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Linear Regression Lab

This lab covers a common supervised learning pipeline, using a subset of the Million Song Dataset from the UCI Machine Learning Repository. Our goal is to train a linear regression model to predict the release year of a song given a set of audio features.

This lab will cover:

• Part 1: Read and parse the initial dataset

  • Visualization 1: Features
  • Visualization 2: Shifting labels

• Part 2: Create and evaluate a baseline model

  • Visualization 3: Predicted vs. actual

• Part 3: Train (via gradient descent) and evaluate a linear regression model

  • Visualization 4: Training error

• Part 4: Train using SparkML and tune hyperparameters via grid search

  • Visualization 5: Best model's predictions
  • Visualization 6: Hyperparameter heat map

• Part 5: Add interactions between features

Note that, for reference, you can look up the details of:

• the relevant Spark methods in Spark's RDD Python API and Spark's DataFrame Python API
• the relevant NumPy methods in the NumPy Reference

labVersion = 'cs120x-lab2-1.0.5'

Part 1: Read and parse the initial dataset

(1a) Load and check the data

The raw data is currently stored in text file. We will start by storing this raw data in as a DataFrame, with each element of the DataFrame representing a data point as a comma-delimited string. Each string starts with the label (a year) followed by numerical audio features. Use the DataFrame count method to check how many data points we have. Then use the take method to create and print out a list of the first 5 data points in their initial string format.

# load testing library
from databricks_test_helper import Test
import os.path
file_name = os.path.join('databricks-datasets', 'cs190', 'data-001', 'millionsong.txt')

raw_data_df = sqlContext.read.load(file_name, 'text')

# TODO: Replace <FILL IN> with appropriate code
num_points =  raw_data_df.count()
print num_points
sample_points =  raw_data_df.take(5)
print sample_points

# TEST Load and check the data (1a)
Test.assertEquals(num_points, 6724, 'incorrect value for num_points')
Test.assertEquals(len(sample_points), 5, 'incorrect length for sample_points')

(1b) Using LabeledPoint

In MLlib, labeled training instances are stored using the LabeledPoint object. Write the parse_points function that takes, as input, a DataFrame of comma-separated strings. We'll pass it the raw_data_df DataFrame.

It should parse each row in the DataFrame into individual elements, using Spark's select and split methods.

For example, split "2001.0,0.884,0.610,0.600,0.474,0.247,0.357,0.344,0.33,0.600,0.425,0.60,0.419" into ['2001.0', '0.884', '0.610', '0.600', '0.474', '0.247', '0.357', '0.344', '0.33', '0.600', '0.425', '0.60', '0.419'].

The first value in the resulting list (2001.0 in the example, above) is the label. The remaining values (0.884, 0.610, etc., in the example) are the features.

After splitting each row, map it to a LabeledPoint. You'll have to step down to an RDD (using .rdd) or use a DataFrame user-defined function to convert to the LabeledPoint object. (See Hint, below.) If you step down to an RDD, you'll have to use toDF() to convert back to a DataFrame.

Use this new parse_points function to parse raw_data_df. Then print out the features and label for the first training point, using the features and label attributes. Finally, calculate the number of features for this dataset.

Hint: Running Arbitrary Lambdas on a DataFrame

To solve this problem, you need a way to run your parse_points function on a DataFrame. There are two ways to do this, which we will illustrate with an extremely simple example.

Suppose you have a DataFrame consisting of a first name and a last name, and you want to add a unique SHA-256 hash to each row.

df = sqlContext.createDataFrame([("John", "Smith"), ("Ravi", "Singh"), ("Julia", "Jones")], ("first_name", "last_name"))

Here's a simple function to calculate such a hash, using Python's built-in hashlib library:

def make_hash(first_name, last_name):
    import hashlib
    m = hashlib.sha256()
    # Join the first name and last name by a blank and hash the resulting
    # string.
    full_name = ' '.join((first_name, last_name))
    m.update(full_name)
    return m.hexdigest()

Okay, that's great. But, how do we use it on our DataFrame? We can use a UDF:

from pyspark.sql.functions import udf
u_make_hash = udf(make_hash)
df2 = df.select(df['*'], u_make_hash(df['first_name'], df['last_name']))
# could run df2.show() here to prove it works

Or we can step down to an RDD, use a lambda to call make_hash and have the lambda return a Row object, which Spark can use to "infer" a new DataFrame.

from pyspark.sql import Row
def make_hash_from_row(row):
    hash = make_hash(row[0], row[1])
    return Row(first_name=row[0], last_name=row[1], hash=hash)

df2 = (df.rdd
         .map(lambda row: make_hash_from_row(row))
         .toDF())

These methods are roughly equivalent. You'll need to do something similar to convert your raw_data_df DataFrame into a new DataFrame of LabeledPoint objects.

from pyspark.mllib.regression import LabeledPoint
import numpy as np

# Here is a sample raw data point:
# '2001.0,0.884,0.610,0.600,0.474,0.247,0.357,0.344,0.33,0.600,0.425,0.60,0.419'
# In this raw data point, 2001.0 is the label, and the remaining values are features

# TODO: Replace <FILL IN> with appropriate code
from pyspark.sql import functions as sql_functions

def parse_points(df):
    """Converts a DataFrame of comma separated unicode strings into a DataFrame of `LabeledPoints`.

    Args:
        df: DataFrame where each row is a comma separated unicode string. The first element in the string
            is the label and the remaining elements are the features.

