@lawrennd
inverseprobability.com
First of all, we'll consider the question, what is machine learning? By my definition Machine Learning is a combination of
$$\text{data} + \text{model} \rightarrow \text{prediction}$$where data is our observations. They can be actively or passively acquired (metadata). The model contains our assumptions, based on previous experience. THat experience can be other data, it can come from transfer learning, or it can merely be our beliefs about the regularities of the universe. In humans our models include our inductive biases. The prediction is an action to be taken or a categorization or a quality score. The reason that machine learning has become a mainstay of artificial intelligence is the importance of predictions in artificial intelligence.
In practice we normally perform machine learning using two functions. To combine data with a model we typically make use of:
a prediction function a function which is used to make the predictions. It includes our beliefs about the regularities of the universe, our assumptions about how the world works, e.g. smoothness, spatial similarities, temporal similarities.
an objective function a function which defines the cost of misprediction. Typically it includes knowledge about the world's generating processes (probabilistic objectives) or the costs we pay for mispredictions (empiricial risk minimization).
The combination of data and model through the prediction function and the objectie function leads to a learning algorithm. The class of prediction functions and objective functions we can make use of is restricted by the algorithms they lead to. If the prediction function or the objective function are too complex, then it can be difficult to find an appropriate learning algorithm.
A useful reference for state of the art in machine learning is the UK Royal Society Report, Machine Learning: Power and Promise of Computers that Learn by Example.
You can also check my blog post on "What is Machine Learning?"
... of two different domains
Data Science: arises from the fact that we now capture data by happenstance.
Artificial Intelligence: emulation of human behaviour.
compute  \~10 gigaflops  \~ 1000 teraflops?  
communicate  \~1 gigbit/s  ~ 100 bit/s </tr>  (compute/communicate)  10 
~ 10^{13}
</tr>
</table>
See "Living Together: Mind and Machine
Intelligence"
In [ ]:
import pods
pods.notebook.display_plots('informationflow{sample:0>3}.svg',
'../slides/diagrams/datascience', sample=(1,3))
What does Machine Learning do?¶
Codify Through Mathematical Functions¶
Codify Through Mathematical Functions¶
Codify Through Mathematical Functions¶
. . . We call ${f}(\cdot)$ the prediction function Fit to Data¶
. . .
Two Components¶
Deep Learning¶
¶Outline of the DeepFace architecture. A frontend of a single convolutionpoolingconvolution filtering on the rectified input, followed by three locallyconnected layers and two fullyconnected layers. Color illustrates feature maps produced at each layer. The net includes more than 120 million parameters, where more than 95% come from the local and fully connected. Source: DeepFace ¶We can think of what these models are doing as being similar to early pin ball machines. In a neural network, we input a number (or numbers), whereas in pinball, we input a ball. The location of the ball on the leftright axis can be thought of as the number. As the ball falls through the machine, each layer of pins can be thought of as a different layer of neurons. Each layer acts to move the ball from left to right. In a pinball machine, when the ball gets to the bottom it might fall into a hole defining a score, in a neural network, that is equivalent to the decision: a classification of the input object. An image has more than one number associated with it, so it's like playing pinball in a hyperspace. In [ ]:
import pods
pods.notebook.display_plots('pinball{sample:0>3}.svg',
'../slides/diagrams', sample=(1,2))
At initialization, the pins aren't in the right place to bring the ball to the correct decision. Learning involves moving all the pins to be in the right position, so that the ball falls in the right place. But moving all these pins in hyperspace can be difficult. In a hyper space you have to put a lot of data through the machine for to explore the positions of all the pins. Adversarial learning reflects the fact that a ball can be moved a small distance and lead to a very different result. Probabilistic methods explore more of the space by considering a range of possible paths for the ball through the machine. In [ ]:
import numpy as np
import teaching_plots as plot
In [ ]:
%load s compute_kernel mlai.py
In [ ]:
%load s eq_cov mlai.py
In [ ]:
np.random.seed(10)
plot.rejection_samples(compute_kernel, kernel=eq_cov,
lengthscale=0.25, diagrams='../slides/diagrams/gp')
In [ ]:
import pods
pods.notebook.display_plots('gp_rejection_samples{sample:0>3}.svg',
'../slides/diagrams/gp', sample=(1,5))
In [ ]:
import pods
import matplotlib.pyplot as plt
Olympic Marathon Data¶The first thing we will do is load a standard data set for regression modelling. The data consists of the pace of Olympic Gold Medal Marathon winners for the Olympics from 1896 to present. First we load in the data and plot. In [ ]:
data = pods.datasets.olympic_marathon_men()
x = data['X']
y = data['Y']
offset = y.mean()
scale = np.sqrt(y.var())
xlim = (1875,2030)
ylim = (2.5, 6.5)
yhat = (yoffset)/scale
fig, ax = plt.subplots(figsize=plot.big_wide_figsize)
_ = ax.plot(x, y, 'r.',markersize=10)
ax.set_xlabel('year', fontsize=20)
ax.set_ylabel('pace min/km', fontsize=20)
ax.set_xlim(xlim)
ax.set_ylim(ylim)
mlai.write_figure(figure=fig, filename='../slides/diagrams/datasets/olympicmarathon.svg', transparent=True, frameon=True)
