# Nonlinear regression for inverse dynamics¶

In this question, we fit a model which can predict what torques a robot needs to apply in order to make its arm reach a desired point in space. The data was collected from a SARCOS robot arm with $7$ degrees of freedom. The input vector $x \in \mathbb{R}^{21}$ encodes the desired position, velocity and accelaration of the $7$ joints. The output vector $y \in \mathbb{R}^7$ encodes the torques that should be applied to the joints to reach that point. The mapping from $x$ to $y$ is highly nonlinear.

We can find the data at http://www.gaussianprocess.org/gpml/data/. Obviously the first step is to read this data.

In :
%matplotlib inline

import matplotlib
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import time


Out:
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 ... x19 x20 x21 y1 y2 y3 y4 y5 y6 y7
0 0.019478 -0.134218 0.027439 1.516401 0.300936 0.058259 0.150134 -0.266791 -0.237134 -0.091272 ... 11.695956 1.210212 -22.119289 50.292652 -36.971897 20.937170 47.821712 -0.424812 -0.907553 8.090739
1 0.017279 -0.137077 0.026999 1.532517 0.301344 0.058259 0.128653 -0.153640 -0.335279 0.006449 ... 14.643369 1.015070 -17.048688 44.104164 -28.851845 16.230194 43.194073 -0.228739 -1.235817 7.762475
2 0.016336 -0.140878 0.027250 1.549670 0.302318 0.059027 0.104104 -0.047313 -0.418732 0.106274 ... 15.467628 0.910548 -11.415526 37.354858 -20.809343 12.379975 39.386017 0.244491 -1.700880 7.289678
3 0.016273 -0.145307 0.029072 1.566855 0.307628 0.059027 0.080321 0.053238 -0.460963 0.188013 ... 10.309203 0.921360 -5.772058 30.676065 -13.963816 7.702940 36.478813 -0.182062 -2.143370 6.410800
4 0.017279 -0.150051 0.031083 1.584416 0.314162 0.059027 0.058840 0.133810 -0.462264 0.263975 ... 2.868096 1.059957 -0.491542 25.920128 -11.178479 5.643934 34.773911 -1.031687 -2.355776 5.792892

5 rows × 28 columns

Let's standardize our data by making each column have mean $0$ and unit variance.

## Linear Regression¶

Now, the first order of business is to fit a standard linear regression. I'll just use the sklearn implementation.

In :
from sklearn import preprocessing
def standardize_data(data):
feature_column_names = [column_name for column_name in all_data.columns if 'x' in column_name]
features = data[feature_column_names]
# only care about the first output for this problem
target = data[TARGET_COLUMN]
X = features.as_matrix()
Y = target.as_matrix()
X_scaled = preprocessing.scale(X, axis = 0, with_mean = True, with_std = True, copy = True)
# only standardize inputs
# Y_scaled = preprocessing.scale(Y, axis = 0, with_mean = True, with_std = True, copy = True)
return X_scaled, Y
# standardize together and separate
all_data = pd.concat([training_data, test_data], axis = 0)
standardize_data(all_data)
X, Y = standardize_data(all_data)
X_scaled, Y_scaled = X[:len(training_data)], Y[:len(training_data)]
X_scaled_test, Y_scaled_test = X[len(training_data):], Y[len(training_data):]

In :
from sklearn import linear_model
## since data is already normalized
linearRegressor = linear_model.LinearRegression(fit_intercept = True, normalize = False)
linearRegressor.fit(X_scaled, Y_scaled)
linearRegressor.score(X_scaled_test, Y_scaled_test)

/usr/local/lib/python3.6/site-packages/scipy/linalg/basic.py:1018: RuntimeWarning: internal gelsd driver lwork query error, required iwork dimension not returned. This is likely the result of LAPACK bug 0038, fixed in LAPACK 3.2.2 (released July 21, 2010). Falling back to 'gelss' driver.
warnings.warn(mesg, RuntimeWarning)

