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# LossFunctions

LossFunctions.jl is a Julia package that provides efficient and well-tested implementations for a diverse set of loss functions that are commonly used in Machine Learning.

Package Status Package Evaluator Build Status

## Available Losses

Distance-based (Regression) Margin-based (Classification)

Others: `PeriodicLoss`, `PoissonLoss`, `ScaledLoss`, `WeightedBinaryLoss`

## Introduction

Typically, the loss functions we work with in Machine Learning fall into the category of supervised losses. These are multivariate functions of two variables, the true target `y`, which represents the "ground truth" (i.e. correct answer), and the predicted output `ŷ`, which is what our model thinks the truth is. A supervised loss function takes these two variables as input and returns a value that quantifies how "bad" our prediction is in comparison to the truth. In other words: the lower the loss, the better the prediction.

This package provides a considerable amount of carefully implemented loss functions, as well as an API to query their properties (e.g. convexity). Furthermore, we expose methods to compute their values, derivatives, and second derivatives for single observations as well as arbitrarily sized arrays of observations. In the case of arrays a user additionally has the ability to define if and how element-wise results are averaged or summed over.

## Example

The following code snippets show a simple "hello world" scenario of how this package can be used to work with loss functions in various ways.

``````using LossFunctions
``````

All the concrete loss "functions" that this package provides are actually defined as immutable types, instead of native Julia functions. We can compute the value of some type of loss using the function `value()`. Let us start with an example of how to compute the loss for a group of three of observations. By default the loss will be computed element-wise.

``````julia> true_targets = [  1,  0, -2];

julia> pred_outputs = [0.5,  2, -1];

julia> value(L2DistLoss(), true_targets, pred_outputs)
# 3-element Array{Float64,1}:
#  0.25
#  4.0
#  1.0
``````

Alternatively, one can also use an instance of a loss just like one would use any other Julia function. This can make the code significantly more readable while not impacting performance, as it is a zero-cost abstraction (i.e. it compiles down to the same code).

``````julia> loss = L2DistLoss()
# LossFunctions.LPDistLoss{2}()

julia> loss(true_targets, pred_outputs)
# 3-element Array{Float64,1}:
#  0.25
#  4.0
#  1.0

julia> loss(1, 0.5f0) # single observation
# 0.25f0
``````

If you are not actually interested in the element-wise results individually, but some accumulation of those (such as mean or sum), you can additionally specify an average mode. This will avoid allocating a temporary array and directly compute the result.

``````julia> value(L2DistLoss(), true_targets, pred_outputs, AvgMode.Sum())
# 5.25

julia> value(L2DistLoss(), true_targets, pred_outputs, AvgMode.Mean())
# 1.75
``````

Aside from these standard unweighted average modes, we also provide weighted alternatives. These expect a weight-factor for each observation in the predicted outputs and so allow to give certain observations a stronger influence over the result.

``````julia> value(L2DistLoss(), true_targets, pred_outputs, AvgMode.WeightedSum([2,1,1]))
# 5.5

julia> value(L2DistLoss(), true_targets, pred_outputs, AvgMode.WeightedMean([2,1,1]))
# 1.375
``````

We do not restrict the targets and outputs to be vectors, but instead allow them to be arrays of any arbitrary shape. The shape of an array may or may not have an interpretation that is relevant for computing the loss. It is possible to explicitly specify which dimension denotes the observations. This is particularly useful for multivariate regression where one could want to accumulate the loss per individual observation.

``````julia> A = rand(2,3)
# 2×3 Array{Float64,2}:
#  0.0939946  0.97639   0.568107
#  0.183244   0.854832  0.962534

julia> B = rand(2,3)
# 2×3 Array{Float64,2}:
#  0.0538206  0.77055  0.996922
#  0.598317   0.72043  0.912274

julia> value(L2DistLoss(), A, B, AvgMode.Sum())
# 0.420741920634

julia> value(L2DistLoss(), A, B, AvgMode.Sum(), ObsDim.First())
# 2-element Array{Float64,1}:
#  0.227866
#  0.192876

julia> value(L2DistLoss(), A, B, AvgMode.Sum(), ObsDim.Last())
# 3-element Array{Float64,1}:
#  0.1739
#  0.060434
#  0.186408
``````

All these function signatures of `value` also apply for computing the derivatives using `deriv` and the second derivatives using `deriv2`.

``````julia> deriv(L2DistLoss(), true_targets, pred_outputs)
# 3-element Array{Float64,1}:
#  -1.0
#   4.0
#   2.0

julia> deriv2(L2DistLoss(), true_targets, pred_outputs)
# 3-element Array{Float64,1}:
#  2.0
#  2.0
#  2.0
``````

For computing the first and second derivatives we additionally expose a convenience syntax which allows for a more math-like look of the code.

``````julia> loss = L2DistLoss()
# LossFunctions.LPDistLoss{2}()

julia> loss'(true_targets, pred_outputs)
# 3-element Array{Float64,1}:
#  -1.0
#   4.0
#   2.0

julia> loss''(true_targets, pred_outputs)
# 3-element Array{Float64,1}:
#  2.0
#  2.0
#  2.0
``````

Additionally, we provide mutating versions for the subset of methods that return an array. These have the same function signatures with the only difference of requiring an additional parameter as the first argument. This variable should always be the preallocated array that is to be used as storage.

``````julia> buffer = zeros(3)
# 3-element Array{Float64,1}:
#  0.0
#  0.0
#  0.0

julia> deriv!(buffer, L2DistLoss(), true_targets, pred_outputs)
# 3-element Array{Float64,1}:
#  -1.0
#   4.0
#   2.0
``````

Note that this only shows a small part of the functionality this package provides. For more information please have a look at the documentation.

## Documentation

Check out the latest documentation

Additionally, you can make use of Julia's native docsystem. The following example shows how to get additional information on `HingeLoss` within Julia's REPL:

``````?HingeLoss
``````
``````search: HingeLoss L2HingeLoss L1HingeLoss SmoothedL1HingeLoss

L1HingeLoss <: MarginLoss

The hinge loss linearly penalizes every predicition where the
resulting agreement a = y⋅ŷ < 1 . It is Lipschitz continuous
and convex, but not strictly convex.

L(a) = \max \{ 0, 1 - a \}

--------------------------------------------------------------------

Lossfunction                     Derivative
┌────────────┬────────────┐      ┌────────────┬────────────┐
3 │'\.                      │    0 │                  ┌------│
│  ''_                    │      │                  |      │
│     \.                  │      │                  |      │
│       '.                │      │                  |      │
L │         ''_             │   L' │                  |      │
│            \.           │      │                  |      │
│              '.         │      │                  |      │
0 │                ''_______│   -1 │------------------┘      │
└────────────┴────────────┘      └────────────┴────────────┘
-2                        2      -2                        2
y ⋅ ŷ                            y ⋅ ŷ
``` html
## Installation

This package is registered in `METADATA.jl` and can be installed
as usual
``````

``````## License

This code is free to use under the terms of the MIT license.
``````

10/11/2015