The main contribution of [Understanding the difficulty of training deep feedforward neural networks](http://jmlr.org/proceedings/papers/v9/glorot10a/glorot10a.pdf) by Glorot et al. is a **normalized weight initialization**

$$W \sim U \left [  - \frac{\sqrt{6}}{\sqrt{n_j + n_{j+1}}}, \frac{\sqrt{6}}{\sqrt{n_j + n_{j+1}}} \right ]$$

where $n_j \in \mathbb{N}^+$ is the number of neurons in the layer $j$.

Showing some ways **how to debug neural networks** might be another reason to read the paper.

The paper analyzed standard multilayer perceptrons (MLPs) on a artificial dataset of $32 \text{px} \times 32 \text{px}$ images with either one or two of the 3 shapes: triangle, parallelogram and ellipse. The MLPs varied in the activation function which was used (either sigmoid, tanh or softsign).

However, no regularization was used and many mini-batch epochs were learned. It might be that batch normalization / dropout might change the influence of initialization very much.

Questions that remain open for me:

* [How is weight initialization done today?](https://www.reddit.com/r/MLQuestions/comments/4jsge9)
* Figure 4: Why is this plot not simply completely dependent on the data?
* Is softsign still used? Why not?
* If the only advantage of softsign is that is has the plateau later, why doesn't anybody use $\frac{1}{1+e^{-0.1 \cdot x}}$ or something similar instead of the standard sigmoid activation function?

Hein and Andriushchenko give a intuitive bound on the robustness of neural networks based on the local Lipschitz constant. With robustness, the authors refer a small $\epsilon$-ball around each sample; this ball is supposed to describe the region where the neural network predicts a constant class. This means that adversarial examples have to compute changes large enough to leave these robust areas. Larger $\epsilon$-balls imply higher robustness to adversarial examples.

When considering a single example $x$, and a classifier $f = (f_1, \ldots, f_K)^T$ (i.e. in a multi-class setting), the bound can be stated as follows. For $q$ and $p$ such that $\frac{1}{q} + \frac{1}{p} = 1$ and $c$ being the class predicted for $x$, the it holds

$x = \arg\max_j f_j(x + \delta)$

for all $\delta$ with

$\|\delta\|_p \leq \max_{R > 0}\min \left\{\min_{j \neq c} \frac{f_c(x) – f_j(x)}{\max_{y \in B_p(x, R)} \|\nabla f_c(y) - \nabla f_j(y)\|_q}, R\right\}$.

Here, $B_p(x, R)$ describes the $R$-ball around $x$ measured using the $p$-norm. Based on the local Lipschitz constant (in the denominator), the bound essentially measures how far we can deviate from the sample $x$ (measured in the $p$-norm) until $f_j(x) > f_c(x)$ for some $j \neq c$. The higher the local Lipschitz constant, the smaller deviations are allowed, i.e. adversarial examples are easier to find. Note that the bound also depends on the confidence, i.e. the edge  $f_c(x)$ has in comparison to all other $f_j(x)$.

In the remaining paper, the authors also provide bounds for simple classifiers including linear classifiers, kernel methods and two-layer perceptrons (i.e. one hidden layer). For the latter, they also propose a new type of regularization called cross-Lipschitz regularization:

$P(f) = \frac{1}{nK^2} \sum_{i = 1}^n \sum_{l,m = 1}^K \|\nabla f_l(x_i) - \nabla f_m(x_i)\|_2^2$.

This regularization term is intended to reduce the Lipschitz constant locally around training examples. They show experimental results using this regularization on MNIST and CIFAR, see the paper for details.

Also view this summary at [davidstutz.de](https://davidstutz.de/category/reading/).

**Dropout for layers** sums it up pretty well. The authors built on the idea of [deep residual networks](http://arxiv.org/abs/1512.03385) to use identity functions to skip layers. 

The main advantages:

* Training speed-ups by about 25%
* Huge networks without overfitting

## Evaluation

* [CIFAR-10](https://www.cs.toronto.edu/~kriz/cifar.html): 4.91% error ([SotA](https://martin-thoma.com/sota/#image-classification): 2.72 %) Training Time: ~15h
* [CIFAR-100](https://www.cs.toronto.edu/~kriz/cifar.html): 24.58% ([SotA](https://martin-thoma.com/sota/#image-classification): 17.18 %) Training time: < 16h
* [SVHN](http://ufldl.stanford.edu/housenumbers/):  1.75% ([SotA](https://martin-thoma.com/sota/#image-classification): 1.59 %) - trained for 50 epochs, begging with a LR of 0.1, divided by 10 after 30 epochs and 35. Training time: < 26h

The paper discusses and empirically investigates by empirical testing the effect of "catastrophic forgetting" (**CF**), i.e. the inability of a model to perform a task it was previously trained to perform if retrained to perform a second task. 

