
$$
<li class=fragment /> Such a network is equivalent to a wide single layer network
$$
\begin{split}
\mathbf{y}^{(2)} &= V \mathbf{x}^{(1)} \\
V &= W^{(2)} W^{(1)}
\end{split}
$$
<li class=fragment /> Non-linearities preserve the composition of layers, and thus have more representational power
$$
\begin{split}
\mathbf{y}^{(2)} &= \sigma(W^{(2)}\sigma(W^{(1)}\mathbf{x}^{(1)} ) )
\end{split}
$$
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
## Deep neural networks
- Before:
<blockquote>
How many hidden layers and how many units per layer do we need? The answer is at most two
</blockquote>
<p class=ref> Hertz, <em>et al.</em> 1991</p>
- Now:
<blockquote>
<img src="images/gist_Szegedy_etal14_90deg.png" style="height:100px" />
</blockquote>
<p class=ref> Szegedy <em>et al.</em> 2014 </p>
<p class=pl> Deeper networks tend to have more representational power </p>
</textarea></section>
<section data-markdown><textarea data-template>
## Credit Assignment Problem
Multilayer (deep) networks are more powerful: how can one train multilayer networks?
<img src=images/mlp_gradient.png />
Two problems:
- Perceptrons' discontinuity
- Credit assignment: which hidden unit weight should we modify to reach a target output?
</textarea></section>
<section data-markdown><textarea data-template>
## Continuous Output Neurons (Sigmoid Neuron)
Neurons in deep neural networks are similar to Perceptrons, but with a continuous activation function
<div class="row">
<div class="column">
<p>Threshold unit (Perceptron)</p>
$$
y = \Theta(a) = \begin{cases} -1 & \mbox{if } a \leq 0 \\ 1 & \mbox{if } a > 0 \end{cases}
$$
</div>
<div class="column">
<p> Sigmoid unit</p>
$$
y = \sigma(\text{a}) = \frac{1}{1+e^{-a}}
$$
</div>
</div>
<div class="row">
<div class="column">
<img src="images/step_function.png" class=small />
</div>
<div class="column">
<img src="images/sigmoid_function.png" class=small />
</div>
</div>
$$
a = \sum_j w_j x_j + b
$$
- A function is continuous if the curve can be drawn "without lifting the pen"
</textarea></section>
<section data-markdown><textarea data-template>
## Single Layer Network with Sigmoid Units
Weight matrix: $W^{(1)} \in \mathbb{R}^{N\times M}$ (meaning $M$ inputs, $N$ outputs)
`$$
\begin{eqnarray}
y^{(1)}_i &=& \sigma(\underbrace{\sum_j W^{(1)}_{ij} x_j}_{a_i^{(1)}}) \\
\end{eqnarray}
$$`
- MSE cost function, assuming a single data sample $\mathbf{x}\in\mathbb{R}^{M} $, and target vector $\mathbf{t}\in\mathbb{R}^{N}$
`$$
C_{MSE} = \frac{1}{2} \sum_i(y^{(1)}_i - t_i)^2
$$`
- Gradient w.r.t. $W^{(1)}$ (in scalar form):
`$$
\frac{\partial }{\partial W^{(1)}_{ij}} C_\text{MSE}= \frac1N (y^{(1)}_i - t_i) \sigma'(a^{(1)}_i) x_j
$$`
</textarea></section>
<section data-markdown><textarea data-template>
## Single Layer Network with Sigmoid Units
<p> Neural networks operations are generally written in Matrix form</p>
<div class=row>
<div class=column>
<center>Scalar Form (one sample)</center>
$$
\begin{split}
a_i^{(1)} &= \sum_j W^{(1)}_{ij} x_j, \quad y^{(1)}_i = \sigma(a_i^{(1)}) \\
\delta_i^{(1)} &= \frac{1}{N^{s}} (y^{(1)}_i - t_i) \sigma'(a^{(1)}_i)\\
\Delta W_{ij} &= -\eta \delta_i x_j
\end{split}
$$
</div>
<div class=column>
<center>Matrix Form ($N^{s}$ samples)</center>
<div class=fragment>
$$
\begin{split}
A^{(1)} & = X W^{(1),T}, \quad
Y^{(1)} = \sigma(A^{(1)}) \\
\end{split}
$$
</div>
<div class=fragment>
$$
\begin{split}
\delta &= \frac{1}{N^{s}} (Y^{(1)} - T) \odot \sigma'(A^{(1)}) \\
\end{split}
$$
</div>
<div class=fragment>
$$
\begin{split}
\Delta W & = - \eta \delta^T \mathbf{X} \\
\end{split}
$$
</div>
</div>
</div>
$$
W \leftarrow W + \Delta W
$$
<ul>
<li /> $\delta$ $\in \mathbb{R}^{N^{s}\times N}$, $X$ $\in \mathbb{R}^{N^{s} \times M}$, and dimension of $\Delta W$ must be same as $W$!
