Linear and logistic regression in Theano
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This blog post shows how to use the theano library to perform linear and logistic regression. I won’t go into details of what linear or logistic regression is, because the purpose of this post is mainly to use the theano library in regression tasks. However, details on linear and logistic regression can be found on the Wikipedia website. For the purpose of this blog post, I created a small python package Regression_theano, which resides in my Github repository. So, assuming that you,
- do want to install it, and you work on a Linux PC (note that it is possible to continue reading the post without installing the package)
- have already installed and configured theano
Then you,
- can download the zip file from the repository
- unzip it
- make the unziped file your current directory (it includes a setup.py file)
- run python setup.py install to install it.
By the way, a tutorial on how to create a package from python code can be found here.
Theano
A few words about theano : Theano is a Python library with a NumPy-like syntax, that lets you define, optimize, and evaluate mathematical expressions. It can be compiled and run efficiently on either CPU or GPU and it can be as fast as the C language on a CPU and orders of magnitude faster when using a GPU. Tutorials on how to start using the theano library can be found here and machine learning implementations like a logistic regression, a Multilayer perceptron or a deep convolutional neural network, here.
Regression in theano
When implementing a simple linear or logistic regression model in theano the first thing to do is to declare the variables and functions. Theano then creates a symbolic graph, which we can use with our inputs to obtain results. In the Regression python Class of the Regression_theano package, first, I define X and y,
Linear regression:
- X : tensor Matrix
- y : tensor vector
Logistic regression:
- X : tensor Matrix
- y : tensor Matrix (in classification tasks the response variable should be one-hot-encoded)
Then, I initialize the weights of the parameters ( here I use a glorot uniform initialization as explained in this jupyter notebook),
init_weights_params = theano.shared(np.asarray(np.random.uniform(low = -np.sqrt(6. / (inp + outp)), high = np.sqrt(6. / (inp + outp)),
size = shape),
dtype=theano.config.floatX)) # shape == dims of the weights matrix
and the weights for the bias ( which default to zero ),
init_weights_bias = theano.shared(np.asarray(0, dtype=theano.config.floatX))
# for linear regression is a single value
init_weights_bias = theano.shared(np.zeros((self.y.shape[1],), dtype=theano.config.floatX))
# for logistic regression is an array of values
Both the init_weights_params and the init_weights_bias are theano-shared-variables, meaning they are shared between different functions and different function calls. In the module, there is also the option to exclude the bias from the calculations.
In the same way, the predictions can be obtained using,
py_x = T.dot(X, init_weights_params) + init_weights_bias # In case of linear regression
py_x = T.nnet.softmax(T.dot(X, init_weights_params) + init_weights_bias) # In case of logistic regression
The aim of training a linear or logistic model in theano is to minimize/maximize the objective function until a global minimum/maximum is reached. In case of the linear model the objective can be for instance the mean-squared-error or the root-mean-squared-error, whereas in logistic regression the objective can be the binary-crossentropy or the categorical crossentropy ( a user-defined-objective is also possible),
cost = T.sqr(py_x - Y)/float(nrows_X) # linear regression
cost = T.nnet.binary_crossentropy(py_x, Y) # logistic regression
Regularization is a method to prevent overfitting and can be added to a model in 3 ways : L1 , L2 or L1 + L2 (elastic-net),
# L1, L2 can be between 0.0 and 1.0 ( 0.0 is no regularization )
reg_param_L1 = abs(T.sum(init_weights_params) + T.sum(init_weights_bias)) # L1 regrularization
reg_param_L2 = T.sum(T.sqr(init_weights_params)) + T.sum(T.sqr(init_weights_bias)) # L2 regularization
cost = cost + L1 * reg_param_L1 + L2 * reg_param_L2
The next step is to create a theano.function for compilation. This function will take the index of all the train data as input and deploy an optimizer (such as sgd, rmsprop, adagrad) to update the parameters at each epoch returning an output (here the cost),
train = theano.function(inputs = [index],
outputs = cost,
updates = sgd(cost, Params, learning_rate),
givens = { X: train_X[0:index], Y: train_y[0:index] },
allow_input_downcast = True)
A nice thing in theano is that the gradients are computed automatically using the function theano.gradient.grad(cost = cost, wrt = Params).
Before proceeding a rule of thumb here is to use the givens argument in combination with an index inside the theano.function as long as the data fits in the GPU memory (if GPU is used). That way the data will reside in the GPU and we can spare unnecessary transfers of data from the CPU to GPU (discussion on this issue).
