@@ -25,30 +25,30 @@ Here we focus on neural networks as a special model class used for function appr
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@@ -25,30 +25,30 @@ Here we focus on neural networks as a special model class used for function appr
To be more precise, we will rely on the following definition.
To be more precise, we will rely on the following definition.
> **Definition** (Neural Network):
> **Definition** (Neural Network):
> For any $`L\in\mathbb{N}`$ and $`d=(d_0,\dots,d_L)\in\mathbb{N}^{L+1}`$ a non-linear map $`\Psi\colon\mathbb{R}^{d_0}\to\mathbb{R}^{d_L}`$ of the form
> For any $`L\in\mathbb{N} \text{ and } d=(d_0,\dots,d_L)\in\mathbb{N}^{L+1}`$ a non-linear map $`\Psi\colon\mathbb{R}^{d_0}\to\mathbb{R}^{d_L}`$ of the form
> is called a _fully connected feed-forward neural network_.
> is called a _fully connected feed-forward neural network_.
Typically, we use the following nomenclature:
Typically, we use the following nomenclature:
- $`L`$ is called the _depth_ of the network with layers $`\ell=0,\dots,L`$.
- $`L`$ is called the _depth_ of the network.
- $`d`$ is called the _width_ of the network, where $`d_\ell`$ is the widths of the layers $`\ell`$.
- $`d`$ is called the _width(s)_ of the network.
- $`W_\ell\in\mathbb{R}^{d_{\ell-1}\times d_\ell}`$ are the _weights_ of layer $`\ell`$.
- $`W_\ell\in\mathbb{R}^{d_{\ell-1}\times d_\ell}`$ are the _weights_ of each layer.
- $`b_\ell\in\mathbb{R}^{d_\ell}`$ is the _biases_ of layer $`\ell`$.
- $`b_\ell\in\mathbb{R}^{d_\ell}`$ are the _biases_ of each layer.
- $`\vartheta=(W_1,b_1,\dots,W_L,b_L)`$ are the _free parameters_ of the neural network.
- $`\vartheta=(W_1,b_1,\dots,W_L,b_L)`$ are the _free parameters_ of the neural network.
Sometimes we write $`\Psi_\vartheta`$ or $`\Psi(x; \vartheta)`$ to indicate the dependence of $`\Psi`$ on the parameters $`\vartheta`$.
Sometimes we write $`\Psi_\vartheta \text{ or } \Psi(x; \vartheta)`$ to indicate the dependence of the neural network on the parameters.
- $`\varphi_\ell`$ is the _activation function_ of layer $`\ell`$.
- $`\varphi_\ell`$ are the _activation functions_ of each layer.
Note that $`\varphi_\ell`$ has to be non-linear and monotone increasing.
Note that the activation functions have to be non-linear and monotone increasing.
Additionally, there exist the following conventions:
Additionally, there exist the following conventions:
- $`x^{(0)}:=x`$ is called the _input (layer)_ of the neural network $`\Psi`$.
- $`x^{(0)}:=x`$ is called the _input (layer)_ of the neural network.
- $`x^{(L)}:=\Psi(x)`$ is called the _output (layer)_ of the neural network $`\Psi`$.
- $`x^{(L)}:=\Psi_\vartheta(x)`$ is called the _output (layer)_ of the neural network.
- Intermediate results $`x^{(\ell)} = \varphi_\ell(W_\ell\, x^{(\ell-1)} + b_\ell)`$ are called _hidden layers_.
- Intermediate results $`x^{(\ell)} = \varphi_\ell(W_\ell\, x^{(\ell-1)} + b_\ell)`$ are called _hidden layers_.
- (debatable) A neural network is called _shallow_ if it has only one hidden layer ($`L=2`$) and deep otherwise.
- (debatable) A neural network is called _shallow_ if it has only one hidden layer and deep otherwise.
**Example:**
**Example:**
Let $`L=3`$, $`d=(6, 10, 10, 3)`$ and $`\varphi_1=\varphi_2=\varphi_3=\tanh`$.
Let $`L=3,\ d=(6, 10, 10, 3) \text{ and } \varphi_1=\varphi_2=\varphi_3=\tanh`$.
