深度学习教程：从 Logistic 回归到卷积神经网络

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"这份资源是LISAlab（蒙特利尔大学）发布的关于深度学习的Theano教程，版本0.1，日期为2013年5月29日。教程涵盖了从入门到深入的各种主题，包括下载Theano、处理数据集、符号表示、监督学习优化简介以及通过Logistic Regression、多层感知机（MLP）和卷积神经网络（LeNet）进行图像分类。此外，还介绍了去噪自编码器（dA）的概念和应用。" 在《theano 深度学习教程》中，读者可以学习到以下关键知识点： 1. **许可证（LICENSE）**：教程可能附带了特定的开源许可证，允许用户自由地使用、修改和分发代码，但具体条件需参照LICENSE文件。 2. **Theano**：Theano是一个Python库，用于定义、优化和评估涉及多维数组的数学表达式，特别适合于构建和训练深度学习模型。Theano能够自动求导，支持GPU计算，并且提供了一个高效的符号表达式系统。 3. **开始使用（Getting Started）**：这部分内容指导用户如何下载Theano，处理数据集，以及理解教程中的符号表示。数据集部分可能涉及获取和预处理常用的数据集，如MNIST手写数字数据集。 4. **监督学习优化基础（APrimeronSupervisedOptimizationforDeepLearning）**：介绍了深度学习中常见的优化算法，如梯度下降及其变种，以及如何在实际问题中调整学习率和正则化参数。 5. **逻辑回归分类MNIST数字（ClassifyingMNISTdigitsusingLogisticRegression）**：通过逻辑回归模型来识别MNIST手写数字，讲解了模型定义、损失函数、模型创建、训练和测试的完整流程。 6. **多层感知机（MultilayerPerceptron, MLP）**：介绍从逻辑回归过渡到MLP，讨论了MLP的结构和训练过程，包括反向传播算法和激活函数的使用。 7. **卷积神经网络（LeNet）**：讲解了LeNet的动机、稀疏连接性、共享权重、卷积运算、最大池化等概念，展示了构建一个完整的LeNet模型并运行代码的步骤。 8. **去噪自编码器（DenoisingAutoencoders, dA）**：介绍了自编码器的基本原理和去噪自编码器的特殊之处，它是通过在输入中引入噪声来提高模型的泛化能力。这个教程是深度学习初学者和进阶者的宝贵资源，它不仅提供了理论知识，还包含了实际操作的代码示例，帮助读者理解和应用深度学习技术。对于希望利用Theano构建和训练各种深度学习模型的人来说，这是一个全面的学习指南。

Deep Learning Tutorial, Release 0.1

L1 and L2 regularization

L1 and L2 regularization involve adding an extra term to the loss function, which penalizes certain parameter

conﬁgurations. Formally, if our loss function is:

NLL(θ, D) = −

|D|

i=0

log P (Y = y

(i)

, θ)

then the regularized loss will be:

E(θ, D) = NLL(θ, D) + λR(θ)

or, in our case

E(θ, D) = NLL(θ, D) + λ||θ||

where

||θ||





|θ|

j=0

|θ





which is the L

norm of θ. λ is a hyper-parameter which controls the relative importance of the regularization

parameter. Commonly used values for p are 1 and 2, hence the L1/L2 nomenclature. If p=2, then the

regularizer is also called “weight decay”.

In principle, adding a regularization term to the loss will encourage smooth network mappings in a neural

network (by penalizing large values of the parameters, which decreases the amount of nonlinearity that

the network models). More intuitively, the two terms (NLL and R(θ)) correspond to modelling the data

well (NLL) and having “simple” or “smooth” solutions (R(θ)). Thus, minimizing the sum of both will, in

theory, correspond to ﬁnding the right trade-off between the ﬁt to the training data and the “generality” of

the solution that is found. To follow Occam’s razor principle, this minimization should ﬁnd us the simplest

solution (as measured by our simplicity criterion) that ﬁts the training data.

Note that the fact that a solution is “simple” does not mean that it will generalize well. Empirically, it

was found that performing such regularization in the context of neural networks helps with generalization,

especially on small datasets. The code block below shows how to compute the loss in python when it

contains both a L1 regularization term weighted by λ

and L2 regularization term weighted by λ

# symbolic Theano variable that represents the L1 regularization term

L1 = T.sum(abs(param))

# symbolic Theano variable that represents the squared L2 term

L2_sqr = T.sum(param

# the loss

loss = NLL + lambda_1

L1 + lambda_2

Early-Stopping

Early-stopping combats overﬁtting by monitoring the model’s performance on a validation set. A validation

set is a set of examples that we never use for gradient descent, but which is also not a part of the test set. The

12 Chapter 3. Getting Started

Deep Learning Tutorial, Release 0.1

validation examples are considered to be representative of future test examples. We can use them during

training because they are not part of the test set. If the model’s performance ceases to improve sufﬁciently

on the validation set, or even degrades with further optimization, then the heuristic implemented here gives

up on much further optimization.

The choice of when to stop is a judgement call and a few heuristics exist, but these tutorials will make use

of a strategy based on a geometrically increasing amount of patience.

