def print_results(test_set, models, probabilities, title, error_unc=None):
"""Prints and returns KLD and error rate on test set.
Args:
test_set: (x_test, y_test, z_test)
models: list of tuples (weights, threshold)
probabilities: list of floats, containing classifier probabilities
title: string, method name to print
error_unc: optional float, the unconstrained classifier's error rate
Returns:
KLD objective, error rate
"""
x_test, y_test, z_test = test_set
error = evaluation.expected_error_rate(x_test, y_test, models,
probabilities)
klds = evaluation.expected_group_klds(x_test, y_test, z_test, models,
probabilities)
if error_unc is None:
print(title + ": %.3f (%.3f, %.3f)" % (sum(klds), error, 1.0))
else:
print(title + ": %.3f (%.3f, %.3f)" %
(sum(klds), error, error / error_unc))
return sum(klds), error
def error_rate_optimizer(train_set, learning_rate, loops):
"""Returns a model that optimizes the hinge loss."""
x_train, y_train, _ = train_set
dimension = x_train.shape[-1]
tf.reset_default_graph()
# Data tensors.
features_tensor = tf.constant(x_train.astype("float32"), name="features")
labels_tensor = tf.constant(y_train.astype("float32"), name="labels")
# Linear model.
weights = tf.Variable(tf.zeros(dimension, dtype=tf.float32),
name="weights")
threshold = tf.Variable(0, name="threshold", dtype=tf.float32)
predictions_tensor = (tf.tensordot(features_tensor, weights, axes=(1, 0))
+ threshold)
# Set up hinge loss objective.
objective = tf.losses.hinge_loss(labels=labels_tensor,
logits=predictions_tensor)
# Set up the optimizer and get `train_op` for gradient updates.
solver = tf.train.AdamOptimizer(learning_rate=learning_rate)
train_op = solver.minimize(objective)
# Start TF session and initialize variables.
session = tf.Session()
session.run(tf.global_variables_initializer())
# We maintain a list of objectives and model weights during training.
objectives = []
models = []
# Perform full gradient updates.
for ii in range(loops):
# Gradient updates.
session.run(train_op)
# Checkpoint once in 10 iterations.
if ii % 10 == 0:
# Model weights.
model = [session.run(weights), session.run(threshold)]
models.append(model)
# Objective.
objective = evaluation.expected_error_rate(
x_train, y_train, [model], [1.0])
objectives.append(objective)
# Use the recorded objectives and constraints to find the best iterate.
best_iterate = np.argmin(objectives)
best_model = models[best_iterate]
return best_model
def run_experiment():
"""Run experiments comparing unconstrained and constrained methods."""
# Range of hyper-parameters for unconstrained and constrained optimization.
lr_range_unc = [0.005, 0.01, 0.05, 0.1, 0.5, 1.0, 5.0, 10.0]
lr_range_con = [0.001, 0.01, 0.1, 1.0]
# Load dataset.
with open(FLAGS.data_file, "rb") as f:
train_set, vali_set, test_set = np.load(f,
allow_pickle=True,
fix_imports=True)
x_vali, y_vali, z_vali = vali_set
##################################################
# Unconstrained Error Optimization.
print("Running unconstrained error optimization")
models_unc = []
param_objectives_unc = []
# Find best learning rate.
for lr_model in lr_range_unc:
model = methods.error_rate_optimizer(train_set,
learning_rate=lr_model,
loops=FLAGS.loops_unc)
error = evaluation.expected_error_rate(x_vali, y_vali, [model], [1.0])
param_objectives_unc.append(error)
models_unc.append(model)
best_param_index_unc = np.argmin(param_objectives_unc)
model_er = models_unc[best_param_index_unc]
print()
##################################################
# Post-shift for Demographic Parity.
print("Running post-shift for demographic parity")
# First train logistic regression model.
models_log = []
param_objectives_log = []
# Find best learning rate.
for lr_model in lr_range_unc:
model = methods.logistic_regression(train_set,
learning_rate=lr_model,
loops=FLAGS.loops_unc)
loss = evaluation.cross_entropy_loss(x_vali, y_vali, model[0],
model[1])
param_objectives_log.append(loss)
models_log.append(model)
best_param_index_log = np.argmin(param_objectives_log)
logreg_model = models_log[best_param_index_log]
# Post-shift logistic regression model for demographic parity.
model_ps, train_set_ps, vali_set_ps, test_set_ps = methods.post_shift_dp(
train_set, vali_set, test_set, logreg_model)
print()
