# Stephen Smith's Blog

Musings on Machine Learning…

with 5 comments

# Introduction

With sophisticated Neural Networks, you are dealing with a quite complicated nonlinear function. When fitting a high degree polynomial to a few data points, the polynomial can go through all the points, but have such steep slopes that it is useless for predicting points between the training points, we get this same sort of behaviour in Neural Networks. In a way you are training the Neural Network to exactly memorize all the training data exactly rather than figuring out the trends and patterns that you can use to predict other values.

We’ve touched upon this problem in other articles like here and here, but glossed over what we are doing about this problem. In this article we’ll explore what we can do about this in more detail.

One solution is to perhaps gather more training data, however this may be impossible or quite expensive. It also might be that the training data is missing some representative samples. Here we’ll concentrate on what we can do with the algorithm rather than trying to improve the data.

# Interpolation and Extrapolation

Here we refer to generalization as wanting to get answers to data that isn’t in the training data. We refer to overfitting as the case where the model works really well for the training data but doesn’t do nearly as well for anything else.

There are two distinct cases we want to worry about. One is interpolation, this is trying to estimate values where the inputs are surrounded by data in the training set. Extrapolation is the process of trying to predict what happens beyond the training data. Our stock market data is an example of extrapolation. Recognizing handwriting is an example of interpolation (assuming you have a good sample of training data)

Extrapolation tends to be a much harder problem than interpolation, but both a strongly affected by overfitting.

# Early Stopping

What we often do is divide our training data into three groups. The largest of these we call the training data and use for training. Another is the test data which we run after training to see how well the algorithm works on data that hasn’t been seen by training. To help with detecting overfitting we create a third group which we run after a certain number of steps during training. The following screenshot shows the results for the training and validation sets (this is for a Kaggle competition so the test set needs to be submitted to Kaggle to get the answer). Here smaller values are better. Notice that the training data gets better starting at 3209.5 and going down to 712.8 which indicates training is working. However the validation data starts at 3014.3 goes down to the 1160s and then starts increasing. This indicates we are overfitting the data.

The approach here is really simple, let’s just stop once the validation data starts increasing. So let’s just stop at this point and say we’re done. This is actually a pretty simple and effective way to prevent overfitting. As an added bonus this is a rare technique that leads to faster training.

# Penalizing Large Weights

A sign of overfitting is that the slope of our function is high at the points in the training data. Since the slope is approximated by the appropriate weights in our matrix, we would want to keep the weights in our weight matrices low. The way we accomplish this is to add a penalty to the loss function based on the size of the weights.

```     loss = (tf.nn.l2_loss( tf.sub(logits, tf_train_labels))

+ tf.nn.l2_loss(layer1_weights)*beta

+ tf.nn.l2_loss(layer2_weights)*beta

+ tf.nn.l2_loss(layer3_weights)*beta

+ tf.nn.l2_loss(layer4_weights)*beta)```

Here we add the sum of the squares of the weights to our loss function. The factor beta is there to let us scale this value to be in the same magnitude as the main loss function. I’ve found that in some problems making the loss due to the weights about equal to the main loss works quite well. In another problem I found choosing beta so that the weights are 10% of the main loss worked quite well.

I have found that combining this with early stopping works quite well. The weight penalty lets us train longer before we start overfitting, which leads to a better overall result.

# Dropout

One property of the Neural Networks in our brain is that brain cells die, but our brain seems to mostly keep on working. In this sense the brain is far more resilient to damage than a computer. The idea behind dropout is to try to add rules to train the Neural Network to be resilient to Neurons being removed. This means the Neural Network can’t be completely reliant on any given Neuron since it could die (be removed from the model).

The way we accomplish this is we add a dropout activation function at some point:

```            if dropout:

hidden = tf.nn.dropout(hidden, 0.5)```

This activation function will remove 50% of the neurons at this layer and scale up its outputs by a matching amount. This is so the sum stays the same which means you can use the same weights whether dropout is present or not.

The reason for the if statement is that you only want to do dropout during training and not during validation, testing or production.

You would do this on each hidden layer. It’s rather surprising that the Neural Network still works as well as it does with this much dropout.

