Click here for a newer version (Knet7) of this tutorial. The code used in this version (KUnet) has been deprecated.

There are a number of deep learning packages out there. However most sacrifice readability for efficiency. This has two disadvantages: (1) It is difficult for a beginner student to understand what the code is doing, which is a shame because sometimes the code can be a lot simpler than the underlying math. (2) Every other day new ideas come out for optimization, regularization, etc. If the package used already has the trick implemented, great. But if not, it is difficult for a researcher to test the new idea using impenetrable code with a steep learning curve. So I started writing KUnet.jl which currently implements backprop with basic units like relu, standard loss functions like softmax, dropout for generalization, L1-L2 regularization, and optimization using SGD, momentum, ADAGRAD, Nesterov's accelerated gradient etc. in less than 500 lines of Julia code. Its speed is competitive with the fastest GPU packages (here is a benchmark). For installation and usage information, please refer to the GitHub repo. The remainder of this post will present (a slightly cleaned up version of) the code as a beginner's neural network tutorial (modeled after Honnibal's excellent parsing example).#### Contents

Layers and netsThe layer function

Prediction

Loss functions

Backpropagation Training

Activation functions

Preprocessing functions

Why Julia?

Future work

#### Layers and nets

A feed forward neural network consists of layers.typealias Net Array{Layer,1}

type Layer w; b; f; fx; ... end

#### The layer function (forw)

Given its parameters and functions, here is how a layer computes its output from its input:function forw(l::Layer, x) l.x = l.fx(l, x) # preprocess the input l.y = l.w * l.x # multiply with weight matrix l.y = l.y .+ l.b # add the bias vector (to every column) l.y = l.f(l,l.y) # apply the activation fn to the output end

#### Prediction

Once we have forw for a single layer, calculating a prediction for the whole network is literally a one-liner:forw(n::Net, x)=(for l=n x=forw(l,x) end; x)

#### Loss functions

In order to train the network, we need a loss function. A loss function takes y, the network output, and dy, the desired output, and returns a loss value which we try to minimize during training. In addition, it calculates the gradient of the loss with respect to y, which tells us how small changes in the network output will effect the loss. As an example, here is the implementation of softmaxloss (albeit a bit low level) in func.jl:function softmaxloss(y, dy) yrows,ycols = size(y) loss = zero(eltype(y)) prob = similar(y, yrows) for j=1:ycols ymax = y[1,j] for i=2:yrows y[i,j] > ymax && (ymax = y[i,j]) end psum = zero(ymax) for i=1:yrows yij = y[i,j] - ymax prob[i] = exp(yij) psum += prob[i] dy[i,j] == 1 && (loss += yij) end loss -= log(psum) for i=1:yrows prob[i] /= psum dy[i,j] = (prob[i] - dy[i,j]) / ycols end end return loss end

`dy[i,j] = (prob[i] - dy[i,j]) / ycols`

. We should note a couple of important design decisions here: (1) y and dy should have the same dimensionality, so use one-of-k encoding for desired classification outputs. (2) The loss function overwrites dy with the gradient of the loss with respect to y.#### Backpropagation

Backpropagation is the algorithm that calculates the gradient of the loss with respect to network weights, given the gradient of the loss with respect to the network output. This can be accomplished taking a backward pass through the layers after the forward pass and the loss calculation:function backprop(net::Net, x, dy, loss=softmaxloss) y = forw(net, x) # y: network output loss(y, dy) # dy: desired output -> gradient back(net, dy) # calculate derivatives end

back(n::Net, dy) = (for i=length(n):-1:1 dy=back(n[i],dy) end)

function back(l::Layer, dy) dy = l.f(l,l.y,dy) l.dw = dy * l.x' l.db = sum!(l.db, dy) l.dx = l.w' * dy l.dx = l.fx(l,l.x,l.dx) end

#### Training

The gradients calculated by backprop, l.dw and l.db, tell us how much small changes in corresponding entries in l.w and l.b will effect the loss (for the last instance, or minibatch). Small steps in the gradient direction will increase the loss, steps in the opposite direction will decrease the loss. This suggests the following update rule:w = w - dw

Over the years, people have noted many subtle problems with this approach and suggested improvements:

**Step size:**If the step sizes are too small, the SGD algorithm will take too long to converge. If they are too big it will overshoot the optimum and start to oscillate. So we scale the gradients with an adjustable parameter called the learning rate:

w = w - learningRate * dw

**Step direction:**More importantly, it turns out the gradient (or its opposite) is often NOT the direction you want to go in order to minimize loss. Let us illustrate with a simple picture:

The two axes are w1 and w2, two parameters of our network, and the contour plot represents the loss with a minimum at x. If we start at x0, the Newton direction (in red) points almost towards the minimum, whereas the gradient (in green), perpendicular to the contours, points to the right.

Unfortunately Newton's direction is expensive to compute. However, it is also probably unnecessary for several reasons: (1) Newton gives us the ideal direction for second degree objective functions, which our neural network loss almost certainly is not, (2) The loss function whose gradient backprop calculated is

*the loss for the last minibatch/instance only*, which at best is a very noisy version of the real loss function, so we shouldn't spend too much effort getting it exactly right.

So people have come up with various approximate methods to improve the step direction. Instead of multiplying each component of the gradient with the same learning rate, these methods scale them separately using their running average (momentum, Nesterov), or RMS (Adagrad, Rmsprop). I realize this necessarily short summary barely covers what has been implemented in KUnet and doesn't do justice to the literature or cover most of the important ideas. The interested reader can start with a standard textbook on numerical optimization, and peruse the latest papers on optimization in deep learning.

