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Resumo
Neste colab, demonstramos como usar os vários otimizadores implementados no TensorFlow Probability.
Dependências e pré-requisitos
Importar
%matplotlib inline
import contextlib
import functools
import os
import time
import numpy as np
import pandas as pd
import scipy as sp
from six.moves import urllib
from sklearn import preprocessing
import tensorflow.compat.v2 as tf
tf.enable_v2_behavior()
import tensorflow_probability as tfp
Otimizadores BFGS e L-BFGS
Os métodos quase Newton são uma classe de algoritmo de otimização de primeira ordem popular. Esses métodos usam uma aproximação definida positiva para o Hessian exato para encontrar a direção da pesquisa.
O algoritmo Broyden-Fletcher-Goldfarb-Shanno ( BFGS ) é uma implementação específica desta ideia geral. É aplicável e é o método de escolha para problemas de tamanho médio, onde o gradiente é contínuo em toda a parte (por exemplo, regressão linear com um \(L_2\) penalidade).
L-BFGS é uma versão de memória limitada de BFGS que é útil para resolver problemas maiores cujas matrizes Hessian não pode ser calculado a um custo razoável ou não são escassos. Em vez de armazenar totalmente densos \(n \times n\) aproximações de matrizes de Hesse, eles só economizar alguns vetores de tamanho \(n\) que representam estas aproximações implicitamente.
Funções auxiliares
CACHE_DIR = os.path.join(os.sep, 'tmp', 'datasets')
def make_val_and_grad_fn(value_fn):
@functools.wraps(value_fn)
def val_and_grad(x):
return tfp.math.value_and_gradient(value_fn, x)
return val_and_grad
@contextlib.contextmanager
def timed_execution():
t0 = time.time()
yield
dt = time.time() - t0
print('Evaluation took: %f seconds' % dt)
def np_value(tensor):
"""Get numpy value out of possibly nested tuple of tensors."""
if isinstance(tensor, tuple):
return type(tensor)(*(np_value(t) for t in tensor))
else:
return tensor.numpy()
def run(optimizer):
"""Run an optimizer and measure it's evaluation time."""
optimizer() # Warmup.
with timed_execution():
result = optimizer()
return np_value(result)
L-BFGS em uma função quadrática simples
# Fix numpy seed for reproducibility
np.random.seed(12345)
# The objective must be supplied as a function that takes a single
# (Tensor) argument and returns a tuple. The first component of the
# tuple is the value of the objective at the supplied point and the
# second value is the gradient at the supplied point. The value must
# be a scalar and the gradient must have the same shape as the
# supplied argument.
# The `make_val_and_grad_fn` decorator helps transforming a function
# returning the objective value into one that returns both the gradient
# and the value. It also works for both eager and graph mode.
dim = 10
minimum = np.ones([dim])
scales = np.exp(np.random.randn(dim))
@make_val_and_grad_fn
def quadratic(x):
return tf.reduce_sum(scales * (x - minimum) ** 2, axis=-1)
# The minimization routine also requires you to supply an initial
# starting point for the search. For this example we choose a random
# starting point.
start = np.random.randn(dim)
# Finally an optional argument called tolerance let's you choose the
# stopping point of the search. The tolerance specifies the maximum
# (supremum) norm of the gradient vector at which the algorithm terminates.
# If you don't have a specific need for higher or lower accuracy, leaving
# this parameter unspecified (and hence using the default value of 1e-8)
# should be good enough.
tolerance = 1e-10
@tf.function
def quadratic_with_lbfgs():
return tfp.optimizer.lbfgs_minimize(
quadratic,
initial_position=tf.constant(start),
tolerance=tolerance)
results = run(quadratic_with_lbfgs)