Considering the previous train function, the givens argument takes as inputs two shared variables, which should be declared at the beginning together with X and y,
index = T.lscalar('index') # tensor variable index should equal the number of rows of the train data
train_X = shared_dataset(X_train_data) # shared variable X_train for the givens argument
train_y = self.shared_dataset(y_train_data) # shared variable y_train for the givens argument
test_X = shared_dataset(X_test) # shared variable, will be used for evaluation
If on the other hand a batch-size is employed due to the data size, then in addition to the previous variables, batch indices should be built,
batch_size = 128 # example of batch_size
n_train_batches = train_X.get_value(borrow=True).shape[0] / batch_size
n_test_batches = test_X.get_value(borrow=True).shape[0] / batch_size
and then the train function will look as follows,
train = theano.function(inputs = [index],
outputs = cost,
updates = sgd(cost, Params, learning_rate),
givens = { X: train_X[index * batch_size: (index + 1) * batch_size],
Y: train_y[index * batch_size: (index + 1) * batch_size]},
allow_input_downcast = True)
The only thing that is different in this function is the givens argument, where train_X, train_y is split into batches.
Another important thing is when to stop train a model, especially in the case of iterative models (which is the case here). Early stopping is implemented in the deeplearning tutorials, however, I found this monitor function, which is meant for gradient boosting, quite useful and I’ve adjusted my Regression class accordingly. So, learning of the model stops when the calculated loss is increased/decreased for a consecutive number of epochs,
predict_valid = theano.function(inputs = [index],
outputs = py_x,
givens = { X: test_X[0:index]}, # returns the predictions when all train data are used
allow_input_downcast = True)
predict_valid = theano.function(inputs = [index],
outputs = py_x,
# returns the predictions when batch-index of train data is used
givens = { X: test_X[index * batch_size: (index + 1) * batch_size]},
allow_input_downcast = True)
A final step is to use a separate prediction function to predict unknown data once training ends,
predict = theano.function(inputs = [X],
outputs = py_x)
This function will take as input the new data set and it will return the predictions.
The explained code-chunks can be found in the Regression_theano.py file.
Example data sets
I’ll test the Theano_regression package using the Boston data for regression and the Mnist data for classification,
from Regression_theano import Regression
from sklearn import preprocessing
from sklearn.datasets import load_boston
from sklearn import metrics
import numpy as np
import random
# REGRESSION
boston = load_boston()
print(boston.data.shape)
X = boston['data']
X = preprocessing.scale(X, axis=0, with_mean=True, with_std=True, copy=False) # scale data
X = X.astype(np.float32)
y = boston['target']
y = y.astype(np.float32)
random.seed(1)
spl = np.random.choice(X.shape[0], int(0.75 * X.shape[0]), replace = False) # split data
not_spl = np.array(list(set(np.array(range(X.shape[0]))) - set(spl)))
print len(spl), len(not_spl)
Xtr = X[spl]
Xte = X[not_spl]
y_tr = y[spl]
y_te = y[not_spl]
# train regression model
# initialize
init = Regression(Xtr, y_tr, Xte, y_te, iters = 10000, learning_rate = 0.5, optimizer = 'sgd', batch_size = None,
L1 = 0.0001, L2 = 0.0001, maximize = False, early_stopping_rounds = 10, weights_initialization = 'glorot_uniform',
objective = 'mean_squared_error', linear_regression = True, add_bias = True, custom_eval = None)
# fit
fit = init.fit()
...
# iter 223 train_loss 0.539 test_loss 171.277
# iter 224 train_loss 0.537 test_loss 170.423
# iter 225 train_loss 0.535 test_loss 169.574
# iter 226 train_loss 0.532 test_loss 168.73
# iter 227 train_loss 0.53 test_loss 167.89
...
# iter 1239 train_loss 0.126 test_loss 13.229
# iter 1240 train_loss 0.126 test_loss 13.225
# iter 1241 train_loss 0.126 test_loss 13.222
# iter 1242 train_loss 0.126 test_loss 13.218
# iter 1243 train_loss 0.126 test_loss 13.214
...