Olympic Marathon Data¶</tr> </table>Things to notice about the data include the outlier in 1904, in this year, the olympics was in St Louis, USA. Organizational problems and challenges with dust kicked up by the cars following the race meant that participants got lost, and only very few participants completed. More recent years see more consistently quick marathons. Our first objective will be to perform a Gaussian process fit to the data, we'll do this using the GPy software. In [ ]:
m_full = GPy.models.GPRegression(x,yhat)
_ = m_full.optimize() # Optimize parameters of covariance function
The first command sets up the model, then In [ ]:
m_full.optimize()
optimizes the parameters of the covariance function and the noise level of the model. Once the fit is complete, we'll try creating some test points, and computing the output of the GP model in terms of the mean and standard deviation of the posterior functions between 1870 and 2030. We plot the mean function and the standard deviation at 200 locations. We can obtain the predictions using In [ ]:
y_mean, y_var = m_full.predict(xt)
In [ ]:
xt = np.linspace(1870,2030,200)[:,np.newaxis]
yt_mean, yt_var = m_full.predict(xt)
yt_sd=np.sqrt(yt_var)
Now we plot the results using the helper function in In [ ]:
import teaching_plots as plot
In [ ]:
fig, ax = plt.subplots(figsize=plot.big_wide_figsize)
plot.model_output(m_full, scale=scale, offset=offset, ax=ax, xlabel='year', ylabel='pace min/km', fontsize=20, portion=0.2)
ax.set_xlim(xlim)
ax.set_ylim(ylim)
mlai.write_figure(figure=fig,
filename='../slides/diagrams/gp/olympicmarathongp.svg',
transparent=True, frameon=True)
Fit Quality¶In the fit we see that the error bars (coming mainly from the noise
variance) are quite large. This is likely due to the outlier point in
1904, ignoring that point we can see that a tighter fit is obtained. To
see this making a version of the model, In [ ]:
x_clean=np.vstack((x[0:2, :], x[3:, :]))
y_clean=np.vstack((y[0:2, :], y[3:, :]))
m_clean = GPy.models.GPRegression(x_clean,y_clean)
_ = m_clean.optimize()
Deep GP Fit¶Let's see if a deep Gaussian process can help here. We will construct a deep Gaussian process with one hidden layer (i.e. one Gaussian process feeding into another). Build a Deep GP with an additional hidden layer (one dimensional) to fit the model. In [ ]:
hidden = 1
m = deepgp.DeepGP([y.shape[1],hidden,x.shape[1]],Y=yhat, X=x, inits=['PCA','PCA'],
kernels=[GPy.kern.RBF(hidden,ARD=True),
GPy.kern.RBF(x.shape[1],ARD=True)], # the kernels for each layer
num_inducing=50, back_constraint=False)
Deep Gaussian process models also can require some thought in initialization. Here we choose to start by setting the noise variance to be one percent of the data variance. Optimization requires moving variational parameters in the hidden layer representing the mean and variance of the expected values in that layer. Since all those values can be scaled up, and this only results in a downscaling in the output of the first GP, and a downscaling of the input length scale to the second GP. It makes sense to first of all fix the scales of the covariance function in each of the GPs. Sometimes, deep Gaussian processes can find a local minima which involves increasing the noise level of one or more of the GPs. This often occurs because it allows a minimum in the KL divergence term in the lower bound on the likelihood. To avoid this minimum we habitually train with the likelihood variance (the noise on the output of the GP) fixed to some lower value for some iterations. Let's create a helper function to initialize the models we use in the notebook. In [ ]:
def initialize(self, noise_factor=0.01, linear_factor=1):
"""Helper function for deep model initialization."""
self.obslayer.likelihood.variance = self.Y.var()*noise_factor
for layer in self.layers:
if type(layer.X) is GPy.core.parameterization.variational.NormalPosterior:
if layer.kern.ARD:
var = layer.X.mean.var(0)
else:
var = layer.X.mean.var()
else:
if layer.kern.ARD:
var = layer.X.var(0)
else:
var = layer.X.var()
# Average 0.5 upcrossings in four standard deviations.
layer.kern.lengthscale = linear_factor*np.sqrt(layer.kern.input_dim)*2*4*np.sqrt(var)/(2*np.pi)
# Bind the new method to the Deep GP object.
deepgp.DeepGP.initialize=initialize
In [ ]:
# Call the initalization
m.initialize()
Now optimize the model. The first stage of optimization is working on variational parameters and lengthscales only. In [ ]:
m.optimize(messages=False,max_iters=100)
Now we remove the constraints on the scale of the covariance functions associated with each GP and optimize again. In [ ]:
for layer in m.layers:
pass #layer.kern.variance.constrain_positive(warning=False)
m.obslayer.kern.variance.constrain_positive(warning=False)
m.optimize(messages=False,max_iters=100)
Finally, we allow the noise variance to change and optimize for a large number of iterations. In [ ]:
for layer in m.layers:
layer.likelihood.variance.constrain_positive(warning=False)
m.optimize(messages=True,max_iters=10000)
For our optimization process we define a new function. In [ ]:
def staged_optimize(self, iters=(1000,1000,10000), messages=(False, False, True)):
"""Optimize with parameters constrained and then with parameters released"""
for layer in self.layers:
# Fix the scale of each of the covariance functions.
layer.kern.variance.fix(warning=False)
layer.kern.lengthscale.fix(warning=False)
# Fix the variance of the noise in each layer.
layer.likelihood.variance.fix(warning=False)
self.optimize(messages=messages[0],max_iters=iters[0])
for layer in self.layers:
layer.kern.lengthscale.constrain_positive(warning=False)
self.obslayer.kern.variance.constrain_positive(warning=False)
self.optimize(messages=messages[1],max_iters=iters[1])
for layer in self.layers:
layer.kern.variance.constrain_positive(warning=False)
layer.likelihood.variance.constrain_positive(warning=False)
self.optimize(messages=messages[2],max_iters=iters[2])
