# AI News, A Neural Network in 13 lines of Python (Part 2 - Gradient Descent) ## A Neural Network in 13 lines of Python (Part 2 - Gradient Descent)

Summary: I learn best with toy code that I can play with.

This tutorial teaches gradient descent via a very simple toy example, a short python implementation.

Followup Post: I intend to write a followup post to this one adding popular features leveraged by state-of-the-art approaches (likely Dropout, DropConnect, and Momentum).

layer_1_delta = layer_2_delta.dot(synapse_1.T) * (layer_1 * (1-layer_1))

It moves the error information from the end of the network to all the weights inside the network so that a different algorithm can optimize those weights to fit our data.

In this tutorial, we will walk through Gradient Descent, which is arguably the simplest and most widely used neural network optimization algorithm.

By learning about Gradient Descent, we will then be able to improve our toy neural network through parameterization and tuning, and ultimately make it a lot more powerful. Part

In our case, the ball is optimizing it's position (from left to right) to find the lowest point in the bucket.

The only information it has is the slope of the side of the bucket at its current position, pictured below with the blue line.

As it gets closer and closer to the bottom, it takes smaller and smaller steps until the slope equals zero, at which point it stops.

What makes this problem so destructive is that overshooting this far means we land at an EVEN STEEPER slope in the opposite direction.

your bucket has a funny shape, and following the slope doesn't take you to the absolute lowest point.

are a myriad of ways in which randomness is used to overcome getting stuck in a local minimum.

For example, if a ball randomly falls within the blue domain, it will converge to the blue minimum.

This is far better than pure random searching, which has to randomly try EVERY space (which could easily be millions of places on this black line depending on the granularity).

Parameterizing this size allows the neural network user to potentially try thousands (or tens of billions) of different local minima in a single neural network. Sidenote

We can search the entire black line above with (in theory) only 5 balls and a handful of iterations.

The current state-of-the-art approaches to avoiding hidden nodes coming up with the same answer (by searching the same space) are Dropout and Drop-Connect, which I intend to cover in a later post. Problem

The ball just drops right into an instant local minimum and ignores the big picture.

So, if we computed the network's error for every possible value of a single weight, it would generate the curve you see above.

We would then pick the value of the single weight that has the lowest error (the lowest part of the curve).

Thus, the x dimension is the value of the weight and the y dimension is the neural network's error when the weight is at that position. Stop

Let's take a look at what this process looks like in a simple 2 layer neural network. 2

[0,1],

[1,0],

[1,0] ])

output dataset y

in this case, we have a single error at the output (single value), which is computed on line 35.

If we take that logic and plot the overall error (a single scalar representing the network error over the entire dataset) for every possible set of weights (from -10 to 10 for x and y), it looks something like this. Don't

It really is as simple as computing every possible set of weights, and the error that the network generates at each set.

Now that we have seen how our neural network leverages Gradient Descent, we can improve our network to overcome these weaknesses in the same way that we improved Gradient Descent in Part 3 (the 3 problems and solutions). Improvement

As described above, the alpha parameter reduces the size of each iteration's update in the simplest way possible.

At the very last minute, right before we update the weights, we multiply the weight update by alpha (usually between 0 and 1, thus reducing the size of the weight update).

We're going to jump back to our 3 layer neural network from the first post and add in an alpha parameter at the appropriate place.

Then, we're going to run a series of experiments to align all the intuition we developed around alpha with its behavior in live code. Improved

[0,1,1],

[1,0,1],

[1,1,1]])

y

# randomly initialize our weights with mean 0

# Feed forward through layers 0, 1, and 2

layer_0 = X

layer_1 = sigmoid(np.dot(layer_0,synapse_0))

layer_2 = sigmoid(np.dot(layer_1,synapse_1))

# how much did we miss the target value?

layer_2_error = layer_2 - y

if (j% 10000) == 0:

print 'Error after '+str(j)+' iterations:' + str(np.mean(np.abs(layer_2_error)))

# in what direction is the target value?

layer_2_delta = layer_2_error*sigmoid_output_to_derivative(layer_2)

# how much did each l1 value contribute to the l2 error (according to the weights)?

# in what direction is the target l1?

layer_1_delta = layer_1_error * sigmoid_output_to_derivative(layer_1)

synapse_1 -= alpha * (layer_1.T.dot(layer_2_delta))

synapse_0 -= alpha * (layer_0.T.dot(layer_1_delta)) Training

= 10Perhaps you were surprised that an alpha that was greater than 1 achieved the best score after only 10,000 iterations!

