def simulate(n=200, p=100, s=10, signal=(0.5, 1), sigma=2, alpha=0.1, B=2000):
# description of statistical problem
X, y, truth = gaussian_instance(n=n,
p=p,
s=s,
equicorrelated=False,
rho=0.5,
sigma=sigma,
signal=signal,
random_signs=True,
scale=False)[:3]
dispersion = sigma**2
S = X.T.dot(y)
covS = dispersion * X.T.dot(X)
splitting_sampler = split_sampler(X * y[:, None], covS)
def meta_algorithm(XTX, XTXi, lam, sampler):
p = XTX.shape[0]
success = np.zeros(p)
loss = rr.quadratic_loss((p, ), Q=XTX)
pen = rr.l1norm(p, lagrange=lam)
scale = 0.
noisy_S = sampler(scale=scale)
loss.quadratic = rr.identity_quadratic(0, 0, -noisy_S, 0)
problem = rr.simple_problem(loss, pen)
soln = problem.solve(max_its=100, tol=1.e-10)
success += soln != 0
return set(np.nonzero(success)[0])
XTX = X.T.dot(X)
XTXi = np.linalg.inv(XTX)
resid = y - X.dot(XTXi.dot(X.T.dot(y)))
dispersion = np.linalg.norm(resid)**2 / (n - p)
lam = 4. * np.sqrt(n)
selection_algorithm = functools.partial(meta_algorithm, XTX, XTXi, lam)
# run selection algorithm
return full_model_inference(X,
y,
truth,
selection_algorithm,
splitting_sampler,
success_params=(1, 1),
B=B,
fit_probability=keras_fit,
fit_args={
'epochs': 10,
'sizes': [100] * 5,
'dropout': 0.,
'activation': 'relu'
})
def simulate(n=200, p=50, s=5, signal=(0.5, 1), sigma=2, alpha=0.1, B=1000):
# description of statistical problem
X, y, truth = gaussian_instance(n=n,
p=p,
s=s,
equicorrelated=False,
rho=0.5,
sigma=sigma,
signal=signal,
random_signs=True,
scale=False)[:3]
XTX = X.T.dot(X)
XTXi = np.linalg.inv(XTX)
resid = y - X.dot(XTXi.dot(X.T.dot(y)))
dispersion = np.linalg.norm(resid)**2 / (n - p)
S = X.T.dot(y)
covS = dispersion * X.T.dot(X)
splitting_sampler = split_sampler(X * y[:, None], covS)
def meta_algorithm(XTX, XTXi, dispersion, lam, sampler):
p = XTX.shape[0]
success = np.zeros(p)
loss = rr.quadratic_loss((p, ), Q=XTX)
pen = rr.l1norm(p, lagrange=lam)
scale = 0.5
noisy_S = sampler(scale=scale)
soln = XTXi.dot(noisy_S)
solnZ = soln / (np.sqrt(np.diag(XTXi)) * np.sqrt(dispersion))
return set(np.nonzero(np.fabs(solnZ) > 2.1)[0])
lam = 4. * np.sqrt(n)
selection_algorithm = functools.partial(meta_algorithm, XTX, XTXi,
dispersion, lam)
# run selection algorithm
return full_model_inference(X,
y,
truth,
selection_algorithm,
splitting_sampler,
success_params=(5, 7),
B=B,
fit_probability=keras_fit,
fit_args={
'epochs': 30,
'sizes': [100, 100],
'activation': 'relu'
})
def simulate(n=200, p=100, s=10, signal=(1.5, 2), sigma=2, alpha=0.1, B=3000):
# description of statistical problem
X, y, truth = gaussian_instance(n=n,
p=p,
s=s,
equicorrelated=False,
rho=0.5,
sigma=sigma,
signal=signal,
random_signs=True,
scale=False)[:3]
dispersion = sigma**2
S = X.T.dot(y)
covS = dispersion * X.T.dot(X)
smooth_sampler = normal_sampler(S, covS)
splitting_sampler = split_sampler(X * y[:, None], covS)
def meta_algorithm(X, XTXi, resid, sampler):
S = sampler(scale=0.) # deterministic with scale=0
ynew = X.dot(XTXi).dot(S) + resid # will be ok for n>p and non-degen X
G = lasso_glmnet(X, ynew, *[None] * 4)
select = G.select()
return set(list(select[0]))
XTX = X.T.dot(X)
XTXi = np.linalg.inv(XTX)
resid = y - X.dot(XTXi.dot(X.T.dot(y)))
