numerical optimization – How can I apply backpropagation with matrix algebra? – In-depth learning

In-depth learning and backpropagation are very poorly taught and often look like a mess, in my opinion.

So I want to start with a simple example on how to use backpropagation:

Suppose we have this network of neurons:

Where the brand `(L)` is the index of the layer and `The` represent `Latest` layer.

In the picture above, we can see two perceptrons $$a ^ {(L-1)} _ {i}$$ and $$a ^ {(L)} _ {i}$$.

Give them initial values!
$$a ^ {(L-1)} _ {i} = 5.7$$
$$W ^ {(L)} _ {i, j} = -23,1$$
$$a ^ {(L)} _ {i} = a ^ {(L-1)} _ {i} W ^ {(L)} _ {i, j} = 5.7 * (-23.1 ) = -131.67$$

Our goal is to find $$W ^ {(L)} _ {i, j}$$ If we want $$a ^ {(L) ^ *} _ {i} = 50.1$$. This is the key idea behind the back propagation. Note that $$a ^ {(L) ^ *} _ {i}$$ is our desire for value.

So what do we do? Well, we first find the error as small as possible.
$$C = (a ^ {(L)} _ {i} – a ^ {(L) ^ *} _ {i}) ^ 2$$

Minimize $$C$$, we have to take into account how much $$C$$ change if $$a ^ {(L)} _ {i}$$ change? Let's write the mathematical expression for that!
$$frac { partial C} { partial a ^ {(L)} _ {i}}$$

But $$a ^ {(L)} _ {i}$$ change if $$W ^ {(L)} _ {i, j}$$ changes. How many $$a ^ {(L)} _ {i}$$ change if $$W ^ {(L)} _ {i, j}$$ changes? Well, let's write a mathematical expression for that too!

$$frac { a ^ {(L)} _ {i}} { partial W ^ {(L)} _ {i, j}}$$

There is a chain connection between
$$frac { partial a ^ {(L)} _ {i}} { partial W ^ {(L)} _ {i, j}$$ and $$frac { partial C} { partial a ^ {(L)} _ {i}}$$. This means that we will apply the chain rule.

$$frac { partial a ^ {(L)} _ {i}} { partial W ^ {(L)} _ {i, j}} frac { partial C} { partial a ^ { L)} _ {i}} = frac { C partial} { partial W ^ {(L)} _ {i, j}}$$

This means that if the weight $$W ^ {(L)} _ {i, j}$$ changes, the error $$C$$ must also change.

To start minimizing this error $$C$$we need to know what weight we are going to use $$W ^ {(L)} _ {i, j}$$. We introduce a learning factor $$R = 0.007$$ and use this formula

$$W ^ {(L)} _ {i, j} (k + 1) = W ^ {(L)} _ {i, j} (k) – R frac { partial C} { partial W ^ {(L)} _ {i, j}}$$

If we repeat this 20 times, we will regain our weight $$W ^ {(L)} _ {i, j}$$. Easy! Let's do it.

$$W ^ {(L)} _ {i, j} (k + 1) = W ^ {(L)} _ {i, j} (k) – R frac { partial a ^ {(L) } _ {i}} { partial W ^ {(L)} _ {i, j}} frac { partial C} { partial a ^ {(L)} _ {i}}$$

$$W ^ {(L)} _ {i, j} (k + 1) = W ^ {(L)} _ {i, j} (k) – R a ^ {(L-1)} _ { i} 2 (a ^ {(L-1)} _ {i} W ^ {(L)} _ {i, j} (k) -a ^ {(L) ^ *} _ {i})$$

Because
$$frac { partial a ^ {(L)} _ {i, j}} { partial W ^ {(L)} _ {i, j}} = frac {a ^ {(L-1) } _ {i} W ^ {(L)} _ {i, j}} {W ^ {(L)} _ {i, j}} = a ^ {(L)} _ {i}$$

and

$$frac { C partial} { partial a ^ {(L)} _ {i, j}} = 2 (a ^ {(L)} _ {i} – a ^ {(L) ^ *} _ {i}) = 2 (a ^ {(L-1)} _ {i} W ^ {_ (L)} _ {i, j} -a ^ {(L) ^ *} _ {i})$$