This means that in the smaller alpha parameters (less than 10), the network's weights were generally headed in the right direction, they just needed to hurry up and get there! Alpha

with an extremely large alpha, we see a textbook example of divergence, with the error increasing instead of decreasing...

[0,1,1],

[1,0,1],

[1,1,1]])

y

# Feed forward through layers 0, 1, and 2

layer_1 = sigmoid(np.dot(layer_0,synapse_0))

layer_2 = sigmoid(np.dot(layer_1,synapse_1))

# how much did we miss the target value?

layer_2_error = y - layer_2

if (j% 10000) == 0:

print 'Error:' + str(np.mean(np.abs(layer_2_error)))

# in what direction is the target value?

layer_2_delta = layer_2_error*sigmoid_output_to_derivative(layer_2)

# how much did each l1 value contribute to the l2 error (according to the weights)?

# in what direction is the target l1?

layer_1_delta = layer_1_error * sigmoid_output_to_derivative(layer_1)

synapse_0_direction_count += np.abs(((synapse_0_weight_update &gt;

synapse_1_direction_count += np.abs(((synapse_1_weight_update &gt;

0) + 0))

synapse_1 += alpha * synapse_1_weight_update

synapse_0 += alpha * synapse_0_weight_update

prev_synapse_0_weight_update = synapse_0_weight_update

prev_synapse_1_weight_update = synapse_1_weight_update

If a slope (derivative) changes direction, it means that it passed OVER the local minimum and needs to go back.

able to increase the size of the hidden layer increases the amount of search space that we converge to in each iteration.

[0,1,1],

[1,0,1],

[1,1,1]])

y

# Feed forward through layers 0, 1, and 2

layer_1 = sigmoid(np.dot(layer_0,synapse_0))

layer_2 = sigmoid(np.dot(layer_1,synapse_1))

# how much did we miss the target value?

layer_2_error = layer_2 - y

if (j% 10000) == 0:

print 'Error after '+str(j)+' iterations:' + str(np.mean(np.abs(layer_2_error)))

# in what direction is the target value?

layer_2_delta = layer_2_error*sigmoid_output_to_derivative(layer_2)

# how much did each l1 value contribute to the l2 error (according to the weights)?

# in what direction is the target l1?

layer_1_delta = layer_1_error * sigmoid_output_to_derivative(layer_1)

synapse_1 -= alpha * (layer_1.T.dot(layer_2_delta))

synapse_0 -= alpha * (layer_0.T.dot(layer_1_delta)) Training

that the best error with 32 nodes is 0.0009 whereas the best error with 4 hidden nodes was only 0.0013.

Even though this is very marginal in this toy problem, this affect plays a huge role when modeling very complex datasets. Part

If you want to be able to create arbitrary architectures based on new academic papers or read and understand sample code for these different architectures, I think that it's a killer exercise.

I worked with neural networks for a couple years before performing this exercise, and it was the best investment of time I've made in the field (and it didn't take long).

5px;}#indJobContent .company_location{font-size: 11px;overflow: hidden;display:block;}#indJobContent.wide .job{display:block;float:left;margin-right: 5px;width: 135px;overflow: hidden}#indeed_widget_wrapper{position: relative;font-family: 'Helvetica Neue',Helvetica,Arial,sans-serif;font-size: 13px;font-weight: normal;line-height: 18px;padding: 10px;height: auto;overflow: hidden;}#indeed_widget_header{font-size:18px;

Followup Post: I intend to write a followup post to this one adding popular features leveraged by state-of-the-art approaches (likely Dropout, DropConnect, and Momentum).

It moves the error information from the end of the network to all the weights inside the network so that a different algorithm can optimize those weights to fit our data.

In this tutorial, we will walk through Gradient Descent, which is arguably the simplest and most widely used neural network optimization algorithm.

In our case, the ball is optimizing it's position (from left to right) to find the lowest point in the bucket.

The only information it has is the slope of the side of the bucket at its current position, pictured below with the blue line.

