Risk Models¶
The prediction model together with a (compatible) loss function constitutes a risk model, which can be expressed as .
In this package, we use a type SupervisedRiskModel
to capture this:
abstract RiskModel
immutable SupervisedRiskModel{PM<:PredictionModel,L<:Loss} <: RiskModel
predmodel::PM
loss::L
end
We also provide a function to construct a risk model:

riskmodel
(pm, loss)¶ Construct a risk model, given the predictio model
pm
and a loss functionloss
.Here,
pm
andloss
need to be compatible, which means that the output of the prediction and the first argument of the loss function should have the same number of dimensions.Actually, the definition of
riskmodel
explicitly enforces this consistency:riskmodel{N,M}(pm::PredictionModel{N,M}, loss::Loss{M}) = SupervisedRiskModel{typeof(pm), typeof(loss)}(pm,loss)
Note
We may provide other risk model in addition to supervised risk model in future. Currently, the supervised risk models, which evaluate the risk by comparing the predictions and the desired responses, are what we focus on.
Common Methods¶
When a set of inputs and the corresponding outputs are given, the risk model can be considered as a function of the parameter .
The package provides methods for computing the total risk and the derivative of the total risk w.r.t. the parameter.

value
(rmodel, theta, x, y)¶ Compute the total risk w.r.t. the risk model rmodel, given
 the prediction parameter
theta
;  the inputs
x
; and  the desired responses
y
.
Here,
x
andy
can be a single sample or matrices comprised of a set of samples.Example:
# constructs a risk model, with a linear prediction # and a squared loss. # # risk := (theta'x  y)^2 / 2 # rmodel = riskmodel(LinearPred(5), SqrLoss()) theta = randn(5) # parameter x = randn(5) # a single input y = randn() # a single output value(rmodel, theta, x, y) # evaluate risk on a single sample (x, y) X = randn(5, 8) # a matrix of 8 inputs Y = randn(8) # corresponding outputs value(rmodel, theta, X, Y) # evaluate the total risk on (X, Y)
 the prediction parameter

value_and_addgrad!(rmodel, beta, g, alpha, theta, x, y)
Compute the total risk on
x
andy
, and its gradient w.r.t. the parametertheta
, and add it tog
in the following manner:Here,
x
andy
can be a single sample or a set of multiple samples. The function returns both the evaluated value andg
as a 2tuple.Note
When
beta
is zero, the computed gradient (or its scaled version) will be written tog
without using the original data ing
(in this case,g
need not be initialized).

value_and_grad
(rmodel, theta, x, y)¶ Compute and return the gradient of the total risk on
x
andy
, w.r.t. the parameterg
.This is just a thin wrapper of
value_and_addgrad!
.
Note that the addgrad!
method is provided for risk model with certain combinations of prediction models and loss functions. Below is a list of combinations that we currently support:
LinearPred
+UnivariateLoss
AffinePred
+UnivariateLoss
MvLinearPred
+MultivariateLoss
MvAffinePred
+MultivariateLoss
If you have a new prediction model that is not defined by the package, you can write your own addgrad!
method, based on the description above.