**Minimize what?**The final problem with gradient descent, other than not telling us the ideal step size or direction, is that it is not even minimizing the right objective! We want small loss on never before seen test data, not just on the training data. The truth is, a sufficiently large neural network with a good optimization algorithm can get arbitrarily low loss on any finite training data (e.g. by just memorizing the answers). And it can typically do so in many different ways (typically many different local minima for training loss in weight space exist). Some of those ways will generalize well to unseen data, some won't. And unseen data is (by definition) not seen, so how will we ever know which weight settings will do well on it? There are at least three ways people deal with this problem: (1) Bayes tells us that we should use all possible networks and weigh their answers by how well they do on training data (see Radford Neal's fbm), (2) New methods like dropout or adding distortions and noise to inputs and weights during training seem to help generalization, (3) Pressuring the optimization to stay in one corner of the weight space (e.g. L1, L2, maxnorm regularization) helps generalization.

KUnet views dropout (and other distortion methods) as a preprocessing step of each layer. The other techniques (learning rate, momentum, regularization etc.) are declared as UpdateParam's for l.w in l.pw and for l.b in l.pb for each layer:

type UpdateParam learningRate; l1reg; l2reg; maxnorm; adagrad; momentum; nesterov; ... end

setparam!(p::UpdateParam,k,v) setparam!(l::Layer,k,v) setparam!(net::Net,k,v)

function update(w, dw, o::UpdateParam) initupdate(w, dw, o) nz(o,:l1reg) && l1reg!(o.l1reg, w, dw) nz(o,:l2reg) && l2reg!(o.l2reg, w, dw) nz(o,:adagrad) && adagrad!(o.adagrad, o.ada, dw) nz(o,:learningRate,1f0) && (dw = dw .* o.learningRate) nz(o,:momentum) && momentum!(o.momentum, o.mom, dw) nz(o,:nesterov) && nesterov!(o.nesterov, o.nes, dw) w = w .- dw nz(o,:maxnorm) && maxnorm!(o.maxnorm, w) end nz(o,n,v=0f0)=(isdefined(o,n) && (o.(n) != v))

The "train" function in net.jl is just a wrapper around "backprop" and "update". It goes through the training set once, splitting it into minibatches, and feeding each minibatch through backprop and update.

#### Activation functions

The activation function follows the linear transformation (wx+b) in a layer and is typically a non-linear element-wise function. Without an activation function, multiple layers in a neural network would be useless because the composition of several linear functions is still a linear function. As an example activation function l.f, here is the definition of the relu (rectified linear unit) from func.jl:relu(l,y)=for i=1:length(y) (y[i]<0) && (y[i]=0) end relu(l,y,dy)=for i=1:length(y) (y[i]==0) && (dy[i]=0) end

relu(l,y::CudaArray)=ccall((:reluforw,libkunet), Void,(Cint,Cmat),length(y),y) relu(l,y::CudaArray,dy::CudaArray)=ccall((:reluback,libkunet), Void,(Cint,Cmat,Cmat),length(dy),y,dy)

#### Preprocessing functions

A preprocessing function precedes the linear transformation (wx+b) modifying the input of the layer. Preprocessing functions typically add some noise to the input to improve generalization. For example dropout can be implemented as a preprocessing function of a layer where each input is dropped (replaced by 0) with a given probability. Adding Gaussian noise, or elastic transformations of image inputs can also be implemented as preprocessing functions. Here is the (simplified) dropout implementation from func.jl:function drop(l, x) rand!(l.xdrop) drop(x, l.xdrop, l.dropout, 1/(1-l.dropout)) end function drop(l, x, dx) drop(dx, l.xdrop, l.dropout, 1/(1-l.dropout)) end function drop(x, xdrop, dropout, scale) for i=1:length(x) x[i] = (xdrop[i] < dropout ? 0 : scale * x[i]) end end function drop(x::CudaArray, xdrop::CudaArray, dropout, scale) ccall((:drop,libkunet),Void,(Cint,Cmat,Cmat,Cfloat,Cfloat), length(x),x,xdrop,dropout,scale) end

#### Why Julia?

I wanted to write something that is concise, easy to understand, easy to extend, and reasonably efficient. There is a subtle trade-off between conciseness and extensibility: If we use a very high level language that already has a "neural_network_train" function, we can write very concise code but we lose the ability to change the training algorithm. If we use a very low level language that only provides primitive arithmetic operations, all the algorithm details are exposed and modifiable but the code is bulky and difficult to understand. For a happy medium, the code should reflect the level at which I think of the problem: e.g. I should be able to say A*B if I want to multiply two matrices, and I shouldn't constantly worry about the location and element types of my arrays. That restricts the playing field to a handful of languages. Efficiency requires being able to work with a GPU. Ideally the code should be generic, i.e. algorithms should be expressed once and work whether the data is on the GPU or the CPU memory. Julia has fledgling GPU support but it does have concise matrix arithmetic and it excels at generic operations. So I struggled a bit with trying to get generic matrix operations to work on the GPU (which is nowhere nearly complete but currently good enough to run the KUnet code), and to express each algorithm in as mathematical and concise a manner as possible (which is still work in progress, largely due to the lack of GPU generics and a standard syntax for in-place operations). But the progress so far (and the invaluable support I got from the Julia community) convinced me that this is a worthwhile endeavor.#### Future work

With very little extra effort (and lines of code) one should be able to extend KUnet to implement convolutional and recurrent nets and new tricks like maxout units, maxnorm regularization, rmsprop optimization etc. You can send me suggestions for improvement (both in coding style and new functionality) using comments to this blog post, or using issues or pull requests on GitHub.Full post...