As it gets closer and closer to the bottom, it takes smaller and smaller steps until the slope equals zero, at which point it stops.

For example, if a ball randomly falls within the blue domain, it will converge to the blue minimum.

This is far better than pure random searching, which has to randomly try EVERY space (which could easily be millions of places on this black line depending on the granularity).

Parameterizing this size allows the neural network user to potentially try thousands (or tens of billions) of different local minima in a single neural network. Sidenote

We can search the entire black line above with (in theory) only 5 balls and a handful of iterations.

The current state-of-the-art approaches to avoiding hidden nodes coming up with the same answer (by searching the same space) are Dropout and Drop-Connect, which I intend to cover in a later post. Problem

It moves the error information from the end of the network to all the weights inside the network so that a different algorithm can optimize those weights to fit our data.

In this tutorial, we will walk through Gradient Descent, which is arguably the simplest and most widely used neural network optimization algorithm.

By learning about Gradient Descent, we will then be able to improve our toy neural network through parameterization and tuning, and ultimately make it a lot more powerful.

In our case, the ball is optimizing it's position (from left to right) to find the lowest point in the bucket.

So, it needs to press the left and right buttons correctly to find the lowest spot So, what information does the ball use to adjust its position to find the lowest point?

The only information it has is the slope of the side of the bucket at its current position, pictured below with the blue line.

As it gets closer and closer to the bottom, it takes smaller and smaller steps until the slope equals zero, at which point it stops.

What makes this problem so destructive is that overshooting this far means we land at an EVEN STEEPER slope in the opposite direction.

For example, if a ball randomly falls within the blue domain, it will converge to the blue minimum.

This is far better than pure random searching, which has to randomly try EVERY space (which could easily be millions of places on this black line depending on the granularity).

Parameterizing this size allows the neural network user to potentially try thousands (or tens of billions) of different local minima in a single neural network.

We can search the entire black line above with (in theory) only 5 balls and a handful of iterations.

The current state-of-the-art approaches to avoiding hidden nodes coming up with the same answer (by searching the same space) are Dropout and Drop-Connect, which I intend to cover in a later post.

So, if we computed the network's error for every possible value of a single weight, it would generate the curve you see above.

Let's take a look at what this process looks like in a simple 2 layer neural network. 2

[0,1],

[1,0],

[1,0] ])

output dataset y

If we take that logic and plot the overall error (a single scalar representing the network error over the entire dataset) for every possible set of weights (from -10 to 10 for x and y), it looks something like this. Don't

Now that we have seen how our neural network leverages Gradient Descent, we can improve our network to overcome these weaknesses in the same way that we improved Gradient Descent in Part 3 (the 3 problems and solutions). Improvement

As described above, the alpha parameter reduces the size of each iteration's update in the simplest way possible.

At the very last minute, right before we update the weights, we multiply the weight update by alpha (usually between 0 and 1, thus reducing the size of the weight update).

We're going to jump back to our 3 layer neural network from the first post and add in an alpha parameter at the appropriate place.

Then, we're going to run a series of experiments to align all the intuition we developed around alpha with its behavior in live code. Improved

[0,1,1],

[1,0,1],

[1,1,1]])

y

# Feed forward through layers 0, 1, and 2

layer_1 = sigmoid(np.dot(layer_0,synapse_0))

layer_2 = sigmoid(np.dot(layer_1,synapse_1))

# how much did we miss the target value?

layer_2_error = layer_2 - y

if (j% 10000) == 0:

print 'Error after '+str(j)+' iterations:' + str(np.mean(np.abs(layer_2_error)))

# in what direction is the target value?

layer_2_delta = layer_2_error*sigmoid_output_to_derivative(layer_2)

# how much did each l1 value contribute to the l2 error (according to the weights)?

# in what direction is the target l1?

layer_1_delta = layer_1_error * sigmoid_output_to_derivative(layer_1)

synapse_1 -= alpha * (layer_1.T.dot(layer_2_delta))

synapse_0 -= alpha * (layer_0.T.dot(layer_1_delta)) Training

= 10Perhaps you were surprised that an alpha that was greater than 1 achieved the best score after only 10,000 iterations!

This means that in the smaller alpha parameters (less than 10), the network's weights were generally headed in the right direction, they just needed to hurry up and get there! Alpha

with an extremely large alpha, we see a textbook example of divergence, with the error increasing instead of decreasing...

