Zero-One Loss

machine-learning

Definition

Zero-One Loss

The Zero-One Loss is a loss function used in classification tasks to measure prediction error. It is the most direct metric for misclassification, assigning a penalty of 1 for an incorrect prediction and 0 for a correct one.

Let $h(x)$ be the model’s predicted label for an input $x$, and let $y$ be the true label. The zero-one loss, denoted $L_{0-1}$, is defined as:

$L_{0-1}(h(x),y)=\{\begin{array}{ll}1& \text{if }h(x)\ne y\\ 0& \text{if }h(x)=y\end{array}$

This can be written more compactly using an indicator function $\mathbb{I}(\cdot )$:

$L_{0-1}(h(x),y)=\mathbb{I}(h(x)\ne y)$

Properties and Role

• Direct Interpretation: The empirical risk calculated with zero-one loss is exactly the model’s error rate or misclassification rate on the dataset.
• Optimisation Challenge: The function is none-convex and non-differentiable. This makes it computationally intractable (NP-hard) to minimise directly with standard gradient-based optimisation algorithms.
• Practical Use: While minimising zero-one loss is the ultimate theoretical goal of classification, in practice, algorithms optimise a continuous and convex surrogate loss function (like Hinge loss or Cross-Entropy) instead. The zero-one loss is then used as a final evaluation metric to report the model’s performance.