I find dropout doesn’t always help, but when it does you can combine it with penalizing the weights and then you can train longer before you need to stop during overfitting. This can sometimes help a network find finer details without overfitting.

When you do dropout, you do have to train for a longer time, so if this is too time prohibitive you might not want to use it.

I think it’s a good sign that Neural Networks can exhibit the same resilience to damage that the brain shows. Perhaps a bit of biological evidence that we are on the correct track.

# Summary

These are a few techniques you can use to avoid overfitting your model. I generally use all three so I can train a bit longer without overfitting. If you can get more good training data that can also help quite a bit. Using a simpler model (with fewer hidden nodes) can also help with overfitting, but perhaps not provide as good a functional approximation as the more complicated model. As with all things in computer science you are always trading off complexity, overfitting and performance.

Written by smist08

October 16, 2016 at 6:49 pm

with 9 comments

# Introduction

Well after a long journey through Linux, Python, Python Libraries, the Stock Market, an Introduction to Neural Networks and training Neural Networks we are now ready to look at a complete Python example to predict the stock market.

I placed the full source code listing on my Google Drive here. As described in the previous articles you will need to run this on a Mac or on Linux (could be a virtual image) with Python and TensorFlow installed. You will also need to have the various libraries that are imported at the top of the source file installed or you will get an error when you go to run it. I would suggest getting the source file to play with, Python is very fussy about indentation, so copy/paste from the article may introduce indentation errors caused by the blog formatting.

The Neural Network we are running here is a simple feed forward network with four hidden layers and uses the hyperbolic tangent as the activation function in each case. This is a very simple model so don’t use it to invest with real money. Hopefully this article gives a flavour for how to create and train a Neural Network using TensorFlow. Then in future articles we can discuss the limitation of this model and how to improve it.

# Import Libraries

First we import all the various libraries we will be using, note tensorflow and numpy as being particularly important.

```# Copyright 2016 Stephen Smith

import time```
```import math
import os
from datetime import date
from datetime import timedelta
import numpy as np
import matplotlib.pyplot as plt
import tensorflow as tf
import pandas as pd
import pandas_datareader as pdr
from pandas_datareader import data, wb
from six.moves import cPickle as pickle
from yahoo_finance import Share```

# Get Stock Market Data

Next we get the stock market data. If the file stocks.pickle exists we assume we’ve previously saved this file and use it. Otherwise we get the data from Yahoo Finance using a Web Service call, made via the Pandas DataReader. We only keep the adjusted close column and we fill in any NaN’s with the first value we saw (this really only applies to Visa in this case). The data will all be in a standard Pandas data frame after this.

```# Choose amount of historical data to use NHistData
NHistData = 30
TrainDataSetSize = 3000

# Load the Dow 30 stocks from Yahoo into a Pandas datasheet

dow30 = ['AXP', 'AAPL', 'BA', 'CAT', 'CSCO', 'CVX', 'DD', 'XOM',
'GE', 'GS', 'HD', 'IBM', 'INTC', 'JNJ', 'KO', 'JPM',
'MCD', 'MMM', 'MRK', 'MSFT', 'NKE', 'PFE', 'PG',
'TRV', 'UNH', 'UTX', 'VZ', 'V', 'WMT', 'DIS']

num_stocks = len(dow30)

trainData = None
loadNew = False

# If stocks.pickle exists then this contains saved stock data, so use this,
# else use the Pandas DataReader to get the stock data and then pickle it.
stock_filename = 'stocks.pickle'
if os.path.exists(stock_filename):
try:
with open(stock_filename, 'rb') as f:
trainData = pickle.load(f)
except Exception as e:
print('Unable to process data from', stock_filename, ':', e)
raise
print('%s already present - Skipping requesting/pickling.' % stock_filename)
else:
# Get the historical data. Make the date range quite a bit bigger than
# TrainDataSetSize since there are no quotes for weekends and holidays. This
# ensures we have enough data.

f = pdr.data.DataReader(dow30, 'yahoo',
date.today()-timedelta(days=TrainDataSetSize*2+5), date.today())
cleanData = f.ix['Adj Close']
trainData = pd.DataFrame(cleanData)
trainData.fillna(method='backfill', inplace=True)
loadNew = True
print('Pickling %s.' % stock_filename)
try:
with open(stock_filename, 'wb') as f:
pickle.dump(trainData, f, pickle.HIGHEST_PROTOCOL)
except Exception as e:
print('Unable to save data to', stock_filename, ':', e)```

# Normalize the Data

We then normalize the data and remember the factor we used so we can de-normalize the results at the end.