dispersion = np.linalg.norm(resid)**2 / (n - p)
selection_algorithm = functools.partial(meta_algorithm, X, XTXi, resid)
# run selection algorithm
return full_model_inference(X,
y,
truth,
selection_algorithm,
splitting_sampler,
success_params=(1, 1),
B=B,
fit_probability=keras_fit,
fit_args={
'epochs': 10,
'sizes': [100] * 5,
'dropout': 0.,
'activation': 'relu'
})
def simulate(s=10, signal=(0.5, 1), sigma=2, alpha=0.1, B=3000, seed=0):
# description of statistical problem
n, p = X_full.shape
if boot_design:
idx = np.random.choice(np.arange(n), n, replace=True)
X = X_full[
idx] # bootstrap X to make it really an IID sample, i.e. don't condition on X throughout
X += 0.1 * np.std(X) * np.random.standard_normal(
X.shape) # to make non-degenerate
else:
X = X_full.copy()
X = X - np.mean(X, 0)[None, :]
X = X / np.std(X, 0)[None, :]
truth = np.zeros(p)
truth[:s] = np.linspace(signal[0], signal[1], s)
np.random.shuffle(truth)
truth *= sigma / np.sqrt(n)
y = X.dot(truth) + sigma * np.random.standard_normal(n)
lam_min, lam_1se = cv_glmnet_lam(X.copy(), y.copy(), seed=seed)
lam_min, lam_1se = n * lam_min, n * lam_1se
XTX = X.T.dot(X)
XTXi = np.linalg.inv(XTX)
resid = y - X.dot(XTXi.dot(X.T.dot(y)))
dispersion = sigma**2
S = X.T.dot(y)
covS = dispersion * X.T.dot(X)
splitting_sampler = split_sampler(X * y[:, None], covS)
def meta_algorithm(X, XTXi, resid, sampler):
S = sampler.center.copy()
ynew = X.dot(XTXi).dot(S) + resid # will be ok for n>p and non-degen X
G = lasso_glmnet(X, ynew, *[None] * 4)
select = G.select(seed=seed)
return set(list(select[0]))
selection_algorithm = functools.partial(meta_algorithm, X, XTXi, resid)
# run selection algorithm
df = highdim_model_inference(
X,
y,
truth,
selection_algorithm,
splitting_sampler,
lam_min,
sigma**2, # dispersion assumed known for now
success_params=(1, 1),
B=B,
fit_probability=keras_fit,
fit_args={
'epochs': 10,
'sizes': [100] * 5,
'dropout': 0.,
'activation': 'relu'
})
if df is not None:
liu_df = liu_inference(X,
y,
1.00001 * lam_min,
dispersion,
truth,
alpha=alpha,
approximate_inverse='BN')
return pd.merge(df, liu_df, on='variable')
def simulate(s=10, signal=(0.5, 1), sigma=2, alpha=0.1, B=5000, seed=0):
# description of statistical problem
n, p = X_full.shape
if boot_design:
idx = np.random.choice(np.arange(n), n, replace=True)
X = X_full[idx] # bootstrap X to make it really an IID sample, i.e. don't condition on X throughout
X += 0.1 * np.std(X) * np.random.standard_normal(X.shape) # to make non-degenerate
else:
X = X_full.copy()
X = X - np.mean(X, 0)[None, :]
X = X / np.std(X, 0)[None, :]
n, p = X.shape
truth = np.zeros(p)
truth[:s] = np.linspace(signal[0], signal[1], s)
np.random.shuffle(truth)
truth /= np.sqrt(n)
truth *= sigma
y = X.dot(truth) + sigma * np.random.standard_normal(n)
XTX = X.T.dot(X)
XTXi = np.linalg.inv(XTX)
resid = y - X.dot(XTXi.dot(X.T.dot(y)))
dispersion = np.linalg.norm(resid)**2 / (n-p)
S = X.T.dot(y)
covS = dispersion * X.T.dot(X)
splitting_sampler = split_sampler(X * y[:, None], covS)
def meta_algorithm(XTX, XTXi, lam, sampler):
p = XTX.shape[0]
success = np.zeros(p)
loss = rr.quadratic_loss((p,), Q=XTX)
pen = rr.l1norm(p, lagrange=lam)
scale = 0.
