Lecture 1.3 — Some simple models of neurons — [ Deep Learning | Geoffrey Hinton | UofT ]
Summary
TLDR视频脚本介绍了几种相对简单的神经元模型,从简单的线性和阈值神经元开始,逐步过渡到更复杂的模型。这些模型虽然比真实的神经元简化,但足以构建能够进行有趣机器学习任务的神经网络。视频中强调了理想化的重要性,即通过简化复杂细节,以便更好地理解和应用数学,同时避免去除关键特性。介绍了几种不同类型的神经元,包括线性神经元、二元阈值神经元、整流线性神经元、S型神经元和随机二元神经元。每种神经元模型都有其特定的计算方式和输出特性,这些特性对于理解神经网络的工作原理至关重要。视频还讨论了逻辑函数在神经元模型中的应用,以及如何通过这些模型来模拟大脑处理信息的方式。
Takeaways
- 🧠 简化模型有助于理解复杂系统:通过简化神经元模型,可以更容易地应用数学和类比,从而理解大脑的工作原理。
- 📈 线性神经元的局限性:线性神经元虽然简单,但在计算上受限,可能无法完全反映真实神经元的复杂性。
- 🔄 阈值神经元的二元输出:McCulloch-Pitts阈值神经元通过计算加权和,超过阈值时输出1,否则输出0,模拟了逻辑命题的真值。
- 📊 激活函数的重要性:激活函数如sigmoid函数提供了平滑且有界的输出,这对于机器学习中的学习过程至关重要。
- 🔢 逻辑神经元与大脑计算:早期认为大脑计算类似于逻辑运算,但后来更关注于大脑如何结合多种不可靠证据。
- ⚡️ 阈值神经元的两种等价表达:总输入Z可以包括或不包括偏置项,两种表达方式在数学上是等价的。
- 📉 ReLU(修正线性单元)的非线性特性:ReLU在输入低于0时输出0,高于0时线性增长,这种特性在神经网络中非常常见。
- 🎢 Sigmoid神经元的平滑导数:Sigmoid神经元的导数连续且平滑,这使得它们在学习算法中表现良好。
- 🎰 随机二进制神经元的概率决策:这些神经元使用逻辑函数计算输出概率,然后基于这个概率进行随机的1或0输出。
- 🔁 ReLU的随机性:ReLU可以决定输出的速率,但实际的尖峰产生时间是随机的,模拟了Poisson过程。
- 🤖 神经网络的实用性:即使某些神经元模型与真实神经元行为不完全一致,它们在机器学习实践中仍然非常有用。
- 🚀 理解简化模型的价值:即使知道某些模型可能是错误的,理解它们仍然有助于我们构建更复杂的模型,更接近现实。
Q & A
视频中提到的最简单的神经元模型是什么?
-视频中提到的最简单的神经元模型是线性神经元(linear neuron),它的输出Y是神经元偏置B和所有输入连接活动的加权和的函数。
McCulloch和Pitts提出的二元阈值神经元模型有哪些特点?
-McCulloch和Pitts提出的二元阈值神经元模型首先计算输入的加权和,然后如果这个加权和超过阈值,就发出一个活动脉冲。它们将脉冲视为命题的真值,每个神经元结合来自其他神经元的真值,产生自己的真值。
什么是修正线性神经元(rectified linear neuron)?
-修正线性神经元首先计算输入的线性加权和,然后根据这个加权和的值给出输出。如果加权和Z小于零,则输出为零;如果大于零,则输出等于Z。它结合了线性神经元和二元阈值神经元的特点。
sigmoid神经元的输出是什么类型的函数?
-sigmoid神经元的输出是一个连续且有界的实数值函数,通常使用logistic函数,其总输入是偏置加上输入线上的加权活动。
为什么sigmoid神经元在机器学习中很有用?
-sigmoid神经元在机器学习中很有用,因为它们的导数是连续变化的,这使得它们在进行学习时表现良好,尤其是在梯度下降等算法中。
随机二元神经元是如何工作的?
-随机二元神经元使用与logistic单元相同的方程计算总输入,并使用logistic函数计算一个实数值,该值是它们输出脉冲的概率。然后,它们不是输出这个概率作为实数,而是进行一个概率决策,实际输出是1或0,具有内在的随机性。
为什么在理解复杂系统时需要理想化?
-理想化是为了简化复杂系统,移除非本质的复杂细节,使我们能够应用数学并将其与其他熟悉的系统进行类比。这有助于我们理解主要原理,并在理解了基本原理后,逐步增加复杂性,使模型更接近现实。
在机器学习中,即使知道某些模型是错误的,为什么仍然值得理解它们?