Then the neural network is given by the concatenation
Then the neural network is given by the concatenation
```math
```math
\Psi\colon \mathbb{R}^6\to\mathbb{R}^3,
\Psi\colon \mathbb{R}^6\to\mathbb{R}^3,
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@@ -63,7 +63,7 @@ A typical graphical representation of the neural network looks like this:
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@@ -63,7 +63,7 @@ A typical graphical representation of the neural network looks like this:
</div>
</div>
<br/>
<br/>
The entries of $`W_\ell`$, $`\ell=1,2,3`$, are depicted as lines connecting nodes in one layer to the subsequent one.
The entries of $`W_\ell,\ \ell=1,2,3`$, are depicted as lines connecting nodes in one layer to the subsequent one.
The color indicates the sign of the entries (blue = "+", magenta = "-") and the opacity represents the absolute value (magnitude) of the values.
The color indicates the sign of the entries (blue = "+", magenta = "-") and the opacity represents the absolute value (magnitude) of the values.
Note that neither the employed actication functions $`\varphi_\ell`$ nor the biases $`b_\ell`$ are represented in this graph.
Note that neither the employed actication functions $`\varphi_\ell`$ nor the biases $`b_\ell`$ are represented in this graph.
...
@@ -101,8 +101,8 @@ For $`\varphi = (\varphi_1, \dots, \varphi_L)`$ we define the model class of a c
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@@ -101,8 +101,8 @@ For $`\varphi = (\varphi_1, \dots, \varphi_L)`$ we define the model class of a c
```
```
If we want to use the neural network to approximate a function $`f`$ the easiest approach would be to conduct a Least-Squares regression in an appropriate norm.
If we want to use the neural network to approximate a function $`f`$ the easiest approach would be to conduct a Least-Squares regression in an appropriate norm.
To make things even easier for the explaination, we assume $`f\colon \mathbb{R}^K \to \mathbb{R}`$, i.e., $`\operatorname{dim}(x^{(0)})=K`$ and $`\operatorname{dim}(x^{(L)})=1`$.
To make things even easier for the explaination, we assume $`f\colon \mathbb{R}^K \to \mathbb{R}, \text{ i.e., }\operatorname{dim}(x^{(0)})=K \text{ and } \operatorname{dim}(x^{(L)})=1`$.
Assuming the function $`f`$ has a second moment, we can use a standard $`L^2`$-norm for our Least-Square problem:
Assuming the function has a second moment, we can use a standard $`L^2`$-norm for our Least-Square problem:
```math
```math
\text{Find}\qquad \Psi_\vartheta
\text{Find}\qquad \Psi_\vartheta
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@@ -112,7 +112,7 @@ Assuming the function $`f`$ has a second moment, we can use a standard $`L^2`$-n
...
@@ -112,7 +112,7 @@ Assuming the function $`f`$ has a second moment, we can use a standard $`L^2`$-n
where we assume $`x\sim\pi`$ for some appropriate probability distribution $`\pi`$ (e.g. uniform or normal).
where we assume $`x\sim\pi`$ for some appropriate probability distribution $`\pi`$ (e.g. uniform or normal).
As computing the integrals above is not feasible for $`K\gg1`$, we consider an empirical version.
As computing the integrals above is not feasible for $`K\gg1`$, we consider an empirical version.
Let $`x^{(1)},\dots,x^{(N)}\sim\pi`$ be independent (random) samples and assume we have access to $`f^{(i)}:=f(x^{(i)})`$, $`i=1,\dots,N`$.
Let $`x^{(1)},\dots,x^{(N)}\sim\pi`$ be independent (random) samples and assume we have access to $`f^{(i)}:=f(x^{(i)}),\ i=1,\dots,N`$.
> **Definition** (training data):
> **Definition** (training data):
> Tuples of the form $`(x^{(i)}, f^{(i)})_{i=1}^N`$ are called _labeled training data_.
> Tuples of the form $`(x^{(i)}, f^{(i)})_{i=1}^N`$ are called _labeled training data_.
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@@ -128,11 +128,9 @@ The empirical regression problem then reads
...
@@ -128,11 +128,9 @@ The empirical regression problem then reads
> **Definition** (loss function):
> **Definition** (loss function):
> A _loss functions_ is any function, which measures how good a neural network approximates the target values.
