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ppi.py
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import numpy as np
import matplotlib.pyplot as plt
import itertools
from scipy.stats import binom, norm
from scipy.special import expit
from scipy.optimize import brentq
from sklearn.linear_model import LogisticRegression
from joblib import delayed, Parallel
import pdb
"""
IID Concentration Bounds
"""
def binomial_iid(N,alpha,muhat):
def invert_upper_tail(mu): return binom.cdf(N*muhat, N, mu) - (alpha/2)
def invert_lower_tail(mu): return binom.cdf(N*muhat, N, mu) - (1-alpha/2)
u = brentq(invert_upper_tail,0,1)
l = brentq(invert_lower_tail,0,1)
return np.array([l,u])
def bentkus_iid(N, alpha, muhat):
return binomial_iid(N, alpha/np.e, muhat)
def clt_iid(x, alpha):
n = x.shape[0]
sigmahat = x.std()
w = norm.ppf(1-alpha/2) * sigmahat / np.sqrt(n)
return np.array([ x.mean() - w, x.mean() + w ])
def wsr_iid(x_n, alpha, grid, num_cpus=10, parallelize: bool = False, intersection: bool = True,
theta: float = 0.5, c: float = 0.75):
n = x_n.shape[0]
t_n = np.arange(1, n + 1)
muhat_n = (0.5 + np.cumsum(x_n)) / (1 + t_n)
sigma2hat_n = (0.25 + np.cumsum(np.power(x_n - muhat_n, 2))) / (1 + t_n)
sigma2hat_tminus1_n = np.append(0.25, sigma2hat_n[: -1])
assert(np.all(sigma2hat_tminus1_n > 0))
lambda_n = np.sqrt(2 * np.log(2 / alpha) / (n * sigma2hat_tminus1_n))
def M(m):
lambdaplus_n = np.minimum(lambda_n, c / m)
lambdaminus_n = np.minimum(lambda_n, c / (1 - m))
return np.maximum(
theta * np.exp(np.cumsum(np.log(1 + lambdaplus_n * (x_n - m)))),
(1 - theta) * np.exp(np.cumsum(np.log(1 - lambdaminus_n * (x_n - m))))
)
if parallelize: # sometimes much slower
M = np.vectorize(M)
M_list = Parallel(n_jobs=num_cpus)(delayed(M)(m) for m in grid)
indicators_gxn = np.array(M_list) < 1 / alpha
else:
indicators_gxn = np.zeros([grid.size, n])
found_lb = False
for m_idx, m in enumerate(grid):
m_n = M(m)
indicators_gxn[m_idx] = m_n < 1 / alpha
if not found_lb and np.prod(indicators_gxn[m_idx]):
found_lb = True
if found_lb and not np.prod(indicators_gxn[m_idx]):
break # since interval, once find a value that fails, stop searching
if intersection:
ci_full = grid[np.where(np.prod(indicators_gxn, axis=1))[0]]
else:
ci_full = grid[np.where(indicators_gxn[:, -1])[0]]
if ci_full.size == 0: # grid maybe too coarse
idx = np.argmax(np.sum(indicators_gxn, axis=1))
if idx == 0:
return np.array([grid[0], grid[1]])
return np.array([grid[idx - 1], grid[idx]])
