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matrix_completion.py
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matrix_completion.py
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# Author: Vlad Niculae <vlad@vene.ro>
# Licence: BSD
from __future__ import division, print_function
import numpy as np
from sklearn.utils import check_random_state
from sklearn.utils.extmath import log_logistic
from sklearn.utils.fixes import expit
class SquaredLoss(object):
def loss(self, y, pred):
return 0.5 * (pred - y) ** 2
def dloss(self, y, pred):
return pred - y
class LogLoss(object):
def loss(self, y, pred):
return (-log_logistic(y * pred))[0][0]
def dloss(self, y, pred):
return (-y * expit(-y * pred))
class EpsilonInsensitive(object):
def __init__(self, eps):
self.eps = eps
def loss(self, y, pred):
loss = np.abs(y - pred) - self.eps
if loss < 0:
return 0
else:
return loss
def dloss(self, y, pred):
raise NotImplemented
class HingeLoss(object):
def loss(self, y, pred):
loss = 1 - y * pred
if loss < 0:
return 0
else:
return loss
def dloss(self, y, pred):
z = pred * y
if z <= 1:
return -y
else:
return 0.0
class MatrixCompletion(object):
"""Online matrix completion by factorization.
Estimates U, V to minimize loss(X, U * V.T).
This is a slow, pure-python implementation.
Basic support for classification when `method='pa'` and
`is_classification=True`.
Parameters
----------
n_components : int, default: 2
Number of components.
method : {'sgd' | 'pa'}, default: 'pa'
Optimization method:
- 'sgd': stochastic gradient descent updates. Defaults to using
the squared loss for regression and the hinge loss
for classification.
- 'pa': passive-aggressive updates. Supports regression (epsilon
insensitive loss) and classification (hinge loss). This
implements PA-I and the aggressiveness parameter is `alpha`.
Supports non-negative constraints only for regression.
is_classification : boolean, default: False
Whether to predict binary (+- 1) values.
classification_loss: {'hingeloss' | 'logloss'}, default: 'hingeloss'
Loss function to use for classification task.
non-negative : boolean, default: False
Whether to enforce non-negativity constraints on the weights. For
`method=='sgd'` this just applies a projection step. For `method=='pa'`
this implements Blondel et al., Online Passive-Aggressive Algorithms for
Non-Negative Matrix Factorization and Completion.
As implemented, only makes sense for regression.
U_init : array, shape=[n_samples, n_components] or None
Initial points for the U factor. If None, initialized to
rand(-0.01, 0.01) + mean(X) / sqrt(n_components)
V_init : array, shape=[n_features, n_components] or None
Initial points for the V factor. If None, initialized to
rand(-0.01, 0.01) + mean(X) / sqrt(n_components)
alpha : float, default: 0.001
If `method='sgd'`, amount of L2 regularization. If `method='pa'`,
aggressiveness factor.
eps : float, default: 0.01
Parameter of the epsilon-insensitive loss. Only used if `method='pa'`
and `is_classification=False`.
initial_learning_rate : float, default: 1e-5
Learning rate to use at the first iteration. Only used if
`method='sgd'`.
pow_t : float, default: 0.5
If `method='sgd'`, the learning rate at iteration t is:
initial_learning_rate / t ** pow_t
tol : float, default: 1e-5
Convergence criterion: absolute difference between consecutive
cumulative losses should be less than it.
max_iter : int, default: 1e5
Maximum number of iterations to perform.
shuffle : boolean, default: False
Whether to shuffle the observations at each iteration.
verbose : int, default: false
Will print debug information at iterations divisible by it.
random_state : int or RandomState
Random number generator seed control.
"""
def __init__(self, n_components=2, method='pa', is_classification=False,
classification_loss='hingeloss', non_negative=False, U_init=None,
V_init=None, alpha=0.001, eps=0.01, initial_learning_rate=1e-5,
pow_t=0.5, tol=1e-5, max_iter=int(1e5), shuffle=False, verbose=False,
random_state=None):
self.n_components = n_components
self.method = method
self.is_classification = is_classification
self.classification_loss = classification_loss
self.non_negative = non_negative
self.U_init = U_init
self.V_init = V_init
self.alpha = alpha
self.eps = eps
self.initial_learning_rate = initial_learning_rate
self.pow_t = pow_t
self.tol = tol
self.max_iter = max_iter
self.shuffle = shuffle
self.verbose = verbose
self.random_state = random_state
def _init(self, X, mask, rng):
n_samples, n_features = X.shape
row_mask, col_mask = mask
if self.is_classification:
mean_x = 0
else:
mean_x = np.sqrt(X[row_mask, col_mask].mean() / self.n_components)
if self.U_init is None:
U = rng.uniform(-0.01, 0.01, size=(n_samples, self.n_components))
U += mean_x
else:
U = self.U_init.copy()
if self.V_init is None:
V = rng.uniform(-0.01, 0.01, size=(n_features, self.n_components))
V += mean_x
else:
V = self.V_init.copy()
if self.non_negative:
# make sure we start in the feasible region
np.clip(U, 0, np.inf, out=U)
np.clip(V, 0, np.inf, out=V)
return U, V
def _update(self, w, x, y, pred=None):
if self.method == 'pa':
return self._pa_update(w, x, y, pred)
else:
return self._sgd_update(w, x, y, pred)
def _sgd_update(self, w, x, y, pred):
"""Squared loss gradient update"""
return -(self.loss_.dloss(y, pred) * x + self.alpha * w)
def _pa_update(self, w, x, y, pred):
# for notation
C = self.alpha
loss = self.loss_.loss(y, pred) # hinge or eps-insensitive
if loss == 0:
return 0
loss /= np.sum(x ** 2)
if loss > C:
loss = C
if self.is_classification:
loss *= y
elif y < pred:
if self.non_negative:
f_C = np.dot(np.clip(w - C * x, 0, np.inf), x) - y - self.eps
if f_C >= 0:
loss = C
loss *= -1
return loss * x
def fit(self, X, y=None, mask=None):
rng = check_random_state(self.random_state)
if not mask:
row_mask, col_mask = X.nonzero()
else:
row_mask, col_mask = mask
# initialize as pyrsvd, so that U_init * V_init.T == X[mask].mean()
U, V = self._init(X, (row_mask, col_mask), rng)
n_nonzero = len(row_mask)
indices = np.arange(n_nonzero)
if 'pa' in self.method:
if self.is_classification:
if 'hingeloss' in self.classification_loss:
self.loss_ = HingeLoss()
else:
self.loss_ = LogLoss() # this works too right?