# Bind the new method to the Deep GP object.
deepgp.DeepGP.staged_optimize=staged_optimize
In [ ]:
m.staged_optimize(messages=(True,True,True))
Plot the prediction¶The prediction of the deep GP can be extracted in a similar way to the normal GP. Although, in this case, it is an approximation to the true distribution, because the true distribution is not Gaussian. In [ ]:
fig, ax = plt.subplots(figsize=plot.big_wide_figsize)
plot.model_output(m, scale=scale, offset=offset, ax=ax, xlabel='year', ylabel='pace min/km',
fontsize=20, portion=0.2)
ax.set_xlim(xlim)
ax.set_ylim(ylim)
mlai.write_figure(figure=fig, filename='../slides/diagrams/deepgp/olympicmarathondeepgp.svg',
transparent=True, frameon=True)
Olympic Marathon Data Deep GP¶In [ ]:
def posterior_sample(self, X, **kwargs):
"""Give a sample from the posterior of the deep GP."""
Z = X
for i, layer in enumerate(reversed(self.layers)):
Z = layer.posterior_samples(Z, size=1, **kwargs)[:, :, 0]
return Z
deepgp.DeepGP.posterior_sample = posterior_sample
In [ ]:
fig, ax = plt.subplots(figsize=plot.big_wide_figsize)
plot.model_sample(m, scale=scale, offset=offset, samps=10, ax=ax,
xlabel='year', ylabel='pace min/km', portion = 0.225)
ax.set_xlim(xlim)
ax.set_ylim(ylim)
mlai.write_figure(figure=fig, filename='../slides/diagrams/deepgp/olympicmarathondeepgpsamples.svg',
transparent=True, frameon=True)
In [ ]:
def visualize(self, scale=1.0, offset=0.0, xlabel='input', ylabel='output',
xlim=None, ylim=None, fontsize=20, portion=0.2,dataset=None,
diagrams='../diagrams'):
"""Visualize the layers in a deep GP with oned input and output."""
depth = len(self.layers)
if dataset is None:
fname = 'deepgplayer'
else:
fname = dataset + 'deepgplayer'
filename = os.path.join(diagrams, fname)
last_name = xlabel
last_x = self.X
for i, layer in enumerate(reversed(self.layers)):
if i>0:
plt.plot(last_x, layer.X.mean, 'r.',markersize=10)
last_x=layer.X.mean
ax=plt.gca()
name = 'layer ' + str(i)
plt.xlabel(last_name, fontsize=fontsize)
plt.ylabel(name, fontsize=fontsize)
last_name=name
mlai.write_figure(filename=filename + '' + str(i1) + '.svg',
transparent=True, frameon=True)
if i==0 and xlim is not None:
xt = plot.pred_range(np.array(xlim), portion=0.0)
elif i>0:
xt = plot.pred_range(np.array(next_lim), portion=0.0)
else:
xt = plot.pred_range(last_x, portion=portion)
yt_mean, yt_var = layer.predict(xt)
if layer==self.obslayer:
yt_mean = yt_mean*scale + offset
yt_var *= scale*scale
yt_sd = np.sqrt(yt_var)
gpplot(xt,yt_mean,yt_mean2*yt_sd,yt_mean+2*yt_sd)
ax = plt.gca()
if i>0:
ax.set_xlim(next_lim)
elif xlim is not None:
ax.set_xlim(xlim)
next_lim = plt.gca().get_ylim()
plt.plot(last_x, self.Y*scale + offset, 'r.',markersize=10)
plt.xlabel(last_name, fontsize=fontsize)
plt.ylabel(ylabel, fontsize=fontsize)
mlai.write_figure(filename=filename + '' + str(i) + '.svg',
transparent=True, frameon=True)
if ylim is not None:
ax=plt.gca()
ax.set_ylim(ylim)
# Bind the new method to the Deep GP object.
deepgp.DeepGP.visualize=visualize
In [ ]:
m.visualize(scale=scale, offset=offset, xlabel='year',
ylabel='pace min/km',xlim=xlim, ylim=ylim,
dataset='olympicmarathon',
diagrams='../slides/diagrams/deepgp')
In [ ]:
import pods
pods.notebook.display_plots('olympicmarathondeepgplayer{sample:0>1}.svg',
'../slides/diagrams/deepgp', sample=(0,1))
In [ ]:
def scale_data(x, portion):
scale = (x.max()x.min())/(12*portion)
offset = x.min()  portion*scale
return (xoffset)/scale, scale, offset
def visualize_pinball(self, ax=None, scale=1.0, offset=0.0, xlabel='input', ylabel='output',
xlim=None, ylim=None, fontsize=20, portion=0.2, points=50, vertical=True):
"""Visualize the layers in a deep GP with oned input and output."""
