深度学习教程：从 Logistic 回归到卷积神经网络

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"这是一份Deep Learning教程的PDF文档，由LISA实验室，蒙特利尔大学在2015年7月30日发布，版本0.1。该教程涵盖了深度学习的基础知识，包括如何入门、下载指南、数据集介绍、符号约定、监督优化的入门、使用Theano和Python的技巧，以及多个深度学习模型的实践应用，如逻辑回归分类MNIST数字、多层感知机和卷积神经网络（LeNet）等。此外，还涉及了去噪自编码器（dA）的内容。" 这篇文档深入浅出地介绍了深度学习的基础概念和实际操作，旨在帮助初学者快速上手。首先，教程提到了LICENSE，意味着文档遵循特定的开源许可协议，允许用户自由使用和分享。接着，教程详细介绍了深度学习的各个方面。"Getting Started"部分指导用户如何下载所需资源，并对数据集进行了概述，这是训练任何机器学习模型的基础。符号约定的解释有助于理解后续复杂的数学表达式，而“APrimeronSupervisedOptimizationforDeepLearning”则引导读者理解监督学习中的优化过程，这对于训练深度学习模型至关重要。文档还提供了Theano和Python的使用提示，这两个工具是实现深度学习的重要编程平台。Theano是一个强大的库，用于定义、优化和评估数学表达式，特别是多维数组操作，而Python是编写深度学习代码的常用语言。教程通过实例介绍了几个深度学习模型。首先，它演示了如何使用逻辑回归分类MNIST手写数字，从模型定义到损失函数、模型训练，再到测试模型，步步为营，使读者能够理解逻辑回归的工作原理。然后，教程引入了多层感知机（MLP），阐述了从逻辑回归到全连接神经网络的转变，以及训练MLP时的技巧。卷积神经网络（LeNet）是图像处理领域的经典模型，教程详述了它的动机、稀疏连接性、共享权重、卷积运算、最大池化等关键组件，以及如何将这些元素整合成完整的LeNet模型。这部分内容对于理解和实现卷积网络非常有用。最后，文档简要介绍了去噪自编码器（dA），这是一种用于无监督学习的深度学习模型，能够帮助数据去噪并学习其潜在结构。这份Deep Learning Tutorial是一份全面且实用的学习资料，覆盖了深度学习的基本概念、核心模型和实战技巧，适合初学者和有一定经验的研究者作为参考。

Deep Learning Tutorial, Release 0.1

# NLL is a symbolic variable ; to get the actual value of NLL, this symbolic

# expression has to be compiled into a Theano function (see the Theano

# tutorial for more details)

NLL = -T.sum(T.log(p_y_given_x)[T.arange(y.shape[0]), y])

# note on syntax: T.arange(y.shape[0]) is a vector of integers [0,1,2,...,len(y)].

# Indexing a matrix M by the two vectors [0,1,...,K], [a,b,...,k] returns the

# elements M[0,a], M[1,b], ..., M[K,k] as a vector. Here, we use this

# syntax to retrieve the log-probability of the correct labels, y.

3.4.2 Stochastic Gradient Descent

What is ordinary gradient descent? it is a simple algorithm in which we repeatedly make small steps down-

ward on an error surface deﬁned by a loss function of some parameters. For the purpose of ordinary gradient

descent we consider that the training data is rolled into the loss function. Then the pseudocode of this algo-

rithm can be described as :

# GRADIENT DESCENT

while True:

loss = f(params)

d_loss_wrt_params = ... # compute gradient

params -= learning_rate

d_loss_wrt_params

if <stopping condition is met>:

return params

Stochastic gradient descent (SGD) works according to the same principles as ordinary gradient descent, but

proceeds more quickly by estimating the gradient from just a few examples at a time instead of the entire

training set. In its purest form, we estimate the gradient from just a single example at a time.

# STOCHASTIC GRADIENT DESCENT

for (x_i,y_i) in training_set:

# imagine an infinite generator

# that may repeat examples (if there is only a finite training set)

loss = f(params, x_i, y_i)

d_loss_wrt_params = ... # compute gradient

params -= learning_rate

d_loss_wrt_params

if <stopping condition is met>:

return params

The variant that we recommend for deep learning is a further twist on stochastic gradient descent using so-

called “minibatches”. Minibatch SGD works identically to SGD, except that we use more than one training

example to make each estimate of the gradient. This technique reduces variance in the estimate of the

gradient, and often makes better use of the hierarchical memory organization in modern computers.

for (x_batch,y_batch) in train_batches:

# imagine an infinite generator

# that may repeat examples

loss = f(params, x_batch, y_batch)

d_loss_wrt_params = ... # compute gradient using theano

params -= learning_rate

d_loss_wrt_params

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Deep Learning Tutorial, Release 0.1

if <stopping condition is met>:

return params

There is a tradeoff in the choice of the minibatch size B. The reduction of variance and use of SIMD

instructions helps most when increasing B from 1 to 2, but the marginal improvement fades rapidly to

nothing. With large B, time is wasted in reducing the variance of the gradient estimator, that time would be

better spent on additional gradient steps. An optimal B is model-, dataset-, and hardware-dependent, and

can be anywhere from 1 to maybe several hundreds. In the tutorial we set it to 20, but this choice is almost

arbitrary (though harmless).

Note: If you are training for a ﬁxed number of epochs, the minibatch size becomes important because it

controls the number of updates done to your parameters. Training the same model for 10 epochs using a

batch size of 1 yields completely different results compared to training for the same 10 epochs but with a

batchsize of 20. Keep this in mind when switching between batch sizes and be prepared to tweak all the

other parameters acording to the batch size used.

All code-blocks above show pseudocode of how the algorithm looks like. Implementing such algorithm in

Theano can be done as follows :

# Minibatch Stochastic Gradient Descent

# assume loss is a symbolic description of the loss function given

# the symbolic variables params (shared variable), x_batch, y_batch;

# compute gradient of loss with respect to params

d_loss_wrt_params = T.grad(loss, params)

# compile the MSGD step into a theano function

updates = [(params, params - learning_rate

d_loss_wrt_params)]

MSGD = theano.function([x_batch,y_batch], loss, updates=updates)

for (x_batch, y_batch) in train_batches:

# here x_batch and y_batch are elements of train_batches and

# therefore numpy arrays; function MSGD also updates the params

print(’Current loss is ’, MSGD(x_batch, y_batch))

if stopping_condition_is_met:

return params

3.4.3 Regularization

There is more to machine learning than optimization. When we train our model from data we are trying

to prepare it to do well on new examples, not the ones it has already seen. The training loop above for

MSGD does not take this into account, and may overﬁt the training examples. A way to combat overﬁtting

is through regularization. There are several techniques for regularization; the ones we will explain here are

L1/L2 regularization and early-stopping.

3.4. A Primer on Supervised Optimization for Deep Learning 11

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

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