[0,1,1],

[1,0,1],

[1,1,1]])

y

# Feed forward through layers 0, 1, and 2

layer_1 = sigmoid(np.dot(layer_0,synapse_0))

layer_2 = sigmoid(np.dot(layer_1,synapse_1))

# how much did we miss the target value?

layer_2_error = y - layer_2

if (j% 10000) == 0:

print 'Error:' + str(np.mean(np.abs(layer_2_error)))

# in what direction is the target value?

layer_2_delta = layer_2_error*sigmoid_output_to_derivative(layer_2)

# how much did each l1 value contribute to the l2 error (according to the weights)?

# in what direction is the target l1?

layer_1_delta = layer_1_error * sigmoid_output_to_derivative(layer_1)

synapse_0_direction_count += np.abs(((synapse_0_weight_update &gt;

synapse_1_direction_count += np.abs(((synapse_1_weight_update &gt;

0) + 0))

synapse_1 += alpha * synapse_1_weight_update

synapse_0 += alpha * synapse_0_weight_update

prev_synapse_0_weight_update = synapse_0_weight_update

prev_synapse_1_weight_update = synapse_1_weight_update

If a slope (derivative) changes direction, it means that it passed OVER the local minimum and needs to go back.

able to increase the size of the hidden layer increases the amount of search space that we converge to in each iteration.

[0,1,1],

[1,0,1],

[1,1,1]])

y

# Feed forward through layers 0, 1, and 2

layer_1 = sigmoid(np.dot(layer_0,synapse_0))

layer_2 = sigmoid(np.dot(layer_1,synapse_1))

# how much did we miss the target value?

layer_2_error = layer_2 - y

if (j% 10000) == 0:

print 'Error after '+str(j)+' iterations:' + str(np.mean(np.abs(layer_2_error)))

# in what direction is the target value?

layer_2_delta = layer_2_error*sigmoid_output_to_derivative(layer_2)

# how much did each l1 value contribute to the l2 error (according to the weights)?

# in what direction is the target l1?

layer_1_delta = layer_1_error * sigmoid_output_to_derivative(layer_1)

synapse_1 -= alpha * (layer_1.T.dot(layer_2_delta))

synapse_0 -= alpha * (layer_0.T.dot(layer_1_delta)) Training

that the best error with 32 nodes is 0.0009 whereas the best error with 4 hidden nodes was only 0.0013.

Even though this is very marginal in this toy problem, this affect plays a huge role when modeling very complex datasets. Part

If you want to be able to create arbitrary architectures based on new academic papers or read and understand sample code for these different architectures, I think that it's a killer exercise.

I worked with neural networks for a couple years before performing this exercise, and it was the best investment of time I've made in the field (and it didn't take long).

5px;}#indJobContent .company_location{font-size: 11px;overflow: hidden;display:block;}#indJobContent.wide .job{display:block;float:left;margin-right: 5px;width: 135px;overflow: hidden}#indeed_widget_wrapper{position: relative;font-family: 'Helvetica Neue',Helvetica,Arial,sans-serif;font-size: 13px;font-weight: normal;line-height: 18px;padding: 10px;height: auto;overflow: hidden;}#indeed_widget_header{font-size:18px;

So, if we computed the network's error for every possible value of a single weight, it would generate the curve you see above.

Thus, the x dimension is the value of the weight and the y dimension is the neural network's error when the weight is at that position.

Let's take a look at what this process looks like in a simple 2 layer neural network.

If we take that logic and plot the overall error (a single scalar representing the network error over the entire dataset) for every possible set of weights (from -10 to 10 for x and y), it looks something like this.

Now that we have seen how our neural network leverages Gradient Descent, we can improve our network to overcome these weaknesses in the same way that we improved Gradient Descent in Part 3 (the 3 problems and solutions). Improvement

As described above, the alpha parameter reduces the size of each iteration's update in the simplest way possible.

At the very last minute, right before we update the weights, we multiply the weight update by alpha (usually between 0 and 1, thus reducing the size of the weight update).

We're going to jump back to our 3 layer neural network from the first post and add in an alpha parameter at the appropriate place.

