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</html>";s:4:"text";s:13087:"Okay, let's suppose we're trying to minimize some function, $C(v)$. It's reassuring because it tells us
that networks of perceptrons can be as powerful as any other computing
device. function usually called by our neural network code. Rather, we
humans are stupendously, astoundingly good at making sense of what our
eyes show us. The first change is to write
$\sum_j w_j x_j$ as a dot product, $w \cdot x \equiv \sum_j w_j x_j$,
where $w$ and $x$ are vectors whose components are the weights and
inputs, respectively. The "training_data" is a list of tuples, "(x, y)" representing the training inputs and the desired, outputs. But to get much higher accuracies it helps to use
established machine learning algorithms. Alternately, you can make a donation by sending me But what's really
exciting about the equation is that it lets us see how to choose
$\Delta v$ so as to make $\Delta C$ negative. Note that I've replaced the $w$ and $b$ notation by $v$
to emphasize that this could be any function - we're not
specifically thinking in the neural networks context any more. They're much closer in spirit to how our
brains work than feedforward networks. This is equivalent to minimizing $\Delta C \approx \nabla C
\cdot \Delta v$. Suppose, for example, that we'd chosen the learning rate to
be $\eta = 0.001$. One way to do this is to choose a weight $w_1 = 6$
for the weather, and $w_2 = 2$ and $w_3 = 2$ for the other conditions. At that
level the performance is close to human-equivalent, and is arguably
better, since quite a few of the MNIST images are difficult even for
humans to recognize with confidence, for example: I trust you'll agree that those are tough to classify! SVMs have a number
of tunable parameters, and it's possible to search for parameters
which improve this out-of-the-box performance. But, in practice gradient
descent often works extremely well, and in neural networks we'll find
that it's a powerful way of minimizing the cost function, and so
helping the net learn. Perceptrons were
developed
in the 1950s and 1960s by the scientist
Frank
  Rosenblatt, inspired by earlier
work
by Warren
  McCulloch and
Walter
  Pitts. In fact, there are
many similarities between perceptrons and sigmoid neurons, and the
algebraic form of the sigmoid function turns out to be more of a
technical detail than a true barrier to understanding. This is useful for, tracking progress, but slows things down substantially. To understand what the problem
is, let's look back at the quadratic cost in
Equation (6)\begin{eqnarray}  C(w,b) \equiv
  \frac{1}{2n} \sum_x \| y(x) - a\|^2 \nonumber\end{eqnarray}$('#margin_636312544623_reveal').click(function() {$('#margin_636312544623').toggle('slow', function() {});});. Using the bias instead of the threshold, the
perceptron rule can be
rewritten:
\begin{eqnarray}
  \mbox{output} = \left\{ 
    \begin{array}{ll} 
      0 & \mbox{if } w\cdot x + b \leq 0 \\
      1 & \mbox{if } w\cdot x + b > 0
    \end{array}
  \right. Some people get hung up thinking: "Hey, I have to be able to
visualize all these extra dimensions". Networks with this kind of many-layer structure - two or more hidden
layers - are called deep neural networks. \tag{13}\end{eqnarray}
Just as for the two variable case, we can
choose
\begin{eqnarray}
  \Delta v = -\eta \nabla C,
\tag{14}\end{eqnarray}
and we're guaranteed that our (approximate)
expression (12)\begin{eqnarray} 
  \Delta C \approx \nabla C \cdot \Delta v \nonumber\end{eqnarray}$('#margin_796021234053_reveal').click(function() {$('#margin_796021234053').toggle('slow', function() {});}); for $\Delta C$ will be negative. Even most professional mathematicians can't visualize four
dimensions especially well, if at all. When meeting the $\nabla C$ notation for the first time,
people sometimes wonder how they should think about the $\nabla$
symbol. shared a post on Instagram: “#anchorchart for teaching students how to write a paragraph. """, """Return the output of the network if ``a`` is input. And so on,
  repeatedly. We can think of stochastic gradient descent as being like political
polling: it's much easier to sample a small mini-batch than it is to
apply gradient descent to the full batch, just as carrying out a poll
is easier than running a full election. In that sense, I've perhaps shown slightly
too simple a function! (After asserting that we'll gain insight by imagining $C$ as a
function of just two variables, I've turned around twice in two
paragraphs and said, "hey, but what if it's a function of many more
than two variables?" Dropping the threshold means
you're more willing to go to the festival. Maybe a clever learning algorithm will find some assignment of weights
that lets us use only $4$ output neurons. Sure enough, this improves the results to $96.59$ percent. That is, given a training input, $x$, we update our
  weights and biases according to the rules $w_k \rightarrow w_k' =
  w_k - \eta \partial C_x / \partial w_k$ and $b_l \rightarrow b_l' =
  b_l - \eta \partial C_x / \partial b_l$. But when doing
detailed comparisons of different work it's worth watching out for. But nearly all that work is done unconsciously. It'll be
convenient to regard each training input $x$ as a $28 \times 28 =
784$-dimensional vector. Supposing the neural network functions in this way, we can give a
plausible explanation for why it's better to have $10$ outputs from
the network, rather than $4$. \tag{22}\end{eqnarray}
There's quite a bit going on in this equation, so let's unpack it
piece by piece. See
  this link for more
  details. But to
understand why sigmoid neurons are defined the way they are, it's
worth taking the time to first understand perceptrons. And, given such principles, can we
do better? Note that production code would be much,
much faster: these Python scripts are intended to help you understand
how neural nets work, not to be high-performance code! An idea called stochastic gradient descent can be used to speed
up learning. To connect this explicitly to learning in neural networks, suppose
$w_k$ and $b_l$ denote the weights and biases in our neural network. Incidentally, when I described the MNIST data earlier, I said it was
split into 60,000 training images, and 10,000 test images. simple, easily readable, and easily modifiable. How can we understand