```# Normalize the data by dividing each price by the first price for a stock.
# This way all the prices start together at 1.
# Remember the normalizing factors so we can go back to real stock prices
# for our final predictions.
factors = np.ndarray(shape=( num_stocks ), dtype=np.float32)
i = 0
for symbol in dow30:
factors[i] = trainData[symbol][0]
trainData[symbol] = trainData[symbol]/trainData[symbol][0]
i = i + 1```

# Re-arrange the Data for TensorFlow

Now we need to build up our training data, test data and validation data. We need to format this as input arrays for the Neural Network. Looking at this code, I think true Python programmers will accuse me of being a C programmer (which I am), since I do this all with loops. I’m sure a more experience Python programmer could accomplish this quicker with more array operations. This part of the code is quite slow so we pickle it, so if we re-run with the saved stock data, we can also use saved training data.

```# Configure how much of the data to use for training, testing and validation.

usableData = len(trainData.index) - NHistData + 1
#numTrainData =  int(0.6 * usableData)
#numValidData =  int(0.2 * usableData
#numTestData = usableData - numTrainData - numValidData - 1
numTrainData = usableData - 1
numValidData = 0
numTestData = 0

train_dataset = np.ndarray(shape=(numTrainData - 1,
num_stocks * NHistData), dtype=np.float32)
train_labels = np.ndarray(shape=(numTrainData - 1, num_stocks),
dtype=np.float32)
valid_dataset = np.ndarray(shape=(max(0, numValidData - 1),
num_stocks * NHistData), dtype=np.float32)
valid_labels = np.ndarray(shape=(max(0, numValidData - 1),
num_stocks), dtype=np.float32)
test_dataset = np.ndarray(shape=(max(0, numTestData - 1),
num_stocks * NHistData), dtype=np.float32)
test_labels = np.ndarray(shape=(max(0, numTestData - 1),
num_stocks), dtype=np.float32)
final_row = np.ndarray(shape=(1, num_stocks * NHistData),
dtype=np.float32)
final_row_prices = np.ndarray(shape=(1, num_stocks * NHistData),
dtype=np.float32)

# Build the taining datasets in the correct format with the matching labels.
# So if calculate based on last 30 stock prices then the desired
# result is the 31st. So note that the first 29 data points can't be used.
# Rather than use the stock price, use the pricing deltas.
pickle_file = "traindata.pickle"
if loadNew == True or not os.path.exists(pickle_file):
for i in range(1, numTrainData):
for j in range(num_stocks):
for k in range(NHistData):
train_dataset[i-1][j * NHistData + k] = (trainData[dow30[j]][i + k]
- trainData[dow30[j]][i + k - 1])
train_labels[i-1][j] = (trainData[dow30[j]][i + NHistData]
- trainData[dow30[j]][i + NHistData - 1])

for i in range(1, numValidData):
for j in range(num_stocks):
for k in range(NHistData):
valid_dataset[i-1][j * NHistData + k] = (trainData[dow30[j]][i + k + numTrainData]
- trainData[dow30[j]][i + k + numTrainData - 1])
valid_labels[i-1][j] = (trainData[dow30[j]][i + NHistData + numTrainData]
- trainData[dow30[j]][i + NHistData + numTrainData - 1])

for i in range(1, numTestData):
for j in range(num_stocks):
for k in range(NHistData):
test_dataset[i-1][j * NHistData + k] = (trainData[dow30[j]][i + k + numTrainData + numValidData]
- trainData[dow30[j]][i + k + numTrainData + numValidData - 1])
test_labels[i-1][j] = (trainData[dow30[j]][i + NHistData + numTrainData + numValidData]
- trainData[dow30[j]][i + NHistData + numTrainData + numValidData - 1])

try:
f = open(pickle_file, 'wb')
save = {
'train_dataset': train_dataset,
'train_labels': train_labels,
'valid_dataset': valid_dataset,
'valid_labels': valid_labels,
'test_dataset': test_dataset,
'test_labels': test_labels,
}
pickle.dump(save, f, pickle.HIGHEST_PROTOCOL)
f.close()
except Exception as e:
print('Unable to save data to', pickle_file, ':', e)
raise

else:
with open(pickle_file, 'rb') as f:
save = pickle.load(f)
train_dataset = save['train_dataset']
train_labels = save['train_labels']
valid_dataset = save['valid_dataset']
valid_labels = save['valid_labels']
test_dataset = save['test_dataset']
test_labels = save['test_labels']
del save  # hint to help gc free up memory

for j in range(num_stocks):
for k in range(NHistData):
final_row_prices[0][j * NHistData + k] = trainData[dow30[j]][k + len(trainData.index - NHistData]
final_row[0][j * NHistData + k] = (trainData[dow30[j]][k + len(trainData.index) - NHistData]
- trainData[dow30[j]][k + len(trainData.index) - NHistData - 1])

print('Training set', train_dataset.shape, train_labels.shape)
print('Validation set', valid_dataset.shape, valid_labels.shape)
print('Test set', test_dataset.shape, test_labels.shape)```

# Accuracy

We now setup an accuracy function that is only used to report how we are doing during training. This isn’t used by the training algorithm. It roughly shows what percentage of predictions are within some tolerance.

