NORMA: NORMA Optimization
In NORMA: Builds General Noise SVRs

Description Usage Arguments Details Value Author(s) References Examples

Computes general noise SVR based on NORMA optimization.

NORMA(x, y, f_0 = 0, beta_0 = 0, lambda = 0, rate = function(t) {     1
  }, kernel = linear_kernel, cost_der = ILF_cost_der,
  cost_name = "ILF_cost_der", gamma = 1, max_iterations = nrow(x),
  stopping_threshold = 0, trace = TRUE, no_beta = TRUE,
  fixed_epsilon = TRUE, ...)

`x`	`matrix` containing training points. Each row must be a point.
`y`	`numeric` containing target for training points x.
`f_0`	initial hypothesis.
`beta_0`	initial value for offset b.
`lambda`	NORMA optimization parameter lambda
`rate`	learning rate for NORMA optimization. Must be a function with one argument.
`kernel`	kernel function to use. Must be a function with three arguments such as `gaussian_kernel`. See also linear_kernel
`cost_der`	Loss function derivative to use. See also ILF_cost_der. Must be "ILF_cost_der" when ILF derivative is used.
`cost_name`	`character` indicating the symbolic name of `cost_der`.
`gamma`	gaussian kernel parameter γ.
`max_iterations`	maximum number of NORMA iterations computed.
`stopping_threshold`	value indicating when to stop NORMA optimization. See also 'Details'.
`trace`	`boolean` indicating if information messages should be printed (TRUE) or not (FALSE).
`no_beta`	`boolean` indicating if an offset b is used (FALSE) or not (TRUE).
`fixed_epsilon`	`boolean` indicating if `epsilon` should be updated (FALSE) or not (TRUE).
`...`	additional arguments to be passed to the low level functions.

Optimization will stop when the sum of the differences between all training predicted values of present iteration versus values from previous iteration does not exceeds stopping_threshold.

Returns a list containing:

alpha: matrix representing α parameters of NORMA optimization in each iteration, one per row.
beta: numeric representing β parameter of NORMA optimization in each iteration.
n_iterations: Number of NORMA iterations performed.

Jesus Prada, jesus.prada@estudiante.uam.es

Link to the scientific paper

Kivinen J., Smola A. J., Williamson R.C.: Online learning with kernels. In: IEEE transactions on signal processing, vol. 52, pp. 2165-2176, IEEE (2004).

with theoretical background for NORMA optimization is provided below.

http://realm.sics.se/papers/KivSmoWil04(1).pdf

1
2
3

NORMA(x=matrix(rnorm(10),nrow=10,ncol=1,byrow=TRUE),y=rnorm(10),kernel=function(x,y,gamma=0){x%*%y},
cost_der=function(phi,sigma_cuad,mu){return((phi-mu)/sigma_cuad)},cost_name="example",
sigma_cuad=1,mu=0)