noisy_S = sampler(scale=scale)
loss.quadratic = rr.identity_quadratic(0, 0, -noisy_S, 0)
problem = rr.simple_problem(loss, pen)
soln = problem.solve(max_its=100, tol=1.e-10)
success += soln != 0
return set(np.nonzero(success)[0])
lam = 4. * np.sqrt(n)
selection_algorithm = functools.partial(meta_algorithm, XTX, XTXi, lam)
# run selection algorithm
df = full_model_inference(X,
y,
truth,
selection_algorithm,
splitting_sampler,
success_params=(1, 1),
B=B,
fit_probability=keras_fit,
fit_args={'epochs':10, 'sizes':[100]*5, 'dropout':0., 'activation':'relu'})
if False: # df is not None:
liu_df = liu_inference(X,
y,
lam,
dispersion,
truth,
alpha=alpha)
return pd.merge(df, liu_df, on='variable')
else:
return df
def simulate(n=1000, p=100, s=20, signal=(2, 4), sigma=2, alpha=0.1, B=2000):
# description of statistical problem
X, y, truth = gaussian_instance(n=n,
p=p,
s=s,
equicorrelated=False,
rho=0.1,
sigma=sigma,
signal=signal,
random_signs=True,
scale=True)[:3]
dispersion = sigma**2
S = X.T.dot(y)
covS = dispersion * X.T.dot(X)
splitting_sampler = split_sampler(X * y[:, None], covS)
def meta_algorithm(XTX, XTXi, sampler):
min_success = 6
ntries = 10
def _alpha_grid(X, y, center, XTX):
n, p = X.shape
alphas, coefs, _ = lasso_path(X, y, Xy=center, precompute=XTX)
nselected = np.count_nonzero(coefs, axis=0)
return alphas[nselected < np.sqrt(0.8 * p)]
alpha_grid = _alpha_grid(X, y, sampler(scale=0.), XTX)
success = np.zeros((p, alpha_grid.shape[0]))
for _ in range(ntries):
scale = 1. # corresponds to sub-samples of 50%
noisy_S = sampler(scale=scale)
_, coefs, _ = lasso_path(X, y, Xy = noisy_S, precompute=XTX, alphas=alpha_grid)
success += np.abs(np.sign(coefs))
selected = np.apply_along_axis(lambda row: any(x>min_success for x in row), 1, success)
vars = set(np.nonzero(selected)[0])
return vars
XTX = X.T.dot(X)
XTXi = np.linalg.inv(XTX)
resid = y - X.dot(XTXi.dot(X.T.dot(y)))
dispersion = np.linalg.norm(resid)**2 / (n-p)
selection_algorithm = functools.partial(meta_algorithm, XTX, XTXi)
# run selection algorithm
return full_model_inference(X,
y,
truth,
selection_algorithm,
splitting_sampler,
success_params=(1, 1),
B=B,
fit_probability=keras_fit,
fit_args={'epochs':10, 'sizes':[100]*5, 'dropout':0., 'activation':'relu'})
def simulate(s=10, signal=(0.5, 1), sigma=2, alpha=0.1, B=3000, seed=0):
# description of statistical problem
n, p = X_full.shape
if boot_design:
idx = np.random.choice(np.arange(n), n, replace=True)
X = X_full[
idx] # bootstrap X to make it really an IID sample, i.e. don't condition on X throughout
X += 0.1 * np.std(X) * np.random.standard_normal(
X.shape) # to make non-degenerate
else:
X = X_full.copy()
X = X - np.mean(X, 0)[None, :]
X = X / np.std(X, 0)[None, :]
n, p = X.shape
truth = np.zeros(p)
truth[:s] = np.linspace(signal[0], signal[1], s)
np.random.shuffle(truth)
truth /= np.sqrt(n)
truth *= sigma
y = X.dot(truth) + sigma * np.random.standard_normal(n)
XTX = X.T.dot(X)
XTXi = np.linalg.inv(XTX)
resid = y - X.dot(XTXi.dot(X.T.dot(y)))
dispersion = np.linalg.norm(resid)**2 / (n - p)
S = X.T.dot(y)
covS = dispersion * X.T.dot(X)
print(dispersion, sigma**2)
splitting_sampler = split_sampler(X * y[:, None], covS)
