 TensorFlow.org'da görüntüleyin |  Google Colab'da çalıştırın |  Kaynağı GitHub'da görüntüleyin |  Not defterini indir |
Soyut
Bu ortak çalışmada, TensorFlow Probability'de uygulanan çeşitli optimize edicilerin nasıl kullanılacağını gösteriyoruz.
Bağımlılıklar ve Ön Koşullar
İçe aktarmak
%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
BFGS ve L-BFGS Optimize Ediciler
Quasi Newton yöntemleri, popüler birinci dereceden optimizasyon algoritmasının bir sınıfıdır. Bu yöntemler, arama yönünü bulmak için kesin Hessian'a pozitif kesin bir yaklaşım kullanır.
Broyden-Fletcher-Goldfarb-Shanno algoritması ( BFGS ) bu genel fikrin belirli uygulamasıdır. Bu uygulanabilir ve gradyan her yerde sürekli ve orta ölçekli sorunlar için tercih edilen bir yöntem (bir, örneğin, doğrusal gerileme \(L_2\) cezası).
L-BFGS olan Hessian matrisler, makul bir maliyetle hesaplanamaz veya seyrek değildir Daha büyük problemleri çözmek için yararlı olan BFGS sınırlı bellek versiyonudur. Yerine tam yoğunlukta depolanması \(n \times n\) Hessian matrislerin yaklaşımları, sadece uzunluk birkaç vektörleri kaydetmek \(n\) örtülü bu yakın temsil eder.
yardımcı fonksiyonlar
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)
Basit bir ikinci dereceden fonksiyon üzerinde L-BFGS
# 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
BFGS ile aynı sorun
@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
L1 cezalı Lineer Regresyon: Prostat Kanseri verileri
Trevor HASTIE Robert Tibshirani ve Jerome Friedman tarafından İstatistiksel Öğrenme Veri Madenciliği, Çıkarım ve Tahmini Elements: Kitabı'ndan örneği.
Bunun L1 cezalı bir optimizasyon sorunu olduğunu unutmayın.
Veri kümesi elde et
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.
Problem tanımı
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
L-BFGS ile Çözme
L-BFGS kullanarak sığdırın. L1 cezası türev süreksizlikler getirse de, uygulamada L-BFGS oldukça iyi çalışıyor.
@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]
Nelder Mead ile Çözme
Nelder Mead yöntemi en popüler türevi serbest minimizasyon yöntemlerden biridir. Bu optimize edici, gradyan bilgisini kullanmaz ve hedef fonksiyonun türevlenebilirliği hakkında hiçbir varsayımda bulunmaz; bu nedenle, örneğin L1 cezalı optimizasyon problemleri gibi düzgün olmayan amaç fonksiyonları için uygundur.
Bir optimizasyon sorunu için \(n\)-Boyutlar bunun bir dizi muhafaza\(n+1\) dejenere olmayan bir simpleks yayılan aday çözümler. Her bir tepe noktasındaki fonksiyon değerlerini kullanarak bir dizi hamleye (yansıma, genişleme, küçülme ve daralma) dayalı olarak simpleksi art arda değiştirir.
# 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]
L2 cezalı Lojistik Regresyon
Bu örnek için, sınıflandırma için sentetik bir veri seti oluşturuyoruz ve parametrelere uyması için L-BFGS optimize edicisini kullanıyoruz.
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]
Gruplama desteği
Hem BFGS hem de L-BFGS, örneğin birçok farklı başlangıç noktasından tek bir işlevi optimize etmek için toplu hesaplamayı destekler; veya tek bir noktadan çoklu parametrik fonksiyonlar.
Tek fonksiyon, çoklu başlangıç noktaları
Himmelblau'nun işlevi, standart bir optimizasyon test durumudur. İşlev şu şekilde verilir:
\[f(x, y) = (x^2 + y - 11)^2 + (x + y^2 - 7)^2\]
Fonksiyonun şu noktada bulunan dört minimumu vardır:
- (3, 2),
- (-2.805118, 3.131312),
- (-3.779310, -3.283186),
- (3.584428, -1.848126).
Tüm bu minimumlara uygun başlangıç noktalarından ulaşılabilir.
# 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]]
Çoklu fonksiyonlar
Gösteri amacıyla, bu örnekte, çok sayıda yüksek boyutlu rastgele oluşturulmuş ikinci dereceden çanakları eşzamanlı olarak optimize ediyoruz.
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