    Returns:
        DataFrame: Each row is converted into a `LabeledPoint`, which consists of a label and
            features. To convert an RDD to a DataFrame, simply call toDF().
    """
    token = df.value.split(',')
    label = float(token[0])
    features = token[1:]
    return LabeledPoint(label,features)

parsed_points_df = raw_data_df.map(lambda x: parse_points(x)).toDF()
first_point_features = parsed_points_df.first().features
first_point_label = parsed_points_df.first().label
print first_point_features, first_point_label

d = len(first_point_features)
print d

# TEST Using LabeledPoint (1b)
Test.assertTrue(isinstance(first_point_label, float), 'label must be a float')
expectedX0 = [0.8841,0.6105,0.6005,0.4747,0.2472,0.3573,0.3441,0.3396,0.6009,0.4257,0.6049,0.4192]
Test.assertTrue(np.allclose(expectedX0, first_point_features, 1e-4, 1e-4),
                'incorrect features for firstPointFeatures')
Test.assertTrue(np.allclose(2001.0, first_point_label), 'incorrect label for firstPointLabel')
Test.assertTrue(d == 12, 'incorrect number of features')

Visualization 1: Features

First we will load and setup the visualization library. Then we will look at the raw features for 50 data points by generating a heatmap that visualizes each feature on a grey-scale and shows the variation of each feature across the 50 sample data points. The features are all between 0 and 1, with values closer to 1 represented via darker shades of grey.

import matplotlib.pyplot as plt
import matplotlib.cm as cm

# takeSample(withReplacement, num, [seed]) randomly selects num elements from the dataset with/without replacement, and has an
# optional seed parameter that one can set for reproducible results

data_values = (parsed_points_df
               .rdd
               .map(lambda lp: lp.features.toArray())
               .takeSample(False, 50, 47))

# You can uncomment the line below to see randomly selected features.  These will be randomly
# selected each time you run the cell because there is no set seed.  Note that you should run
# this cell with the line commented out when answering the lab quiz questions.
# data_values = (parsedPointsDF
#                .rdd
#                .map(lambda lp: lp.features.toArray())
#                .takeSample(False, 50))

def prepare_plot(xticks, yticks, figsize=(10.5, 6), hideLabels=False, gridColor='#999999',
                 gridWidth=1.0):
    """Template for generating the plot layout."""
    plt.close()
    fig, ax = plt.subplots(figsize=figsize, facecolor='white', edgecolor='white')
    ax.axes.tick_params(labelcolor='#999999', labelsize='10')
    for axis, ticks in [(ax.get_xaxis(), xticks), (ax.get_yaxis(), yticks)]:
        axis.set_ticks_position('none')
        axis.set_ticks(ticks)
        axis.label.set_color('#999999')
        if hideLabels: axis.set_ticklabels([])
    plt.grid(color=gridColor, linewidth=gridWidth, linestyle='-')
    map(lambda position: ax.spines[position].set_visible(False), ['bottom', 'top', 'left', 'right'])
    return fig, ax

# generate layout and plot
fig, ax = prepare_plot(np.arange(.5, 11, 1), np.arange(.5, 49, 1), figsize=(8,7), hideLabels=True,
                       gridColor='#eeeeee', gridWidth=1.1)
image = plt.imshow(data_values,interpolation='nearest', aspect='auto', cmap=cm.Greys)
for x, y, s in zip(np.arange(-.125, 12, 1), np.repeat(-.75, 12), [str(x) for x in range(12)]):
    plt.text(x, y, s, color='#999999', size='10')
plt.text(4.7, -3, 'Feature', color='#999999', size='11'), ax.set_ylabel('Observation')
display(fig)

(1c) Find the range

Now let's examine the labels to find the range of song years. To do this, find the smallest and largest labels in the parsed_points_df.

We will use the min and max functions that are native to the DataFrames, and thus can be optimized using Spark's Catalyst Optimizer and Project Tungsten (don't worry about the technical details). This code will run faster than simply using the native min and max functions in Python. Use selectExpr to retrieve the min and max label values.

# TODO: Replace <FILL IN> with appropriate code
content_stats = (parsed_points_df
                 .selectExpr('label'))

min_year = content_stats.selectExpr('min(label)').first()[0]
max_year = content_stats.selectExpr('max(label)').first()[0]

print min_year, max_year, max_year-min_year

# TEST Find the range (1c)
Test.assertEquals(len(parsed_points_df.first().features), 12,
                  'unexpected number of features in sample point')
sum_feat_two = parsed_points_df.rdd.map(lambda lp: lp.features[2]).sum()
Test.assertTrue(np.allclose(sum_feat_two, 3158.96224351), 'parsedPointsDF has unexpected values')
year_range = max_year - min_year
Test.assertTrue(year_range == 89, 'incorrect range for minYear to maxYear')

(1d) Shift labels

As we just saw, the labels are years in the 1900s and 2000s. In learning problems, it is often natural to shift labels such that they start from zero. Starting with parsed_points_df, create a new DataFrame in which the labels are shifted such that smallest label equals zero (hint: use select). After, use withColumnRenamed to rename the appropriate columns to features and label.

# TODO: Replace <FILL IN> with appropriate code
parsed_data_df = parsed_points_df.map(lambda data: LabeledPoint(data.label - min_year, data.features)).toDF()

# View the first point
print '\n{0}'.format(parsed_data_df.first())

# TEST Shift labels (1d)
old_sample_features = parsed_points_df.first().features
new_sample_features = parsed_data_df.first().features
Test.assertTrue(np.allclose(old_sample_features, new_sample_features),
                'new features do not match old features')
sum_feat_two = parsed_data_df.rdd.map(lambda lp: lp.features[2]).sum()
Test.assertTrue(np.allclose(sum_feat_two, 3158.96224351), 'parsed_data_df has unexpected values')
min_year_new = parsed_data_df.groupBy().min('label').first()[0]
max_year_new = parsed_data_df.groupBy().max('label').first()[0]
Test.assertTrue(min_year_new == 0, 'incorrect min year in shifted data')
Test.assertTrue(max_year_new == 89, 'incorrect max year in shifted data')

Visualization 2: Shifting labels

We will look at the labels before and after shifting them. Both scatter plots below visualize tuples storing:

• a label value and
• the number of training points with this label.

