Lab 7: Neural Nets

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This lab is due by Monday, November 23 at 10:00pm.   
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This lab is due by Wednesday, November 2 at 10:00pm.   
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To work on this lab, you will need to get the code:
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To work on this lab, you will need to get the code, much like you did for the previous labs.  You can:
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* You can view the files '''and change log''' at: http://web.mit.edu/6.034/www/labs/lab6/
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* Use Git on your computer: <tt>git clone username@athena.dialup.mit.edu:/mit/6.034/www/labs/lab6</tt>
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* Use Git on Athena: <tt>git clone /mit/6.034/www/labs/lab6</tt>
* Download it as a ZIP file: http://web.mit.edu/6.034/www/labs/lab6/lab6.zip
* Download it as a ZIP file: http://web.mit.edu/6.034/www/labs/lab6/lab6.zip
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* Or, on Athena, <tt>add 6.034</tt> and copy it from <tt>/mit/6.034/www/labs/lab6/</tt>.
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* View the files individually: http://web.mit.edu/6.034/www/labs/lab6/
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The online tester is now available! <!--Online tests will be made available by the end of Tuesday, November 10.  In the meantime, the local tester provides thorough unit tests for each section of the lab.-->
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<!--The online tester is now available! Online tests will be made available by the end of Tuesday, November 10.  In the meantime, the local tester provides thorough unit tests for each section of the lab.-->
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Your answers for this lab belong in the main file <tt>lab6.py</tt>.  
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Your answers for this lab belong in the main file <tt>lab6.py</tt>.
= Problems: Neural Nets =
= Problems: Neural Nets =

Revision as of 23:33, 12 October 2016

Contents


This lab is due by Wednesday, November 2 at 10:00pm.

To work on this lab, you will need to get the code, much like you did for the previous labs. You can:


Your answers for this lab belong in the main file lab6.py.

Problems: Neural Nets

Neural Net Subroutines

Wiring a neural net

A neural net is composed of individual neurons, which generally take this form:


Image:Lab6_SimpleNeuron.png


We form the net by combining the neurons into a structure, such as the example shown below.


Image:Lab6_SimpleNeuralNet.png‎


In a neural net with two inputs x and y, each input-layer neuron draws a line and shades one side of it, satisfying the equation ax + by >= T. The remaining neurons in the later layers of the neural net perform logic functions on the shadings.

Each of the following pictures can be produced by a neural net with two inputs x and y. For each one, determine the minimum number of neurons necessary to produce the picture. Express your answer as a list indicating the number of nodes per layer.

As an example, the neural net shown above would be represented by the list [3, 2, 3, 1].

Image:Neural_net_pictures_2015.jpg

The last picture (nn_grid) is optional. To test it locally, change the boolean flag TEST_NN_GRID to True.

Note: If you downloaded the lab files in the first two hours after it was released, there was a mistake in the answer for nn_grid, which has since been corrected.

Neural net equations (reference only)

For reference, here are the fundamental equations that define a neural net:

Image:Lab6_NN_Eqns.png

Helper functions

First, you'll code helper functions for the neural nets. The stairstep and sigmoid functions are threshold functions; each neuron in a neural net uses a threshold function to determine whether its input stimulation is large enough for it to emit a non-zero output. Fill in each of the functions below.

stairstep: Computes the output of the stairstep function using the given threshold (T)

def stairstep(x, threshold=0):

sigmoid: Computes the output of the sigmoid function using the given steepness (S) and midpoint (M)

def sigmoid(x, steepness=1, midpoint=0):

For your convenience, the constant e is defined in lab6.py.


The accuracy function is used when training the neural net with back propagation. It measures the performance of the neural net as a function of its desired output and its actual output (given some set of inputs). Note that the accuracy function is symmetric -- that is, it doesn't care which argument is the desired output and which is the actual output.

accuracy: Computes accuracy using desired_output and actual_output. If the neurons in the network are using the stairstep threshold function, the accuracy will range from -0.5 to 0

def accuracy(desired_output, actual_output):

Forward propagation

Next, you'll code forward propagation, which takes in a dictionary of inputs and computes the output of every neuron in a neural net. As part of coding forward propagation, you should understand how a single neuron computes its output as a function of its input: each input into the neuron is multiplied by the weight on the wire, the weighted inputs are summed together, and the sum is passed through a specified threshold function to produce the output.

