"""
============================================
Comparison of solvers for Poisson regression
============================================

In this example, we explain how to try out different solvers for the Poisson
regression model, using :math:`\ell_2^2` penalization, namely ridge (which is
the default value for the ``penalty`` parameter in
`tick.inference.PoissonRegression`).

Note that for this learner, the ``step`` of the solver cannot be tuned
automatically. So, the default value might work, or not.
We therefore urge users to try out different values of the ``step`` parameter
until getting good concergence properties.

Other penalizations are available in `tick.inference.PoissonRegression`:

* no penalization, using ``penalty='none'``
* :math:`\ell_1` penalization, using ``penalty='l1'``
* elastic-net penalization (combination of :math:`\ell_1` and :math:`\ell_2^2`,
  using ``penalty='elasticnet'``, where in this case the ``elastic_net_ratio``
  parameter should be used as well
* total-variation penalization, using ``penalty='tv'``

**Remark**: we don't use in this example ``solver='sgd'`` (namely vanilla
stochastic gradient descent, see `tick.solver.SGD`) since it performs
too poorly.

The plot given below compares the distance to the minimum of each solver along
iterations, on a logarithmic scale.
"""

import numpy as np
import matplotlib.pyplot as plt

from tick.simulation import weights_sparse_gauss
from tick.linear_model import SimuPoisReg, PoissonRegression
from tick.plot import plot_history

n_samples = 50000
n_features = 100
np.random.seed(123)
weight0 = weights_sparse_gauss(n_features, nnz=int(n_features - 1)) / 20.
intercept0 = -0.1
X, y = SimuPoisReg(weight0, intercept0, n_samples=n_samples, verbose=False,
                   seed=123).simulate()

opts = {'verbose': False, 'record_every': 1, 'tol': 1e-8, 'max_iter': 40}

poisson_regressions = [
    PoissonRegression(solver='gd', **opts),
    PoissonRegression(solver='agd', **opts),
    PoissonRegression(solver='svrg', random_state=1234, **opts),
    PoissonRegression(solver='bfgs', **opts)
]

for poisson_regression in poisson_regressions:
    poisson_regression.fit(X, y)

plot_history(poisson_regressions, log_scale=True, dist_min=True)
plt.title('Solvers comparison for Poisson regression', fontsize=16)
plt.tight_layout()