return np.array([ci_full.min(), ci_full.max()]) # only output the interval
"""
Mean estimation confidence intervals
"""
def pp_mean_iid_asymptotic(Y_labeled, Yhat_labeled, Yhat_unlabeled, alpha):
n = Y_labeled.shape[0]
N = Yhat_unlabeled.shape[0]
tildethetaf = Yhat_unlabeled.mean()
rechat = (Yhat_labeled - Y_labeled).mean()
thetahatPP = tildethetaf - rechat
sigmaftilde = np.std(Yhat_unlabeled)
sigmarec = np.std(Yhat_labeled - Y_labeled)
hw = norm.ppf(1-alpha/2)*np.sqrt((sigmaftilde**2/N) + (sigmarec**2/n))
return [thetahatPP - hw, thetahatPP + hw]
"""
OLS algorithm with sandwich variance estimator
"""
def ols(features, outcome):
ols_coeffs = np.linalg.pinv(features).dot(outcome)
return ols_coeffs
def classical_ols_interval(X, Y, alpha, return_stderr=False, sandwich=True):
n = X.shape[0]
thetahat = ols(X, Y)
Sigmainv = np.linalg.inv(1/n * X.T@X)
if sandwich:
M = 1/n * (X.T*((Y - X@thetahat)**2)[None,:])@X
else:
M = 1/n * ((Y - X@thetahat)**2).mean() * X.T@X
V = Sigmainv@M@Sigmainv
stderr = np.sqrt(np.diag(V))
if return_stderr:
return stderr
halfwidth = norm.ppf(1-alpha/2) * stderr/np.sqrt(n)
return thetahat - halfwidth, thetahat + halfwidth
def pp_ols_interval(X_labeled, X_unlabeled, Y_labeled, Yhat_labeled, Yhat_unlabeled, alpha, sandwich=True):
n = X_labeled.shape[0]
N = X_unlabeled.shape[0]
thetatildef = ols(X_unlabeled, Yhat_unlabeled)
rectifierhat = ols(X_labeled, Y_labeled - Yhat_labeled)
pp_thetahat = thetatildef + rectifierhat
stderr_tildef = classical_ols_interval(X_unlabeled, Yhat_unlabeled, 0.001*alpha, return_stderr=True, sandwich=sandwich)
stderr_rec = classical_ols_interval(X_labeled, Y_labeled-Yhat_labeled, 0.999*alpha, return_stderr=True, sandwich=sandwich)
halfwidth = norm.ppf(1-alpha/2)*np.sqrt(((stderr_rec**2)/n) + ((stderr_tildef**2)/N))
return pp_thetahat - halfwidth, pp_thetahat + halfwidth
"""
Logistic regression algorithm
"""
def logistic(X, y):
clf = LogisticRegression(penalty='none', solver='lbfgs', max_iter=10000, tol=1e-15, fit_intercept=False).fit(X,y)
return clf.coef_.squeeze()
def product(*args, **kwds):
# product('ABCD', 'xy') --> Ax Ay Bx By Cx Cy Dx Dy
# product(range(2), repeat=3) --> 000 001 010 011 100 101 110 111
pools = map(tuple, args)
result = [[]]
for pool in pools:
result = [x+[y] for x in result for y in pool]
for prod in result:
yield tuple(prod)
def classical_logistic_interval(X, Y, alpha, num_grid=500):
n = X.shape[0]
d = X.shape[1]
Y = (Y >= 0.5).astype(int)
point_estimate = logistic(X, Y) # Used for setting the grid.