if self.non_negative:
raise NotImplementedError("Non-negativity constraints for "
"passive-aggressive with hinge "
"loss are not implemented "
"(do they even make sense?)")
else:
self.loss_ = EpsilonInsensitive(eps=self.eps)
else:
if self.is_classification:
if 'hingeloss' in self.classification_loss:
self.loss_ = HingeLoss()
else:
self.loss_ = LogLoss()
else:
self.loss_ = SquaredLoss()
old_cumulative_loss = np.inf
numpy_error_settings = np.seterr(all='raise')
for ii in range(self.max_iter):
if 'pa' in self.method:
learning_rate = 1.0
else:
# inverse scaling learning rate, as in scikit-learn SGDRegressor
learning_rate = self.initial_learning_rate / ((1 + ii) **
self.pow_t)
cumulative_loss = 0
if self.shuffle:
rng.shuffle(indices)
for idx in indices:
row, col = row_mask[idx], col_mask[idx]
pred = np.dot(U[row], V[col])
y = X[row, col]
# Keep track of sum of squared errors for all data
loss = self.loss_.loss(y, pred)
cumulative_loss += loss
u_update = self._update(U[row], V[col], y, pred)
v_update = self._update(V[col], U[row], y, pred)
U[row] += learning_rate * u_update
V[col] += learning_rate * v_update
if self.non_negative:
np.clip(U[row], 0, np.inf, out=U[row])
np.clip(V[col], 0, np.inf, out=V[col])
if np.abs(old_cumulative_loss - cumulative_loss) < self.tol:
if self.verbose:
print("Converged")
break
if self.verbose and not ii % self.verbose:
print("Iteration", ii, ", loss:", cumulative_loss)
old_cumulative_loss = cumulative_loss
np.seterr(**numpy_error_settings)
if self.verbose:
print("Iteration", ii, ", loss:", cumulative_loss)
self.U_, self.V_ = U, V
return self
def example_regression():
rng = np.random.RandomState(0)
U_true = rng.rand(50, 2) * 2
V_true = rng.rand(40, 2) * 2
X = np.dot(U_true, V_true.T)
print(X[:5, :5])
# Put aside 1500 entries for testing
row_mask = rng.randint(0, 50, size=1500)
col_mask = rng.randint(0, 40, size=1500)
# Get a mask for the remaining (training) observations
mask = np.zeros((50, 40), dtype=np.bool)
mask[row_mask, col_mask] = 1
fit_mask = (~mask).nonzero()
MF = MatrixCompletion(method='pa', n_components=2, non_negative=True,
random_state=0, alpha=0.1, eps=0.01, verbose=1,
shuffle=True)
MF.fit(X, mask=fit_mask)
U, V = MF.U_, MF.V_
X_pred = np.dot(U, V.T)
test_err = np.sum((X_pred[row_mask, col_mask] - X[row_mask, col_mask]) ** 2)
print("Test resid. norm ", np.sqrt(test_err))
print(np.dot(U, V.T)[:5, :5])
def example_classification(method='pa', classification_loss='hingeloss', initial_learning_rate=1e-4,
tol=0.01):
noise = 0.1
rng = np.random.RandomState(0)
U_true = rng.randn(50, 2)
V_true = rng.randn(40, 2)
X = np.sign(np.dot(U_true, V_true.T) + noise * rng.randn(50, 40))
print(X[:5, :5])
# Put aside 1500 entries for testing
row_mask = rng.randint(0, 50, size=1500)
col_mask = rng.randint(0, 40, size=1500)
# Get a mask for the remaining (training) observations
mask = np.zeros((50, 40), dtype=np.bool)
mask[row_mask, col_mask] = 1
fit_mask = (~mask).nonzero()
MF = MatrixCompletion(method=method, is_classification=True,
classification_loss = classification_loss,
initial_learning_rate=initial_learning_rate,
n_components=2,random_state=0, alpha=0.1,
verbose=10, shuffle=True, tol=tol)
MF.fit(X, mask=fit_mask)
U, V = MF.U_, MF.V_
X_pred = np.dot(U, V.T)
test_acc = np.mean(np.sign(X_pred[row_mask, col_mask]) ==
X[row_mask, col_mask])
print("Test accuracy:", test_acc)
print(np.dot(U, V.T)[:5, :5])
if __name__ == '__main__':
example_classification()