def meta_algorithm(X, XTXi, resid, sampler):
S = sampler(scale=0.5) # deterministic with scale=0
ynew = X.dot(XTXi).dot(S) + resid # will be ok for n>p and non-degen X
G = lasso_glmnet(X, ynew, *[None] * 4)
select = G.select(seed=seed)
return set(list(select[0]))
selection_algorithm = functools.partial(meta_algorithm, X, XTXi, resid)
# run selection algorithm
df = full_model_inference(X,
y,
truth,
selection_algorithm,
splitting_sampler,
success_params=(6, 10),
B=B,
fit_probability=keras_fit,
fit_args={
'epochs': 10,
'sizes': [100] * 5,
'dropout': 0.,
'activation': 'relu'
})
return df
def simulate(s=10, signal=(0.5, 1), sigma=2, alpha=0.1, B=3000, seed=0):
# description of statistical problem
n, p = X_full.shape
if boot_design:
idx = np.random.choice(np.arange(n), n, replace=True)
X = X_full[
idx] # bootstrap X to make it really an IID sample, i.e. don't condition on X throughout
X += 0.1 * np.std(X) * np.random.standard_normal(
X.shape) # to make non-degenerate
else:
X = X_full.copy()
X = X - np.mean(X, 0)[None, :]
X = X / np.std(X, 0)[None, :]
n, p = X.shape
truth = np.zeros(p)
truth[:s] = np.linspace(signal[0], signal[1], s)
np.random.shuffle(truth)
truth /= np.sqrt(n)
truth *= sigma
y = X.dot(truth) + sigma * np.random.standard_normal(n)
XTX = X.T.dot(X)
XTXi = np.linalg.inv(XTX)
resid = y - X.dot(XTXi.dot(X.T.dot(y)))
dispersion = np.linalg.norm(resid)**2 / (n - p)
S = X.T.dot(y)
covS = dispersion * X.T.dot(X)
print(dispersion, sigma**2)
splitting_sampler = split_sampler(X * y[:, None], covS)
def meta_algorithm(XTX, XTXi, sampler):
min_success = 6
ntries = 10
def _alpha_grid(X, y, center, XTX):
n, p = X.shape
alphas, coefs, _ = lasso_path(X.copy(),
y.copy(),
Xy=center.copy(),
precompute=XTX.copy())
nselected = np.count_nonzero(coefs, axis=0)
alphas = alphas[nselected < 20]
return alphas
alpha_grid = _alpha_grid(X, y, sampler.center, XTX)
success = np.zeros((p, alpha_grid.shape[0]))
for _ in range(ntries):
scale = 1. # corresponds to sub-samples of 50%
noisy_S = sampler(scale=scale)
_, coefs, _ = lasso_path(X,
y,
Xy=noisy_S,
precompute=XTX,
alphas=alpha_grid)
success += np.abs(np.sign(coefs))
selected = np.apply_along_axis(
lambda row: any(x > min_success for x in row), 1, success)
vars = set(np.nonzero(selected)[0])
return vars
selection_algorithm = functools.partial(meta_algorithm, X, XTXi)
# run selection algorithm
df = full_model_inference(X,
y,
truth,
selection_algorithm,
splitting_sampler,
success_params=(6, 10),
B=B,
fit_probability=keras_fit,
fit_args={
'epochs': 10,
'sizes': [100] * 5,
'dropout': 0.,
'activation': 'relu'
})
return df
def simulate(n=1000, p=60, s=15, signal=3, sigma=2, alpha=0.1):
# description of statistical problem
X, y, truth = gaussian_instance(n=n,
p=p,
s=s,
equicorrelated=False,
rho=0.5,
sigma=sigma,
signal=signal,
random_signs=True)[:3]
dispersion = sigma**2
S = X.T.dot(y)
covS = dispersion * X.T.dot(X)
smooth_sampler = normal_sampler(S, covS)
splitting_sampler = split_sampler(X * y[:, None], covS / n)
def meta_algorithm(XTX, XTXi, dispersion, sampler):
min_success = 3
ntries = 7
p = XTX.shape[0]
success = np.zeros(p)
for _ in range(ntries):
scale = 0.5
frac = 1. / (scale**2 + 1.)