<li /> Implement it here (Optional):
</ul>
[](https://colab.research.google.com/drive/1xQiNcjf-O88vi_09R5Zntv_gK2zB7MMh?usp=sharing)
</textarea></section>
<section data-markdown><textarea data-template>
## Two Layer Network with Sigmoid Units
- Two layers means we have two weight matrices $W^{(1)}$ and $W^{(2)}$
- $W^{(1)} \in \mathbb{R}^{N^{(1)}\times M}$, $W^{(2)} \in \mathbb{R}^{N^{(2)}\times N^{(1)}}$
- The output is a composition of two functions:
$$
\begin{eqnarray}
\mathbf{y}^{(1)} &=& \sigma(W^{(1)} \mathbf{x} ) \\\\
\mathbf{y}^{(2)} &=& \sigma(W^{(2)} \mathbf{y}^{(1)} ) \\\\
\end{eqnarray}
$$
<ul>
<li class=fragment /> Cost function $ C_{MSE} = \frac{1}{2N^{s}} \sum_{i=1}^{N^{(2)}}(y^{(2)}_i - t_i)^2 $
<li class=fragment /> Gradient wrt $W^{(2)}$ is: $ \frac{\partial }{\partial W^{(2)}_{ij}} C_\text{MSE}= \underbrace{(y^{(2)}_i - t_i) \sigma'(a^{(2)}_i)}_{\delta^{(2)}_i} y^{(1)}_j $
<li class=fragment /> Gradient wrt $W^{(1)}$ is: $ \frac{\partial}{\partial { W_{jk}^{(1)}}} C_{\text{MSE}} = \underbrace{(\sum_i \delta_i^{(2)} W^{(2)}_{ij}) \sigma'(a^{(1)}_j)}_{backpropagated\, error\, \delta^{(1)}_{j}} x_k $
<li class=fragment /> This is a special case of the gradient backpropagation algorithm
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
## The Gradient Back-Propagation Algorithm
1. Forward-propagate to compute $y^{(k)}$ for all layers $k$
2. Compute loss and error
3. Back-propagate error through network, *i.e* compute all $\mathbf{\delta}^{(k)}$
</textarea></section>
<section data-markdown data-background-color=#BBBBBB><textarea data-template>
## The Gradient BP algorithm (for your reference)
<div style='font-size:24px;text-align:left;' >
<p>The task of learning is to minimize a cost function $\mathcal{L}$ over the entire dataset.
In a neural network, this can be achieved by gradient descent, which modifies the network parameters $\mathbf{W}$ in the direction opposite to the gradient:
$$
\begin{split}
W_{ij} \leftarrow W_{ij} - \eta \Delta W_{ij}, & \text{where } \Delta W_{ij} =
\Dp{\mathcal{L}}{W_{ij}} =
\Dp{\mathcal{L}}{y_i}
\Dp{y_i}{ a_i }
\Dp{a_i}{W_{ij}}
\end{split}
$$
with $a_i = \sum_j W_{ij} x_j$ the total input to the neuron, $y_i$ is the output of neuron $i$, and $\eta$ a small learning rate.
The first term is the error of neuron $i$ and the second term reflects the sensitivity of the neuron output to changes in the parameter.
In multilayer networks, gradient descent is expressed as the BP of the errors starting from the prediction (output) layer to the inputs.