```# This accuracy function is used for reporting progress during training, it isn't actually
# used for training.
def accuracy(predictions, labels):
err = np.sum( np.isclose(predictions, labels, 0.0, 0.005) ) / (predictions.shape[0] * predictions.shape[1])
return (100.0 * err)```

# TensorFlow Variables

We now start setting up TensorFlow by creating our graph and defining our datasets and variables.

```batch_size = 4
num_hidden = 16
num_labels = num_stocks

graph = tf.Graph()

# input is 30 days of dow 30 prices normalized to be between 0 and 1.
# output is 30 values for normalized next day price change of dow stocks
# use a 4 level neural network to compute this.

with graph.as_default():

# Input data.
tf_train_dataset = tf.placeholder(
tf.float32, shape=(batch_size, num_stocks * NHistData))
tf_train_labels = tf.placeholder(tf.float32, shape=(batch_size, num_labels))
tf_valid_dataset = tf.constant(valid_dataset)
tf_test_dataset = tf.constant(test_dataset)
tf_final_dataset = tf.constant(final_row)

# Variables.
layer1_weights = tf.Variable(tf.truncated_normal(
[NHistData * num_stocks, num_hidden], stddev=0.05))
layer1_biases = tf.Variable(tf.zeros([num_hidden]))
layer2_weights = tf.Variable(tf.truncated_normal(
[num_hidden, num_hidden], stddev=0.05))
layer2_biases = tf.Variable(tf.constant(1.0, shape=[num_hidden]))
layer3_weights = tf.Variable(tf.truncated_normal(
[num_hidden, num_hidden], stddev=0.05))
layer3_biases = tf.Variable(tf.constant(1.0, shape=[num_hidden]))
layer4_weights = tf.Variable(tf.truncated_normal(
[num_hidden, num_labels], stddev=0.05))
layer4_biases = tf.Variable(tf.constant(1.0, shape=[num_labels]))```

# TensorFlow Model

We now define our Neural Network model. Hyperbolic Tangent is our activation function and rest is matrix algebra as we described in previous articles.

```  # Model.
def model(data):
hidden = tf.tanh(tf.matmul(data, layer1_weights) + layer1_biases)
hidden = tf.tanh(tf.matmul(hidden, layer2_weights) + layer2_biases)
hidden = tf.tanh(tf.matmul(hidden, layer3_weights) + layer3_biases)
return tf.matmul(hidden, layer4_weights) + layer4_biases```

# Training Model

Now we setup the training model and the optimizer to use, namely gradient descent. We also define what are the correct answers to compare against.

```  # Training computation.
logits = model(tf_train_dataset)
loss = tf.nn.l2_loss( tf.sub(logits, tf_train_labels))

# Optimizer.
optimizer = tf.train.GradientDescentOptimizer(0.01).minimize(loss)
# Predictions for the training, validation, and test data.
train_prediction = logits
valid_prediction = model(tf_valid_dataset)
test_prediction = model(tf_test_dataset)
next_prices = model(tf_final_dataset)```

# Run the Model

So far we have setup TensorFlow ready to go, but we haven’t calculated anything. This next set of code executes the training run. It will use the data we’ve provided in the configured batch size to train our network while printing out some intermediate information.

