Update NN basics

doc/basics.md

@@ -87,11 +87,11 @@ Typically, these networks use different types of activation functions, such as:
 | --- | --- | --- |
 | <img src="argmax.png" title="argmax" alt="argmax" height=200 /> | <img src="softmax.png" title="softmax" alt="softmax" height=200 /> | <img src="maxpool.png" title="maxpool" alt="maxpool" height=200 /> |
 
-Training
---------
+Minimization Problem
+--------------------
 
 In this section we focus on training a fully-connected network for a regression task.
-The principles stay the same of any other objective, such as classification, but may be more complicated at different points.
+The principles stay the same of any other objective, such as classification, but may be more complicated in some aspects.
 
 Let $`M = \sum_{\ell=1,\dots,L} d_\ell(d_{\ell-1}+1)`$ denote the number of degrees of freedom encorporated in $`\vartheta`$.
 For $`\varphi = (\varphi_1, \dots, \varphi_L)`$ we define the model class of a certain (fully connected) network topology by
@@ -100,31 +100,114 @@ For $`\varphi = (\varphi_1, \dots, \varphi_L)`$ we define the model class of a c
 \mathcal{M}_{d, \varphi} = \{ \Psi_\vartheta \,\vert\, \vartheta \in \mathbb{R}^M \text{ and activation functions } \varphi\}.
 ```
 
-If we want to use the neural network to approximate a function $f$ in some appropriate norm, we can use Least-Squares.
-The problem then reads:
+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 $`\operatorname{dim}(x^{(L)})=1`$.
+Assuming the function $`f`$ has a second moment, we can use a standard $`L^2`$-norm for our Least-Square problem:
 