noisy_S = sampler(scale=scale)
noisy_beta = XTXi.dot(noisy_S)
noisy_Z = noisy_beta / np.sqrt(dispersion * np.diag(XTXi) * frac)
success += np.fabs(noisy_Z) > 2
return set(np.nonzero(success >= min_success)[0])
XTX = X.T.dot(X)
XTXi = np.linalg.inv(XTX)
resid = y - X.dot(XTXi.dot(X.T.dot(y)))
dispersion = np.linalg.norm(resid)**2 / (n-p)
selection_algorithm = functools.partial(meta_algorithm, XTX, XTXi, dispersion)
# run selection algorithm
observed_set = selection_algorithm(splitting_sampler)
# find the target, based on the observed outcome
# we just take the first target
pivots, covered, lengths, naive_lengths = [], [], [], []
for idx in observed_set:
print(idx, len(observed_set))
true_target = truth[idx]
(pivot,
interval) = infer_full_target(selection_algorithm,
observed_set,
[idx],
splitting_sampler,
dispersion,
hypothesis=[true_target],
fit_probability=probit_fit,
success_params=(1, 1),
alpha=alpha,
B=1000)[0][:2]
pivots.append(pivot)
covered.append((interval[0] < true_target) * (interval[1] > true_target))
lengths.append(interval[1] - interval[0])
target_sd = np.sqrt(dispersion * XTXi[idx, idx])
naive_lengths.append(2 * ndist.ppf(1 - 0.5 * alpha) * target_sd)
return pivots, covered, lengths, naive_lengths
def simulate(n=200, p=100, s=10, signal=(0.5, 1), sigma=2, alpha=0.1, B=2000):
# description of statistical problem
X, y, truth = gaussian_instance(n=n,
p=p,
s=s,
equicorrelated=False,
rho=0.5,
sigma=sigma,
signal=signal,
random_signs=True,
scale=False)[:3]
dispersion = sigma**2
S = X.T.dot(y)
covS = dispersion * X.T.dot(X)
smooth_sampler = normal_sampler(S, covS)
splitting_sampler = split_sampler(X * y[:, None], covS)
def meta_algorithm(X, XTXi, resid, sampler):
S = sampler(scale=0.) # deterministic with scale=0
ynew = X.dot(XTXi).dot(S) + resid # will be ok for n>p and non-degen X
G = lasso_glmnet(X, ynew, *[None] * 4)
select = G.select()
return set(list(select[0]))
XTX = X.T.dot(X)
XTXi = np.linalg.inv(XTX)
resid = y - X.dot(XTXi.dot(X.T.dot(y)))
dispersion = np.linalg.norm(resid)**2 / (n - p)
selection_algorithm = functools.partial(meta_algorithm, X, XTXi, resid)
# run selection algorithm
success_params = (1, 1)
observed_set = repeat_selection(selection_algorithm, smooth_sampler,
*success_params)
# find the target, based on the observed outcome
# we just take the first target
pivots, covered, lengths, pvalues = [], [], [], []
lower, upper = [], []
naive_pvalues, naive_pivots, naive_covered, naive_lengths = [], [], [], []
targets = []
observed_list = sorted(observed_set)
np.random.shuffle(observed_list)
for idx in observed_list[:1]:
print("variable: ", idx, "total selected: ", len(observed_set))
true_target = [truth[idx]]
targets.extend(true_target)
(pivot, interval,
pvalue) = infer_full_target(selection_algorithm,
observed_set, [idx],
splitting_sampler,
dispersion,
hypothesis=true_target,
fit_probability=probit_fit,
success_params=success_params,
alpha=alpha,
B=B,
single=True)[0][:3]
pvalues.append(pvalue)
pivots.append(pivot)
covered.append(
(interval[0] < true_target[0]) * (interval[1] > true_target[0]))
lengths.append(interval[1] - interval[0])
target_sd = np.sqrt(dispersion * XTXi[idx, idx])
observed_target = np.squeeze(XTXi[idx].dot(X.T.dot(y)))
quantile = ndist.ppf(1 - 0.5 * alpha)
naive_interval = (observed_target - quantile * target_sd,
observed_target + quantile * target_sd)
naive_pivot = (1 - ndist.cdf(
(observed_target - true_target[0]) / target_sd))