# The optimization results contain multiple pieces of information. The most
# important fields are: 'converged' and 'position'.
# Converged is a boolean scalar tensor. As the name implies, it indicates
# whether the norm of the gradient at the final point was within tolerance.
# Position is the location of the minimum found. It is important to check
# that converged is True before using the value of the position.
print('L-BFGS Results')
print('Converged:', results.converged)
print('Location of the minimum:', results.position)
print('Number of iterations:', results.num_iterations)
Evaluation took: 0.014586 seconds L-BFGS Results Converged: True Location of the minimum: [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.] Number of iterations: 10
Mesmo problema com BFGS
@tf.function
def quadratic_with_bfgs():
return tfp.optimizer.bfgs_minimize(
quadratic,
initial_position=tf.constant(start),
tolerance=tolerance)
results = run(quadratic_with_bfgs)
print('BFGS Results')
print('Converged:', results.converged)
print('Location of the minimum:', results.position)
print('Number of iterations:', results.num_iterations)
Evaluation took: 0.010468 seconds BFGS Results Converged: True Location of the minimum: [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.] Number of iterations: 10
Regressão linear com penalidade L1: dados do câncer de próstata
Exemplo do livro: The Elements of Statistical Aprendizagem, Data Mining, inferência e predição por Trevor Hastie, Robert Tibshirani e Jerome Friedman.
Observe que este é um problema de otimização com penalidade L1.
Obter conjunto de dados
def cache_or_download_file(cache_dir, url_base, filename):
"""Read a cached file or download it."""
filepath = os.path.join(cache_dir, filename)
if tf.io.gfile.exists(filepath):
return filepath
if not tf.io.gfile.exists(cache_dir):
tf.io.gfile.makedirs(cache_dir)
url = url_base + filename
print("Downloading {url} to {filepath}.".format(url=url, filepath=filepath))
urllib.request.urlretrieve(url, filepath)
return filepath
def get_prostate_dataset(cache_dir=CACHE_DIR):
"""Download the prostate dataset and read as Pandas dataframe."""
url_base = 'http://web.stanford.edu/~hastie/ElemStatLearn/datasets/'
return pd.read_csv(
cache_or_download_file(cache_dir, url_base, 'prostate.data'),
delim_whitespace=True, index_col=0)
prostate_df = get_prostate_dataset()
Downloading http://web.stanford.edu/~hastie/ElemStatLearn/datasets/prostate.data to /tmp/datasets/prostate.data.
Definição de problema
np.random.seed(12345)
feature_names = ['lcavol', 'lweight', 'age', 'lbph', 'svi', 'lcp',
'gleason', 'pgg45']
# Normalize features
scalar = preprocessing.StandardScaler()
prostate_df[feature_names] = pd.DataFrame(
scalar.fit_transform(
prostate_df[feature_names].astype('float64')))
# select training set
prostate_df_train = prostate_df[prostate_df.train == 'T']
# Select features and labels
features = prostate_df_train[feature_names]
labels = prostate_df_train[['lpsa']]
# Create tensors
feat = tf.constant(features.values, dtype=tf.float64)
lab = tf.constant(labels.values, dtype=tf.float64)
dtype = feat.dtype
regularization = 0 # regularization parameter
dim = 8 # number of features
# We pick a random starting point for the search
start = np.random.randn(dim + 1)
def regression_loss(params):
"""Compute loss for linear regression model with L1 penalty
Args:
params: A real tensor of shape [dim + 1]. The zeroth component
is the intercept term and the rest of the components are the
beta coefficients.
Returns:
The mean square error loss including L1 penalty.
"""
params = tf.squeeze(params)
intercept, beta = params[0], params[1:]
pred = tf.matmul(feat, tf.expand_dims(beta, axis=-1)) + intercept
mse_loss = tf.reduce_sum(
tf.cast(
tf.losses.mean_squared_error(y_true=lab, y_pred=pred), tf.float64))
l1_penalty = regularization * tf.reduce_sum(tf.abs(beta))
total_loss = mse_loss + l1_penalty
return total_loss
Resolvendo com L-BFGS
Ajuste usando L-BFGS. Mesmo que a penalidade L1 introduza descontinuidades derivadas, na prática, L-BFGS funciona muito bem ainda.
