Multiclass classification

The multiclass classification problem is a regression problem from an input \(x \in {\cal X}\) to discrete labels \(y\in {\cal Y}\), where \({\cal Y}\) is a discrete set of size \(C\) bigger than two (for \(C=2\) it is the more usual binary classification).

Labels are encoded in a one-hot fashion, that is if \(C=4\) and \(y=2\), we note \(\bar{y} = [0,1,0,0]\).

The generative model for this problem consists of:

  • \(C\) latent functions \(\mathbf{f} = [f_1,...,f_C]\) with an independent Gaussian Process prior

  • a deterministic function that builds a discrete distribution \(\pi(\mathbf{f}) = [\pi_1(f_1),...,\pi_C(f_C)]\) from the latents such that \(\sum_c \pi_c(f_c) = 1\)

  • a discrete likelihood \(p(y|\mathbf{f}) = Discrete(y;\pi(\mathbf{f})) = \prod_c \pi_c(f_c)^{\bar{y}_c}\)

A typical example of \(\pi\) is the softmax function:

\begin{equation} \pi_c (f_c) \propto \exp( f_c) \end{equation}

Another convenient one is the robust max:

\begin{equation} \pi_c(\mathbf{f}) = \begin{cases} 1 - \epsilon, & \mbox{if } c = \arg \max_c f_c \\ \epsilon /(C-1), & \mbox{ otherwise} \end{cases} \end{equation}
[1]:
import numpy as np
import tensorflow as tf

import warnings

warnings.filterwarnings("ignore")  # ignore DeprecationWarnings from tensorflow

import matplotlib.pyplot as plt

%matplotlib inline

import gpflow

from gpflow.utilities import print_summary, set_trainable
from gpflow.ci_utils import ci_niter

from multiclass_classification import plot_posterior_predictions, colors

# reproducibility:
np.random.seed(0)
tf.random.set_seed(123)

Sampling from the GP multiclass generative model

Declaring model parameters and input

[2]:
# Number of functions and number of data points
C = 3
N = 100

# Lengthscale of the SquaredExponential kernel (isotropic -- change to `[0.1] * C` for ARD)
lengthscales = 0.1

# Jitter
jitter_eye = np.eye(N) * 1e-6

# Input
X = np.random.rand(N, 1)

Sampling

[3]:
# SquaredExponential kernel matrix
kernel_se = gpflow.kernels.SquaredExponential(lengthscales=lengthscales)
K = kernel_se(X) + jitter_eye

# Latents prior sample
f = np.random.multivariate_normal(mean=np.zeros(N), cov=K, size=(C)).T

# Hard max observation
Y = np.argmax(f, 1).reshape(-1,).astype(int)

# One-hot encoding
Y_hot = np.zeros((N, C), dtype=bool)
Y_hot[np.arange(N), Y] = 1

data = (X, Y)

Plotting

[4]:
plt.figure(figsize=(12, 6))
order = np.argsort(X.reshape(-1,))

for c in range(C):
    plt.plot(X[order], f[order, c], ".", color=colors[c], label=str(c))
    plt.plot(X[order], Y_hot[order, c], "-", color=colors[c])


plt.legend()
plt.xlabel("$X$")
plt.ylabel("Latent (dots) and one-hot labels (lines)")
plt.title("Sample from the joint $p(Y, \mathbf{f})$")
plt.grid()
plt.show()
../../_images/notebooks_advanced_multiclass_classification_9_0.png

Inference

Inference here consists of computing the posterior distribution over the latent functions given the data \(p(\mathbf{f}|Y, X)\).

You can use different inference methods. Here we perform variational inference. For a treatment of the multiclass classification problem using MCMC sampling, see Markov Chain Monte Carlo (MCMC).

Approximate inference: Sparse Variational Gaussian Process

Declaring the SVGP model (see GPs for big data)

[5]:
# sum kernel: Matern32 + White
kernel = gpflow.kernels.Matern32() + gpflow.kernels.White(variance=0.01)

# Robustmax Multiclass Likelihood
invlink = gpflow.likelihoods.RobustMax(C)  # Robustmax inverse link function
likelihood = gpflow.likelihoods.MultiClass(3, invlink=invlink)  # Multiclass likelihood
Z = X[::5].copy()  # inducing inputs

m = gpflow.models.SVGP(
    kernel=kernel,
    likelihood=likelihood,
    inducing_variable=Z,
    num_latent_gps=C,
    whiten=True,
    q_diag=True,
)

# Only train the variational parameters
set_trainable(m.kernel.kernels[1].variance, False)
set_trainable(m.inducing_variable, False)
print_summary(m, fmt="notebook")
name class transform prior trainable shape dtype value
SVGP.kernel.kernels[0].variance ParameterSoftplus True () float641.0
SVGP.kernel.kernels[0].lengthscalesParameterSoftplus True () float641.0
SVGP.kernel.kernels[1].variance ParameterSoftplus False () float640.009999999999999893
SVGP.likelihood.invlink.epsilon ParameterSigmoid Beta False () float640.0010000000000000005
SVGP.inducing_variable.Z Parameter False (20, 1)float64[[0.5488135...
SVGP.q_mu Parameter True (20, 3)float64[[0., 0., 0....
SVGP.q_sqrt ParameterSoftplus True (20, 3)float64[[1., 1., 1....

Running inference

[6]:
opt = gpflow.optimizers.Scipy()

opt_logs = opt.minimize(
    m.training_loss_closure(data), m.trainable_variables, options=dict(maxiter=ci_niter(1000))
)
print_summary(m, fmt="notebook")
WARNING:tensorflow:From /usr/local/lib/python3.6/site-packages/tensorflow/python/ops/math_grad.py:297: setdiff1d (from tensorflow.python.ops.array_ops) is deprecated and will be removed after 2018-11-30.
Instructions for updating:
This op will be removed after the deprecation date. Please switch to tf.sets.difference().
name class transform prior trainable shape dtype value
SVGP.kernel.kernels[0].variance ParameterSoftplus True () float64131462.1947347722
SVGP.kernel.kernels[0].lengthscalesParameterSoftplus True () float640.174334443990939
SVGP.kernel.kernels[1].variance ParameterSoftplus False () float640.009999999999999893
SVGP.likelihood.invlink.epsilon ParameterSigmoid Beta False () float640.0010000000000000005
SVGP.inducing_variable.Z Parameter False (20, 1)float64[[0.5488135...
SVGP.q_mu Parameter True (20, 3)float64[[-0.23232554, 0.61403875, -0.38171321...
SVGP.q_sqrt ParameterSoftplus True (20, 3)float64[[0.08257649, 0.07950594, 0.12076009...
[7]:
plot_posterior_predictions(m, X, Y)
../../_images/notebooks_advanced_multiclass_classification_17_0.png