that? Comparing a deep network to a shallow network is a
bit like comparing a programming language with the ability to make
function calls to a stripped down language with no ability to make
such calls. The first thing we need is to get the MNIST data. And it's possible that
recurrent networks can solve important problems which can only be
solved with great difficulty by feedforward networks. And so throughout the
book we'll return repeatedly to the problem of handwriting
recognition. If we had $4$ outputs, then the first
output neuron would be trying to decide what the most significant bit
of the digit was. ): If it were true that a small change in a weight (or bias) causes only
a small change in output, then we could use this fact to modify the
weights and biases to get our network to behave more in the manner we
want. This helps give us confidence that our system can
recognize digits from people whose writing it didn't see during
training. If there are a
million such $v_j$ variables then we'd need to compute something like
a trillion (i.e., a million squared) second partial
derivatives*
*Actually, more like half a trillion, since
  $\partial^2 C/ \partial v_j \partial v_k = \partial^2 C/ \partial
  v_k \partial v_j$. And so on, until we've exhausted the training
inputs, which is said to complete an
epoch of training. How should we interpret the output from a sigmoid neuron? For example, if a particular training image, $x$, depicts a $6$, then
$y(x) = (0, 0, 0, 0, 0, 0, 1, 0, 0, 0)^T$ is the desired output from
the network. For example, such heuristics can be used
to help determine how to trade off the number of hidden layers against
the time required to train the network. Now customize the name of a clipboard to store your clips. When you try to make such rules precise, you quickly get lost in a
morass of exceptions and caveats and special cases. convenient for use in our implementation of neural networks. Of course, when testing our network we'll ask it to
recognize images which aren't in the training set! To quantify how well we're achieving this goal
we define a cost function*
*Sometimes referred to as a
  loss or objective function. I suggest $5, but you can choose the amount. It seems
hopeless. and then develop a system which can learn from those training
examples. Obviously, it'd be easiest to do this if the
output was a $0$ or a $1$, as in a perceptron. Now that we have a design for our neural network, how can it learn to
recognize digits? Maybe
we can only see part of the face, or the face is at an angle, so some
of the facial features are obscured. To put these questions more starkly, suppose that a few decades hence
neural networks lead to artificial intelligence (AI). Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Finally, we'll use stochastic gradient descent to learn from the MNIST
training_data over 30 epochs, with a mini-batch size of 10, and a
learning rate of $\eta = 3.0$. Since 2006, a set of techniques has been developed that enable
learning in deep neural nets. please cite this book as: Michael A. Nielsen, "Neural Networks and Having defined neural networks, let's return to handwriting
recognition. In preparation for
that, it helps to explain some terminology that lets us name different
parts of a network. If the optional argument test_data is supplied, then
the program will evaluate the network after each epoch of training,
and print out partial progress. With some luck that
might work when $C$ is a function of just one or a few variables. We humans solve this segmentation
  problem with ease, but it's challenging
for a computer program to correctly break up the image. And so we'll take
Equation (10)\begin{eqnarray} 
  \Delta v = -\eta \nabla C \nonumber\end{eqnarray}$('#margin_129183303476_reveal').click(function() {$('#margin_129183303476').toggle('slow', function() {});}); to define the "law of motion"
for the ball in our gradient descent algorithm. The output layer of the network contains 10 neurons. Unknown. This is a simple procedure,
and is easy to code up, so I won't explicitly write out the code -
if you're interested it's in the
GitHub
  repository. It turns
out that we can devise learning
  algorithms which can
automatically tune the weights and biases of a network of artificial
neurons. We'll get to sigmoid neurons shortly. Notice that this cost
function has the form $C = \frac{1}{n} \sum_x C_x$, that is, it's an
average over costs $C_x \equiv \frac{\|y(x)-a\|^2}{2}$ for individual
training examples. The simplest baseline of
all, of course, is to randomly guess the digit. The condition $\sum_j
w_j x_j > \mbox{threshold}$ is cumbersome, and we can make two
notational changes to simplify it. So while your "9" might now be
classified correctly, the behaviour of the network on all the other
images is likely to have completely changed in some hard-to-control
way. Isn't this a rather ad
  hoc choice? We could do this simulation
simply by computing derivatives (and perhaps some second derivatives)
of $C$ - those derivatives would tell us everything we need to know
about the local "shape" of the valley, and therefore how our ball
should roll. If
we don't, we might end up with $\Delta C > 0$, which obviously would
not be good! In some sense, the moral of
both our results and those in more sophisticated papers, is that for
some problems: While our neural network gives impressive performance, that
performance is somewhat mysterious. The reason is that the NAND gate is universal for
computation, that is, we can build any computation up out of
NAND gates. We'll use the test data to
evaluate how well our neural network has learned to recognize digits. These ball-mimicking variations have some advantages, but also have a
major disadvantage: it turns out to be necessary to compute second
partial derivatives of $C$, and this can be quite costly. But for now
I just want to mention one problem. Note that I have focused on making the code. Perhaps we can use this idea as a way to find a
minimum for the function? Is there some special ability they're missing, some
ability that "real" supermathematicians have? If you don't use git then you can download the data and code
here. What seems easy when we do it ourselves suddenly becomes
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