@tf.function
def l1_regression_with_lbfgs():
return tfp.optimizer.lbfgs_minimize(
make_val_and_grad_fn(regression_loss),
initial_position=tf.constant(start),
tolerance=1e-8)
results = run(l1_regression_with_lbfgs)
minimum = results.position
fitted_intercept = minimum[0]
fitted_beta = minimum[1:]
print('L-BFGS Results')
print('Converged:', results.converged)
print('Intercept: Fitted ({})'.format(fitted_intercept))
print('Beta: Fitted {}'.format(fitted_beta))
Evaluation took: 0.017987 seconds L-BFGS Results Converged: True Intercept: Fitted (2.3879985744556484) Beta: Fitted [ 0.68626215 0.28193532 -0.17030254 0.10799274 0.33634988 -0.24888523 0.11992237 0.08689026]
Resolvendo com Nelder Mead
O método Nelder Mead é um dos métodos mais populares de minimização livres derivados. Este otimizador não usa informações de gradiente e não faz suposições sobre a diferenciabilidade da função de destino; é, portanto, apropriado para funções objetivo não suaves, por exemplo, problemas de otimização com penalidade de L1.
Por um problema de otimização em \(n\)-Dimensions mantém um conjunto de\(n+1\) soluções candidatos que se estendem por um simplex não-degenerada. Ele modifica sucessivamente o simplex com base em um conjunto de movimentos (reflexão, expansão, encolhimento e contração) usando os valores da função em cada um dos vértices.
# Nelder mead expects an initial_vertex of shape [n + 1, 1].
initial_vertex = tf.expand_dims(tf.constant(start, dtype=dtype), axis=-1)
@tf.function
def l1_regression_with_nelder_mead():
return tfp.optimizer.nelder_mead_minimize(
regression_loss,
initial_vertex=initial_vertex,
func_tolerance=1e-10,
position_tolerance=1e-10)
results = run(l1_regression_with_nelder_mead)
minimum = results.position.reshape([-1])
fitted_intercept = minimum[0]
fitted_beta = minimum[1:]
print('Nelder Mead Results')
print('Converged:', results.converged)
print('Intercept: Fitted ({})'.format(fitted_intercept))
print('Beta: Fitted {}'.format(fitted_beta))
Evaluation took: 0.325643 seconds Nelder Mead Results Converged: True Intercept: Fitted (2.387998456121595) Beta: Fitted [ 0.68626266 0.28193456 -0.17030291 0.10799375 0.33635132 -0.24888703 0.11992244 0.08689023]
Regressão logística com penalidade L2
Para este exemplo, criamos um conjunto de dados sintéticos para classificação e usamos o otimizador L-BFGS para ajustar os parâmetros.
np.random.seed(12345)
dim = 5 # The number of features
n_obs = 10000 # The number of observations
betas = np.random.randn(dim) # The true beta
intercept = np.random.randn() # The true intercept
features = np.random.randn(n_obs, dim) # The feature matrix
probs = sp.special.expit(
np.matmul(features, np.expand_dims(betas, -1)) + intercept)
labels = sp.stats.bernoulli.rvs(probs) # The true labels
regularization = 0.8
feat = tf.constant(features)
lab = tf.constant(labels, dtype=feat.dtype)
@make_val_and_grad_fn
def negative_log_likelihood(params):
"""Negative log likelihood for logistic model with L2 penalty
Args:
params: A real tensor of shape [dim + 1]. The zeroth component
is the intercept term and the rest of the components are the
beta coefficients.
Returns:
The negative log likelihood plus the penalty term.