naive_pivot = 2 * min(naive_pivot, 1 - naive_pivot)
naive_pivots.append(naive_pivot)
naive_pvalue = (1 - ndist.cdf(observed_target / target_sd))
naive_pvalue = 2 * min(naive_pivot, 1 - naive_pivot)
naive_pvalues.append(naive_pvalue)
naive_covered.append((naive_interval[0] < true_target[0]) *
(naive_interval[1] > true_target[0]))
naive_lengths.append(naive_interval[1] - naive_interval[0])
lower.append(interval[0])
upper.append(interval[1])
if len(pvalues) > 0:
return pd.DataFrame({
'pivot': pivots,
'target': targets,
'pvalue': pvalues,
'coverage': covered,
'length': lengths,
'naive_pivot': naive_pivots,
'naive_coverage': naive_covered,
'naive_length': naive_lengths,
'upper': upper,
'lower': lower
})
def simulate(n=200, p=100, s=10, signal=(0.5, 1), sigma=2, alpha=0.1, B=1000):
# description of statistical problem
X, y, truth = gaussian_instance(n=n,
p=p,
s=s,
equicorrelated=False,
rho=0.5,
sigma=sigma,
signal=signal,
random_signs=True,
scale=False)[:3]
XTX = X.T.dot(X)
XTXi = np.linalg.inv(XTX)
resid = y - X.dot(XTXi.dot(X.T.dot(y)))
dispersion = np.linalg.norm(resid)**2 / (n-p)
S = X.T.dot(y)
covS = dispersion * X.T.dot(X)
smooth_sampler = normal_sampler(S, covS)
splitting_sampler = split_sampler(X * y[:, None], covS)
def meta_algorithm(XTX, XTXi, dispersion, sampler):
p = XTX.shape[0]
success = np.zeros(p)
scale = 0.
noisy_S = sampler(scale=scale)
soln = XTXi.dot(noisy_S)
solnZ = soln / (np.sqrt(np.diag(XTXi)) * np.sqrt(dispersion))
pval = ndist.cdf(solnZ)
pval = 2 * np.minimum(pval, 1 - pval)
return set(BHfilter(pval, q=0.2))
selection_algorithm = functools.partial(meta_algorithm, XTX, XTXi, dispersion)
# run selection algorithm
success_params = (1, 1)
observed_set = repeat_selection(selection_algorithm, smooth_sampler, *success_params)
# find the target, based on the observed outcome
# we just take the first target
targets = []
idx = sorted(observed_set)
np.random.shuffle(idx)
idx = idx[:1]
if len(idx) > 0:
print("variable: ", idx, "total selected: ", len(observed_set))
true_target = truth[idx]
results = infer_full_target(selection_algorithm,
observed_set,
idx,
splitting_sampler,
dispersion,
hypothesis=true_target,
fit_probability=logit_fit,
fit_args={'df':20},
success_params=success_params,
alpha=alpha,
B=B,
single=True)
pvalues = [r[2] for r in results]
covered = [(r[1][0] < t) * (r[1][1] > t) for r, t in zip(results, true_target)]
pivots = [r[0] for r in results]
target_sd = np.sqrt(np.diag(dispersion * XTXi)[idx])
observed_target = XTXi[idx].dot(X.T.dot(y))
quantile = ndist.ppf(1 - 0.5 * alpha)
naive_interval = np.vstack([observed_target - quantile * target_sd, observed_target + quantile * target_sd])
naive_pivots = (1 - ndist.cdf((observed_target - true_target) / target_sd))
naive_pivots = 2 * np.minimum(naive_pivots, 1 - naive_pivots)
naive_pvalues = (1 - ndist.cdf(observed_target / target_sd))
naive_pvalues = 2 * np.minimum(naive_pvalues, 1 - naive_pvalues)
naive_covered = (naive_interval[0] < true_target) * (naive_interval[1] > true_target)
naive_lengths = naive_interval[1] - naive_interval[0]
lower = [r[1][0] for r in results]
upper = [r[1][1] for r in results]
lengths = np.array(upper) - np.array(lower)
return pd.DataFrame({'pivot':pivots,
'pvalue':pvalues,
'coverage':covered,
'length':lengths,
'naive_pivot':naive_pivots,
'naive_coverage':naive_covered,
'naive_length':naive_lengths,
'upper':upper,
'lower':lower,
'targets':true_target,
'batch_size':B * np.ones(len(idx), np.int)})