Out:
0.92494581827142286

According to the problem, we should be able to get a standardized mean squared error (SMSE) of 0.075. We define

\begin{equation} SMSE = \left(\frac{1}{N_{test}}\sum_{i=1}^{N_{test}}\left(y_i - \hat{y}_i\right)^2\right)/\sigma^2, \end{equation}

where

\begin{equation} \sigma^2 = \frac{1}{N_{train}}\sum_{i=1}^{N_{train}}\left(y_i - \bar{y}\right)^2. \end{equation}
In :
def compute_mse(prediction, labels):
return (sum((labels - prediction)**2)/len(prediction))
def compute_model_mse(model, X, y):
predicted_y = model.predict(X)
return compute_mse(predicted_y, y)
def log_mse(mse):
print('MSE: {:.10f}'.format(mse))
print('SMSE: {:.10f}'.format(mse/np.var(Y_scaled, ddof = 0)))

test_mse = compute_model_mse(linearRegressor, X_scaled_test, Y_scaled_test)

log_mse(test_mse)

MSE: 31.0857592861
SMSE: 0.0742260930


Looks like we got the correct result. Yay!

Next, we'll try using a radial basis function (RBF) network. First, we need to do some K-means clustering. We'll pick the number of clusters with the qualitative kink, knee, or elbow method (three different names for the same method). The book says to use cross-validation, but that's nonsense since the book itself says that using cross-validation is not possible for non-probablistic methods. I guess we could use a Gaussian Mixture Model if we want to do cross-validation?

We'll fit the model with various values of $k$ and look at the reconstruction error.

In :
from sklearn import cluster
from joblib import Parallel, delayed

k_range = np.arange(1, 107, 3)
def score_k(k):
kMeansModel = cluster.KMeans(n_clusters = k, init = 'k-means++', algorithm = 'elkan', precompute_distances = True)
kMeansModel.fit(X_scaled)
return -kMeansModel.score(X_scaled_test)
start = time.perf_counter() ## time how long it takes
reconstruction_error = Parallel(n_jobs = 2, backend='threading')(delayed(score_k)(k) for k in k_range)
print('Time elapsed: ' + str(time.perf_counter() - start))
reconstruction_error

Time elapsed: 963.612736572999

Out:
[93511.741947211151,
65971.803182832591,
52906.722708804191,
45695.425134108205,
40914.984232328658,
37248.225253985787,
34558.310657963739,
32687.881212347915,
30928.954367901584,
29095.062690792329,
28426.793310216104,
26914.31468947696,
25975.490225306414,
25435.360392172126,
24836.868461234892,
24133.903034204865,
23626.584694583617,
23060.447629759739,
22521.141339972521,
21986.600362484041,
21708.632679338189,
21322.262079304437,
20787.919953330376,
20624.359384467432,
20029.795155720603,
19853.067069977682,
19440.537303560875,
19233.056525067761,
18875.450036315928,
18604.189801324806,
18443.502619621235,
18096.416501089247,
17887.377666428139,
17637.052858147334,
17348.111435998861,
17166.150898743901]

Let's plot the reconstruction error to find the kink.

In :
reconstruction_error = np.array(reconstruction_error, dtype = np.float64)
fig = plt.figure(figsize=(10,6))
ax = plt.gca()
ax.plot(k_range, reconstruction_error, '.k-')
ax.grid(True)
ax.set_ylabel('Reconstruction error')
ax.set_xlabel('$k$ - the number of clusters')
ax.set_title('Test Data Reconstruction Error versus $k$')
ax.get_yaxis().set_major_formatter(
matplotlib.ticker.FuncFormatter(lambda y, pos : format(int(y), ',')))
fig.show()

/usr/local/lib/python3.6/site-packages/matplotlib/figure.py:397: UserWarning: matplotlib is currently using a non-GUI backend, so cannot show the figure
"matplotlib is currently using a non-GUI backend, " Qualitatively, $100$ looks good to me. Now, let's fit our RBF network, which is a super simple neural network. We have 1 hidden layer of 100 units with the RBF activation function, and we run a linear regression on that hidden layer.