```num_steps = 2052

with tf.Session(graph=graph) as session:
tf.initialize_all_variables().run()
print('Initialized')
for step in range(num_steps):
offset = (step * batch_size) % (train_labels.shape[0] - batch_size)
batch_data = train_dataset[offset:(offset + batch_size), :]
batch_labels = train_labels[offset:(offset + batch_size), :]
feed_dict = {tf_train_dataset : batch_data, tf_train_labels : batch_labels}
_, l, predictions = session.run(
[optimizer, loss, train_prediction], feed_dict=feed_dict)
acc = accuracy(predictions, batch_labels)
if (step % 100 == 0):
print('Minibatch loss at step %d: %f' % (step, l))
print('Minibatch accuracy: %.1f%%' % acc)
if numValidData > 0:
print('Validation accuracy: %.1f%%' % accuracy(
valid_prediction.eval(), valid_labels))
if numTestData > 0:
print('Test accuracy: %.1f%%' % accuracy(test_prediction.eval(), test_labels))```

# Make a Prediction

The final bit of code uses our trained model to make a prediction based on the last set of data we have (where we don’t know the right answer). If you get fresh stock market data for today, then the prediction will be for tomorrow’s price changes. If you run this late enough that Yahoo has updated its prices for the day, then you will get some real errors for comparison. Note that Yahoo is very slow and erratic about doing this, so be careful when reading this table.

```predictions = next_prices.eval() * factors
print("Stock    Last Close  Predict Chg   Predict Next      Current     Current Chg       Error")
i = 0
for x in dow30:
yhfeed = Share(x)
currentPrice = float(yhfeed.get_price())
print( "%-6s  %9.2f  %9.2f       %9.2f       %9.2f     %9.2f     %9.2f" % (x,
final_row_prices[0][i * NHistData + NHistData - 1] * factors[i],
predictions[0][i],
final_row_prices[0][i * NHistData + NHistData - 1] * factors[i] + predictions[0][i],
currentPrice,
currentPrice - final_row_prices[0][i * NHistData + NHistData - 1] * factors[i],
abs(predictions[0][i] - (currentPrice - final_row_prices[0][i * NHistData + NHistData - 1] * factors[i]))) )
i = i + 1```

# Results

Below is a screenshot of one run predicting the stock changes for Sept. 22. Basically it didn’t do very well. We’ll talk about why and what to do about this in a future article. As you can see it is very conservative in its predictions.

# Summary

This article shows the code for training and executing a very simple Neural Network using TensorFlow. Definitely don’t bet on the stock market based on this model, it is very simple at this point. We still need to add a number of elements to start making this into a useful model which we’ll look at in future articles.

Written by smist08

September 23, 2016 at 4:17 pm

with 3 comments

# Introduction

Last time we looked at the matrix equation that would be our Neural Network which is:

Output of Layer = ActivationFunction( A x (Input of Layer) + b )

We also specified that our input vector would be 900 elements large (the 30 Dow stocks times the last 30 price changes) and the output vector would be 30 elements (then next price change for each of the Dow 30 stocks). This means that if we have just one hidden layer of say 100 Neurons then we need a 900×100 matrix and a 100×30 matrix plus a 100 element bias vector and a 30 element bias vector. This means we need 900×100 + 100×30 + 100 + 30 = 93,130 values. Where do these all come from? In this article we’ll look at where we get these.

# Training

What we want to do is use some sort of known or historical data to train the Neural Network. When Neural Networks were first proposed, Computer Scientists tuned these by hand which resulted in taking a long time to get a very small Neural Network that didn’t work well. Later on many methods were developed to calculate these from databases of known cases, however until recently these databases were too small to be effective and led to extreme over-fitting. With the advent of big data, shared cloud resources and automated data collection, a large number of high quality extremely large databases are available to train Neural Networks for well know problems like hand writing recognition or shape identification. Notice that in the introduction to find 93,130 values requires far more than 93,130 bits of data, since this will lead to over-fitting (which we’ll talk a lot about in a future article).

If you remember back to basic statistics and linear regression, we found the best fit for a straight line through a number of data points by minimizing the squares of the distance from the line to each data point. This is why its often called least squares regression. Basically we are formulating the problem as an optimization problem where we are trying to minimize an error function. The linear regression problem is then easily solvable by first year linear algebra. For the Neural Network case it’s a little bit more complicated, but the basic idea is the same.