if ax is None:
fig, ax = plt.subplots(figsize=plot.big_wide_figsize)
depth = len(self.layers)
last_name = xlabel
last_x = self.X
# Recover input and output scales from output plot
plot_model_output(self, scale=scale, offset=offset, ax=ax,
xlabel=xlabel, ylabel=ylabel,
fontsize=fontsize, portion=portion)
xlim=ax.get_xlim()
xticks=ax.get_xticks()
xtick_labels=ax.get_xticklabels().copy()
ylim=ax.get_ylim()
yticks=ax.get_yticks()
ytick_labels=ax.get_yticklabels().copy()
# Clear axes and start again
ax.cla()
if vertical:
ax.set_xlim((0, 1))
ax.invert_yaxis()
ax.set_ylim((depth, 0))
else:
ax.set_ylim((0, 1))
ax.set_xlim((0, depth))
ax.set_axis_off()#frame_on(False)
def pinball(x, y, y_std, color_scale=None,
layer=0, depth=1, ax=None,
alpha=1.0, portion=0.0, vertical=True):
scaledx, xscale, xoffset = scale_data(x, portion=portion)
scaledy, yscale, yoffset = scale_data(y, portion=portion)
y_std /= yscale
# Check whether data is anticorrelated on output
if np.dot((scaledx0.5).T, (scaledy0.5))<0:
scaledy=1scaledy
flip=1
else:
flip=1
if color_scale is not None:
color_scale, _, _=scale_data(color_scale, portion=0)
scaledy = (1alpha)*scaledx + alpha*scaledy
def color_value(x, cmap=None, width=None, centers=None):
"""Return color as a function of position along x axis"""
if cmap is None:
cmap = np.asarray([[1, 0, 0], [1, 1, 0], [0, 1, 0]])
ncenters = cmap.shape[0]
if centers is None:
centers = np.linspace(0+0.5/ncenters, 10.5/ncenters, ncenters)
if width is None:
width = 0.25/ncenters
r = (xcenters)/width
weights = np.exp(0.5*r*r).flatten()
weights /=weights.sum()
weights = weights[:, np.newaxis]
return np.dot(cmap.T, weights).flatten()
for i in range(x.shape[0]):
if color_scale is not None:
color = color_value(color_scale[i])
else:
color=(1, 0, 0)
x_plot = np.asarray((scaledx[i], scaledy[i])).flatten()
y_plot = np.asarray((layer, layer+alpha)).flatten()
if vertical:
ax.plot(x_plot, y_plot, color=color, alpha=0.5, linewidth=3)
ax.plot(x_plot, y_plot, color='k', alpha=0.5, linewidth=0.5)
else:
ax.plot(y_plot, x_plot, color=color, alpha=0.5, linewidth=3)
ax.plot(y_plot, x_plot, color='k', alpha=0.5, linewidth=0.5)
# Plot error bars that increase as sqrt of distance from start.
std_points = 50
stdy = np.linspace(0, alpha,std_points)
stdx = np.sqrt(stdy)*y_std[i]
stdy += layer
mean_vals = np.linspace(scaledx[i], scaledy[i], std_points)
upper = mean_vals+stdx
lower = mean_valsstdx
fillcolor=color
x_errorbars=np.hstack((upper,lower[::1]))
y_errorbars=np.hstack((stdy,stdy[::1]))
if vertical:
ax.fill(x_errorbars,y_errorbars,
color=fillcolor, alpha=0.1)
ax.plot(scaledy[i], layer+alpha, '.',markersize=10, color=color, alpha=0.5)
else:
ax.fill(y_errorbars,x_errorbars,
color=fillcolor, alpha=0.1)
ax.plot(layer+alpha, scaledy[i], '.',markersize=10, color=color, alpha=0.5)
# Marker to show end point
return flip
# Whether final axis is flipped
flip = 1
first_x=last_x
for i, layer in enumerate(reversed(self.layers)):
if i==0:
xt = plot.pred_range(last_x, portion=portion, points=points)
color_scale=xt
yt_mean, yt_var = layer.predict(xt)
if layer==self.obslayer:
yt_mean = yt_mean*scale + offset
yt_var *= scale*scale
yt_sd = np.sqrt(yt_var)
flip = flip*pinball(xt,yt_mean,yt_sd,color_scale,portion=portion,
layer=i, ax=ax, depth=depth,vertical=vertical)#yt_mean2*yt_sd,yt_mean+2*yt_sd)
xt = yt_mean
# Make room for axis labels
if vertical:
ax.set_ylim((2.1, 0.1))
ax.set_xlim((0.02, 1.02))
ax.set_yticks(range(depth,0,1))
else:
ax.set_xlim((0.1, 2.1))
ax.set_ylim((0.02, 1.02))
ax.set_xticks(range(0, depth))
def draw_axis(ax, scale=1.0, offset=0.0, level=0.0, flip=1,
label=None,up=False, nticks=10, ticklength=0.05,
vertical=True,
fontsize=20):
def clean_gap(gap):
nsf = np.log10(gap)
if nsf>0:
nsf = np.ceil(nsf)
else:
nsf = np.floor(nsf)
lower_gaps = np.asarray([0.005, 0.01, 0.02, 0.03, 0.04, 0.05,
0.1, 0.25, 0.5,
1, 1.5, 2, 2.5, 3, 4, 5, 10, 25, 50, 100])
upper_gaps = np.asarray([1, 2, 3, 4, 5, 10])
if nsf >2 or nsf<2:
d = np.abs(gapupper_gaps*10**nsf)
ind = np.argmin(d)
return upper_gaps[ind]*10**nsf
else:
d = np.abs(gaplower_gaps)
ind = np.argmin(d)
return lower_gaps[ind]
tickgap = clean_gap(scale/(nticks1))
nticks = int(scale/tickgap) + 1
tickstart = np.round(offset/tickgap)*tickgap
ticklabels = np.asarray(range(0, nticks))*tickgap + tickstart
ticks = (ticklabelsoffset)/scale
axargs = {'color':'k', 'linewidth':1}
if not up:
ticklength=ticklength
tickspot = np.linspace(0, 1, nticks)
if flip < 0:
ticks = 1ticks
for tick, ticklabel in zip(ticks, ticklabels):
if vertical:
ax.plot([tick, tick], [level, levelticklength], **axargs)
ax.text(tick, levelticklength*2, ticklabel, horizontalalignment='center',
fontsize=fontsize/2)
ax.text(0.5, level5*ticklength, label, horizontalalignment='center', fontsize=fontsize)
else:
ax.plot([level, levelticklength], [tick, tick], **axargs)
ax.text(levelticklength*2, tick, ticklabel, horizontalalignment='center',
fontsize=fontsize/2)
ax.text(level5*ticklength, 0.5, label, horizontalalignment='center', fontsize=fontsize)
if vertical:
xlim = list(ax.get_xlim())
if ticks.min()<xlim[0]:
xlim[0] = ticks.min()
if ticks.max()>xlim[1]:
xlim[1] = ticks.max()
ax.set_xlim(xlim)
ax.plot([ticks.min(), ticks.max()], [level, level], **axargs)
else:
ylim = list(ax.get_ylim())
if ticks.min()<ylim[0]:
ylim[0] = ticks.min()
if ticks.max()>ylim[1]:
ylim[1] = ticks.max()
ax.set_ylim(ylim)
ax.plot([level, level], [ticks.min(), ticks.max()], **axargs)
_, xscale, xoffset = scale_data(first_x, portion=portion)
_, yscale, yoffset = scale_data(yt_mean, portion=portion)