-即使知道某些模型是错误的,理解这些模型仍然有价值,因为它们可以帮助我们理解基本原理,并且这些简化的模型在实践中可能非常有用,尤其是在机器学习的某些应用中。
如何理解线性神经元的输出Y与输入的关系?
-在线性神经元中,输出Y是输入的加权和与偏置的函数。如果我们将偏置加上加权输入活动作为x轴,输出Y作为y轴,那么可以得到一条通过原点的直线。
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二元阈值神经元的输入输出函数如何表示?
-二元阈值神经元的输入输出函数可以表示为:如果加权输入Z超过阈值,则输出为1;否则输出为0。也可以表示为包含偏置项的总输入,如果总输入超过零,则输出为1,否则为0。
修正线性神经元的输入输出曲线有什么特点?
-修正线性神经元的输入输出曲线在Z小于零时输出为零,当Z大于零时输出等于Z,因此它在零点处有一个硬决策,但在Z大于零时是线性的。
为什么说逻辑可能不再是大脑工作的最佳范式?
-因为现代的神经科学研究认为大脑在处理信息时,更多地是结合了多种不同来源的不可靠证据,而不仅仅是逻辑运算。这意味着大脑的工作方式可能更复杂,涉及概率和不确定性的处理。
Outlines
😀 简单神经模型介绍
本段介绍了几种相对简单的神经元模型,从基本的线性和阈值神经元开始,逐步过渡到更复杂的模型。这些模型虽然比真实的神经元简单,但足以构建能够进行有趣机器学习任务的神经网络。作者强调了理想化的重要性,即通过简化复杂细节来理解主要原理,并通过数学和类比来加深理解。同时,作者提醒我们,在理想化过程中不应忽略关键特性。此外,还讨论了线性神经元、二元阈值神经元和修正线性神经元的工作原理,以及它们的输入输出函数。
😉 激活函数与随机二进制神经元
第二段深入讨论了不同类型的激活函数,包括sigmoid神经元和随机二进制神经元。Sigmoid神经元提供连续且有界的实数值输出,通常使用logistic函数,其导数平滑且连续,便于学习。随机二进制神经元使用与logistic神经元相同的方程式,但最终输出是1或0,取决于基于logistic函数计算出的概率进行的随机决策。此外,还提到了修正线性单元(ReLU),它在输入大于0时以确定性的方式产生输出,而输出的具体时间则是随机过程,模拟了Poisson过程。
Mindmap
Keywords
💡神经元模型
💡线性神经元
💡阈值神经元
💡修正线性神经元(ReLU)
💡sigmoid神经元
💡随机二进制神经元
💡理想化
💡机器学习
💡权重
💡偏置
💡激活函数
💡逻辑函数
Highlights
视频介绍了一些相对简单的神经元模型,包括线性神经元和阈值神经元等。
通过简化模型,可以去除不必要的复杂细节,帮助我们理解主要原理。
即使模型被认为错误,理解这些模型仍然有价值,只要我们记得它们的错误。
线性神经元的输出是神经元偏置和输入连接活动的加权和的函数。
阈值神经元由McCulloch和Pitts提出,它计算输入的加权和,超过阈值时发出活动脉冲。
阈值神经元将输入的逻辑值组合起来,产生自己的逻辑值。
20世纪40年代,逻辑是理解大脑如何工作的主要范式。
人们开始对大脑如何结合多种不同来源的不可靠证据感兴趣,逻辑不再是一个好的范式。
阈值神经元的输入-输出函数是,如果加权输入超过阈值,则输出1,否则输出0。
整流线性神经元结合了线性神经元和阈值神经元的特性,它首先计算输入的线性加权和,然后给出非线性输出。
当输入大于等于零时,整流线性神经元的输出等于输入值,小于零时输出为零。
Sigmoid神经元提供连续有界的实数值输出,通常使用logistic函数。
当总输入很大且为正时,sigmoid神经元的输出接近1;当总输入很大且为负时,输出接近0。
Sigmoid神经元的导数连续且易于处理,便于学习。
随机二进制神经元使用与logistic单元相同的方程,计算总输入和概率,然后随机决定输出1或0。
如果输入非常正,随机二进制神经元几乎总是产生1;如果输入非常负,则几乎总是产生0。
整流线性单元可以决定产生脉冲的速率,而实际产生脉冲的时间则是一个随机过程。
Transcripts
in this video I'm going to describe some
relatively simple models of neurons I'll
describe a number of different models
starting with simple linear and
threshold neurons and then described in
slightly more complicated models these
are much simpler than real neurons but
they're still complicated enough to
allow us to make neural nets that do
some very interesting kinds of machine
learning in order to understand anything
complicated we have to idealize it that
is we have to make simplifications that
allow us to get a handle on how it might
work with atoms for example we
simplified them as behaving like little
solar systems idealization removes the
complicated details that are not
essential for understanding the main
principles it allows us to apply
mathematics and to make analogies to
other familiar systems and once we
understand the basic principles it's
easy to add complexity and make the
model more faithful to reality of course
we have to be careful when we idealize
something not to remove the thing that's
giving it is main properties it's often
worth understanding models that are
known to be wrong as long as we don't
forget that wrong so for example a lot
of work on neural networks uses neurons
that communicate real values rather than
discrete spikes of activity and we know
cortical neurons don't behave like that
but is still worth understanding systems
like that and in practice they can be
very useful for machine learning the
first kind of neuron I want to tell you
about is the simplest it's a linear
neuron it's simple it's computationally
limited in what it can do it may allow
us to get insights into more complicated
neurons but it may be somewhat
misleading so in a linear neuron the
output Y is a function of a bias of the
neuron B and the sum over all its
incoming connections of the activity on
an input
time's the weight on that wine that's
the synaptic weight on the input line
and if we plot that as a curve then if
we plot on the x-axis the bias plus the
weighted activities on the input lines
we get a straight line that goes through
zero very different from linear neurons
our binary threshold neurons that were
introduced by McCulloch and Pitt's they
actually influenced von Lohmann when he
was thinking about how to design a
universal computer in a binary threshold
neuron you first compute a weighted sum
of the inputs and then you send out a
spike of activity if that weighted sum
exceeds the threshold McCulloch and
Pitt's thought that the spikes were like
the truth values of propositions so each