##################################################
# Surrogate-based Lagrangian Optimizer for Convex Rate Metrics (Algorithm 2).
print("Running constrained Lagrangian optimization (Algorithm 2)")
# Set additive slack to unconstrained error * epsilon.
x_train, y_train, _ = train_set
error_unc_train = evaluation.expected_error_rate(x_train, y_train,
[model_er], [1.0])
additive_slack = error_unc_train * FLAGS.epsilon
# Maintain list of models, objectives and violations for hyper-parameters.
stochastic_models_list = []
deterministic_models_list = []
param_objectives_con = []
param_violations_con = []
# Find best learning rates for model parameters and Lagrange multipliers.
for lr_model in lr_range_con:
for lr_constraint in lr_range_con:
stochastic_model, deterministic_model = (
methods.lagrangian_optimizer_kld(
train_set,
learning_rate=lr_model,
learning_rate_constraint=lr_constraint,
loops=FLAGS.loops_con,
additive_slack=additive_slack))
stochastic_models_list.append(stochastic_model)
deterministic_models_list.append(deterministic_model)
# Record objective and constraint violations for stochastic model.
klds = evaluation.expected_group_klds(x_vali, y_vali, z_vali,
stochastic_model[0],
stochastic_model[1])
param_objectives_con.append(sum(klds))
error = evaluation.expected_error_rate(x_vali, y_vali,
stochastic_model[0],
stochastic_model[1])
param_violations_con.append([error - additive_slack])
print("Parameters (%.3f, %.3f): %.3f (%.3f)" %
(lr_model, lr_constraint, param_objectives_con[-1],
max(param_violations_con[-1])))
# Best param.
best_param_index_con = tfco.find_best_candidate_index(
np.array(param_objectives_con), np.array(param_violations_con))
stochastic_model_con = stochastic_models_list[best_param_index_con]
deterministic_model_con = deterministic_models_list[best_param_index_con]
print()
# Print summary of performance on test set.
results = {}
results["UncError"] = print_results(test_set, [model_er], [1.0],
"UncError")
error_unc = results["UncError"][1]
results["PostShift"] = print_results(test_set_ps, [model_ps], [1.0],
"PostShift", error_unc)
results["Stochastic"] = print_results(test_set, stochastic_model_con[0],
stochastic_model_con[1],
"Constrained (Stochastic)",
error_unc)
results["Deterministic"] = print_results(test_set,
[deterministic_model_con], [1.0],
"Constrained (Deterministic)",
error_unc)
print()
# Print summary of performance on train set.
results = {}
results["UncError"] = print_results(train_set, [model_er], [1.0],
"UncError")
error_unc = results["UncError"][1]
results["PostShift"] = print_results(train_set_ps, [model_ps], [1.0],
"PostShift", error_unc)
results["Stochastic"] = print_results(train_set, stochastic_model_con[0],
stochastic_model_con[1],
"Constrained (Stochastic)",
error_unc)
results["Deterministic"] = print_results(train_set,
[deterministic_model_con], [1.0],
"Constrained (Deterministic)",
error_unc)
print()
# Print summary of performance on vali set.
results = {}
results["UncError"] = print_results(vali_set, [model_er], [1.0],
"UncError")
error_unc = results["UncError"][1]
results["PostShift"] = print_results(vali_set_ps, [model_ps], [1.0],
"PostShift", error_unc)
results["Stochastic"] = print_results(vali_set, stochastic_model_con[0],
stochastic_model_con[1],
"Constrained (Stochastic)",
error_unc)
results["Deterministic"] = print_results(vali_set,
[deterministic_model_con], [1.0],
"Constrained (Deterministic)",
error_unc)
def lagrangian_optimizer_kld(
train_set, additive_slack, learning_rate, learning_rate_constraint, loops):
"""Implements surrogate-based Lagrangian optimizer (Algorithm 2).
Specifically solves:
min_{theta} sum_{G = 0, 1} KLD(p, pprG(theta))
s.t. error_rate <= additive_slack,
where p is the overall proportion of positives and pprG is the positive
prediction rate for group G.