Now, we should iterate this equation:
$$W ^ {(L)} _ {i, j} (k + 1) = W ^ {(L)} _ {i, j} (k) – R a ^ {(L-1)} _ { i} 2 (a ^ {(L-1)} _ {i} W ^ {(L)} _ {i, j} (k) -a ^ {(L) ^ *} _ {i})$$

``````# Simple neural network
r = 0.007;
w = -23.1;
a = 5.7;
y = 50.1;
Warray = [];
for i = 1:20
w = w - r * a * 2 * (a * w - y)
Warray (i) = w;
end

% Ground
intrigue (1:20, Warray);
grid on

w = -8.5948
w = -0.68736
w = 3.6233
w = 5.9732
w = 7.2542
w = 7.9525
w = 8.3332
w = 8.5408
w = 8.6539
w = 8.7156
w = 8.7492
w = 8.7675
w = 8.7775
w = 8.7829
w = 8.7859
w = 8.7875
w = 8.7884
w = 8.7889
w = 8.7892
w = 8.7893
>>
``````

And the result:

$$a * w = 5.7 * 8.7893 = 50.099 about a ^ {(L) ^ *} _ {i}$$

And if I extend this network of neurons too.

I have just extended the formula to:

$$frac { partial a ^ {(L-1)} _ {i}} { partial W ^ {(L-1)} _ {i, j} frac { partial W ^ {(L)} _ {i, j}} { partial a ^ {(L-1)} _ {i}} frac { partial a ^ {(L)} _ { i}} { partial W ^ {(L)} _ {i, j}} frac { partial C} { partial a ^ {(L)} _ {i}} = frac { partial C} { partial W ^ {(L)} _ {i, j}}$$

And then solve for
$$W ^ {(L-1)} _ {i, j} (k + 1) = W ^ {(L-1)} _ {i, j} (k) – R frac { partial a ^ { L-1)} _ {i}} { W partial {^ (L-1)} _ {i, j}} frac { partial W ^ {(L)} _ {i, j}} { partial a ^ {(L-1)} _ {i}} frac { partial a ^ {(L)} _ { i}} { partial W ^ {(L)} _ {i, j}} frac { partial C} { partial a ^ {(L)} _ {i}}$$

Beacse we found $$W ^ {(L)} _ {i, j}$$ already, we can simply write.
$$W ^ {(L-1)} _ {i, j} (k + 1) = W ^ {(L-1)} _ {i, j} (k) – R frac { partial a ^ { L-1)} _ {i}} { W partial {^ (L-1)} _ {i, j}} W ^ {(L)} _ {i, j} frac { C partial} { partial a ^ {(L)} _ {i}}$$

Simplify:
$$W ^ {(L-1)} _ {i, j} (k + 1) = W ^ {(L-1)} _ {i, j} (k) – R a ^ {(L-2)} _{I} W ^ {(L)} _ {i, j} 2 (a ^ {(L-1)} _ {i} W ^ {(L)} _ {i, j} -a ^ {(L) ^ * } _ {i})$$

Question:

How can I do backpropagation if I have a model like this:

Amazon Web Services – Removing and then reinstalling Anaconda on an AWS Ubuntu Deep Learning EC2 instance and unable to enter in-depth learning environments

I just set up an Ubuntu Deep Learning AMI EC2 instance. I am a beginner on AWS / Packet Processing.

My goal is to use the instance to run a Python deep learning script. This script uses a variety of packages.

When installing some of these packages with conda, an error has occurred indicating inconsistencies in the environment for more than 100 packages. After several attempts to solve this problem, I thought that removing Anaconda and reinstalling it could do the trick. After that, I realized that I had perhaps further spoiled my instance. I can no longer use the predefined deep learning environments for which the AMI has been configured because they have been accessed using conda commands, which seems to have been removed (IMO).

I've tried repeating the commands, but I get an error stating that these environments no longer exist. A tutorial using these commands is mentioned here:
https://docs.aws.amazon.com/dlami/latest/devguide/tutorial-conda.html

``````active source tensorflow_p36
``````

I was expecting the above to enter the tensorflow_p36 environment. A sin:

``````(tensorflow_p36) ubuntu @ ip-172-31-45-96: ~ / scripts
``````

However, this gives an error message:

``````impossible to find the environment: tensorflow_p36
``````

I realize that the uninstallation of conda was a major rookie error that seems to have totally disabled my instance. If anyone has any ideas to get it back, it would be very appreciated!

thank you so much

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Views: 239

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Recommended method for combining subcategory selection and in-depth exploration in mobile navigation.

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