In :
n_clusters = 100
kMeansModel = cluster.KMeans(n_clusters = n_clusters, init = 'k-means++', algorithm = 'elkan', precompute_distances = True)
kMeansModel.fit(X_scaled)
prototypes = kMeansModel.cluster_centers_


The RBF kerneal is

\begin{equation} \kappa(\mathbf{x}, \mathbf{x}^\prime) = \exp\left(-\frac{\lVert\mathbf{x}-\mathbf{x}^\prime\rVert^2}{2\sigma^2}\right), \end{equation}

so we'll perform our linear regression with design matrix $\Phi$, where rows of $\Phi$ are

\begin{equation} \phi(\mathbf{x}) = \begin{pmatrix} \kappa(\mathbf{x}, \boldsymbol\mu_1) & \kappa(\mathbf{x}, \boldsymbol\mu_2) & \cdots & \kappa(\mathbf{x}, \boldsymbol\mu_{k}) \end{pmatrix}, \end{equation}

where the $\boldsymbol\mu_j$ for $j = 1,2,\ldots,k$ are our cluster centers. $\sigma^2$ is parameter called bandwidth that we'll have to find with cross-validation.

In :
from sklearn import model_selection

class RBFNetwork:
"""
Implements RBFNetwork as a special case of linear regression.
"""
def __init__(self, mu, sigma2):
self.linearRegressor = linear_model.LinearRegression(fit_intercept = True, normalize = False)
self.mu = mu
self.sigma2 = sigma2

def fit(self, X, y):
Phi = self.rbf_transform(X, self.mu, self.sigma2)
linearRegressor.fit(Phi, y)
return self

def predict(self, X):
Phi = self.rbf_transform(X, self.mu, self.sigma2)
return linearRegressor.predict(Phi)

def score(self, X, y):
predicted_y = self.predict(X)
return -(sum((y - predicted_y)**2)/len(y))

def get_params(self, deep = True):
return {'mu': self.mu, 'sigma2': self.sigma2}

def set_params(self, **params):
if 'mu' in params:
self.mu = params['mu']
if 'sigma2' in params:
self.sigma2 = params['sigma2']
return self

@staticmethod
def rbf(x1, x2, sigma2):
deltaX = (x1 - x2)
distX = sum(deltaX*deltaX)
return np.exp(-distX/2/sigma2)

@staticmethod
def phi(x, mu, sigma2):
return np.apply_along_axis(func1d = RBFNetwork.rbf, axis = 1, arr = mu, x2 = x, sigma2 = sigma2)

@staticmethod
def rbf_transform(X, mu, sigma2):
return np.apply_along_axis(func1d = RBFNetwork.phi, axis = 1, arr = X, mu = mu, sigma2 = sigma2)

bandwidths = [1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
gridSearchCV = model_selection.GridSearchCV(RBFNetwork(mu = prototypes, sigma2 = 1024),
param_grid = {'sigma2': bandwidths},
cv = model_selection.KFold(n_splits = 5, shuffle = True, random_state = 2016),
n_jobs = 1) ## parallel jobs are broken for a reason that I don't feel like debugging
gridSearchCV.fit(X_scaled, Y_scaled)
cv_results = pd.DataFrame({'bandwidth': bandwidths,
'mean_test_score': gridSearchCV.cv_results_['mean_test_score'],
'std_test_score': gridSearchCV.cv_results_['std_test_score'],
'mean_train_score': gridSearchCV.cv_results_['mean_train_score'],
'std_train_score': gridSearchCV.cv_results_['std_train_score']})
cv_results

Out:
bandwidth mean_test_score mean_train_score std_test_score std_train_score
0 1 -267.152902 -265.932425 15.279617 3.813405
1 2 -188.522188 -187.648077 11.168259 2.790824
2 4 -122.894823 -122.112400 6.091327 1.522200
3 8 -72.431290 -71.858192 2.480018 0.620141
4 16 -40.163809 -39.825444 0.767083 0.193418
5 32 -25.264711 -25.057333 0.227940 0.056388
6 64 -20.167603 -20.004292 0.097739 0.025807
7 128 -18.716779 -18.568064 0.192818 0.049968
8 256 -18.422612 -18.277355 0.228230 0.058791
9 512 -18.432984 -18.287904 0.238813 0.061415
In :
fig = plt.figure(figsize=(10, 6))
ax = fig.gca()
test_error_bars = ax.errorbar(cv_results['bandwidth'], cv_results['mean_test_score'], cv_results['std_test_score'],
linestyle='-', marker='.', color='b', capsize=5)
train_error_bars = ax.errorbar(cv_results['bandwidth'], cv_results['mean_train_score'], cv_results['std_train_score'],
linestyle='-', marker='.', color='r', capsize=5)
ax.grid(True)
ax.legend([test_error_bars, train_error_bars], ['Test Set', 'Training Set'], loc='lower right')
ax.set_xscale('log', basex = 2)
ax.set_xlabel('Bandwidth ($\sigma^2$)')
ax.set_ylabel('Score (negative MSE)')
ax.set_title('Score versus Bandwidth')
fig.show()