> A _loss functions_ is any function, which measures how good a neural network approximates the target values.
**TODO: Is there a maximum number of inline math?**
Typical loss functions for regression and classification tasks are
Typical loss functions for regression and classification tasks are
- mean-square error (MSE, standard $`L^2`$-error)
- mean-square error (MSE, standard $`L^2`$-error)
- weighted $`L^p`$- or $`H^k`$-norms (solutions of PDEs)
- weighted $`L^p \text{- or } H^k\text{-}`$norms (solutions of PDEs)
- cross-entropy (difference between distributions)
- cross-entropy (difference between distributions)
Solving this equation requires the evaluation of the Jacobian (gradient) of the neural network $`\Psi_\vartheta`$ with respect to the network parameters $`\vartheta`$.
Solving this equation requires the evaluation of the Jacobian (gradient) of the neural network with respect to the network parameters $`\vartheta`$.
As $`\vartheta\in\mathbb{R}^M`$ with $`M\gg1`$ (millions of degrees of freedom), computation of the gradient w.r.t. all parameters for each training data point is infeasible.
As $`\vartheta\in\mathbb{R}^M \text{ with } M\gg1`$ (millions of degrees of freedom), computation of the gradient w.r.t. all parameters for each training data point is infeasible.
Optimization (Training)
Optimization (Training)
-----------------------
-----------------------
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@@ -162,9 +160,8 @@ The easiest and most well known approach is gradient descent (Euler's method), i
...
@@ -162,9 +160,8 @@ The easiest and most well known approach is gradient descent (Euler's method), i
where the step size $`\eta>0`$ is typically called the _learning rate_ and $`\vartheta^{(0)}`$ is a random initialization of the weights and biases.
where the step size $`\eta>0`$ is typically called the _learning rate_ and $`\vartheta^{(0)}`$ is a random initialization of the weights and biases.
The key why gradient descent is more promising then first-order optimality criterion is the iterative character.
The key why gradient descent is more promising then first-order optimality criterion is the iterative character.
In particular, we can use the law of large numbers and restrict the number of summands in $`\mathcal{L}_N`$ to a random subset of fixed size in each iteration step, which is called _stochastic gradient descent_ (SGD).
In particular, we can use the law of large numbers and restrict the number of summands in our loss to a random subset of fixed size in each iteration step, which is called _stochastic gradient descent_ (SGD).
Convergence of SGD can be shown by convex minimization and stochastic approximation theory and only requires that the learning rate $`\eta`$ with an appropriate rate.
Convergence of SGD can be shown by convex minimization and stochastic approximation theory and only requires that the learning rate decays with an appropriate rate.
**(see ?? for mor information)**
Here, however, I want to focus more on the difference between "normal" GD and SGD (in an intuitive level).
Here, however, I want to focus more on the difference between "normal" GD and SGD (in an intuitive level).
In principle, SGD trades gradient computations of a large number of term against the convergence rate of the algorithm.
In principle, SGD trades gradient computations of a large number of term against the convergence rate of the algorithm.
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@@ -180,8 +177,8 @@ The best metaphor to remember the difference (I know of) is the following:
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@@ -180,8 +177,8 @@ The best metaphor to remember the difference (I know of) is the following:
What remains is the computation of $`\operatorname{\nabla}_\vartheta\Psi_{\vartheta^{(i)}}`$ for $`i\in\Gamma_j\subset\{1,\dots,N\}`$ in each step.
What remains is the computation of $`\operatorname{\nabla}_\vartheta\Psi_{\vartheta^{(i)}} \text{ for } i\in\Gamma_j\subset\{1,\dots,N\}`$ in each step.
Lucky for us, we know that $`\Psi_\vartheta`$ is a simple concatenation of activation functions $`\varphi_\ell`$ and affine maps $`A_\ell(x^{(\ell-1)}) = W_\ell x^{(\ell-1)} + b_\ell`$ with derivative
Lucky for us, we know that our neural network is a simple concatenation of activation functions and affine maps $`A_\ell(x^{(\ell-1)}) = W_\ell x^{(\ell-1)} + b_\ell`$ with derivative