The first scatter plot uses the initial labels, while the second one uses the shifted labels. Note that the two plots look the same except for the labels on the x-axis.

# get data for plot
old_data = (parsed_points_df
             .rdd
             .map(lambda lp: (lp.label, 1))
             .reduceByKey(lambda x, y: x + y)
             .collect())
x, y = zip(*old_data)

# generate layout and plot data
fig, ax = prepare_plot(np.arange(1920, 2050, 20), np.arange(0, 150, 20))
plt.scatter(x, y, s=14**2, c='#d6ebf2', edgecolors='#8cbfd0', alpha=0.75)
ax.set_xlabel('Year'), ax.set_ylabel('Count')
display(fig)

# get data for plot
new_data = (parsed_points_df
             .rdd
             .map(lambda lp: (lp.label, 1))
             .reduceByKey(lambda x, y: x + y)
             .collect())
x, y = zip(*new_data)

# generate layout and plot data
fig, ax = prepare_plot(np.arange(0, 120, 20), np.arange(0, 120, 20))
plt.scatter(x, y, s=14**2, c='#d6ebf2', edgecolors='#8cbfd0', alpha=0.75)
ax.set_xlabel('Year (shifted)'), ax.set_ylabel('Count')
display(fig)
pass

(1e) Training, validation, and test sets

We're almost done parsing our dataset, and our final task involves spliting the dataset into training, validation and test sets. Use the randomSplit method with the specified weights and seed to create DataFrames storing each of these datasets. Next, cache each of these DataFrames, as we will be accessing them multiple times in the remainder of this lab. Finally, compute the size of each dataset and verify that the sum of their sizes equals the value computed in Part (1a).

# TODO: Replace <FILL IN> with appropriate code
weights = [.8, .1, .1]
seed = 42
parsed_train_data_df, parsed_val_data_df, parsed_test_data_df = parsed_data_df.randomSplit(weights, seed)
parsed_train_data_df.cache()
parsed_val_data_df.cache()
parsed_test_data_df.cache()
n_train = parsed_train_data_df.count()
n_val = parsed_val_data_df.count()
n_test = parsed_test_data_df.count()

print n_train, n_val, n_test, n_train + n_val + n_test
print parsed_data_df.count()

# TEST Training, validation, and test sets (1e)
Test.assertEquals(len(parsed_train_data_df.first().features), 12,
                  'parsed_train_data_df has wrong number of features')
sum_feat_two = (parsed_train_data_df
                 .rdd
                 .map(lambda lp: lp.features[2])
                 .sum())
sum_feat_three = (parsed_val_data_df
                  .rdd
                  .map(lambda lp: lp.features[3])
                  .reduce(lambda x, y: x + y))
sum_feat_four = (parsed_test_data_df
                  .rdd
                  .map(lambda lp: lp.features[4])
                  .reduce(lambda x, y: x + y))
Test.assertTrue(np.allclose([sum_feat_two, sum_feat_three, sum_feat_four],
                            2526.87757656, 297.340394298, 184.235876654),
                'parsed Train, Val, Test data has unexpected values')
Test.assertTrue(n_train + n_val + n_test == 6724, 'unexpected Train, Val, Test data set size')
Test.assertEquals(n_train, 5382, 'unexpected value for nTrain')
Test.assertEquals(n_val, 672, 'unexpected value for nVal')
Test.assertEquals(n_test, 670, 'unexpected value for nTest')

Part 2: Create and evaluate a baseline model

(2a) Average label

A very simple yet natural baseline model is one where we always make the same prediction independent of the given data point, using the average label in the training set as the constant prediction value. Compute this value, which is the average (shifted) song year for the training set. Use selectExpr and first() from the DataFrame API.

# TODO: Replace <FILL IN> with appropriate code
average_train_year = (parsed_train_data_df
                        .selectExpr('mean(label)')).first()[0]
print average_train_year

# TEST Average label (2a)
Test.assertTrue(np.allclose(average_train_year, 54.0403195838),
                'incorrect value for average_train_year')

(2b) Root mean squared error

We naturally would like to see how well this naive baseline performs. We will use root mean squared error (RMSE) for evaluation purposes. Using Regression Evaluator, compute the RMSE given a dataset of (prediction, label) tuples.

# TODO: Replace <FILL IN> with appropriate code
from pyspark.ml.evaluation import RegressionEvaluator

preds_and_labels = [(1., 3.), (2., 1.), (2., 2.)]
preds_and_labels_df = sqlContext.createDataFrame(preds_and_labels, ["prediction", "label"])

evaluator = RegressionEvaluator()
def calc_RMSE(dataset):
    """Calculates the root mean squared error for an dataset of (prediction, label) tuples.

    Args:
        dataset (DataFrame of (float, float)): A `DataFrame` consisting of (prediction, label) tuples.

    Returns:
        float: The square root of the mean of the squared errors.
    """
    return evaluator.evaluate(dataset)

example_rmse = calc_RMSE(preds_and_labels_df)
print example_rmse
# RMSE = sqrt[((1-3)^2 + (2-1)^2 + (2-2)^2) / 3] = 1.291

# TEST Root mean squared error (2b)
Test.assertTrue(np.allclose(example_rmse, 1.29099444874), 'incorrect value for exampleRMSE')

(2c) Training, validation and test RMSE

Now let's calculate the training, validation and test RMSE of our baseline model. To do this, first create DataFrames of (prediction, label) tuples for each dataset, and then call calc_RMSE(). Note that each RMSE can be interpreted as the average prediction error for the given dataset (in terms of number of years). You can use createDataFrame to make a DataFrame with the column names of "prediction" and "label" from an RDD.