To train our Neural Network we will use historical data where we provide 30 days of price changes for the Dow 30 stocks and then we know the next change so we can provide the error for our error function. To define an error function, we are going to start by just using the square of the difference, so basically just doing least square minimization just like least squares regression. In TensorFlow we can define our loss function as:

loss = tf.nn.l2_loss( tf.sub(logits, tf_train_labels))

Now that we have the data and an error function how do we go about training our network. First we start by seeding the matrix weights with normally distributed random numbers. TensorFlow provides some help here

layer1_weights = tf.Variable(tf.truncated_normal(
[NHistData * num_stocks, num_hidden], stddev=0.05))

to define our matrix and initialize it with normalized random numbers.

There are a number of optimization algorithms that can be used to solve this problem. The one we are going to use is called Gradient Descent which is a form of Back Propagation. The key property of back propagation algorithms is that they can be applied to Neural Networks with multiple hidden layers. The basic idea is that you take the partial derivative of the loss function with respect to each weight. This gives you a gradient with respect to each weight and then based on whether the gradient is positive or negative you can increase or decrease the weight by a little bit. This little bit is the learning rate which is a parameter to the algorithm (or can be changed dynamically by another algorithm). You then run the training data through this algorithm and hopefully observe your error function decreasing as you go.

This is then the basis of training your network. Once you have the weights you can calculate all the values as you like.

Example of Gradient Descent where it could lead to 2 different minimums

# Testing

A big danger here is that you are overfitting. You reduce the error function to next than nothing and the network works well for all your training data. You then try it on something else and it produces very bad results. This is similar to fitting a 10th degree polynomial through 11 data points. It fits all those points exactly, but has no predictive value outside of those exact points.

A common technique is to divide the training data into three buckets: actual training data, testing data and final validation data. You use the training data to train and as you train, you use the testing data to see how you are doing. Then when everything is finished you use the final validation data to do a final test (where the training process has never seen this data). This then gives you an idea of how well the network will fare out in the real world. We will make the size of these three buckets configurable.

# Local Versus Global Minimums

During the training process a few different things could happen. The solution could diverge, the error could just keep getting larger and larger. The solution could get stuck in a valley and just orbit a minimum value without converging to it. The solution could converge, but to a local minimum rather than the global minimum. These are all things that need to be watched out for.

Since the initial values are random, re-running the training can lead to quite different solutions. For some problems you want to train repeatedly to get the best solution. Or perhaps compare different optimization algorithms to see which gives the best result. Another idea is to use a combination of algorithms, perhaps start with one that gets into the correct neighborhood, and then another that can zero in on it.

There are quite a few tricks to get out of local minimums and to escape valleys using various random numbers. One is to change the learning rate to occasionally take a bigger jump. Others are to try some random perturbations to see if you can start converging to another solution.

# Batch Versus Single

A lot of time we process the training data in batches where we take the average of the partial derivatives to adjust the weights. This can greatly speed up training and avoids the problem of one bad data point sending us in the wrong direction. Again the batch size is a meta-parameter to the training algorithm that we can tune to get the best results.

# Summary

This was a really quick introduction to training a Neural Network. There are many optimization algorithms that can be applied to solve this problem, but we are starting with gradient descent. A number of the algorithms chosen, are done so to facilitate using a GPU or distributed network to parallelize and hence speed up the training process.

Next time we’ll start looking at the TensorFlow code for a simple Neural Network model, then we will start enhancing it to get better results.

Written by smist08

September 21, 2016 at 11:32 pm