Then, we're going to run a series of experiments to align all the intuition we developed around alpha with its behavior in live code. Improved

[0,1,1],

[1,0,1],

[1,1,1]])

y

# Feed forward through layers 0, 1, and 2

layer_1 = sigmoid(np.dot(layer_0,synapse_0))

layer_2 = sigmoid(np.dot(layer_1,synapse_1))

# how much did we miss the target value?

layer_2_error = layer_2 - y

if (j% 10000) == 0:

print 'Error after '+str(j)+' iterations:' + str(np.mean(np.abs(layer_2_error)))

# in what direction is the target value?

layer_2_delta = layer_2_error*sigmoid_output_to_derivative(layer_2)

# how much did each l1 value contribute to the l2 error (according to the weights)?

# in what direction is the target l1?

layer_1_delta = layer_1_error * sigmoid_output_to_derivative(layer_1)

synapse_1 -= alpha * (layer_1.T.dot(layer_2_delta))

synapse_0 -= alpha * (layer_0.T.dot(layer_1_delta)) Training

= 10Perhaps you were surprised that an alpha that was greater than 1 achieved the best score after only 10,000 iterations!

This means that in the smaller alpha parameters (less than 10), the network's weights were generally headed in the right direction, they just needed to hurry up and get there! Alpha

with an extremely large alpha, we see a textbook example of divergence, with the error increasing instead of decreasing...

[0,1,1],

[1,0,1],

[1,1,1]])

y

# Feed forward through layers 0, 1, and 2

layer_1 = sigmoid(np.dot(layer_0,synapse_0))

layer_2 = sigmoid(np.dot(layer_1,synapse_1))

# how much did we miss the target value?

layer_2_error = y - layer_2

if (j% 10000) == 0:

print 'Error:' + str(np.mean(np.abs(layer_2_error)))

# in what direction is the target value?

layer_2_delta = layer_2_error*sigmoid_output_to_derivative(layer_2)

# how much did each l1 value contribute to the l2 error (according to the weights)?

# in what direction is the target l1?

layer_1_delta = layer_1_error * sigmoid_output_to_derivative(layer_1)

synapse_0_direction_count += np.abs(((synapse_0_weight_update &gt;

synapse_1_direction_count += np.abs(((synapse_1_weight_update &gt;

0) + 0))

synapse_1 += alpha * synapse_1_weight_update

synapse_0 += alpha * synapse_0_weight_update

prev_synapse_0_weight_update = synapse_0_weight_update

prev_synapse_1_weight_update = synapse_1_weight_update

If a slope (derivative) changes direction, it means that it passed OVER the local minimum and needs to go back.

able to increase the size of the hidden layer increases the amount of search space that we converge to in each iteration.

[0,1,1],

[1,0,1],

[1,1,1]])

y

# Feed forward through layers 0, 1, and 2

layer_1 = sigmoid(np.dot(layer_0,synapse_0))

layer_2 = sigmoid(np.dot(layer_1,synapse_1))

# how much did we miss the target value?

layer_2_error = layer_2 - y

if (j% 10000) == 0:

print 'Error after '+str(j)+' iterations:' + str(np.mean(np.abs(layer_2_error)))

# in what direction is the target value?

layer_2_delta = layer_2_error*sigmoid_output_to_derivative(layer_2)

# how much did each l1 value contribute to the l2 error (according to the weights)?

# in what direction is the target l1?

layer_1_delta = layer_1_error * sigmoid_output_to_derivative(layer_1)

synapse_1 -= alpha * (layer_1.T.dot(layer_2_delta))

synapse_0 -= alpha * (layer_0.T.dot(layer_1_delta)) Training

that the best error with 32 nodes is 0.0009 whereas the best error with 4 hidden nodes was only 0.0013.

Even though this is very marginal in this toy problem, this affect plays a huge role when modeling very complex datasets. Part

If you want to be able to create arbitrary architectures based on new academic papers or read and understand sample code for these different architectures, I think that it's a killer exercise.

I worked with neural networks for a couple years before performing this exercise, and it was the best investment of time I've made in the field (and it didn't take long).

