# Models¶

Models in Shark can be seen as an abstract concept of a function, transforming an input into an output (or: producing an input given an output). In a machine learning context, models often correspond to hypotheses. Models represent the solutions to machine learning problems. For example, in classification we want to learn a model assigning classes to input points. The models are often parameterized, and then the process of learning corresponds to optimizing model parameters. After learning, the model with the optimized parameters represents the solution.

A simple model is a linear model, which can for example map vectorial input to a lower dimensional subspace:

In this case, we can say that all entries of the matrix *A* and of the
vector *b* form the parameters of the model *f*.
Optimizing parameters often requires derivatives
which requires the model to be differentiable
with respect to its own parameters.

## The base class ‘AbstractModel’¶

The base class for models in Shark is the templated class
`AbstractModel<InputTypeT,OutputTypeT>`

. For an in-depth description
of its methods, check the doxygen documentation of
shark::AbstractModel. Here, we describe how the concepts
introduced above are represented by the interface, and how models can
be used in Shark.

In general, most routines are optimized for batch computation (see the tutorial on Data Batches), that is, for processing many elements at one time. For example, models support to be evaluated on a batch of inputs and to compute their weighted derivatives for a batch of inputs at once (also see Shark Conventions for Derivatives).

The AbstractModel class is templatized on the input type as well as
the output type. For a classification model, the input type is likely
to be a vector type like `RealVector`

, and the output type to be an
`unsigned int`

for a class label. From these types, the model
infers the rest of the types needed for the interface and made public by
the model:

Types | Description |
---|---|

`InputType` |
Shortcut for the input type |

`OutputType` |
Shortcut for the output type |

`BatchInputType` |
A Batch of inputs as returned by Batch<InputType>::type |

`BatchOutputType` |
A Batch of outputs as returned by Batch<OutputType>::type |

The basic capabilities of a model are managed through a set of flags. If a model
can for example calculate the first input derivative, the flag
`HAS_FIRST_INPUT_DERIVATIVE`

is set. If the flag is not set and a function relying on
it is called, an exception is thrown. Flags can be queried via
convenience functions summarized in the table below:

Flag and accessor function name | Description |
---|---|

`HAS_FIRST_PARAMETER_DERIVATIVE` , `hasFirstParameterDerivative()` |
First derivative w.r.t. the parameters is available |

`HAS_FIRST_INPUT_DERIVATIVE` , `hasFirstInputDerivative()` |
First derivative w.r.t. the inputs is available |

To evaluate a model, there exist several variants of `eval`

and
`operator()`

. The most notable exception is the stateful valuated version of `eval`

.
The state allows the model to store computation results during `eval`

which then can be reused
in the computation of the derivative to save computation time.
In general, if the state is not required, it is a matter of taste which functions
are called. We recommend using `operator()`

for convenience.
The list of evaluation functions is:

Method | Description |
---|---|

`eval(InputType const&,OutputType&)` |
Evaluates the model’s response to a single input and stores it in the output |

`eval(BatchInputType const&, BatchOutputType&)` |
Evaluates the model’s response to a batch of inputs and stores them, in corresponding order, in the output batch type |

`eval(BatchInputType const&, BatchOutputType&, State& state)` |
Same as the batch version of eval, but also stores intermediate results which can be reused in computing the derivative |

`OutputType operator()(InputType)` |
Calls eval(InputType, OutputType) and returns the result |

`BatchOutputType operator()(BatchInputType)` |
Calls eval(BatchInputType, BatchOutputType) and returns the result |

`Data<OutputType> operator()(Data<InputType>)` |
Evaluates the model’s response for a whole dataset and returns the result |

The only method required to be implemented in a model is the stateful batch input version of eval. All other evaluation methods are inferred from this routine. It can also make sense to implement the single-input version of eval, because the default implementation would otherwise copy the input into a batch of size 1 and then call the batch variant. However, the single-input variant will usually not be called when performance is important, so not implementing it should not have critical drawbacks from the point of view of the standard Shark code base. If a model indicates by its flags that it offers first derivatives, then the following methods also need to be implemented:

Method | Description |
---|---|

`weightedParameterDerivative` |
Computes first or second drivative w.r.t the parameters for every output value and input and weights these results together |

`weightedInputDerivative` |
Computes first or second drivative w.r.t the inputs for every output value and input and weights these results together |

`weightedDerivatives` |
Computes first input and parameter derivative at the same time, making it possible to share calculations of both derivatives. Can be omitted. |

The parameter list of these methods is somewhat lengthy, and thus we recommend looking up their exact signature in the doxygen documentation. However, all versions require the state computed during eval. Example code to evaluate the first derivative of a model with respect to its parameters thus looks like this:

```
BatchInputType inputs; //batch of inputs
BatchOutputType outputs; //batch of model evaluations
MyModel model; //the differentiable model
// evaluate the model for the inputs and store the intermediate values in the state
boost::shared_ptr<State> state = model.createState();
model.eval(inputs,outputs,*state);
// somehow compute some weights and calculate the parameter derivative
RealMatrix weights = someFunction(inputs,outputs);
RealVector derivative;
modl.weightedParameterDerivative(inputs, outputs, weights,*state,derivative);
```

There are a few more methods which result from the fact that AbstractModel implements several higher-level interfaces, namely IParameterizable, INameable, and ISerializable. For example, models are parameterizable and serialized to store results:

Method | Description |
---|---|

`numberOfParameters` |
Number of parameters which can be optimized |

`parameterVector` |
Returns the current parameter vector of the model |

`setParameterVector` |
Sets the parameter vector to new values |

`read` , `write` |
Loads and saves a serializable object |

`createState` |
Returns a newly created State object holding the state to be stored in eval |

## List of Models¶

We end this tutorial with a list of some models currently implemented in Shark, together with a brief description.

We start with general purpose models:

Model | Description |
---|---|

LinearModel | A simple linear model mapping an n-dimensional input to an m-dimensional output It offers the possibility to add an activation function |

Conv2DModel | A simple linear model mapping an n-dimensional input to an m-dimensional output It offers the possibility to add an activation function |

ConcatenatedModel | Chains two models together by using the output of one model as the input to the second. |

NeuronLayer | Implements a nonlinear activation function. |

RBFLayer | Implements a layer of a radial basis function network using gaussian distributions |

CMACMap | Discretizes the space using several randomized tile maps and calculates a weighted sum of the discretized activation |

KernelExpansion | linear combination of outputs of AbstractKernelFunction, given points of a dataset and the point to be evaluated (input point) |

Some models for Classification or Regression:

Model | Description |
---|---|

Classifier | Wraps another model with 1(for binary) or n (for multi-class) output. Returns the index of the class with largest value for the given point. |

LinearClassifier | Classifier based on the prediction of a LinearModel |

KernelClassifier | Classifier based on the prediction of a KernelExpansion |

OneVersusOneClassifier | Multi-class classifier which does majority voting using binary classifiers for every class combination |

NearestNeighborModel | Nearest neighbor search for classification and regression using a (weighted) majority vote system. |

RFClassifier | Random Forest based on a collection of decision trees. Can be used for classification and regression |

Models for Clustering:

Model | Description |
---|---|

ClusteringModel | Base class for all clustering models, requires an AbstractClustering to work. |

SoftClusteringModel | Returns for a given point \(x\) a vector of propabilities \(p(c_i|x)\) indicating the propability of the point to be in the cluster \(c_i\) |

HardClusteringModel | Returns the index of the cluster with highest probability for a given point, \(\arg \max_i p(c_i|x)\). |

Special purpose models:

Model | Description |
---|---|

MissingFeaturesKernelExpansion | KernelExpansion with support for missing input values. |

MeanModel | Computes the mean output of a set of models. |

Normalizer | Special case of the LinearModel which only has a diagonal matrix and an optional offset. Used for normalisation |

DropoutLayer | Implements dropout of inputs |