We frame this as a constrained optimization problem:
min_{theta, xi_pos0, xi_pos1, xi_neg0, xi_neg1} {
-p log(xi_pos0) - (1-p) log(xi_neg0) - p log(xi_pos1)
-(1-p) log(xi_neg1)}
s.t.
error_rate <= additive_slack,
xi_pos0 <= ppr0(theta), xi_neg0 <= npr0(theta),
xi_pos1 <= ppr1(theta), xi_neg1 <= npr1(theta),
and formulate the Lagrangian:
max_{lambda's >= 0} min_{xi's} {
-p log(xi_pos0) - (1-p) log(xi_neg0) - p log(xi_pos1)
-(1-p) log(xi_neg1)
+ lambda_pos0 (xi_pos0 - ppr0(theta))
+ lambda_neg0 (xi_neg0 - npr0(theta))
+ lambda_pos1 (xi_pos1 - ppr1(theta))
+ lambda_neg1 (xi_neg1 - npr1(theta))}
s.t.
error_rate <= additive_slack.
We do best response for the slack variables xi:
BR for xi_pos0 = p / lambda_pos0
BR for xi_neg0 = (1 - p) / lambda_neg0
BR for xi_pos1 = p / lambda_pos1
BR for xi_neg1 = (1 - p) / lambda_neg1
We do gradient ascent on the lambda's, where
Gradient w.r.t. lambda_pos0
= BR for xi_pos0 - ppr0(theta)
= p / lambda_pos0 - ppr0(theta)
= Gradient w.r.t. lambda_pos0 of
(p log(lambda_pos0) - lambda_pos0 ppr0(theta))
Gradient w.r.t. lambda_neg0
= Gradient w.r.t. lambda_neg0 of
((1 - p) log(lambda_neg0) - lambda_neg0 npr0(theta))
Gradient w.r.t. lambda_pos1
= Gradient w.r.t. lambda_pos1 of
(p log(lambda_pos1) - lambda_pos1 ppr1(theta))
Gradient w.r.t. lambda_neg1
= Gradient w.r.t. lambda_neg1 of
((1 - p) log(lambda_neg1) - lambda_neg1 npr1(theta)).
We do gradient descent on thetas's, with ppr's and npr's replaced with hinge
surrogates. We use concave lower bounds on ppr's and npr's, so that when they
get negated in the updates, we get convex upper bounds.
See Appendix D.1 in the paper for more details.
Args:
train_set: (features, labels, groups)
additive_slack: float, additive slack on error rate constraint
learning_rate: float, learning rate for model parameters
learning_rate_constraint: float, learning rate for Lagrange multipliers
loops: int, number of iterations
Returns:
stochastic_model containing list of models and probabilities,
deterministic_model.
"""
x_train, y_train, z_train = train_set
dimension = x_train.shape[-1]
tf.reset_default_graph()
# Data tensors.
features_tensor = tf.constant(x_train.astype("float32"), name="features")
labels_tensor = tf.constant(y_train.astype("float32"), name="labels")
# Linear model.
weights = tf.Variable(tf.zeros(dimension, dtype=tf.float32),
name="weights")
threshold = tf.Variable(0, name="threshold", dtype=tf.float32)
predictions_tensor = (tf.tensordot(features_tensor, weights, axes=(1, 0))
+ threshold)
# Group-specific predictions.
predictions_group0 = tf.boolean_mask(predictions_tensor, mask=(z_train < 1))
num_examples0 = np.sum(z_train < 1)
predictions_group1 = tf.boolean_mask(predictions_tensor, mask=(z_train > 0))
num_examples1 = np.sum(z_train > 0)
# We use the TF Constrained Optimization (TFCO) library to set up the
# constrained optimization problem. The library doesn't currently support best
# responses for slack variables. So we maintain explicit Lagrange multipliers
# for the slack variables, and let the library deal with the Lagrange
# multipliers for the error rate constraint.
# Since we need to perform a gradient descent update on the model parameters,
# and an ascent update on the Lagrange multipliers on the slack variables, we
# create a single "minimization" objective using stop gradients, where a
# descent gradient update has the effect of minimizing over the model
# parameters and maximizing over the Lagrange multipliers for the slack
# variables. As noted above, the ascent update on the Lagrange multipliers for
# the error rate constraint is done by the library internally.
# Placeholders for Lagrange multipliers for the four slack variables.
lambda_pos0 = tf.Variable(0.5, dtype=tf.float32, name="lambda_pos0")
lambda_neg0 = tf.Variable(0.5, dtype=tf.float32, name="lambda_neg0")
lambda_pos1 = tf.Variable(0.5, dtype=tf.float32, name="lambda_pos1")
lambda_neg1 = tf.Variable(0.5, dtype=tf.float32, name="lambda_neg1")
# Set up prediction rates and surrogate relaxations on them.
p = np.mean(y_train) # Proportion of positives.