5px;}#indJobContent .company_location{font-size: 11px;overflow: hidden;display:block;}#indJobContent.wide .job{display:block;float:left;margin-right: 5px;width: 135px;overflow: hidden}#indeed_widget_wrapper{position: relative;font-family: 'Helvetica Neue',Helvetica,Arial,sans-serif;font-size: 13px;font-weight: normal;line-height: 18px;padding: 10px;height: auto;overflow: hidden;}#indeed_widget_header{font-size:18px;

Now that we have seen how our neural network leverages Gradient Descent, we can improve our network to overcome these weaknesses in the same way that we improved Gradient Descent in Part 3 (the 3 problems and solutions).

As described above, the alpha parameter reduces the size of each iteration's update in the simplest way possible.

At the very last minute, right before we update the weights, we multiply the weight update by alpha (usually between 0 and 1, thus reducing the size of the weight update).

We're going to jump back to our 3 layer neural network from the first post and add in an alpha parameter at the appropriate place.

Then, we're going to run a series of experiments to align all the intuition we developed around alpha with its behavior in live code.

:) Alpha = 10Perhaps you were surprised that an alpha that was greater than 1 achieved the best score after only 10,000 iterations!

This means that in the smaller alpha parameters (less than 10), the network's weights were generally headed in the right direction, they just needed to hurry up and get there!

with an extremely large alpha, we see a textbook example of divergence, with the error increasing instead of decreasing...

This is a more extreme version of Problem 3 where it overcorrectly whildly and ends up very far away from any local minimums.

If a slope (derivative) changes direction, it means that it passed OVER the local minimum and needs to go back.

Being able to increase the size of the hidden layer increases the amount of search space that we converge to in each iteration.

Consider the network and output Notice that the best error with 32 nodes is 0.0009 whereas the best error with 4 hidden nodes was only 0.0013.

Even though this is very marginal in this toy problem, this affect plays a huge role when modeling very complex datasets.

If you want to be able to create arbitrary architectures based on new academic papers or read and understand sample code for these different architectures, I think that it's a killer exercise.

I worked with neural networks for a couple years before performing this exercise, and it was the best investment of time I've made in the field (and it didn't take long).

5px;}#indJobContent .company_location{font-size: 11px;overflow: hidden;display:block;}#indJobContent.wide .job{display:block;float:left;margin-right: 5px;width: 135px;overflow: hidden}#indeed_widget_wrapper{position: relative;font-family: 'Helvetica Neue',Helvetica,Arial,sans-serif;font-size: 13px;font-weight: normal;line-height: 18px;padding: 10px;height: auto;overflow: hidden;}#indeed_widget_header{font-size:18px;

## play Discovering new knowledge

It is able to do this by using a novel form of reinforcement learning, in which AlphaGo Zero becomes its own teacher.

This updated neural network is then recombined with the search algorithm to create a new, stronger version of AlphaGo Zero, and the process begins again.

In each iteration, the performance of the system improves by a small amount, and the quality of the self-play games increases, leading to more and more accurate neural networks and ever stronger versions of AlphaGo Zero.

## On the relevance of the alpha frequency oscillation’s small-world network architecture for cognitive flexibility

A total of n = 55 healthy young participants (24 females) between 18 and 30 years (M = 23.75 ± 0.49) took part in the experiment.

During task execution, participants were seated in a separate room (to prevent any disturbances or distractions) in front of a monitor and a regular keyboard.

The paradigm consisted of two conditions: switching and repetition trials, which were separated into 12 experimental blocks (6 cue- and 6 memory-based blocks, with 198 trials per condition, resulting in a total of 396 trials).

Participants had to respond using their index finger (which was either the right or the left one, based on response-hand mapping and respective experimental version), if the digit was smaller than five, had small font size or was uneven in addition they had to respond using their right index finger, if the digit was larger than five, had a large font size or was even.

In case a participant lost count and failed to apply the correct rule in 3 consecutive trials, the regular cues (i.e., NUM, GER, or SG) replaced the dummy cue in the following 3 trials.

To eliminate the reference potential from the data and to re-reference the data, we applied a current source density (CSD) transformation (4 splines and 10 polynominals)65 which works as a spatial filter66,67, suppresses volume conduction and accentuates electrode sites and makes it easier to identify electrode sites that best reflect relevant neuronal activity.

Through the application of a specific cue- and target-locked baseline, we meet the needs of possible shifts of ERPs, and ensure that neural activity which might be increased due to attentive cues has no impact on the analysis of target-locked data analysis All ERP amplitudes were quantified against this baseline period at the single-subject level.