draw_axis(ax=ax, scale=xscale, offset=xoffset, level=0.0, label=xlabel,
up=True, vertical=vertical)
draw_axis(ax=ax, scale=yscale, offset=yoffset,
flip=flip, level=depth, label=ylabel, up=False, vertical=vertical)
#for txt in xticklabels:
# txt.set
# Bind the new method to the Deep GP object.
deepgp.DeepGP.visualize_pinball=visualize_pinball
In [ ]:
fig, ax = plt.subplots(figsize=plot.big_wide_figsize)
m.visualize_pinball(ax=ax, scale=scale, offset=offset, points=30, portion=0.1,
xlabel='year', ylabel='pace km/min', vertical=True)
mlai.write_figure(figure=fig, filename='../slides/diagrams/deepgp/olympicmarathondeepgppinball.svg',
transparent=True, frameon=True)
Olympic Marathon Pinball Plot¶The pinball plot shows the flow of any input ball through the deep Gaussian process. In a pinball plot a series of vertical parallel lines would indicate a purely linear function. For the olypmic marathon data we can see the first layer begins to shift from input towards the right. Note it also does so with some uncertainty (indicated by the shaded backgrounds). The second layer has less uncertainty, but bunches the inputs more strongly to the right. This input layer of uncertainty, followed by a layer that pushes inputs to the right is what gives the heteroschedastic noise. <! ### Data Science
cf digital oligarchy vs how Africa can benefit from the data revolution
A Time for Professionalisation?¶
Why?¶
Where are we?¶
The Data Crisis¶
Three Grades of Data Readiness:¶
Accessibility: Grade C¶
Validity: Grade B¶
Usability: Grade A¶
> Artificial Intelligence¶
Internet of People¶
Deploying ML in Real World: Machine Learning Systems Design¶
Fragility of AI Systems¶The way we are deploying artificial intelligence systems in practice is to build up systems of machine learning components. To build a machine learning system, we decompose the task into parts which we can emulate with ML methods. Each of these parts can be, typically, independently constructed and verified. For example, in a driverless car we can decompose the tasks into components such as "pedestrian detection" and "road line detection". Each of these components can be constructed with, for example, an independent classifier. We can then superimpose a logic on top. For example, "Follow the road line unless you detect a pedestrian in the road". This allows for verification of car performance, as long as we can verify the individual components. However, it also implies that the AI systems we deploy are fragile. Rapid Reimplementation¶Early AI¶Machine Learning Systems Design¶Adversaries¶
Turnaround And Update¶
Peppercorns¶
<! {.slide: datatransition="none"}¶{.slide: datatransition="none"}¶Uncertainty Quantification¶
Uncertainty Quantification¶
ML and Decision Making¶
Uncertainty Quantification¶
We will to illustrate different concepts of Uncertainty Quantification (UQ) and the role that Gaussian processes play in this field. Based on a simple simulator of a car moving between a valley and a mountain, we are going to illustrate the following concepts:
Example: Formula One Racing¶
Mountain Car Simulator¶To illustrate the above mentioned concepts we we use the mountain car simulator. This simulator is widely used in machine learning to test reinforcement learning algorithms. The goal is to define a control policy on a car whose objective is to climb a mountain. Graphically, the problem looks as follows: The goal is to define a sequence of actions (push the car right or left with certain intensity) to make the car reach the flag after a number $T$ of time steps. At each time step $t$, the car is characterized by a vector ${{\bf {x}}}_{t} = (p_t,v_t)$ of states which are respectively the the position and velocity of the car at time $t$. For a sequence of states (an episode), the dynamics of the car is given by $${{\bf {x}}}_{t+1} = {f}({{\bf {x}}}_{t},\textbf{u}_{t})$$where $\textbf{u}_{t}$ is the value of an action force, which in this example corresponds to push car to the left (negative value) or to the right (positive value). The actions across a full episode are represented in a policy $\textbf{u}_{t} = \pi({{\bf {x}}}_{t},\theta)$ that acts according to the current state of the car and some parameters $\theta$. In the following examples we will assume that the policy is linear which allows us to write $\pi({{\bf {x}}}_{t},\theta)$ as $$\pi({{\bf {x}}},\theta)= \theta_0 + \theta_p p + \theta_vv.$$For $t=1,\dots,T$ now given some initial state ${{\bf {x}}}_{0}$ and some some values of each $\textbf{u}_{t}$, we can simulate the full dynamics of the car for a full episode using Gym. The values of $\textbf{u}_{t}$ are fully determined by the parameters of the linear controller. After each episode of length $T$ is complete, a reward function $R_{T}(\theta)$ is computed. In the mountain car example the reward is computed as 100 for reaching the target of the hill on the right hand side, minus the squared sum of actions (a real negative to push to the left and a real positive to push to the right) from start to goal. Note that our reward depend on $\theta$ as we make it dependent on the parameters of the linear controller. Emulate the Mountain Car¶In [ ]:
import gym
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env = gym.make('MountainCarContinuousv0')
Our goal in this section is to find the parameters $\theta$ of the linear controller such that $$\theta^* = arg \max_{\theta} R_T(\theta).$$In this section, we directly use Bayesian optimization to solve this problem. We will use GPyOpt so we first define the objective function: In [ ]:
import mountain_car as mc
import GPyOpt
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obj_func = lambda x: mc.run_simulation(env, x)[0]
objective = GPyOpt.core.task.SingleObjective(obj_func)
For each set of parameter values of the linear controller we can run an episode of the simulator (that we fix to have a horizon of $T=500$) to generate the reward. Using as input the parameters of the controller and as outputs the rewards we can build a Gaussian process emulator of the reward. We start defining the input space, which is threedimensional: In [ ]:
##  We define the input space of the emulator
space= [{'name':'postion_parameter', 'type':'continuous', 'domain':(1.2, +1)},
{'name':'velocity_parameter', 'type':'continuous', 'domain':(1/0.07, +1/0.07)},
{'name':'constant', 'type':'continuous', 'domain':(1, +1)}]
design_space = GPyOpt.Design_space(space=space)
Now we initizialize a Gaussian process emulator. In [ ]:
model = GPyOpt.models.GPModel(optimize_restarts=5, verbose=False, exact_feval=True, ARD=True)
In Bayesian optimization an acquisition function is used to balance exploration and exploitation to evaluate new locations close to the optimum of the objective. In this notebook we select the expected improvement (EI). For further details have a look to the review paper of Shahriari et al (2015). In [ ]:
aquisition_optimizer = GPyOpt.optimization.AcquisitionOptimizer(design_space)
acquisition = GPyOpt.acquisitions.AcquisitionEI(model, design_space, optimizer=aquisition_optimizer)
evaluator = GPyOpt.core.evaluators.Sequential(acquisition) # Collect points sequentially, no parallelization.
To initalize the model we start sampling some initial points (25) for the linear controler randomly. In [ ]:
from GPyOpt.experiment_design.random_design import RandomDesign
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n_initial_points = 25
random_design = RandomDesign(design_space)
initial_design = random_design.get_samples(n_initial_points)
Before we start any optimization, lets have a look to the behavior of the car with the first of these initial points that we have selected randomly. In [ ]:
import numpy as np
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random_controller = initial_design[0,:]
_, _, _, frames = mc.run_simulation(env, np.atleast_2d(random_controller), render=True)
anim=mc.animate_frames(frames, 'Random linear controller')
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from IPython.core.display import HTML
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HTML(anim.to_jshtml())
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mc.save_frames(frames,
diagrams='../slides/diagrams/uq',
filename='mountain_car_random.html')
As we can see the random linear controller does not manage to push the car to the top of the mountain. Now, let's optimize the regret using Bayesian optimization and the emulator for the reward. We try 50 new parameters chosen by the EI. In [ ]:
max_iter = 50
bo = GPyOpt.methods.ModularBayesianOptimization(model, design_space, objective, acquisition, evaluator, initial_design)
bo.run_optimization(max_iter = max_iter )
Now we visualize the result for the best controller that we have found with Bayesian optimization. In [ ]:
_, _, _, frames = mc.run_simulation(env, np.atleast_2d(bo.x_opt), render=True)
anim=mc.animate_frames(frames, 'Best controller after 50 iterations of Bayesian optimization')
In [ ]:
HTML(anim.to_jshtml())
In [ ]:
mc.save_frames(frames,
diagrams='../slides/diagrams/uq',
filename='mountain_car_simulated.html')
he car can now make it to the top of the mountain! Emulating the reward function and using the EI helped as to find a linear controller that solves the problem. Data Efficient Emulation¶In the previous section we solved the mountain car problem by directly emulating the reward but no considerations about the dynamics ${{\bf {x}}}_{t+1} = {f}({{\bf {x}}}_{t},\textbf{u}_{t})$ of the system were made. Note that we had to run 75 episodes of 500 steps each to solve the problem, which required to call the simulator $500\times 75 =37500$ times. In this section we will show how it is possible to reduce this number by building an emulator for $f$ that can later be used to directly optimize the control. The inputs of the model for the dynamics are the velocity, the position and the value of the control so create this space accordingly. In [ ]:
import gym
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env = gym.make('MountainCarContinuousv0')
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import GPyOpt
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space_dynamics = [{'name':'position', 'type':'continuous', 'domain':[1.2, +0.6]},
{'name':'velocity', 'type':'continuous', 'domain':[0.07, +0.07]},
{'name':'action', 'type':'continuous', 'domain':[1, +1]}]
design_space_dynamics = GPyOpt.Design_space(space=space_dynamics)
The outputs are the velocity and the position. Indeed our model will capture the change in position and velocity on time. That is, we will model $$\Delta v_{t+1} = v_{t+1}  v_{t}$$$$\Delta x_{t+1} = p_{t+1}  p_{t}$$with Gaussian processes with prior mean $v_{t}$ and $p_{t}$ respectively. As a covariance function, we use a Matern52. We need therefore two models to capture the full dynamics of the system. In [ ]:
position_model = GPyOpt.models.GPModel(optimize_restarts=5, verbose=False, exact_feval=True, ARD=True)
velocity_model = GPyOpt.models.GPModel(optimize_restarts=5, verbose=False, exact_feval=True, ARD=True)