# early-stopping parameters

patience = 5000 # look as this many examples regardless

patience_increase = 2 # wait this much longer when a new best is

# found

improvement_threshold = 0.995 # a relative improvement of this much is

# considered significant

validation_frequency = min(n_train_batches, patience/2)

# go through this many

# minibatches before checking the network

# on the validation set; in this case we

# check every epoch

best_params = None

best_validation_loss = numpy.inf

test_score = 0.

start_time = time.clock()

done_looping = False

epoch = 0

while (epoch < n_epochs) and (not done_looping):

# Report "1" for first epoch, "n_epochs" for last epoch

epoch = epoch + 1

for minibatch_index in xrange(n_train_batches):

d_loss_wrt_params = ... # compute gradient

params -= learning_rate

d_loss_wrt_params # gradient descent

# iteration number. We want it to start at 0.

iter = (epoch - 1)

n_train_batches + minibatch_index

# note that if we do ‘iter % validation_frequency‘ it will be

# true for iter = 0 which we do not want. We want it true for

# iter = validation_frequency - 1.

if (iter + 1) % validation_frequency == 0:

this_validation_loss = ... # compute zero-one loss on validation set

if this_validation_loss < best_validation_loss:

# improve patience if loss improvement is good enough

if this_validation_loss < best_validation_loss

improvement_threshold:

patience = max(patience, iter

patience_increase)

best_params = copy.deepcopy(params)

best_validation_loss = this_validation_loss

if patience <= iter:

3.4. A Primer on Supervised Optimization for Deep Learning 13

Deep Learning Tutorial, Release 0.1

done_looping = True

break

# POSTCONDITION:

# best_params refers to the best out-of-sample parameters observed during the optimization

If we run out of batches of training data before running out of patience, then we just go back to the beginning

of the training set and repeat.

Note: The validation_frequency should always be smaller than the patience. The code should

check at least two times how it performs before running out of patience. This is the reason we used the

formulation validation_frequency = min( value, patience/2.)

Note: This algorithm could possibly be improved by using a test of statistical signiﬁcance rather than the

simple comparison, when deciding whether to increase the patience.

3.4.4 Testing

After the loop exits, the best_params variable refers to the best-performing model on the validation set. If

we repeat this procedure for another model class, or even another random initialization, we should use the

same train/valid/test split of the data, and get other best-performing models. If we have to choose what the

best model class or the best initialization was, we compare the best_validation_loss for each model. When

we have ﬁnally chosen the model we think is the best (on validation data), we report that model’s test set

performance. That is the performance we expect on unseen examples.

3.4.5 Recap

That’s it for the optimization section. The technique of early-stopping requires us to partition the set of

examples into three sets (training D

train

, validation D

valid

, test D

test

). The training set is used for minibatch

stochastic gradient descent on the differentiable approximation of the objective function. As we perform

this gradient descent, we periodically consult the validation set to see how our model is doing on the real

objective function (or at least our empirical estimate of it). When we see a good model on the validation set,

we save it. When it has been a long time since seeing a good model, we abandon our search and return the

best parameters found, for evaluation on the test set.

3.5 Theano/Python Tips

3.5.1 Loading and Saving Models

When you’re doing experiments, it can take hours (sometimes days!) for gradient-descent to ﬁnd the best

parameters. You will want to save those weights once you ﬁnd them. You may also want to save your

current-best estimates as the search progresses.

Pickle the numpy ndarrays from your shared variables

14 Chapter 3. Getting Started

Deep Learning Tutorial, Release 0.1

The best way to save/archive your model’s parameters is to use pickle or deepcopy the ndarray objects. So

for example, if your parameters are in shared variables w, v, u, then your save command should look

something like:

>>> import cPickle

>>> save_file = open(’path’, ’wb’) # this will overwrite current contents

>>> cPickle.dump(w.get_value(borrow=True), save_file, -1) # the -1 is for HIGHEST_PROTOCOL

>>> cPickle.dump(v.get_value(borrow=True), save_file, -1) # .. and it triggers much more efficient

>>> cPickle.dump(u.get_value(borrow=True), save_file, -1) # .. storage than numpy’s default

>>> save_file.close()

Then later, you can load your data back like this:

>>> save_file = open(’path’)

>>> w.set_value(cPickle.load(save_file), borrow=True)

>>> v.set_value(cPickle.load(save_file), borrow=True)

>>> u.set_value(cPickle.load(save_file), borrow=True)

This technique is a bit verbose, but it is tried and true. You will be able to load your data and render it in

matplotlib without trouble, years after saving it.

Do not pickle your training or test functions for long-term storage

Theano functions are compatible with Python’s deepcopy and pickle mechanisms, but you should not nec-

essarily pickle a Theano function. If you update your Theano folder and one of the internal changes, then

you may not be able to un-pickle your model. Theano is still in active development, and the internal APIs

are subject to change. So to be on the safe side – do not pickle your entire training or testing functions for

long-term storage. The pickle mechanism is aimed at for short-term storage, such as a temp ﬁle, or a copy

to another machine in a distributed job.

Read more about serialization in Theano, or Python’s pickling.

3.5.2 Plotting Intermediate Results

Visualizations can be very powerful tools for understanding what your model or training algorithm is doing.

You might be tempted to insert matplotlib plotting commands, or PIL image-rendering commands

into your model-training script. However, later you will observe something interesting in one of those pre-

rendered images and want to investigate something that isn’t clear from the pictures. You’ll wished you had

saved the original model.

If you have enough disk space, your training script should save intermediate models and a visualiza-

tion script should process those saved models.

You already have a model-saving function right? Just use it again to save these intermediate models.

Libraries you’ll want to know about: Python Image Library (PIL), matplotlib.

3.5. Theano/Python Tips 15

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