An illuminating example is what happens in ML systems with convex objectives: regardless of the initialization (i.e. of what was learnt by doing the first task), the training of the second task will always end in the global minimum, thus totally "forgetting" the first one. 

Neuroscientific evidence (and common sense) suggest that the outcome of the experiment is deeply influenced by the similarity of the tasks involved. Namely, if (i) the two tasks are *functionally identical but input is presented in a different format* or if (ii)  *tasks are similar* and the third case for (iii) *dissimilar tasks*. 

Relevant examples may be provided respectively by (i) performing the same image classification task starting from two different image representations as RGB or HSL, (ii) performing image classification tasks with semantically similar as classifying two similar animals and (iii) performing a text classification followed by image classification. 

The problem is investigated by an empirical study covering two methods of training ("SGD" and "dropout") combined with 4 activations functions (logistic sigmoid, RELU, LWTA, Maxout). A random search is carried out on these parameters. 

From a practitioner's point of view, it is interesting to note that dropout has been set to 0.5 in hidden units and 0.2 in the visible one since this is a reasonably well-known parameter. 

## Why the paper is important
It is apparently the first to provide a systematic empirical analysis of CF. Establishes a framework and baselines to face the problem.

## Key conclusions, takeaways and modelling remarks
* dropout helps in preventing CF
* dropout seems to increase the optimal model size with respect to the model without dropout 
* choice of activation function has a less consistent effect than dropout\no dropout choice
* dissimilar task experiment provides a  notable exception of then dissimilar task experiment
* the previous hypothesis that LWTA activation is particularly resistant to CF is rejected  (even if it performs best in the new task in the dissimilar task pair the behaviour is inconsistent)
* choice of activation function should always be cross-validated
* If computational resources are insufficient for cross-validation the combination dropout + maxout activation function is recommended.

FaceNet directly maps face images to $\mathbb{R}^{128}$ where distances directly correspond to a measure of face similarity. They use a triplet loss function. The triplet is (face of person A, other face of person A, face of person which is not A). Later, this is called (anchor, positive, negative).

The loss function is learned and inspired by LMNN. The idea is to minimize the distance between the two images of the same person and maximize the distance to the other persons image.

## LMNN

Large Margin Nearest Neighbor (LMNN) is learning a pseudo-metric

$$d(x, y) = (x -y) M  (x -y)^T$$

where $M$ is a positive-definite matrix. The only difference between a pseudo-metric and a metric is that $d(x, y) = 0 \Leftrightarrow x = y$ does not hold.

## Curriculum Learning: Triplet selection

Show simple examples first, then increase the difficulty. This is done by selecting the triplets.

They use the triplets which are *hard*. For the positive example, this means the distance between the anchor and the positive example is high. For the negative example this means the distance between the anchor and the negative example is low.

They want to have

$$||f(x_i^a) - f(x_i^p)||_2^2 + \alpha < ||f(x_i^a) - f(x_i^n)||_2^2$$

where $\alpha$ is a margin and $x_i^a$ is the anchor, $x_i^p$ is the positive face example and $x_i^n$ is the negative example. They increase $\alpha$ over time. It is crucial that $f$ maps the images not in the complete $\mathbb{R}^{128}$, but on the unit sphere. Otherwise one could double $\alpha$ by simply making $f' = 2 \cdot f$.

## Tasks

* **Face verification**: Is this the same person?
* **Face recognition**: Who is this person?

## Datasets

* 99.63% accuracy on Labeled FAces in the Wild (LFW)
* 95.12% accuracy on YouTube Faces DB

## Network

Two models are evaluated: The [Zeiler & Fergus model](http://www.shortscience.org/paper?bibtexKey=journals/corr/ZeilerF13)  and an architecture based on the [Inception model](http://www.shortscience.org/paper?bibtexKey=journals/corr/SzegedyLJSRAEVR14).

## See also

* [DeepFace](http://www.shortscience.org/paper?bibtexKey=conf/cvpr/TaigmanYRW14#martinthoma)