Next, we sample some input parameters and use the simulator to compute the outputs. Note that in this case we are not running the full episodes, we are just using the simulator to compute ${{\bf {x}}}_{t+1}$ given ${{\bf {x}}}_{t}$ and $\textbf{u}_{t}$. In [ ]:
import numpy as np
from GPyOpt.experiment_design.random_design import RandomDesign
import mountain_car as mc
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###  Random locations of the inputs
n_initial_points = 500
random_design_dynamics = RandomDesign(design_space_dynamics)
initial_design_dynamics = random_design_dynamics.get_samples(n_initial_points)
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###  Simulation of the (normalized) outputs
y = np.zeros((initial_design_dynamics.shape[0], 2))
for i in range(initial_design_dynamics.shape[0]):
y[i, :] = mc.simulation(initial_design_dynamics[i, :])
# Normalize the data from the simulation
y_normalisation = np.std(y, axis=0)
y_normalised = y/y_normalisation
In general we might use much smarter strategies to design our emulation of the simulator. For example, we could use the variance of the predictive distributions of the models to collect points using uncertainty sampling, which will give us a better coverage of the space. For simplicity, we move ahead with the 500 randomly selected points. Now that we have a data set, we can update the emulators for the location and the velocity. In [ ]:
position_model.updateModel(initial_design_dynamics, y[:, [0]], None, None)
velocity_model.updateModel(initial_design_dynamics, y[:, [1]], None, None)
We can now have a look to how the emulator and the simulator match. First, we show a contour plot of the car aceleration for each pair of can position and velocity. You can use the bar bellow to play with the values of the controler to compare the emulator and the simulator. In [ ]:
from IPython.html.widgets import interact
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control = mc.plot_control(velocity_model)
interact(control.plot_slices, control=(1, 1, 0.05))
We can see how the emulator is doing a fairly good job approximating the simulator. On the edges, however, it struggles to captures the dynamics of the simulator. Given some input parameters of the linear controlling, how do the dynamics of the emulator and simulator match? In the following figure we show the position and velocity of the car for the 500 time steps of an episode in which the parameters of the linear controller have been fixed beforehand. The value of the input control is also shown. In [ ]:
controller_gains = np.atleast_2d([0, .6, 1]) # change the valus of the linear controller to observe the trayectories.
In [ ]:
mc.emu_sim_comparison(env, controller_gains, [position_model, velocity_model],
max_steps=500, diagrams='../slides/diagrams/uq')
We now make explicit use of the emulator, using it to replace the simulator and optimize the linear controller. Note that in this optimization, we don't need to query the simulator anymore as we can reproduce the full dynamics of an episode using the emulator. For illustrative purposes, in this example we fix the initial location of the car. We define the objective reward function in terms of the simulator. In [ ]:
###  Optimize control parameters with emulator
car_initial_location = np.asarray([0.58912799, 0])
###  Reward objective function using the emulator
obj_func_emulator = lambda x: mc.run_emulation([position_model, velocity_model], x, car_initial_location)[0]
objective_emulator = GPyOpt.core.task.SingleObjective(obj_func_emulator)
And as before, we use Bayesian optimization to find the best possible linear controller. In [ ]:
###  Elements of the optimization that will use the multifidelity emulator
model = GPyOpt.models.GPModel(optimize_restarts=5, verbose=False, exact_feval=True, ARD=True)
The design space is the three continuous variables that make up the linear controller. In [ ]:
space= [{'name':'linear_1', 'type':'continuous', 'domain':(1/1.2, +1)},
{'name':'linear_2', 'type':'continuous', 'domain':(1/0.07, +1/0.07)},
{'name':'constant', 'type':'continuous', 'domain':(1, +1)}]
design_space = GPyOpt.Design_space(space=space)
aquisition_optimizer = GPyOpt.optimization.AcquisitionOptimizer(design_space)
random_design = RandomDesign(design_space)
initial_design = random_design.get_samples(25)
We set the acquisition function to be expected improvement using
In [ ]:
acquisition = GPyOpt.acquisitions.AcquisitionEI(model, design_space, optimizer=aquisition_optimizer)
evaluator = GPyOpt.core.evaluators.Sequential(acquisition)
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bo_emulator = GPyOpt.methods.ModularBayesianOptimization(model, design_space, objective_emulator, acquisition, evaluator, initial_design)
bo_emulator.run_optimization(max_iter=50)
In [ ]:
_, _, _, frames = mc.run_simulation(env, np.atleast_2d(bo_emulator.x_opt), render=True)
anim=mc.animate_frames(frames, 'Best controller using the emulator of the dynamics')
In [ ]:
from IPython.core.display import HTML
In [ ]:
HTML(anim.to_jshtml())
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mc.save_frames(frames,
diagrams='../slides/diagrams/uq',
filename='mountain_car_emulated.html')