theta_grid = np.concatenate([
np.linspace(-3*point_estimate, point_estimate, num_grid//2),
np.linspace(point_estimate, 3*point_estimate, num_grid//2)[1:]
])
mu = expit(X@theta_grid.T)
g = 1/n * X.T@(mu - Y[:, None])
sigmahat_err = np.std(X[:,:,None]*(mu - Y[:,None])[:,None,:], axis=0)
grad_halfwidth = norm.ppf(1-alpha/(2*d)) * sigmahat_err/np.sqrt(n)
condition = np.all(np.abs(g) <= grad_halfwidth, axis=0)
Cpp = theta_grid[condition]
# TODO: If all positive, make grid wider
assert (condition[0] == False) & (condition[-1] == False)
return [ Cpp.min(axis=0), Cpp.max(axis=0) ]
def pp_logistic_interval(X_labeled, X_unlabeled, Y_labeled, Yhat_labeled, Yhat_unlabeled, alpha, num_grid=500):
X = np.concatenate([X_labeled, X_unlabeled], axis=0)
n = X_labeled.shape[0]
d = X_labeled.shape[1]
N = X_unlabeled.shape[0]
Yhat_labeled = np.clip(Yhat_labeled, 0, 1)
Yhat_unlabeled = np.clip(Yhat_unlabeled, 0, 1)
Yhat = np.concatenate([Yhat_labeled, Yhat_unlabeled], axis=0)
point_estimate = logistic(X_labeled, (Y_labeled > 0.5).astype(int))
rechat = 1/n * X_labeled.T @ (Yhat_labeled - Y_labeled)
sigmahat_rec = np.std(X_labeled * (Yhat_labeled - Y_labeled)[:,None], axis=0)
theta_grid = np.concatenate([
np.linspace(-3*point_estimate, point_estimate, num_grid//2),
np.linspace(point_estimate, 3*point_estimate, num_grid//2)[1:]
])
mu = expit(X_unlabeled@theta_grid.T)
g = 1/N * X_unlabeled.T@(mu - Yhat_unlabeled[:, None])
sigmahat_err = np.std(X_unlabeled[:,:,None]*(mu - Yhat_unlabeled[:,None])[:,None,:], axis=0)
halfwidth = norm.ppf(1-alpha/(2*d)) * np.sqrt( sigmahat_rec[:,None]**2/n + sigmahat_err**2/N)
condition = np.all( np.abs(g + rechat[:,None]) <= halfwidth, axis=0)
Cpp = theta_grid[condition]
# TODO: If all positive, make grid wider
assert (condition[0] == False) & (condition[-1] == False)
return [ Cpp.min(axis=0), Cpp.max(axis=0) ]
"""
DISCRETE L_p ESTIMATION RATES
"""
def linfty_dkw(N, K, alpha):
return np.sqrt(2/N * np.log(2 / alpha))
def linfty_binom(N, K, alpha, qhat):
epsilon = 0
for k in np.arange(K):
bci = binomial_iid(N, alpha/K, qhat[k])
epsilon = np.maximum(epsilon, np.abs(bci-qhat[k]).max())
return epsilon
"""
SAMPLING WITHOUT REPLACEMENT
"""
def clt_swr(x,N,alpha):
n = x.shape[0]
point_estimate = x.mean()
fluctuations = x.std()*norm.cdf(1-alpha/2)*np.sqrt((N-n)/(N*n))
return np.array([point_estimate-fluctuations, point_estimate+fluctuations])
def wsr_swr(x,N,alpha,grid,num_cpus=10, intersection=True): # x is a [0,1] bounded sequence
n = x.shape[0]
def mu(m,i): return (N*m - np.concatenate([np.array([0,]), np.cumsum(x[:i-1])]))/(N - (np.arange(i)+1) + 1 )
muhats = (1/2 + np.cumsum(x))/(np.arange(n)+1)
sigmahat2s = (1/4 + np.cumsum((x-muhats)**2))/(np.arange(n)+1)
lambdas = np.concatenate([np.array([1,]), np.sqrt(2*np.log(2/alpha)/(n*sigmahat2s))[:-1]]) # can't use last entry
def M(m,i): return 1/2*np.maximum(
np.prod(1+np.minimum(lambdas[:i], 0.5/mu(m,i))*(x[:i]-mu(m,i))),
np.prod(1-np.minimum(lambdas[:i], 0.5/(1-mu(m,i)))*(x[:i]-mu(m,i)))
)
M = np.vectorize(M)
if intersection:
M_list = Parallel(n_jobs=num_cpus)(delayed(M)(grid,i) for i in range(1,n+1))
else:
M_list =[M(grid, n),]
ci_full = grid[np.where(np.prod(np.stack(M_list, axis=1) < 1/alpha , axis=1))[0]]
return np.array([ci_full.min(), ci_full.max()]) # only output the interval