Using superscripts $l=0,...,L$ to denote the layer ($0$ is input, $L$ is output):
`$$
\frac{\mathrm{\partial}}{\mathrm{\partial} W^{(l)}_{ij}} \mathcal{L} =
\delta_{i}^{(l)} y^{(l-1)}_j,\text{ where }\delta_{i}^{(l)} = \sigma'\left(
a_i^{(l)} \right) \sum_k \delta_{k}^{(l+1)} W_{ik}^{\top,(l)},
$$`
where $\sigma'$ is the derivative of the activation function, and $\delta_{i}^{(L)}=\Dp{\mathcal{L}}{y_i^{(L)}}$ is the error of
output neuron $i$ and $y_{i}^{(0)}=x_i$ and $\top$ indicates the transpose.
Learning is typically carried out in forward passes (evaluation of the neural network activities) and backward passes (evaluation of $\delta$s).
</p>
</div>
</textarea></section>
</section>
<section>
<iframe data-src="http://playground.tensorflow.org/" width=95% style='max-height:720px;height:720px;max-width:100%;'></iframe>
</section>
<section>
<section data-markdown><textarea data-template>
## Neural Networks with PyTorch
- PyTorch is a machine learning framework that facilitates the implementation of Deep Learning
- Other tools such as Tensorflow or Theano have a similar purpose, but PyTorch is more versatile and easier to use.
- To use pytorch, remember to import it:
<pre><code class="Python" data-trim data-noescape> import torch </code></pre>
- Follow-along notebook
[](https://drive.google.com/open?id=1botgh24In3QFyMc9AmezPszgt2ntJ-OL)
- PyTorch is one out of several other ML frameworks (PyTorch, Tensorflow/Keras, Jax). PyTorch arguably the easiest to use and is <a href="https://pytorch.org/blog/PyTorchfoundation/?utm_source=twitter&utm_medium=organic_social&utm_campaign=foundation3a" >belongs to the linux foundation</a> (open source)
</textarea></section>
<section data-markdown><textarea data-template>
## Tensor
- The tensor is the basic data type of Pytorch (and many other tools likes tensorflow)
- A tensor is a multi-dimensional array
> For example, a color image could be encoded as a 3D tensor with dimensions of width, height, and color plane.
- Apart from dimensions, a tensor is characterized by the type of its elements (integer, float, double, byte, boolean etc.).
</textarea></section>
<section data-markdown><textarea data-template>
## Tensor Creation
Tensors can be created like numpy arrays
<ul>
<li > Numpy
</ul>
<pre><code class="Python" data-trim data-noescape>
a = np.array([[1,2,3],[3,2,1]])
</code></pre>
<div class=fragment >
<ul>
<li > PyTorch
</ul>
<pre><code class="Python" data-trim data-noescape>
b = torch.FloatTensor([[1,2,3],[3,2,1]])
b = torch.FloatTensor(a) #equivalent
</code></pre>
</div>
</textarea></section>
<section data-markdown><textarea data-template>
## Tensor Creation
- Tensors can also be converted from a numpy array
<pre><code class="Python" data-trim data-noescape>
n = np.zeros(shape=(3, 2))
c= torch.tensor(n)
</code></pre>
</textarea></section>
<section data-markdown><textarea data-template>
## Tensor Operations
There are many operations that can be performed on tensors. See http://pytorch.org/docs/. Some examples below
- Addition
<pre><code class="Python" data-trim data-noescape>
b + b
</code></pre>
- Multiplication
<pre><code class="Python" data-trim data-noescape>
b*b
</code></pre>
</textarea></section>
<section data-markdown><textarea data-template>
## Tensor Operations (continued)
There are many operations that can be performed on tensors. See http://pytorch.org/docs/. Some examples below
<ul>
<li/> Transpose
<pre><code class="Python" data-trim data-noescape>
b.transpose(0,1) #flips axes 0 and 1
</code></pre>
<li/> Matrix multiplication
<pre><code class="Python" data-trim data-noescape>
torch.mm(b,b.transpose(0,1))
[email protected](0,1)
</code></pre>
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
## Graph Representation of an Expression
Pytorch and other frameworks represent a function using a graph
<img src="images/slides2_compgraph.png" id="fwimg" style="height:150px"/>
<pre><code class="Python" data-trim data-noescape>
v1 = torch.tensor([1.0, 1.0], requires_grad=True)
v2 = torch.tensor([2.0, 2.0])
v_sum = v1 + v2
v_res = (v_sum*2).sum()
</code></pre>
</textarea></section>
<section data-markdown><textarea data-template>
## Automatic Differentiation
<ul>
<li/> A key feature of machine learning frameworks is automatic differentiation
<li/> With automatic differentation, the gradients are computed automatically across a set of operations
<li/> Once a backward operation is called on a node, the gradients of all <i>leaf</i> nodes and parameters in the expression are numerically computed
<pre><code class="Python" data-trim data-noescape>
v_res.backward() #this command computes the gradient
v1.grad # returns tensor([2., 2.])
