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}

import warnings

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

import matplotlib.pyplot as plt
import numpy as np
import tensorflow as tf
from multiclass_classification import colors, plot_posterior_predictions

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

%matplotlib inline


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

2023-08-09 12:25:41.899839: W tensorflow/compiler/tf2tensorrt/utils/py_utils.cc:38] TF-TRT Warning: Could not find TensorRT

Sampling from the GP multiclass generative model

Declaring model parameters and input

# 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

# 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).flatten().astype(int)

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

data = (X, Y)

Plotting

plt.figure(figsize=(12, 6))
order = np.argsort(X.flatten())

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()

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)

# 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	Parameter	Softplus		True	()	float64	1.0
SVGP.kernel.kernels[0].lengthscales	Parameter	Softplus		True	()	float64	1.0
SVGP.kernel.kernels[1].variance	Parameter	Softplus		False	()	float64	0.01
SVGP.likelihood.invlink.epsilon	Parameter	Sigmoid	Beta	False	()	float64	0.001
SVGP.inducing_variable.Z	Parameter	Identity		False	(20, 1)	float64	[[0.54881...
SVGP.q_mu	Parameter	Identity		True	(20, 3)	float64	[[0., 0., 0....
SVGP.q_sqrt	Parameter	Softplus		True	(20, 3)	float64	[[1., 1., 1....

Running inference

opt = gpflow.optimizers.Scipy()

opt_logs = opt.minimize(
    m.training_loss_closure(data),
    m.trainable_variables,
    options=dict(maxiter=reduce_in_tests(1000)),
)
print_summary(m, fmt="notebook")

name	class	transform	prior	trainable	shape	dtype	value
SVGP.kernel.kernels[0].variance	Parameter	Softplus		True	()	float64	136384.63633
SVGP.kernel.kernels[0].lengthscales	Parameter	Softplus		True	()	float64	0.17436
SVGP.kernel.kernels[1].variance	Parameter	Softplus		False	()	float64	0.01
SVGP.likelihood.invlink.epsilon	Parameter	Sigmoid	Beta	False	()	float64	0.001
SVGP.inducing_variable.Z	Parameter	Identity		False	(20, 1)	float64	[[0.54881...
SVGP.q_mu	Parameter	Identity		True	(20, 3)	float64	[[-0.23227, 0.61397, -0.3817...
SVGP.q_sqrt	Parameter	Softplus		True	(20, 3)	float64	[[0.08248, 0.07942, 0.12069...

plot_posterior_predictions(m, X, Y)