# iter 1739 train_loss 0.124 test_loss 12.747
# iter 1740 train_loss 0.124 test_loss 12.747
# iter 1741 train_loss 0.124 test_loss 12.747
# iter 1742 train_loss 0.124 test_loss 12.747
# iter 1743 train_loss 0.124 test_loss 12.747
# regression stopped after 10 consecutive increases of loss and 1743 Epochs
# predictions for train, test
pred_tr = init.PREDICT(Xtr)
pred_te = init.PREDICT(Xte)
# mse train - test
print 'mean_squared_error on train data is', str(metrics.mean_squared_error(y_tr, pred_tr))
# mean_squared_error on train data is 27.0498
print 'mean_squared_error on test data is', str(metrics.mean_squared_error(y_te, pred_te))
# mean_squared_error on test data is 12.7468
For a random train-test split on the data the sgd-linear-regression gives after 1740 iterations a mse-error of 27.04 on train-data and 12.74 on test-data. As a comparison I’ve run also an sklearn-linear-regression, which gives comparable results,
from sklearn.linear_model import LinearRegression
lr = LinearRegression()
lr.fit(Xtr, y_tr)
pred_tr_skl = lr.predict(Xtr)
pred_te_skl = lr.predict(Xte)
# mse train - test
print 'mean_squared_error on train data is', str(metrics.mean_squared_error(y_tr, pred_tr_skl))
# mean_squared_error on train data is 24.8352
print 'mean_squared_error on test data is', str(metrics.mean_squared_error(y_te, pred_te_skl))
# mean_squared_error on test data is 14.51736
A few words about the parameters in the Regression() function,
| parameter | notes |
|---|---|
| X | train data |
| Y | train response |
| X_test | test data |
| Y_test | test response |
| iters | number of iterations |
| learning_rate | learning rate |
| optimizer | ‘sgd’ |
| batch_size | if None then in each iteration all data will be used for training, if integer value then a mini-batch will be used for training |
| L1 | L1-regularization |
| L2 | L2-regularization |
| maximize | if the evaluation metric should be maximized/minimized (used in early-stopping) |
| early_stopping_rounds | after how many iters of increasing/decreasing loss should the training-process stop |
| weights_initialization | one of ‘uniform’, ‘normal’, ‘glorot_uniform’ |
| objective | the objective function |
| linear_regression | boolean [True, False] (linear_regression or logistic_regression) |
| add_bias | if bias should be added to the model |
| custom_eval | use a custom evaluation function in form of a tuple (function, ‘name_of_function’) to evaluate the test data during training, otherwise None |
The second example is about classification using the mnist data set (digit recognition),
from Regression_theano import Regression
from sklearn import preprocessing
import pandas as pd
import random
from sklearn import metrics
from scikits.statsmodels.tools import categorical
df = pd.read_csv('train.csv') # assuming the train data is downloaded in the current working directory
X = np.array(df.iloc[:,1:df.shape[1]].as_matrix(columns=None))
X = X.astype(np.float32)
X /= 255 # divide pixels to the maximum to get values between 0 and 1
target = np.array(df.iloc[:, 0].as_matrix(columns=None))
random.seed(1)
spl = np.random.choice(X.shape[0], int(0.75 * X.shape[0]), replace = False) # split data
not_spl = np.array(list(set(np.array(range(X.shape[0]))) - set(spl)))
Xtr = X[spl]
Xte = X[not_spl]
y_tr = target[spl]
y_te = target[not_spl]
y_categ = categorical(y_tr, drop = True) # response should be one-hot-encoded
y_categ_te = categorical(y_te, drop = True)
print Xtr.shape, Xte.shape, y_categ.shape, y_categ_te.shape
# (31500, 784) (10500, 784) (31500, 10) (10500, 10)
def evaluation_func(y_test, pred_test): # customized evaluation metric [ accuracy ]
out = np.mean(np.argmax(pred_test, axis = 1) == np.argmax(y_test, axis = 1))
return out
# initialize data
init = Regression(Xtr, y_categ, Xte, y_categ_te, iters = 1000, learning_rate = 0.5, optimizer = 'sgd', batch_size = 512,
L1 = 0.00001, L2 = 0.00001, maximize = False, early_stopping_rounds = 10, weights_initialization = 'normal',
objective = 'categorical_crossentropy', linear_regression = False, add_bias = True,
custom_eval = (evaluation_func, 'accuracy'))
# fit data
fit = init.fit()
# iter 1 train_loss 0.497 test_accuracy 0.877
# iter 2 train_loss 0.441 test_accuracy 0.894
# iter 3 train_loss 0.418 test_accuracy 0.901
# iter 4 train_loss 0.404 test_accuracy 0.904
# iter 5 train_loss 0.394 test_accuracy 0.906
# iter 6 train_loss 0.386 test_accuracy 0.907
# iter 7 train_loss 0.38 test_accuracy 0.909
# iter 8 train_loss 0.374 test_accuracy 0.91
# iter 9 train_loss 0.37 test_accuracy 0.911
# iter 10 train_loss 0.366 test_accuracy 0.912
# iter 11 train_loss 0.363 test_accuracy 0.913
# regression stopped after 10 consecutive increases of loss and 11 Epochs
# predictions
pred_tr = init.PREDICT(Xtr)
pred_te = init.PREDICT(Xte)
# accuracy train - test
print 'mean_squared_error on train data is', str(metrics.accuracy_score(y_tr, np.argmax(pred_tr, axis = 1)))
# mean_squared_error on train data is 0.9173
print 'mean_squared_error on test data is', str(metrics.accuracy_score(y_te, np.argmax(pred_te, axis = 1)))
# mean_squared_error on test data is 0.9123
Nowadays, digit recognition using convolutional neural networks approaches 0.0 %, so I think that approximately 91.0 % accuracy using a single sgd-logistic-regression model is not that bad.
final word
A linear or logistic regression model in theano can be thought of as a neural network with a single hidden layer. It can be used as a basis to build a neural network by adding, for instance, a certain number of hidden layers, dropout or batch-normalization.
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