</code></pre>
<li class=fragment /> The gradient of v2 is None because we did not enable requires_grad
<pre><code class="Python" data-trim data-noescape>
v2.grad # returns None
</code></pre>
<pre><code class="Python" data-trim data-noescape>
v2.requires_grad #returns False
</code></pre>
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
## GPUs and Tensors
<ul>
<li/> By default, tensors are stored in the CPU (main memory)
<pre><code class="Python" data-trim data-noescape>
b.device
</code></pre>
<li/> GPUs can be >100x faster on certain operations compared to CPUs
<li/> Tensors can be moved to the GPU using the cuda() function
<pre><code class="Python" data-trim data-noescape>
b_cuda = b.cuda(); b_cuda = b.to('cuda') #Both have the same effect
</code></pre>
<li/> If you get a runtime error, it means you didn't request for a gpu. go to Runtime> Change Runtime Type>Hardware accelerator and choose GPU, then SAVE
<img src=images/colab_runtime_cuda.png class=small />
<li/> All operations must occur on the same device. For example, you cannot add a cuda tensor to a cpu tensor (you need to cast it to cpu first).
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
## Neural Network Building Blocks: Modules
<ul>
Up to now, nothing was specific for neural networks:
<li/> Machine learning frameworks can be used for all types of computations
PyTorch has predefined modules for constructing neural networks
<li/> All neural network building blocks are PyTorch modules
<pre><code class="Python" data-trim data-noescape>
torch.nn.Module
</code></pre>
<li/> Modules are containers for functions, tensors and parameters
<li/> Modules can be called like functions. (but to call them, you need to implement the forward function)
<pre><code class="Python" data-trim data-noescape>
my_module = torch.nn.Module()
my_module() #returns NotImplementedError
my_module.forward() #same as above
</code></pre>
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
## Example: the linear module
- nn.Linear is a class that implements a module
- For example the nn.Linear module defines the linear transformation $$y = Wx + b$$
<pre><code class="Python" data-trim data-noescape>
lin = torch.nn.Linear(in_features = 2, out_features = 5)
x = torch.FloatTensor([1, 2])
y = lin(x)
</code></pre>
- The .parameters() function returns all parameters of the module
</textarea></section>
<section data-markdown><textarea data-template>
## Fully-connected Feedforward Networks (MLP)
<img src="images/mlp.png" id="fwimg" style="height:200px"/>
- Consists of fully connected (dense) layer.
- Implements the function: $$ \mathbf{y} = \sigma \left( W \mathbf{x} + \mathbf{b}\right) $$
- $W$ are trainable weights
- $\mathbf{b}$ are trainable biases
- $\sigma$ is an activation function
<pre><code class="Python" data-trim data-noescape>
a = torch.nn.Linear(in_channels=10, out_channels=5)
y = torch.sigmoid(a)
</code></pre>
</textarea></section>
<section data-markdown><textarea data-template>
## PyTorch Neural Network Bulding Block: Module
Modules can be composed to build a neural network.
The simplest method is the "sequential" mode that chains the operations
<pre><code class="py" data-trim data-noescape>
my_first_nn = torch.nn.Sequential(
torch.nn.Linear(2, 5),
torch.nn.Sigmoid(), #this is an activation function
torch.nn.Linear(5, 20),
torch.nn.Sigmoid(),
torch.nn.Linear(20, 2))
</code></pre>
Sequential returns a module, so it can be called as a function
<pre><code class="py" data-trim data-noescape>
my_first_nn(x)
</code></pre>
- Note that output dimensions of layer l-1 must match input dimensions of current layer l!