"""
intercept, beta = params[0], params[1:]
logit = tf.matmul(feat, tf.expand_dims(beta, -1)) + intercept
log_likelihood = tf.reduce_sum(tf.nn.sigmoid_cross_entropy_with_logits(
labels=lab, logits=logit))
l2_penalty = regularization * tf.reduce_sum(beta ** 2)
total_loss = log_likelihood + l2_penalty
return total_loss
start = np.random.randn(dim + 1)
@tf.function
def l2_regression_with_lbfgs():
return tfp.optimizer.lbfgs_minimize(
negative_log_likelihood,
initial_position=tf.constant(start),
tolerance=1e-8)
results = run(l2_regression_with_lbfgs)
minimum = results.position
fitted_intercept = minimum[0]
fitted_beta = minimum[1:]
print('Converged:', results.converged)
print('Intercept: Fitted ({}), Actual ({})'.format(fitted_intercept, intercept))
print('Beta:\n\tFitted {},\n\tActual {}'.format(fitted_beta, betas))
Evaluation took: 0.056751 seconds
Converged: True
Intercept: Fitted (1.4111415084244365), Actual (1.3934058329729904)
Beta:
Fitted [-0.18016612 0.53121578 -0.56420632 -0.5336374 2.00499675],
Actual [-0.20470766 0.47894334 -0.51943872 -0.5557303 1.96578057]
Suporte de lote
Tanto o BFGS quanto o L-BFGS oferecem suporte à computação em lote, por exemplo, para otimizar uma única função a partir de muitos pontos de partida diferentes; ou múltiplas funções paramétricas de um único ponto.
Função única, vários pontos de partida
A função de Himmelblau é um caso de teste de otimização padrão. A função é dada por:
\[f(x, y) = (x^2 + y - 11)^2 + (x + y^2 - 7)^2\]
A função possui quatro mínimos localizados em:
- (3, 2),
- (-2,805118, 3,131312),
- (-3,779310, -3,283186),
- (3,584428, -1,848126).
Todos esses mínimos podem ser alcançados a partir de pontos de partida apropriados.
# The function to minimize must take as input a tensor of shape [..., n]. In
# this n=2 is the size of the domain of the input and [...] are batching
# dimensions. The return value must be of shape [...], i.e. a batch of scalars
# with the objective value of the function evaluated at each input point.
@make_val_and_grad_fn
def himmelblau(coord):
x, y = coord[..., 0], coord[..., 1]
return (x * x + y - 11) ** 2 + (x + y * y - 7) ** 2
starts = tf.constant([[1, 1],
[-2, 2],
[-1, -1],
[1, -2]], dtype='float64')
# The stopping_condition allows to further specify when should the search stop.
# The default, tfp.optimizer.converged_all, will proceed until all points have
# either converged or failed. There is also a tfp.optimizer.converged_any to
# stop as soon as the first point converges, or all have failed.
@tf.function
def batch_multiple_starts():
return tfp.optimizer.lbfgs_minimize(
himmelblau, initial_position=starts,
stopping_condition=tfp.optimizer.converged_all,
tolerance=1e-8)
results = run(batch_multiple_starts)
print('Converged:', results.converged)
print('Minima:', results.position)
Evaluation took: 0.019095 seconds Converged: [ True True True True] Minima: [[ 3. 2. ] [-2.80511809 3.13131252] [-3.77931025 -3.28318599] [ 3.58442834 -1.84812653]]
Múltiplas funções
Para fins de demonstração, neste exemplo, otimizamos simultaneamente um grande número de tigelas quadráticas de alta dimensão geradas aleatoriamente.
np.random.seed(12345)
dim = 100
batches = 500
minimum = np.random.randn(batches, dim)
scales = np.exp(np.random.randn(batches, dim))
@make_val_and_grad_fn
def quadratic(x):
return tf.reduce_sum(input_tensor=scales * (x - minimum)**2, axis=-1)
# Make all starting points (1, 1, ..., 1). Note not all starting points need
# to be the same.
start = tf.ones((batches, dim), dtype='float64')
@tf.function
def batch_multiple_functions():
return tfp.optimizer.lbfgs_minimize(
quadratic, initial_position=start,
stopping_condition=tfp.optimizer.converged_all,
max_iterations=100,
tolerance=1e-8)
results = run(batch_multiple_functions)
print('All converged:', np.all(results.converged))
print('Largest error:', np.max(results.position - minimum))
Evaluation took: 1.994132 seconds All converged: True Largest error: 4.4131473142527966e-08