neuron is combining the truth values it
gets from other neurons to produce the
truth value of its own and that's like
combining some propositions to compute
the truth value of another proposition
at the time in the 1940s logic was the
main paradigm for how the mind might
work since then people thinking about
how the brain computes but become much
more interested in the idea that brain
is combining lots of different sources
of unreliable evidence and so logic
isn't such a good paradigm for what the
brains up to for a binary threshold
neuron you can think of his input-output
function as if the weighted input is
above the threshold it gives an output
of 1 otherwise it gives an output of 0
they're actually two equivalent ways to
write the equations for a binary
threshold neuron we can say that the
total input Z is just the activities on
the input lines times the weights and
then the output Y is 1 if that Z is
above the threshold and 0 otherwise
alternatively we could say that the
total input includes a bias term so the
total input is what comes in on the
input lines times the weights plus this
bias term
and then we can say the output is one if
that total input is above zero and is
zero otherwise
then the equivalence is simply that the
threshold in the first formulation is
equal to the negative of the bias in the
second formulation a kind of neuron that
combines the properties of both linear
neurons and binary threshold neurons is
a rectified linear neuron it first
computes a linear weighted sum of its
inputs but then it gives an output
there's a nonlinear function of this
weighted sum so we compute Z in the same
way as before if Z is below zero we give
an output of zero otherwise we give an
output that's equal to Z so bub zero is
linear an ad zero it makes a hard
decision so the input/output curve looks
like this it's definitely not linear but
above zero is linear so in the neuron
like this we can get a lot of the nice
properties of linear systems when it's
above zero we can also get the ability
to make decisions at zero the neurons
that we'll use a lot in this course and
are probably the commonest kinds of
neurons to use in artificial neural Nets
a sigmoid neurons they give a
real-valued output that is a smooth and
bounded function of their total input
it's typical to use the logistic
function where the total input is
computed as before as a bias plus what
comes in on the input lines weighted the
output for a logistic neuron is 1 over 1
plus e to the minus the total input if
you think about that if the total input
is big and positive e to the minus a big
positive number is zero and so the
output will be 1 if the total input is
big in negative e to the minus a big
negative number is a large number and so
the output will be
so the input-output function looks like
this when the total input 0 e to the
minus 0 is 1
so the actors are 1/2 and the nice thing
about a sigmoid is it has smooth
derivatives the derivatives change
continuously and so they're nicely
behaved and they make it easy to do
learning as we'll see in lecture 3
finally the stochastic binary neurons
they use just the same equations as
logistic units they compute their total
input the same way and they use the
logistic function to compute a real
value which is the probability that they
will output a spike but then instead of
outputting that probability as a real
number they actually make a
probabilistic decision and so what they
actually output is either a 1 or a 0
they intrinsically random so they're
treating the P is the probability of
producing a 1 not as a real number of
course if the input is very big and
positive they will almost always produce
a 1 and if the inputs bigger negative
that almost always produces 0 we can do
a similar trick with rectified linear
units we can say that the output this
real value that comes out of a rectified
linear unit if its input is above 0 is
the rate of producing spikes
so that's deterministic but once we
figured out this rate of producing
spikes the actual times of which spikes
are produced is a random process as a
Poisson process so the rectified linear
unit determines the rate but intrinsic
randomness in the unit determines when
the spikes are actually produced
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