# tranches¶

Non-optimised implementation of neural networks for educational purposes.

## User’s Guide¶

### Requirements¶

tranches is build on top of the following libraries:

### Issues¶

Should you encounter any issue with the library you can raise them here: https://github.com/RonsenbergVI/tranches/issues

### Installing tranches¶

#### Installation from GitHub¶

tranches releases are also available on github (https://github.com/ronsenbergVI/tranches).

You first need to clone (or fork if you want to modify it) and

$git clone https://github.com/ronsenbergVI/tranches.git$ cd tranches
$python setup.py build$ python setup.py install


## Machine Learning Basics¶

Deep learning is a vibrant subfield of machine learning.. The field has attracted a lot of attention over the past few years. It is a consequence of the continuous progress made in data collection, processing and storage and of the advances in parallelisation allowing to state of the art models.

Even though deep learning has its specificities, it is based on learning theories and it is worth going over some of the fundamentals. This chapter is a brief introduction to the theory of learning. It is the bare minimum needed to go forward but is in no case complete. Feel free to skip to the next chapter.

### The supervised learning problem¶

We assume that we have at hand a set of examples $$(\mathbf{x}_i,y_i)_{i \in \{0,...,N\}}$$ where $$\mathbf{x}_i$$ is a vector of features and $$y_i$$ are labels that are either draw from a discrete set (classification) or the real line (regression). In other words, we have a set of inputs and their desired output.

The goal of supervised learning is to find a function $$f$$ that ultimately is able to predict the desired output for new examples by learning from the examples at our disposal. Ideally, to solve this problem, we would like to be able to find the joint probability:

$F(x,y) = \mathbb{P}(\mathbf{x},y)$

The learning problem is usually defined over a parameter set and thus we are looking for a function $$f_{\theta}(x)$$ with $$x \in \Theta$$ and $$\Theta$$ is the parameter set. The learning procedure will define the parameter update rules. The parameters are updated so that the model output $$\hat{y_n}=f_{\theta}(x_n)$$ are the closest possible to the expected outputs $$y_n$$. We need to define metrics to define how close is the model.

### Loss and Risk Functions¶

In order to learn the optimal parameters $$\theta^{*}$$ supervised learning algorithms want to minimise what is referred to as a loss function.

A loss function is defined a map taking as input a pair of elements from the possible of labels to a real number. Ideally we would like to have a function that:

1. is regular enough (continuous and convex is the jackpot bu hardly ever the case for any interesting model);
2. sensitive to magnitude (this allows to distinguish between the types of error).

The type of loss function chosen depends on the supervised task. For classification the most common loss function is the cross entropy defined as:

$C(y,\hat{y}) = \sum_{k} \mathbb{P}(y=k) log \mathbb{P}(\hat{y}=k)$

and for regression it is the mean square root (MSE) defined as:

$C(y,\hat{y}) = ||y - \hat{y}=k||$

The loss function computes the “cost” of the learner’s error for one example (intuitively a zero cost means a correct prediction). We want to minimise the cost for the whole dataset. For this we define the (empirical) risk function as:

$J(\theta) = \dfrac{1}{N} \sum_{n} C(y_n,\hat{y}_n)$

### Gradient Descent¶

Gradient based optimisation methods are quite popular in machine learning and particularly in deep learning. The gradient of a function gives meaningful information about how to update its parameters to reduce its value. Gradient based methods are iterative algorithms that produce a sequence (theta_n) of parameters where each new iteration improves the risk function.

1. defining an initial value $$x_0$$
2. an iteration process to get $$x_{n+1}$$ given $$x_{n}$$
3. a convergence criteria to stop the training when reached

The simplest gradient based optimisation method is the gradient (or steepest) descent algorithm where each new iteration is defined by going in small steps in the opposite direction of the risk function gradient:

1. initialise $$x_0$$
2. $$x_{n+1}=x_{n} - \gamma \nabla f(x_{n})$$
3. until $$||x_{n+1}-x_{n}||<\epsilon$$

The interesting feature of this method is that it is guaranteed to reach a local minima for a convex function on a convex set.

The hyperparameter $$\gamma$$ is referred to as the learning rate. Choosing a learning rate is no easy task (well nothing is really in deep learning) and is critical for the performance of the training. Choosing a too large learning learning rate prevents the algorithm to target precisely enough regions of local minima and thus oscillates. A large enough can event prevent the algorithm to converge. Too small a convergence makes the algorithm slow as it takes too many steps towards a local minima.

A common method is to start with a large learning rate when the model starts training and is (arguably) far from the optimal solution and as training goes to decay the rate as the model gets closer to the solution we need to take more precaution and not miss it. The popular learning rate schedules used in practice are:

1. exponential decay:
$\gamma_r = \gamma_0 \exp(-H \dfrac{r}{R})$
1. power decay:

$\gamma_r = \dfrac{\gamma_0}(1+\dfrac{r}{R})^H}$
2. power decay:

$\gamma_r = \gamma_0(1-\dfrac{r}{R})^H)$

## Neural Networks¶

<intro to chapter here>

## The MNIST dataset (yet again)¶

<intro to chapter here>

## API Reference¶

If you are looking for information on a specific function, class or method, this part of the documentation is for you.

## tranches changelog¶

We detail here the changes made to the library

### Version 0.1.1¶

Release day: September 27 2018

• Initial public release.

## License¶

tranches is licensed under the Apache 2.0 Licence. It means that the source code provided in the binaries can be used, modified, or distributed freely for commercial or personal use with conditions only requiring preservation of copyright and license notices.

The full license text can be found below (tranches-license).

### Authors¶

tranches is written and maintained by Rene-Jean Corneille

### Contributing¶

Contribution will be welcomed once a first stable release is ready.

### License Definitions¶

The following section contains the full license texts for tranches and the documentation.

• “AUTHORS” hereby refers to all the authors listed in the authors section.
• The “tranches-license” applies to all the source code shipped as part of tranches (tranches itself as well as the examples and the unittests) as well as documentation.

### tranches License¶

Apache License

Version 2.0, January 2004

http://www.apache.org/licenses/

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