# TODO: Replace <FILL IN> with appropriate code
preds_and_labels_train = parsed_train_data_df.map(lambda x:(x.label,average_train_year))
preds_and_labels_train_df = sqlContext.createDataFrame(preds_and_labels_train, ["prediction", "label"])
rmse_train_base =calc_RMSE(preds_and_labels_train_df)


preds_and_labels_val = parsed_val_data_df.map(lambda x: (x.label,average_train_year))
preds_and_labels_val_df = sqlContext.createDataFrame(preds_and_labels_val, ["prediction", "label"])
rmse_val_base = calc_RMSE(preds_and_labels_val_df)

preds_and_labels_test = parsed_test_data_df.map(lambda x: (x.label,average_train_year))
preds_and_labels_test_df = sqlContext.createDataFrame(preds_and_labels_test, ["prediction", "label"])
rmse_test_base = calc_RMSE(preds_and_labels_test_df)
                           
print 'Baseline Train RMSE = {0:.3f}'.format(rmse_train_base)
print 'Baseline Validation RMSE = {0:.3f}'.format(rmse_val_base)
print 'Baseline Test RMSE = {0:.3f}'.format(rmse_test_base)

# TEST Training, validation and test RMSE (2c)
Test.assertTrue(np.allclose([rmse_train_base, rmse_val_base, rmse_test_base],
                            [21.4303303309, 20.9179691056, 21.828603786]), 'incorrect RMSE values')

Visualization 3: Predicted vs. actual

We will visualize predictions on the validation dataset. The scatter plots below visualize tuples storing i) the predicted value and ii) true label. The first scatter plot represents the ideal situation where the predicted value exactly equals the true label, while the second plot uses the baseline predictor (i.e., average_train_year) for all predicted values. Further note that the points in the scatter plots are color-coded, ranging from light yellow when the true and predicted values are equal to bright red when they drastically differ.

from matplotlib.colors import ListedColormap, Normalize
from matplotlib.cm import get_cmap
cmap = get_cmap('YlOrRd')
norm = Normalize()

def squared_error(label, prediction):
    """Calculates the squared error for a single prediction."""
    return float((label - prediction)**2)

actual = np.asarray(parsed_val_data_df
                    .select('label')
                    .collect())
error = np.asarray(parsed_val_data_df
                   .rdd
                   .map(lambda lp: (lp.label, lp.label))
                   .map(lambda (l, p): squared_error(l, p))
                   .collect())
clrs = cmap(np.asarray(norm(error)))[:,0:3]

fig, ax = prepare_plot(np.arange(0, 100, 20), np.arange(0, 100, 20))
plt.scatter(actual, actual, s=14**2, c=clrs, edgecolors='#888888', alpha=0.75, linewidths=0.5)
ax.set_xlabel('Predicted'), ax.set_ylabel('Actual')
display(fig)

def squared_error(label, prediction):
    """Calculates the squared error for a single prediction."""
    return float((label - prediction)**2)

predictions = np.asarray(parsed_val_data_df
                         .rdd
                         .map(lambda lp: average_train_year)
                         .collect())
error = np.asarray(parsed_val_data_df
                   .rdd
                   .map(lambda lp: (lp.label, average_train_year))
                   .map(lambda (l, p): squared_error(l, p))
                   .collect())
norm = Normalize()
clrs = cmap(np.asarray(norm(error)))[:,0:3]

fig, ax = prepare_plot(np.arange(53.0, 55.0, 0.5), np.arange(0, 100, 20))
ax.set_xlim(53, 55)
plt.scatter(predictions, actual, s=14**2, c=clrs, edgecolors='#888888', alpha=0.75, linewidths=0.3)
ax.set_xlabel('Predicted'), ax.set_ylabel('Actual')
display(fig)

Part 3: Train (via gradient descent) and evaluate a linear regression model

(3a) Gradient summand

Now let's see if we can do better via linear regression, training a model via gradient descent (we'll omit the intercept for now). Recall that the gradient descent update for linear regression is: \[ \scriptsize \mathbf{w}_{i+1} = \mathbf{w}_i - \alpha_i \sum_j (\mathbf{w}_i^\top\mathbf{x}_j - y_j) \mathbf{x}_j \,.\] where \( \scriptsize i \) is the iteration number of the gradient descent algorithm, and \( \scriptsize j \) identifies the observation.

First, implement a function that computes the summand for this update, i.e., the summand equals \( \scriptsize (\mathbf{w}^\top \mathbf{x} - y) \mathbf{x} \, ,\) and test out this function on two examples. Use the DenseVector dot method.

from pyspark.mllib.linalg import DenseVector

# TODO: Replace <FILL IN> with appropriate code
def gradient_summand(weights, lp):
    """Calculates the gradient summand for a given weight and `LabeledPoint`.

    Note:
        `DenseVector` behaves similarly to a `numpy.ndarray` and they can be used interchangably
        within this function.  For example, they both implement the `dot` method.

    Args:
        weights (DenseVector): An array of model weights (betas).
        lp (LabeledPoint): The `LabeledPoint` for a single observation.

    Returns:
        DenseVector: An array of values the same length as `weights`.  The gradient summand.
    """
    return (weights.dot(lp.features) - lp.label) * lp.features

example_w = DenseVector([1, 1, 1])
example_lp = LabeledPoint(2.0, [3, 1, 4])
# gradient_summand = (dot([1 1 1], [3 1 4]) - 2) * [3 1 4] = (8 - 2) * [3 1 4] = [18 6 24]
summand_one = gradient_summand(example_w, example_lp)
print summand_one

example_w = DenseVector([.24, 1.2, -1.4])
example_lp = LabeledPoint(3.0, [-1.4, 4.2, 2.1])
summand_two = gradient_summand(example_w, example_lp)
print summand_two

# TEST Gradient summand (3a)
Test.assertTrue(np.allclose(summand_one, [18., 6., 24.]), 'incorrect value for summand_one')
Test.assertTrue(np.allclose(summand_two, [1.7304,-5.1912,-2.5956]), 'incorrect value for summand_two')

(3b) Use weights to make predictions

Next, implement a get_labeled_predictions function that takes in weights and an observation's LabeledPoint and returns a (prediction, label) tuple. Note that we can predict by computing the dot product between weights and an observation's features.

