GRAM

GRAM (Gradient-based RAdius-Margin optimization) is a TwoStepMKL algorithm finding the kernels combination that simultaneously maximizes the margin between classes while minimizes the radius of the resulting kernel.

The algorithm learns a convex combination of base kernels with form

$$k_{\mu}(x,z)=\sum_r^P\mu_rk_r(x,z),\quad \mu_r\ge 0 \land\|\mu\|_1=1,$$

where $\mu$ is the weights vector learned by GRAM.

Paper

If you need additional details about GRAM, please refer to the following paper:
Ivano Lauriola, Mirko Polato, and Fabio Aiolli: "Radius-Margin Ratio Optimization for Dot-Product Boolean Kernel Learning". ICANN (2017)

MKLpy.algorithms.GRAM(
    **kwargs,
    )

Parameters (init)

Parameter	Type	Description
**kwargs	args	TwoStepMKL parameters, see here

Attributes

Attribute	Type	Description
n_kernels	int	number of combined kernels
KL	list	the training kernels list
func_form	callable	the combination function (e.g. summation, average...)
solution	dict	the solution of the optimization
direction	str	direction of the optimization, min or max
convergence	bool	True iff the algorithm reaches the convergence
cache	dict	a dictionary containing intermediate results and data structures used to speed-up the computation

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Methods

See standard TwoStepMKL methods here

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Examples

from MKLpy.algorithms import GRAM
from MKLpy.scheduler  import ReduceOnWorsening
from MKLpy.callbacks  import EarlyStopping

earlystop = EarlyStopping(
    KLva, Yva,      #validation data, KL is a validation kernels list
    patience=5,     #max number of acceptable negative steps
    cooldown=1,     #how ofter we run a measurement, 1 means every optimization step
    metric='roc_auc',   #the metric we monitor
)

#ReduceOnWorsening automatically redure the learning rate when a worsening solution occurs
scheduler = ReduceOnWorsening()

mkl = GRAM(
    max_iter=1000,          
    learning_rate=.01,      
    callbacks=[earlystop],
    scheduler=ReduceOnWorsening()).fit(KLtr, Ytr)

Warning

If the learning rate is too high or if the initial solution (i.e. the simple average) is not separable, you may have numerical errors.

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Citing GRAM

If you use GRAM in your scientific project, please cite the following pubblication:

@inproceedings{lauriola2017radius,
  title={Radius-margin ratio optimization for dot-product boolean kernel learning},
  author={Lauriola, Ivano and Polato, Mirko and Aiolli, Fabio},
  booktitle={International conference on artificial neural networks},
  pages={183--191},
  year={2017},
  organization={Springer}
}