To compute the output of each neuron in a neural net, iterate over each neuron in the network in order, starting from the input neurons and working toward the output neuron. (Hint: The function net.topological_sort() may be useful; see the API for details). The algorithm is called forward propagation because the outputs you calculate for earlier neurons will be propagated forward through the network and used to calculate outputs for later neurons.


Implement the method forward_prop:

def forward_prop(net, input_values, threshold_fn=stairstep):

Here, net is a neural network, input_values is a dictionary mapping input variables to their values, and threshold_fn is a function that each neuron will use to decide what value to output. This function should return a tuple containing

  1. The overall output value of the network, i.e. the output value associated with the output neuron of the network.
  2. A dictionary mapping neurons to their immediate outputs.

The dictionary of outputs is permitted to contain extra keys (for example, the input values). The function should not modify the neural net in any way.

Backward propagation

Backward propagation is the process of training a neural network using a particular training point to modify the weights of the network, with the goal of improving the network's performance.

To perform back propagation on a given training point, or set of inputs:

  1. Use forward propagation (with the sigmoid threshold function) to compute the output of each neuron in the network.
  2. Compute the update coefficient delta_B for each neuron in the network, starting from the output neuron and working backward toward the input neurons. (The function net.topological_sort() can be useful here; see the API for details).
  3. Use the update coefficients delta_B to compute new weights for the network.
  4. Update all of the weights in the network.


You have already coded the forward_propagation routine. To complete the definition of back propagation, you'll define a helper function calculate_deltas for computing the update coefficients delta_B of each neuron in the network, and a function update_weights that retrieves the list of update coefficients using calculate_deltas, then modifies the weights of the network accordingly.

Implement calculate_deltas to return a dictionary mapping neurons to update coefficients (delta_B values):

def calculate_deltas(net, input_values, desired_output):


Next, use calculate_deltas to implement update_weights, which performs a single step of back propagation. The function should compute delta_B values and weight updates for entire neural net, then update all weights. The function update_weights should return the modified neural net with appropriately updated weights. (Hint: You can update the weight of a Wire instance by setting its wire.weight to a new value.)

def update_weights(net, input_values, desired_output, r=1):


Now you're ready to complete the back_prop function, which continues to update weights in the neural net until the accuracy surpasses the accuracy threshold. back_prop should return a tuple containing:

  1. The modified neural net, with trained weights
  2. The number of iterations (that is, the number of weight updates)
def back_prop(net, input_values, desired_output, r=1, accuracy_threshold=-.001):


If your code is failing the back_prop tests but passing everything else:

  • Make sure you have the current version of tests.py and neural_net_api.py, which were updated on 11/7.
  • Make sure you're using out_A in the weight update formula (r * out_A * delta_B), not out_B.

Training a Neural Net

Train a neural net on a dataset

[TODO]


In practice, we would want to use multiple training points to train a neural net, not just one. There are many possible implementations -- for instance, you could put all the training points into a queue and perform back propagation with each point in turn. Alternatively, you could use a multidimensional accuracy function and try to train with multiple training points simultaneously.

Here are some examples of datasets that you could use to train a 2-input neural net:

1. The letter L

4 + -
3 + - 
2 + -
1 + - - - -
0 - + + + +
  0 1 2 3 4

2. This moat-like shape:

4 - - - - -
3 -       - 
2 -   +   -
1 -       -
0 - - - - -
  0 1 2 3 4

3. This patchy shape:

4 - -   + +
3 - -   + +
2        
1 + +   - -
0 + +   - -
  0 1 2 3 4

It should be possible to train a 6-neuron neural net to classify any one of the three shapes.