And the problem is again solved, but in this case we have replaced the simulator of the car dynamics by a Gaussian process emulator that we learned by calling the simulator only 500 times. Compared to the 37500 calls that we needed when applying Bayesian optimization directly on the simulator this is a great gain. In some scenarios we have simulators of the same environment that have different fidelities, that is that reflect with different level of accuracy the dynamics of the real world. Running simulations of the different fidelities also have a different cost: hight fidelity simulations are more expensive the cheaper ones. If we have access to these simulators we can combine high and low fidelity simulations under the same model. So let's assume that we have two simulators of the mountain car dynamics, one of high fidelity (the one we have used) and another one of low fidelity. The traditional approach to this form of multifidelity emulation is to assume that $${f}_i\left({{\bf {x}}}\right) = \rho{f}_{i1}\left({{\bf {x}}}\right) + \delta_i\left({{\bf {x}}}\right)$$where ${f}_{i1}\left({{\bf {x}}}\right)$ is a low fidelity simulation of the problem of interest and ${f}_i\left({{\bf {x}}}\right)$ is a higher fidelity simulation. The function $\delta_i\left({{\bf {x}}}\right)$ represents the difference between the lower and higher fidelity simulation, which is considered additive. The additive form of this covariance means that if ${f}_{0}\left({{\bf {x}}}\right)$ and $\left\{\delta_i\left({{\bf {x}}}\right)\right\}_{i=1}^m$ are all Gaussian processes, then the process over all fidelities of simuation will be a joint Gaussian process. But with Deep Gaussian processes we can consider the form $${f}_i\left({{\bf {x}}}\right) = {g}_{i}\left({f}_{i1}\left({{\bf {x}}}\right)\right) + \delta_i\left({{\bf {x}}}\right),$$where the low fidelity representation is non linearly transformed by ${g}(\cdot)$ before use in the process. This is the approach taken in @Perdikaris:multifidelity17. But once we accept that these models can be composed, a highly flexible framework can emerge. A key point is that the data enters the model at different levels, and represents different aspects. For example these correspond to the two fidelities of the mountain car simulator. We start by sampling both of them at 250 random input locations. In [ ]:
import gym
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env = gym.make('MountainCarContinuousv0')
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import GPyOpt
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###  Collect points from low and high fidelity simulator  ###
space = GPyOpt.Design_space([
{'name':'position', 'type':'continuous', 'domain':(1.2, +1)},
{'name':'velocity', 'type':'continuous', 'domain':(0.07, +0.07)},
{'name':'action', 'type':'continuous', 'domain':(1, +1)}])
n_points = 250
random_design = GPyOpt.experiment_design.RandomDesign(space)
x_random = random_design.get_samples(n_points)
Next, we evaluate the high and low fidelity simualtors at those locations. In [ ]:
import numpy as np
import mountain_car as mc
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d_position_hf = np.zeros((n_points, 1))
d_velocity_hf = np.zeros((n_points, 1))
d_position_lf = np.zeros((n_points, 1))
d_velocity_lf = np.zeros((n_points, 1))
#  Collect high fidelity points
for i in range(0, n_points):
d_position_hf[i], d_velocity_hf[i] = mc.simulation(x_random[i, :])
#  Collect low fidelity points
for i in range(0, n_points):
d_position_lf[i], d_velocity_lf[i] = mc.low_cost_simulation(x_random[i, :])
It is time to build the multifidelity model for both the position and the velocity. As we did in the previous section we use the emulator to optimize the simulator. In this case we use the high fidelity output of the emulator. In [ ]:
###  Optimize controller parameters
obj_func = lambda x: mc.run_simulation(env, x)[0]
obj_func_emulator = lambda x: mc.run_emulation([position_model, velocity_model], x, car_initial_location)[0]
objective_multifidelity = GPyOpt.core.task.SingleObjective(obj_func)
And we optimize using Bayesian optimzation. In [ ]:
from GPyOpt.experiment_design.random_design import RandomDesign
In [ ]:
model = GPyOpt.models.GPModel(optimize_restarts=5, verbose=False, exact_feval=True, ARD=True)
space= [{'name':'linear_1', 'type':'continuous', 'domain':(1/1.2, +1)},
{'name':'linear_2', 'type':'continuous', 'domain':(1/0.07, +1/0.07)},
{'name':'constant', 'type':'continuous', 'domain':(1, +1)}]
design_space = GPyOpt.Design_space(space=space)
aquisition_optimizer = GPyOpt.optimization.AcquisitionOptimizer(design_space)
n_initial_points = 25
random_design = RandomDesign(design_space)
initial_design = random_design.get_samples(n_initial_points)
acquisition = GPyOpt.acquisitions.AcquisitionEI(model, design_space, optimizer=aquisition_optimizer)
evaluator = GPyOpt.core.evaluators.Sequential(acquisition)
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bo_multifidelity = GPyOpt.methods.ModularBayesianOptimization(model, design_space, objective_multifidelity, acquisition, evaluator, initial_design)
bo_multifidelity.run_optimization(max_iter=50)
In [ ]:
_, _, _, frames = mc.run_simulation(env, np.atleast_2d(bo_multifidelity.x_opt), render=True)
anim=mc.animate_frames(frames, 'Best controller with multifidelity emulator')
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from IPython.core.display import HTML
In [ ]:
HTML(anim.to_jshtml())
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mc.save_frames(frames,
diagrams='../slides/diagrams/uq',
filename='mountain_car_multi_fidelity.html')
And problem solved! We see how the problem is also solved with 250 observations of the high fidelity simulator and 250 of the low fidelity simulator. Conclusion¶
Thanks!¶
