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Astratto
In questa collaborazione mostriamo come utilizzare i vari ottimizzatori implementati in TensorFlow Probability.
Dipendenze e prerequisiti
Importare
%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
Ottimizzatori BFGS e L-BFGS
I metodi Quasi Newton sono una classe di popolari algoritmi di ottimizzazione del primo ordine. Questi metodi utilizzano un'approssimazione definita positiva all'esatta dell'Assia per trovare la direzione di ricerca.
L'algoritmo Broyden-Fletcher-Goldfarb-Shanno ( BFGS ) è un'implementazione specifica di questa idea generale. È applicabile ed è il metodo di scelta per problemi di medie dimensioni, dove la pendenza è continua ovunque (es regressione lineare con un \(L_2\) penalità).
L-BFGS è una versione limitata memoria di BFGS che è utile per risolvere i problemi più grandi cui matrici Hesse non può essere calcolato a un costo ragionevole o non sono sparse. Invece di memorizzare completamente densi \(n \times n\) approssimazioni di matrici Hessian, salvano solo pochi vettori di lunghezza \(n\) che rappresentano queste approssimazioni implicitamente.
Funzioni di aiuto
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 su una semplice funzione quadratica
# 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
Stesso problema con 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
Regressione lineare con penalità L1: dati sul cancro alla prostata
Esempio dal libro: The Elements of Learning statistica, data mining, inferenza, e previsione da Trevor Hastie, Robert Tibshirani e Jerome Friedman.
Nota che questo è un problema di ottimizzazione con penalità L1.
Ottieni set di dati
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.
Definizione del 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
Risolvere con L-BFGS
Montare utilizzando L-BFGS. Anche se la penalità L1 introduce discontinuità derivate, in pratica, L-BFGS funziona ancora abbastanza bene.
@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]
Risolvere con Nelder Mead
Il metodo di Nelder Mead è uno dei più popolari metodi di minimizzazione liberi derivati. Questo ottimizzatore non utilizza informazioni sul gradiente e non fa ipotesi sulla differenziabilità della funzione target; è quindi appropriato per funzioni obiettivo non uniformi, ad esempio problemi di ottimizzazione con penalità L1.
Per un problema di ottimizzazione in \(n\)-Dimensions mantiene un insieme di\(n+1\) soluzioni candidate che si estendono su una simplex non degenere. Modifica successivamente il simplesso in base a un insieme di movimenti (riflessione, espansione, restringimento e contrazione) utilizzando i valori della funzione in ciascuno dei vertici.
# 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]
Regressione logistica con penalità L2
Per questo esempio, creiamo un set di dati sintetico per la classificazione e utilizziamo l'ottimizzatore L-BFGS per adattare i parametri.
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]
Supporto per il dosaggio
Sia BFGS che L-BFGS supportano il calcolo batch, ad esempio per ottimizzare una singola funzione da molti punti di partenza diversi; o più funzioni parametriche da un unico punto.
Singola funzione, più punti di partenza
La funzione di Himmelblau è un test case di ottimizzazione standard. La funzione è data da:
\[f(x, y) = (x^2 + y - 11)^2 + (x + y^2 - 7)^2\]
La funzione ha quattro minimi situati in:
- (3, 2),
- (-2.805118, 3.131312),
- (-3.779310, -3.283186),
- (3.584428, -1.848126).
Tutti questi minimi possono essere raggiunti da punti di partenza appropriati.
# 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]]
Molteplici funzioni
A scopo dimostrativo, in questo esempio ottimizziamo contemporaneamente un gran numero di ciotole quadratiche ad alta dimensione generate casualmente.
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