# TODO: Replace <FILL IN> with appropriate code
def get_labeled_prediction(weights, observation):
    """Calculates predictions and returns a (prediction, label) tuple.

    Note:
        The labels should remain unchanged as we'll use this information to calculate prediction
        error later.

    Args:
        weights (np.ndarray): An array with one weight for each features in `trainData`.
        observation (LabeledPoint): A `LabeledPoint` that contain the correct label and the
            features for the data point.

    Returns:
        tuple: A (prediction, label) tuple. Convert the return type of the label and prediction to a float.
    """
    return (float(weights.dot(observation.features)), float(observation.label))

weights = np.array([1.0, 1.5])
prediction_example = sc.parallelize([LabeledPoint(2, np.array([1.0, .5])),
                                     LabeledPoint(1.5, np.array([.5, .5]))])
preds_and_labels_example = prediction_example.map(lambda lp: get_labeled_prediction(weights, lp))
print preds_and_labels_example.collect()

# TEST Use weights to make predictions (3b)
Test.assertTrue(isinstance(preds_and_labels_example.first()[0], float), 'prediction must be a float')
Test.assertEquals(preds_and_labels_example.collect(), [(1.75, 2.0), (1.25, 1.5)],
                  'incorrect definition for getLabeledPredictions')

(3c) Gradient descent

Next, implement a gradient descent function for linear regression and test out this function on an example.

# TODO: Replace <FILL IN> with appropriate code
def linreg_gradient_descent(train_data, num_iters):
    """Calculates the weights and error for a linear regression model trained with gradient descent.

    Note:
        `DenseVector` behaves similarly to a `numpy.ndarray` and they can be used interchangably
        within this function.  For example, they both implement the `dot` method.

    Args:
        train_data (RDD of LabeledPoint): The labeled data for use in training the model.
        num_iters (int): The number of iterations of gradient descent to perform.

    Returns:
        (np.ndarray, np.ndarray): A tuple of (weights, training errors).  Weights will be the
            final weights (one weight per feature) for the model, and training errors will contain
            an error (RMSE) for each iteration of the algorithm.
    """
    # The length of the training data
    n = train_data.count()
    # The number of features in the training data
    d = len(train_data.first().features)
    w = np.zeros(d)
    alpha = 1.0
    # We will compute and store the training error after each iteration
    error_train = np.zeros(num_iters)
    for i in range(num_iters):
        # Use get_labeled_prediction from (3b) with trainData to obtain an RDD of (label, prediction)
        # tuples.  Note that the weights all equal 0 for the first iteration, so the predictions will
        # have large errors to start.
        preds_and_labels_train =train_data.map(lambda lp: get_labeled_prediction(w, lp))
        preds_and_labels_train_df = sqlContext.createDataFrame(preds_and_labels_train, ["prediction", "label"])
        error_train[i] = calc_RMSE(preds_and_labels_train_df)

        # Calculate the `gradient`.  Make use of the `gradient_summand` function you wrote in (3a).
        # Note that `gradient` should be a `DenseVector` of length `d`.
        gradient= train_data.map(lambda lp: gradient_summand(w, lp)).sum()

        # Update the weights
        alpha_i = alpha / (n * np.sqrt(i+1))
        w -= alpha_i*gradient
    return w, error_train

# create a toy dataset with n = 10, d = 3, and then run 5 iterations of gradient descent
# note: the resulting model will not be useful; the goal here is to verify that
# linreg_gradient_descent is working properly
example_n = 10
example_d = 3
example_data = (sc
                 .parallelize(parsed_train_data_df.take(example_n))
                 .map(lambda lp: LabeledPoint(lp.label, lp.features[0:example_d])))
print example_data.take(2)
example_num_iters = 5
example_weights, example_error_train = linreg_gradient_descent(example_data, example_num_iters)
print example_weights

# TEST Gradient descent (3c)
expected_output = [22.68915382, 46.210194, 51.74336678]
Test.assertTrue(np.allclose(example_weights, expected_output), 'value of example_weights is incorrect')
expected_error = [66.32269596, 45.61098865, 38.6123992, 35.28952945, 33.4708604]
Test.assertTrue(np.allclose(example_error_train, expected_error),
                'value of exampleErrorTrain is incorrect')

(3d) Train the model

Now let's train a linear regression model on all of our training data and evaluate its accuracy on the validation set. Note that the test set will not be used here. If we evaluated the model on the test set, we would bias our final results.

We've already done much of the required work: we computed the number of features in Part (1b); we created the training and validation datasets and computed their sizes in Part (1e); and, we wrote a function to compute RMSE in Part (2b).

# TODO: Replace <FILL IN> with appropriate coe
num_iters = 50
weights_LR0, error_train_LR0 = linreg_gradient_descent(parsed_train_data_df, num_iters)

preds_and_labels = parsed_val_data_df.map(lambda lp: get_labeled_prediction(weights_LR0, lp))

preds_and_labels_df = sqlContext.createDataFrame(preds_and_labels, ["prediction", "label"])
rmse_val_LR0 = calc_RMSE(preds_and_labels_df)

print 'Validation RMSE:\n\tBaseline = {0:.3f}\n\tLR0 = {1:.3f}'.format(rmse_val_base,
                                                                       rmse_val_LR0)

# TEST Train the model (3d)
expected_output = [ 22.2588534,   20.26005774,   0.01539014,   8.69071379,   5.63536339, -4.19700345,
                    15.54525224,   3.88968175,   9.76633157,   5.9276698,   11.41170336,   3.7525027 ]
Test.assertTrue(np.allclose(weights_LR0, expected_output), 'incorrect value for weights_LR0')

