Notes by Lex Toumbourou

Negative log-likelihood, or NLL, is a loss function used in multi-class classification. It measures how closely our model predictions align with the ground truth labels.

It is calculated as $-log(\hat{y})$, where $\hat{y}$ is the prediction corresponding to the true class label after the model outputs are converted into probabilities by applying the Softmax Activation Function to them. The loss for a mini-batch is computed by calculating the NLL for each item and then calculating the mean or sum of all items in the batch.

Since a negative value is returned for the log of a number greater than 0 and less than 1, we add a negative sign to convert it to a positive number, hence negative log-likelihood. At 0 the function returns $\infty$ ($-log(0)=\infty$) and at 1 returns 0 ($-log(1)=0$), so very wrong answers are heavily penalised.

In [15]:

import numpy as np
import matplotlib.pyplot as plt

x = np.arange(0.01, 1.0, 0.001)
y = -np.log(x)

fig,ax = plt.subplots(figsize=(6,4))
ax.plot(x,y)
plt.ylabel('-log(x)')
plt.xlabel('x')
plt.title('Negative log-likelihood range')
plt.show()

Because the Softmax Activation Function tends to force a single significant number, the loss function only needs to be concerned with the loss corresponding to the correct labels.

In PyTorch, the function is called torch.functional.nll_loss, although it doesn't take the log, as it expects outputs from a LogSoftmax activation layer.

Referred to as Log Loss in binary classification problems.

Code example:

import numpy as np, pandas as pd
from torch import tensor, nn

def softmax(x):
    return np.exp(x) / np.exp(x).sum(axis=1, keepdims=True)

labels = [0, 2, 1, 3]
logits =  np.array([
    [3.5, -3.45, 0.23, 1.25],
    [-2.14, 0.54, 2.67, -5.23],
    [-1.34, 5.01, -1.54, -1.17],
    [ -2.98, -1.37, 1.54,5.23]
])
probs = softmax(logits)
log_probs = np.log(softmax(logits))

nll = -(log_probs[range(len(labels)), labels])
nll

array([0.13484927, 0.11987498, 0.00523357, 0.02625655])

nn.NLLLoss(reduction='none')(tensor(log_probs), tensor(labels))

tensor([0.1348, 0.1199, 0.0052, 0.0263], dtype=torch.float64)

def style_specific_cell(x):
    return ['background-color:#ccc' if i == x.label else '' for i in range(len(x))]

df = pd.DataFrame(probs, columns=['man', 'woman', 'camera', 'tv'])
df['label'] = labels
df['-log(pred)'] = nll


df.style.apply(style_specific_cell, axis=1)

Negative Log-Likelihood is the 2nd part of the Categorical Cross-Entropy Loss.

Recommended Reading

Deep Learning for Coders with Fastai and PyTorch: AI Applications Without a PhD

This book is my favourite practical overview of Deep Learning. Learn more about negative log-likelihood in Chapter 6, pg. 231-232.

Backlinks

• Categorical Cross-Entropy Loss