/usr/local/lib/python3.6/site-packages/matplotlib/figure.py:397: UserWarning: matplotlib is currently using a non-GUI backend, so cannot show the figure
"matplotlib is currently using a non-GUI backend, " It looks like the best model has a bandwidth of about $2^{8}$. Let's compute its SMSE.

In :
rBFNetwork = RBFNetwork(mu = prototypes, sigma2 = 2**8)
rBFNetwork.fit(X_scaled, Y_scaled)
-rBFNetwork.score(X_scaled_test, Y_scaled_test)/np.var(Y_scaled, ddof=0)

Out:
0.042844306703931787

Seems that we can only get down to an SMSE of 0.043. The next step is to try a neural network. This has more parameters to choose. In particular, we're interested in the number of hidden units, layers, and the L2 regularization parameter.

## Neural Networks with TensorFlow¶

Since I work at Google, I thought that I would demonstrate constructing a neural network in TensorFlow. Since I'm doing this with the intent of learning the details of the models, I'm going to eschew the high-level APIs and construct the graph mostly by hand. Much of the code is imported from sarco_train.py.

First, we'll set up our features which need to normalized. We make it so that the input has mean 0 and variance 1.

In :
import tensorflow as tf
from sarco_train import create_feature_columns

feature_columns = create_feature_columns(pd.concat([training_data, test_data]))

def get_feature_data(data, feature_columns):
graph = tf.Graph()
with graph.as_default():
features = {column: tf.constant(column_data, dtype=tf.float32)
for column, column_data in data.to_dict('list').items()}
input_layer = tf.feature_column.input_layer(
features=features, feature_columns=feature_columns)
with tf.Session(graph=graph) as sess:
input_data = sess.run(input_layer)
return input_data, data[TARGET_COLUMN].astype(np.float32)

training_input_data, training_target_data = get_feature_data(training_data, feature_columns)
test_input_data, test_target_data = get_feature_data(test_data, feature_columns)


## Building the Graph¶

Everything in TensorFlow is a graph. We build build it layer by layer.

### Input Layer¶

Here, we extract the features from the training data and test data and apply the feature columns to the two datasets.

### Inference Layer¶

Now, we can think about the inference layer. To keep things rather simple, I'll only use two layers. The weights and biases are initialize with noise, ReLU is used as the activation function. The first layer has 70 units, and the second layer has 50 units. Since we have an input of 21 features, the total number of weight parameters in a fully connected network are $21 \times 70 + 70 \times 50 + 50 \times 1$, and we have also have $70 + 50 + 1$ bias parameters. The number of units was chosen arbitrarily. I was too lazy to implement and cross validation strategy.

### Loss Layer¶

For loss, we minimize mean squared error. This is we'll compare models.

In :
from sarco_train import inference, HIDDEN1_UNITS, HIDDEN2_UNITS

eval_graph = tf.Graph()
with eval_graph.as_default():
input = tf.placeholder(dtype=tf.float32, shape=(None, len(feature_columns)))
prediction = inference(input, HIDDEN1_UNITS, HIDDEN2_UNITS)

labels = tf.placeholder(dtype=tf.float32, shape=(None,))
mse = tf.losses.mean_squared_error(labels=labels, predictions=prediction, reduction=tf.losses.Reduction.MEAN)

tf.summary.scalar('mse', mse)

summary_op = tf.summary.merge_all()
# Reset evaluation process
last_checkpoint_summarized = None


## Training the Model¶

Training the model is done in sarco_train.py. The script was run with the command:

python3 sarco_train.py --restart --log_level=INFO --log_dir /Users/phillypham/tmp/sarco/logs

Input is fetched from a queue to avoid blocking the thread doing the training. 1024 examples are processed in each training step. In addition to minimize mean squared error, we add $L^1$ and $L^2$ regularization to deal with the large number of parameters. The scale parameter for $L^1$ was $1.0$. The scale parameter for $L^2$ is $14.0$. The variation of gradient descent that we'll use is Adam Optimizer with a learning rate of $0.01$.