</textarea></section>
<section data-markdown><textarea data-template>
## Sequential Module
Modules can be composed to build a neural network.
The simplest method is the "sequential" mode that chains the operations
<pre><code class="py" data-trim data-noescape>
my_first_nn = torch.nn.Sequential(torch.nn.Linear(2, 5),
torch.nn.Sigmoid(), #this is an activation function
torch.nn.Linear(5, 20),
torch.nn.Sigmoid(),
torch.nn.Linear(20, 2))
</code></pre>
Sequential returns a module, so it can be called as a function
<pre><code class="py" data-trim data-noescape>
my_first_nn(x)
</code></pre>
- Note that output dimensions of layer l-1 must match input dimensions of current layer l!
[](https://drive.google.com/open?id=1botgh24In3QFyMc9AmezPszgt2ntJ-OL)
<pre><code class="py" data-trim data-noescape>
my_first_nn
Sequential(
(0): Linear(in_features=3, out_features=5, bias=True)
(1): Sigmoid()
(2): Linear(in_features=5, out_features=20, bias=True)
(3): Sigmoid()
(4): Linear(in_features=20, out_features=2, bias=True)
)
</code></pre>
</textarea></section>
<section data-markdown><textarea data-template>
## PyTorch Neural Network Building Block: Module
- Sequential works well for fully feedforward networks.
- In most cases, however, neural networks are implemented explicitely:
<pre><code class="Python" data-trim data-noescape>
class MySecondNetwork(torch.nn.Module):
def __init__(self, n1, n2, n3, num_classes):
super(MySecondNetwork, self).__init__()
self.layer1 = torch.nn.Linear(n1,n2)
self.layer2 = torch.nn.Linear(n2,n3)
self.layer3 = torch.nn.Linear(n3,num_classes)
self.sigmoid = torch.nn.Sigmoid()
def forward(self, data):
y1 = self.sigmoid(self.layer1(data))
y2 = self.sigmoid(self.layer2(y1))
y3 = self.sigmoid(self.layer3(y2))
return y3
my_second_net = MySecondNetwork(3,10,5,2)
my_second_net(x) # x has shape [M,3]
</code></pre>
</textarea></section>
<section data-markdown><textarea data-template>
## Loss functions and optimizers
- The loss function defines our objective. It is generally a **scalar** function.
- An optimizer defines the strategy to minimize the loss function
- Common loss functions for regression and classification:
- nn.MSELoss: Mean-Squared Error, default for regression tasks
$$ L_{MSE} = \frac1N \sum_{n} \sum_i (y_{ni}-t_{ni})^2 $$
- nn.CrossEntropyLoss: Default for classification tasks
$$L_{XENT} = - \frac1N \sum_n \sum_i t_{ni} \log y_{ni}$$
</textarea></section>
<section data-markdown><textarea data-template>
## Loss Function Example
- Mean-Squared Error (MSE)
<pre><code class="Python" data-trim data-noescape>
mse_loss = torch.nn.MSELoss()
target = torch.FloatTensor([[1.,0.,0.],[0.,0.,1.]])
loss = mse_loss(my_first_nn(data), target)
</code></pre>
- Cross Entropy
<pre><code class="Python" data-trim data-noescape>
xent_loss = torch.nn.CrossEntropyLoss()
target = torch.LongTensor([0,2])
loss = xent_loss(my_first_nn(data), target)
</code></pre>
</textarea></section>
<section data-markdown><textarea data-template>
## Loss functions and optimizers
- The loss function defines our objective
- An optimizer that defines the strategy to minimize the loss function (thus reach the objective)
- Common optimizers:
<ul>
<li /> SGD : A vanilla stochastic gradient descent algorithm
<pre><code class="Python" data-trim data-noescape> torch.optim.SGD </code></pre>
<li /> RMSprop : An optimizer that normalizes the gradients using moving root-mean-square averages
<pre><code class="Python" data-trim data-noescape> torch.optim.RMSProp </code></pre>
<li /> Adam : An adaptive gradients optimizer, works best in many cases
<pre><code class="Python" data-trim data-noescape> torch.optim.Adam </code></pre>
</ul>