# Positive and negative prediction rates for group 0 and group 1.
ppr_group0 = tf.reduce_sum(tf.cast(
tf.greater(predictions_group0, tf.zeros(num_examples0, dtype="float32")),
"float32")) / num_examples0
npr_group0 = 1 - ppr_group0
ppr_group1 = tf.reduce_sum(tf.cast(
tf.greater(predictions_group1, tf.zeros(num_examples1, dtype="float32")),
"float32")) / num_examples1
npr_group1 = 1 - ppr_group1
# Hinge concave lower bounds on the positive and negative prediction rates.
# In the gradient updates, these get negated and become convex upper bounds.
# For group 0:
ppr_hinge_group0 = tf.reduce_sum(
1 - tf.nn.relu(1 - predictions_group0)) * 1.0 / num_examples0
npr_hinge_group0 = tf.reduce_sum(
1 - tf.nn.relu(1 + predictions_group0)) * 1.0 / num_examples0
# For group 1:
ppr_hinge_group1 = tf.reduce_sum(
1 - tf.nn.relu(1 - predictions_group1)) * 1.0 / num_examples1
npr_hinge_group1 = tf.reduce_sum(
1 - tf.nn.relu(1 + predictions_group1)) * 1.0 / num_examples1
# Set up KL-divergence objective for constrained optimization.
# We use stop gradients to ensure that a single descent gradient update on the
# objective has the effect of minimizing over the model parameters and
# maximizing over the Lagrange multipliers for the slack variables.
# KL-divergence for group 0.
kld_hinge_pos_group0 = (
- tf.stop_gradient(lambda_pos0) * ppr_hinge_group0
- p * tf.log(lambda_pos0) + lambda_pos0 * tf.stop_gradient(ppr_group0))
kld_hinge_neg_group0 = (
- tf.stop_gradient(lambda_neg0) * npr_hinge_group0
- (1 - p) * tf.log(lambda_neg0)
+ lambda_neg0 * tf.stop_gradient(npr_group0))
kld_hinge_group0 = kld_hinge_pos_group0 + kld_hinge_neg_group0
# KL-divergence for group 1.
kld_hinge_pos_group1 = (
- tf.stop_gradient(lambda_pos1) * ppr_hinge_group1
- p * tf.log(lambda_pos1) + lambda_pos1 * tf.stop_gradient(ppr_group1))
kld_hinge_neg_group1 = (
- tf.stop_gradient(lambda_neg1) * npr_hinge_group1
- (1 - p) * tf.log(lambda_neg1)
+ lambda_neg1 * tf.stop_gradient(npr_group1))
kld_hinge_group1 = kld_hinge_pos_group1 + kld_hinge_neg_group1
# Wrap the objective into a rate object.
objective = tfco.wrap_rate(kld_hinge_group0 + kld_hinge_group1)
# Set up error rate constraint for constrained optimization.
context = tfco.rate_context(predictions_tensor, labels_tensor)
error = tfco.error_rate(context)
constraints = [error <= additive_slack]
# Cretae rate minimization problem object.
problem = tfco.RateMinimizationProblem(objective, constraints)
# Set up optimizer.
optimizer = tfco.LagrangianOptimizerV1(
tf.train.AdamOptimizer(learning_rate=learning_rate),
constraint_optimizer=tf.train.AdamOptimizer(
learning_rate=learning_rate_constraint))
train_op = optimizer.minimize(problem)
# Start TF session and initialize variables.
session = tf.Session()
session.run(tf.global_variables_initializer())
# We maintain a list of objectives and model weights during training.
objectives = []
violations = []
models = []
# Perform full gradient updates.
for ii in range(loops):
# Gradient updates.
session.run(train_op)
# Checkpoint once in 10 iterations.
if ii % 10 == 0:
# Model weights.
model = [session.run(weights), session.run(threshold)]
models.append(model)
# Objective.
klds = evaluation.expected_group_klds(
x_train, y_train, z_train, [model], [1.0])
objectives.append(sum(klds))
# Violation.
error = evaluation.expected_error_rate(
x_train, y_train, [model], [1.0])
violations.append([error - additive_slack])
# Use the recorded objectives and constraints to find the best iterate.
best_iterate = tfco.find_best_candidate_index(
np.array(objectives), np.array(violations))
deterministic_model = models[best_iterate]
# Use shrinking to find a sparse distribution over iterates.
probabilities = tfco.find_best_candidate_distribution(
np.array(objectives), np.array(violations))
models_pruned = [models[i] for i in range(len(models)) if
probabilities[i] > 0.0]
probabilities_pruned = probabilities[probabilities > 0.0]
return (models_pruned, probabilities_pruned), deterministic_model