Visualization 4: Training error

We will look at the log of the training error as a function of iteration. The first scatter plot visualizes the logarithm of the training error for all 50 iterations. The second plot shows the training error itself, focusing on the final 44 iterations.

norm = Normalize()
clrs = cmap(np.asarray(norm(np.log(error_train_LR0))))[:,0:3]

fig, ax = prepare_plot(np.arange(0, 60, 10), np.arange(2, 6, 1))
ax.set_ylim(2, 6)
plt.scatter(range(0, num_iters), np.log(error_train_LR0), s=14**2, c=clrs, edgecolors='#888888', alpha=0.75)
ax.set_xlabel('Iteration'), ax.set_ylabel(r'$\log_e(errorTrainLR0)$')
display(fig)

norm = Normalize()
clrs = cmap(np.asarray(norm(error_train_LR0[6:])))[:,0:3]

fig, ax = prepare_plot(np.arange(0, 60, 10), np.arange(17, 22, 1))
ax.set_ylim(17.8, 21.2)
plt.scatter(range(0, num_iters-6), error_train_LR0[6:], s=14**2, c=clrs, edgecolors='#888888', alpha=0.75)
ax.set_xticklabels(map(str, range(6, 66, 10)))
ax.set_xlabel('Iteration'), ax.set_ylabel(r'Training Error')
display(fig)

Part 4: Train using SparkML and perform grid search

(4a) LinearRegression

We're already doing better than the baseline model, but let's see if we can do better by adding an intercept, using regularization, and (based on the previous visualization) training for more iterations. SparkML's LinearRegression essentially implements the same algorithm that we implemented in Part (3b), albeit more efficiently and with various additional functionality, such as including an intercept in the model and allowing L1, L2, or elastic net regularization. Elastic net regularization is a linear combination of L1 and L2 regularization. For alpha = 0, the penalty is an L2 penalty. For alpha = 1, it is an L1 penalty.

First use LinearRegression to train a model with elastic net regularization and an intercept. This method returns a LinearRegressionModel. Next, use the model's coefficients (weights) and intercept attributes to print out the model's parameters.

from pyspark.ml.regression import LinearRegression
# Values to use when training the linear regression model

num_iters = 500  # iterations
reg = 1e-1  # regParam
alpha = .2  # elasticNetParam
use_intercept = True  # intercept

# TODO: Replace <FILL IN> with appropriate code
lin_reg = LinearRegression().setMaxIter(num_iters).setRegParam(reg).setElasticNetParam(alpha)

first_model = lin_reg.fit(parsed_train_data_df)


# coeffsLR1 stores the model coefficients; interceptLR1 stores the model intercept
coeffs_LR1 =first_model.weights
intercept_LR1 = first_model.intercept
print coeffs_LR1, intercept_LR1

# TEST LinearRegression (4a)
expected_intercept = 64.2456893425
expected_weights = [21.8238800212, 27.6186877074, -66.4789086231, 54.191182811, -14.2978518435, -47.0287067393,35.1372526918,
                   -20.0165577186, 0.737339261177, -3.8022145328, -7.62277095338, -15.9836308238]
Test.assertTrue(np.allclose(intercept_LR1, expected_intercept), 'incorrect value for intercept_LR1')
Test.assertTrue(np.allclose(coeffs_LR1, expected_weights), 'incorrect value for weights_LR1')

(4b) Transform

Now use the LinearRegressionModel.transform() method to make predictions on the parsed_train_data_df.

# TODO: Replace <FILL IN> with appropriate code
sample_prediction = first_model.transform(parsed_train_data_df)

display(sample_prediction)

# TEST Predict (4b)
Test.assertTrue(np.allclose(sample_prediction.first().prediction, 38.63757807530045),
                'incorrect value for sample_prediction')

(4c) Evaluate RMSE

Next evaluate the accuracy of this model on the validation set. Use the transform() method to create predictions, and then use the calc_RMSE() function from Part (2b).

# TODO: Replace <FILL IN> with appropriate code
val_pred_df = first_model.transform(parsed_val_data_df)

rmse_val_LR1 = calc_RMSE(val_pred_df)
print ('Validation RMSE:\n\tBaseline = {0:.3f}\n\tLR0 = {1:.3f}' +
       '\n\tLR1 = {2:.3f}').format(rmse_val_base, rmse_val_LR0, rmse_val_LR1)

# TEST Evaluate RMSE (4c)
Test.assertTrue(np.allclose(rmse_val_LR1, 15.3130800661), 'incorrect value for rmseValLR1')

(4d) Grid search

We're already outperforming the baseline on the validation set by almost 2 years on average, but let's see if we can do better. Perform grid search to find a good regularization parameter. Try regParam values 1e-10, 1e-5, and 1.0.

# TODO: Replace <FILL IN> with appropriate code
best_RMSE = rmse_val_LR1
best_reg_param = reg
best_model = first_model

num_iters = 500  # iterations
alpha = .2  # elasticNetParam
use_intercept = True  # intercept

for reg in [1e-10, 1e-5, 1.0]:
    lin_reg = LinearRegression(maxIter=num_iters, regParam=reg, elasticNetParam=alpha, fitIntercept=use_intercept)
    model = lin_reg.fit(parsed_train_data_df)
    val_pred_df = model.transform(parsed_val_data_df)

    rmse_val_grid = calc_RMSE(val_pred_df)
    print rmse_val_grid

    if rmse_val_grid < best_RMSE:
        best_RMSE = rmse_val_grid
        best_reg_param = reg
        best_model = model

rmse_val_LR_grid = best_RMSE

print ('Validation RMSE:\n\tBaseline = {0:.3f}\n\tLR0 = {1:.3f}\n\tLR1 = {2:.3f}\n' +
       '\tLRGrid = {3:.3f}').format(rmse_val_base, rmse_val_LR0, rmse_val_LR1, rmse_val_LR_grid)