 ```math
 \text{Find}\qquad \Psi_\vartheta
-= \operatorname*{arg\, min}_{\Psi_\theta\in\mathcal{M}_{d,\varphi}} \Vert f - \Psi_\theta \Vert^2
-= \operatorname*{arg\, min}_{\theta\in\mathbb{R}^{M}} \Vert f - \Psi_\theta \Vert^2.
+= \operatorname*{arg\, min}_{\Psi_\theta\in\mathcal{M}_{d,\varphi}} \Vert f - \Psi_\theta \Vert_{L^2(\pi)}^2
+= \operatorname*{arg\, min}_{\theta\in\mathbb{R}^{M}} \int_{\mathbb{R}^K} \bigl(f(x) - \Psi_\theta(x)\bigr)^2 \ \mathrm{d}\pi(x),
+```
+
+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.
+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):
+> Tuples of the form $`(x^{(i)}, f^{(i)})_{i=1}^N`$ are called _labeled training data_.
+
+The empirical regression problem then reads
+
+```math
+\text{Find}\qquad \Psi_\vartheta
+= \operatorname*{arg\, min}_{\Psi_\theta\in\mathcal{M}_{d,\varphi}} \frac{1}{N} \sum_{i=1}^N \bigl(f^{(i)} - \Psi_\theta(x^{(i)})\bigr)^2
+=: \operatorname*{arg\, min}_{\Psi_\theta\in\mathcal{M}_{d,\varphi}} \mathcal{L}_N(\Psi_\theta)
+```
+
+> **Definition** (loss function):
+> A _loss functions_ is any function, which measures how good a neural network approximates the target values.
+
+Typical loss functions for regression and classification tasks are
+- mean-square error (MSE, standard $`L^2`$-error)
+- weighted $`L^p`$- or $`H^k`$-norms (solutions of PDEs)
+- cross-entropy (difference between distributions)
+- Kullback-Leibler divergence, Hellinger distance, Wasserstein metrics
+- Hinge loss (SVM)
+
+To find a minimizer of our loss function $`\mathcal{L}_N`$, we want to use the first-order optimality criterion
+
+```math
+0
+= \operatorname{\nabla}_\vartheta \mathcal{L}_N(\Psi_\vartheta)
+= -\frac{2}{N} \sum_{i=1}^N \bigl(f^{(i)} - \Psi_\vartheta(x^{(i)}\bigr) \operatorname{\nabla}_\vartheta \Psi_\vartheta.
 ```
-The first order optimality criterion then gives us the linear system
 
-$$
-\langle f-\Psi_\vartheta,\, \operatorname{\nabla_\vartheta} \Psi_\vartheta \rangle = 0
-$$
+Solving this equation requires the evaluation of the Jacobian (gradient) of the neural network $`\Psi_\vartheta`$ 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.
+ 
+Optimization (Training)
+-----------------------
+
+Instead of solving the minimization problem explicitly, we can use iterative schemes to approximate the solution.
+The easiest and most well known approach is gradient descent (Euler's method), i.e.
+
+```math
+\vartheta^{(j+1)} = \vartheta^{(j)} - \eta \operatorname{\nabla}_{\vartheta}\mathcal{L}_N(\Psi_{\vartheta^{(j)}}),
+\qquad j=0, 1, 2, \dots
+```
+
+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.
+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).
+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.
+**(see ?? for mor information)**
+
+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.
+The best metaphor to remember the difference (I know of) is the following:
+
+> **Metaphor (SGD):**
+> Assume you and a friend of yours have had a party on the top of a mountain.
+> As the party has come to an end, you both want to get back home somewhere in the valley.
+> You, scientist that you are, plan the most direct way down the mountain, following the steepest descent, planning each step carefully as the terrain is very rough.
+> Your friend, however, drank a little to much and is not capable of planning anymore.
+> So they stagger down the mountain in a more or less random direction.
+> Each step they take is with little thought, but it takes them a long time overall to get back home (or at least close to it).
+> <img src="sgd.png" title="sgd" alt="sgd" height=400 />
+
+What remains is the computation of $`\operatorname{\nabla}_\vartheta\Psi_{\vartheta^{(i)}}`$ 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
+
+```
+\partial_{W^{(m)}_{\alpha,\beta}} A^{(\ell)} = 
+\begin{cases}
+W^{(\ell)}_{\alpha,\beta} & \text{if }m=\ell,\\
+0 & \text{if }m\neq\ell,
+\end{cases}
+\qquad\text{and}\qquad
+\partial_{b^{(m)}_{\alpha}} A^{(\ell)} = 
+\begin{cases}
+b^{(\ell)}_{\alpha} & \text{if }m=\ell,\\
+0 & \text{if }m\neq\ell.
+\end{cases}
+```
 
-- LS regression (and differentiation -> derivative w.r.t. $\vartheta$)
-- loss function
-- back-prob (computing gradient w.r.t. $\vartheta$ by chain-rule)
-- examples of loss functions
+The gradient $`\operatorname{\nabla}_\vartheta\Psi_{\vartheta^{(i)}}`$ can then be computed using the chain rule due to the compositional structure of the neural network.
+Computing the gradient through the chain rule is still very inefficient and most probably infeasible if done in a naive fashion.
+The so called _Backpropagation_ is esentially a way to compute the partial derivatives layer-wise storting only the necessary information to prevent repetitive computations, rendering the computation manaeable. 
 
 Types of Neural Networks
 ------------------------
 
-| Fully Connected Neural Network | Convolutional Neural Network |
+| Name | Graph |
 | --- | --- |
-| <img src="nn_fc.png" title="nn_fc" alt="nn_fc" height=250 /> | <img src="nn_conv.png" title="nn_conv" alt="nn_conv" height=250/> |
+| Fully Connected Neural Network | <img src="nn_fc.png" title="nn_fc" alt="nn_fc" height=250 /> |
+| Convolutional Neural Network | <img src="nn_conv.png" title="nn_conv" alt="nn_conv" height=250/> |
+| U-Net | <img src="u_net.png" title="u_net" alt="u_net" height=250/> |
+| Residual Neural Network | <img src="res_net.png" title="res_net" alt="res_net" height=250/> |
+| Invertible Neural Network | <img src="inn.png" title="inn" alt="inn" height=250/> |
 
 Further Reading
 ---------------