For some unspecified amount of extra credit, extend your code and/or the API to train a neural net on multiple inputs. The steps include:

  • Wire up a neural net (see nn_problems.py for some examples)
  • Encode the training data (you can use one of the example datasets above, a quiz problem, or make up your own)
  • Develop an algorithm (e.g. extend your back_prop function) for training your neural net on multiple training points
  • Initialize your neural net with random weights
  • Train the neural net and see if it works!

Survey

Please answer these questions at the bottom of your lab6.py file:

  • NAME: What is your name? (string)
  • COLLABORATORS: Other than 6.034 staff, whom did you work with on this lab? (string, or empty string if you worked alone)
  • HOW_MANY_HOURS_THIS_LAB_TOOK: Approximately how many hours did you spend on this lab? (number or string)
  • WHAT_I_FOUND_INTERESTING: Which parts of this lab, if any, did you find interesting? (string)
  • WHAT_I_FOUND_BORING: Which parts of this lab, if any, did you find boring or tedious? (string)
  • (optional) SUGGESTIONS: What specific changes would you recommend, if any, to improve this lab for future years? (string)


(We'd ask which parts you find confusing, but if you're confused you should really ask a TA.)

When you're done, run the online tester to submit your code.

API

Neural Nets

The file neural_net_api.py defines the Wire and NeuralNet classes, described below.

NeuralNet

A neural net is represented as a directed graph whose edges are Wires and nodes can be neurons, inputs, or the output OUT.

A NeuralNet has the following attributes:

  • net.inputs, a list of named inputs to the network. An input can be either variable or constant: Variable inputs are represented by strings denoting their names, while constant inputs are represented as numbers (eg -1).
  • net.neurons, a list of neurons. Neurons are represented as strings denoting their names.
  • net.wires, a list of the wires (edges) that connect the nodes in the network. Each wire is a Wire object (see below.)

In this lab, inputs are supplied to neural nets in the form of a dictionary input_values that associates each named (variable) input with an input value.


The constant string NeuralNet.OUT represents the final output of a neural net. It is not a neuron, but it is connected to the output neuron by a wire with weight 1.


You can retrieve the nodes (neurons, inputs, or NeuralNet.OUT) in a network:

  • net.get_incoming_neighbors(node). Return a list of the nodes which are connected as inputs to node
  • net.get_outgoing_neighbors(node). Return a list of the nodes to which node sends its output.
  • net.get_output_neuron(). Return the output neuron of the network, which is the neuron that leads to NeuralNet.OUT. (In this lab, each neural net has exactly one output neuron.)
  • net.topological_sort(). Return a sorted list of all the neurons in the network. The list is "topologically" sorted, which means that each neuron appears in the list after all the neurons that provide its inputs. Thus, the input layer neurons are first, the output neuron is last, etc.


You can also retrieve the wires:

  • net.get_wires(startNode=None, endNode=None). Return a list of all the wires in the network. If startNode or endNode are provided, returns only wires that start/end at particular nodes. Note that there is a wire leading from the output neuron to NeuralNet.OUT whose weight should always be 1.
  • net.get_incoming_wires(node). Return a list of wires that feed in to the given node.
  • net.get_outgoing_wires(node). Return a list of wires that the node feeds into. (That is, wires that lead out of the node.)


Finally, you can query the parts of the network:

  • net.has_incoming_neuron(node). Returns True if the node has at least one incoming neuron, otherwise False. (Note: this method was added on 11/7)
  • net.is_output_neuron(neuron). Return True if the neuron is the final, output neuron in the network, otherwise False.
  • net.is_connected(startNode, endNode). Return True if there is a wire from startNode to endNode in the network, otherwise False.


Wire

A Wire is represented as a weighted, directed edge in a graph. A wire can connect an input to a neuron, a neuron to a neuron, or a neuron to NeuralNet.OUT. A wire's attributes are:

  • wire.startNode, the input or neuron at which the wire starts. An input can be either a string or a number (eg -1).
  • wire.endNode, the neuron (or NeuralNet.OUT) at which the wire ends.
  • wire.weight, the weight on the wire (a float or int).
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