# TEST Grid search (4d)
Test.assertTrue(np.allclose(15.3052663831, rmse_val_LR_grid), 'incorrect value for rmseValLRGrid')

Visualization 5: Best model's predictions

Next, we create a visualization similar to 'Visualization 3: Predicted vs. actual' from Part 2 using the predictions from the best model from Part (4d) on the validation dataset. Specifically, we create a color-coded scatter plot visualizing tuples storing i) the predicted value from this model and ii) true label.

parsed_val_df = best_model.transform(parsed_val_data_df)
predictions = np.asarray(parsed_val_df
                         .select('prediction')
                         .collect())
actual = np.asarray(parsed_val_df
                      .select('label')
                      .collect())
error = np.asarray(parsed_val_df
                     .rdd
                     .map(lambda lp: squared_error(lp.label, lp.prediction))
                     .collect())

norm = Normalize()
clrs = cmap(np.asarray(norm(error)))[:,0:3]

fig, ax = prepare_plot(np.arange(0, 120, 20), np.arange(0, 120, 20))
ax.set_xlim(15, 82), ax.set_ylim(-5, 105)
plt.scatter(predictions, actual, s=14**2, c=clrs, edgecolors='#888888', alpha=0.75, linewidths=.5)
ax.set_xlabel('Predicted'), ax.set_ylabel(r'Actual')
display(fig)

Visualization 6: Hyperparameter heat map

Next, we perform a visualization of hyperparameter search using a larger set of hyperparameters (with precomputed results). Specifically, we create a heat map where the brighter colors correspond to lower RMSE values. The first plot has a large area with brighter colors. In order to differentiate within the bright region, we generate a second plot corresponding to the hyperparameters found within that region.

from matplotlib.colors import LinearSegmentedColormap

# Saved parameters and results, to save the time required to run 36 models
num_iters = 500
reg_params = [1.0, 2.0, 4.0, 8.0, 16.0, 32.0]
alpha_params = [0.0, .1, .2, .4, .8, 1.0]
rmse_val = np.array([[ 15.317156766552452, 15.327211561989827, 15.357152971253697, 15.455092206273847, 15.73774335576239,
                       16.36423857334287, 15.315019185101972, 15.305949211619886, 15.355590337955194, 15.573049001631558,
                       16.231992712117222, 17.700179790697746, 15.305266383061921, 15.301104931027034, 15.400125020566225,
                       15.824676190630191, 17.045905140628836, 19.365558346037535, 15.292810983243772, 15.333756681057828,
                       15.620051033979871, 16.631757941340428, 18.948786862836954, 20.91796910560631, 15.308301384150049,
                       15.522394576046239, 16.414106221093316, 18.655978799189178, 20.91796910560631, 20.91796910560631,
                       15.33442896030322, 15.680134490745722, 16.86502909075323, 19.72915603626022, 20.91796910560631,
                       20.91796910560631 ]])

num_rows, num_cols = len(alpha_params), len(reg_params)
rmse_val = np.array(rmse_val)
rmse_val.shape = (num_rows, num_cols)

fig, ax = prepare_plot(np.arange(0, num_cols, 1), np.arange(0, num_rows, 1), figsize=(8, 7), hideLabels=True,
                       gridWidth=0.)
ax.set_xticklabels(reg_params), ax.set_yticklabels(alpha_params)
ax.set_xlabel('Regularization Parameter'), ax.set_ylabel('Alpha')

colors = LinearSegmentedColormap.from_list('blue', ['#0022ff', '#000055'], gamma=.2)
image = plt.imshow(rmse_val,interpolation='nearest', aspect='auto',
                    cmap = colors)
display(fig)

# Zoom into the top left
alpha_params_zoom, reg_params_zoom = alpha_params[1:5], reg_params[:4]
rmse_val_zoom = rmse_val[1:5, :4]

num_rows, num_cols = len(alpha_params_zoom), len(reg_params_zoom)

fig, ax = prepare_plot(np.arange(0, num_cols, 1), np.arange(0, num_rows, 1), figsize=(8, 7), hideLabels=True,
                       gridWidth=0.)
ax.set_xticklabels(reg_params_zoom), ax.set_yticklabels(alpha_params_zoom)
ax.set_xlabel('Regularization Parameter'), ax.set_ylabel('Alpha')

colors = LinearSegmentedColormap.from_list('blue', ['#0022ff', '#000055'], gamma=.2)
image = plt.imshow(rmse_val_zoom, interpolation='nearest', aspect='auto',
                    cmap = colors)
display(fig)

Part 5: Add interactions between features

(5a) Add 2-way interactions

So far, we've used the features as they were provided. Now, we will add features that capture the two-way interactions between our existing features. Write a function two_way_interactions that takes in a LabeledPoint and generates a new LabeledPoint that contains the old features and the two-way interactions between them.

Note:

• A dataset with three features would have nine ( \( \scriptsize 3^2 \) ) two-way interactions.
• You might want to use itertools.product to generate tuples for each of the possible 2-way interactions.
• Remember that you can combine two DenseVector or ndarray objects using np.hstack.

# TODO: Replace <FILL IN> with appropriate code
import itertools

def two_way_interactions(lp):
    """Creates a new `LabeledPoint` that includes two-way interactions.

    Note:
        For features [x, y] the two-way interactions would be [x^2, x*y, y*x, y^2] and these
        would be appended to the original [x, y] feature list.

    Args:
        lp (LabeledPoint): The label and features for this observation.