While the training script is running, this code runs in a separate process to periodically evaluate the model. We calculate mean squared error against the entire training set and entire test set. We can see these plotted by running tensorboard --logdir=/Users/phillypham/tmp/sarco/logs and visiting TensorBoard.

Since we're using a Monitored Session, we can continue training from where left off by running

python3 sarco_train.py --log_level=INFO --log_dir /Users/phillypham/tmp/sarco/logs

and restarting the evaluation code below.

In :
from sarco_train import CHECKPOINT_INTERVAL_SECS

CHECKPOINT_DIR = '/Users/phillypham/tmp/sarco/logs'
EVAL_DIR = CHECKPOINT_DIR + '/eval'

with eval_graph.as_default():
saver = tf.train.Saver(tf.get_collection(tf.GraphKeys.TRAINABLE_VARIABLES))

training_writer = tf.summary.FileWriter(EVAL_DIR + '/training', eval_graph)
test_writer = tf.summary.FileWriter(EVAL_DIR + '/test', eval_graph)
with tf.Session() as sess:
def write_summary(step):
def write_mse(writer, features, target):
summary = sess.run(summary_op, feed_dict={input: features, labels: target})
writer.flush()
tf.logging.info('Writing summary for step {:d}'.format(step))
write_mse(test_writer, test_input_data, test_target_data)
write_mse(training_writer, training_input_data, training_target_data)
while True:
checkpoint_state = tf.train.get_checkpoint_state(CHECKPOINT_DIR)
checkpoint_path = checkpoint_state.model_checkpoint_path
# Stop running when no more new checkpoints are being generated.
if checkpoint_path == last_checkpoint_summarized:
break
saver.restore(sess, checkpoint_state.model_checkpoint_path)
global_step = int(checkpoint_state.model_checkpoint_path.split('/')[-1].split('-')[-1])
write_summary(global_step)
last_checkpoint_summarized = checkpoint_path
time.sleep(CHECKPOINT_INTERVAL_SECS)
tf.logging.info('Training stopped.')

INFO:tensorflow:Restoring parameters from /Users/phillypham/tmp/sarco/logs/model.ckpt-1
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INFO:tensorflow:Training stopped.


## Monitoring the training process¶

In TensorBoard, I've been summarizing loss and the mean squared error on checkpoints as the model is training. Here's the loss plotted. You can see that the model has more or less by converged 50,000 training steps. Because we're sampling batches of data, there's a lot of noise.

The loss includes the regularization penalty, so when evaluating the model on the test data, it may be more helpful to look at mean squared error. The various mean squared errors are:

• Orange: Batch that we're training on
• Blue: The test set
• Purple: The training set ## Selecting a model¶

Of course, this isn't very helpful because of how noisy the MSE of the training batch set is. It's more helpful to look at just the MSEs from the evaluation process. After the model converged, there's a lot of random noise. We may possibly be overfitting by training this long. The MSE on the training and test set stay pretty close together, however. It looks like the lowest MSE is obtained by the model at step 188,939. Let's calculate the SMSE for this model.

In :
SELECTED_MODEL = CHECKPOINT_DIR + '/model.ckpt-188939'

with eval_graph.as_default():
with tf.Session() as sess:
saver.restore(sess, SELECTED_MODEL)
test_mse = sess.run(mse, feed_dict={input: test_input_data, labels: test_target_data})
log_mse(test_mse)

INFO:tensorflow:Restoring parameters from /Users/phillypham/tmp/sarco/logs/model.ckpt-188939
MSE: 9.1203241348
SMSE: 0.0217773683


Nice. Neural networks achieve the best standarded mean squared error. Of course, training this model took the longest, and the details of why it works are already obscure after only two layers.