This choice of electrode positions and time windows was validated using the methods proposed by69: Briefly, the above time intervals were taken and the mean amplitude within the defined search intervals was determined for each of the 60 electrode positions.

t = time, A = (σt √π)−1/2, σt = wavelet duration, and i = √−1.

Mean alpha band power (pooled across electrodes) was analyzed from the entire cue-target (−1300 ms – 0 ms) interval, as well as in an interval from 800 ms to 1500 ms after target presentation, in which strong alpha band activity was evident (refer Fig. 3).

Therefore, only the imaginary part of the coherence spectrum for all possible pairs of nodes was calculated to effectively suppress spurious coherence driven by volume conduction71 For the coherence values, three different individual percentile thresholds of P = 85, P = 90, P = 95 was set for each subject, so only the 15, 10 or 5 percent of each individual’s highest coherence values were included in the analysis.

A binary 60 × 60 adjacency network matrix (because of 60 electrodes being used) was then calculated with 1 representing an un-weighted and undirected connection between any pair of electrodes and 0 representing no connection.

In order to study small world networks, the method by72 was used and applied to each single-subject: Starting from a one-dimensional network, where each node in the network is only connected to its k nearest neighbors on either side, representing a ‘regular’ network with randomness ϐ = 0, a ring lattice with N nodes of mean degree 2k is created.

According to the Watts and Strogatz model, a network has small-world network properties if it demonstrates properties from both lattice networks, with clustered interconnectivity within groups of nodes sharing many nearest neighbors in common (high clustering coefficient, ‘C’), and properties from random networks, represented by a short geodetic distance (average path length, ‘L’) between any two nodes in the network.

real

) and average clustering coefficient (C

real

) were calculated.

rand and C

rand and C

latt were computed.

Small-world values of ω are restricted to the interval −1 to 1 regardless of network size.

## Highlights from our work in 2016

During the games, AlphaGo played a handful of highly inventive winning moves, several of which - including move 37 in game two - were so surprising they overturned hundreds of years of received wisdom, and have since been examined extensively by players of all levels.

Unlike the earlier versions of AlphaGo which learnt how to play the game using thousands of human amateur and professional games, AlphaGo Zero learnt to play the game of Go simply by playing games against itself, starting from completely random play.  In doing so, it surpassed the performance of all previous versions, including those which beat the World Go Champions Lee Sedol and Ke Jie, becoming arguably the strongest Go player of all time.

## AlphaGo Zero

By playing games against itself, AlphaGo Zero surpassed the strength of AlphaGo Lee in three days by winning 100 games to 0, reached the level of AlphaGo Master in 21 days, and exceeded all the old versions in 40 days.&#91;2&#93;

Training artificial intelligence (AI) without datasets derived from human experts has significant implications for the development of AI with superhuman skills because expert data is 'often expensive, unreliable or simply unavailable.'&#91;3&#93;

In December 2017, a generalized version of AlphaGo Zero, named AlphaZero, beat the 3-day version of AlphaGo Zero by winning 60 games to 40, and with 8 hours of training it outperformed AlphaGo Lee on an Elo scale, as well as a top chess program (Stockfish) and a top Shōgi program (Elmo).&#91;6&#93;&#91;7&#93;

According to Hassabis, AlphaGo's algorithms are likely to be of the most benefit to domains that require an intelligent search through an enormous space of possibilities, such as protein folding or accurately simulating chemical reactions.&#91;14&#93;

Gary Marcus, a psychologist at New York University, has cautioned that for all we know, AlphaGo may contain 'implicit knowledge that the programmers have about how to construct machines to play problems like Go' and will need to be tested in other domains before being sure that its base architecture is effective at much more than playing Go.

Jin-seok, who directs the South Korean national Go team, said the Go world has already been imitating the playing styles of previous versions of AlphaGo and creating new ideas from them, and he is hopeful that new ideas will come out from AlphaGo Zero.

On 5 December 2017, DeepMind team released a preprint on arXiv, introducing AlphaZero, a program using generalized AlphaGo Zero's approach, which achieved within 24 hours a superhuman level of play in chess, shogi, and Go, defeating world-champion programs, Stockfish, Elmo, and 3-day version of AlphaGo Zero in each case.&#91;6&#93;

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