    Returns:
        LabeledPoint: The new `LabeledPoint` should have the same label as `lp`.  Its features
            should include the features from `lp` followed by the two-way interaction features.
    """
    cartesian_product_iterator = [item for item in itertools.product(lp.features, lp.features)]
    cartesian_product = map(lambda x: np.prod(x), cartesian_product_iterator)
    return LabeledPoint(lp.label, np.hstack((lp.features, cartesian_product)))


print two_way_interactions(LabeledPoint(0.0, [2, 3]))

# Transform the existing train, validation, and test sets to include two-way interactions.
# Remember to convert them back to DataFrames at the end.
train_data_interact_df = parsed_train_data_df.map(two_way_interactions).toDF()
val_data_interact_df = parsed_val_data_df.map(two_way_interactions).toDF()
test_data_interact_df = parsed_test_data_df.map(two_way_interactions).toDF()

# TEST Add two-way interactions (5a)
two_way_example = two_way_interactions(LabeledPoint(0.0, [2, 3]))
Test.assertTrue(np.allclose(sorted(two_way_example.features),
                            sorted([2.0, 3.0, 4.0, 6.0, 6.0, 9.0])),
                'incorrect features generatedBy two_way_interactions')
two_way_point = two_way_interactions(LabeledPoint(1.0, [1, 2, 3]))
Test.assertTrue(np.allclose(sorted(two_way_point.features),
                            sorted([1.0,2.0,3.0,1.0,2.0,3.0,2.0,4.0,6.0,3.0,6.0,9.0])),
                'incorrect features generated by twoWayInteractions')
Test.assertEquals(two_way_point.label, 1.0, 'incorrect label generated by two_way_interactions')
Test.assertTrue(np.allclose(sum(train_data_interact_df.first().features), 28.623429648737346),
                'incorrect features in train_data_interact_df')
Test.assertTrue(np.allclose(sum(val_data_interact_df.first().features), 23.582959172640948),
                'incorrect features in val_data_interact_df')
Test.assertTrue(np.allclose(sum(test_data_interact_df.first().features), 26.045820467171758),
                'incorrect features in test_data_interact_df')

(5b) Build interaction model

Now, let's build the new model. We've done this several times now. To implement this for the new features, we need to change a few variable names.

Note:

• Remember that we should build our model from the training data and evaluate it on the validation data.
• You should re-run your hyperparameter search after changing features, as using the best hyperparameters from your prior model will not necessary lead to the best model.
• For this exercise, we have already preset the hyperparameters to reasonable values.

# TODO: Replace <FILL IN> with appropriate code
num_iters = 500
reg = 1e-10
alpha = .2
use_intercept = True

lin_reg = LinearRegression(maxIter=num_iters, regParam=reg, elasticNetParam=alpha, fitIntercept=use_intercept)

model_interact = lin_reg.fit(train_data_interact_df)
preds_and_labels_interact_df = model_interact.transform(val_data_interact_df)
rmse_val_interact = calc_RMSE(preds_and_labels_interact_df)

print ('Validation RMSE:\n\tBaseline = {0:.3f}\n\tLR0 = {1:.3f}\n\tLR1 = {2:.3f}\n\tLRGrid = ' +
       '{3:.3f}\n\tLRInteract = {4:.3f}').format(rmse_val_base, rmse_val_LR0, rmse_val_LR1,
                                                 rmse_val_LR_grid, rmse_val_interact)

# TEST Build interaction model (5b)
Test.assertTrue(np.allclose(rmse_val_interact, 14.3495530997), 'incorrect value for rmse_val_interact')

(5c) Evaluate interaction model on test data

Our next step is to evaluate the new model on the test dataset. Note that we haven't used the test set to evaluate any of our models. Because of this, our evaluation provides us with an unbiased estimate for how our model will perform on new data. If we had changed our model based on viewing its performance on the test set, our estimate of RMSE would likely be overly optimistic.

We'll also print the RMSE for both the baseline model and our new model. With this information, we can see how much better our model performs than the baseline model.

# TODO: Replace <FILL IN> with appropriate code
preds_and_labels_test_df = model_interact.transform(test_data_interact_df)
rmse_test_interact = calc_RMSE(preds_and_labels_test_df)

print ('Test RMSE:\n\tBaseline = {0:.3f}\n\tLRInteract = {1:.3f}'
       .format(rmse_test_base, rmse_test_interact))

# TEST Evaluate interaction model on test data (5c)
Test.assertTrue(np.allclose(rmse_test_interact, 14.9990015721),
                'incorrect value for rmse_test_interact')

(5d) Use a pipeline to create the interaction model

Our final step is to create the interaction model using a Pipeline. Note that Spark contains the PolynomialExpansion transformer which will automatically generate interactions for us. In this section, you'll need to generate the PolynomialExpansion transformer and set the stages for the Pipeline estimator. Make sure to use a degree of 2 for PolynomialExpansion, set the input column appropriately, and set the output column to "polyFeatures". The pipeline should contain two stages: the polynomial expansion and the linear regression.

# TODO: Replace <FILL IN> with appropriate code
from pyspark.ml import Pipeline
from pyspark.ml.feature import PolynomialExpansion

num_iters = 500
reg = 1e-10
alpha = .2
use_intercept = True

polynomial_expansion = PolynomialExpansion(degree=2,inputCol="features", outputCol='polyFeatures')
linear_regression = LinearRegression(maxIter=num_iters, regParam=reg, elasticNetParam=alpha,
                                     fitIntercept=use_intercept, featuresCol='polyFeatures')

pipeline = Pipeline(stages=[polynomial_expansion, linear_regression])
pipeline_model = pipeline.fit(parsed_train_data_df)

predictions_df = pipeline_model.transform(parsed_test_data_df)

evaluator = RegressionEvaluator()
rmse_test_pipeline = evaluator.evaluate(predictions_df, {evaluator.metricName: "rmse"})
print('RMSE for test data set using pipelines: {0:.3f}'.format(rmse_test_pipeline))

# TEST Use a pipeline to create the interaction model (5d)
Test.assertTrue(np.allclose(rmse_test_pipeline, 14.99415450247963),
                'incorrect value for rmse_test_pipeline')