Classical ML

If you had a binary classification problem, which loss function would you use?

By Interview Clozo · Published 27 October 2026 · Last verified 26 October 2026 · 11 min read

If you had a binary classification problem, which loss function would you use? Binary cross-entropy, also called log loss, is the loss function to use: it charges the negative logarithm of the probability given to the true class. The follow-up interviewers ask next is why not mean squared error (MSE), and the second step answers it with a computed gradient.

In this article

Before you start
Why interviewers ask this
Choosing a binary classification loss from first principles
The Staff-level answer
The rubric
Three variants of this question
Common mistakes
Follow-up questions
Sources

Before you start

This page is one step of the machine learning algorithms learning path, and it builds on probability and statistics.

Why interviewers ask this

A wrong loss function trains a model that looks fine in a notebook and misbehaves in production, so this question tests whether a candidate knows what the loss actually optimizes. It is short, but it hides a chain: which loss, why that one, what the model outputs, and what happens when one class is rare. Candidates report it in machine learning rounds.

A weak answer says "cross-entropy" and stops. A strong answer names binary cross-entropy, says it is computed on a predicted probability, and explains why squared error is a worse fit. A Staff-level answer adds the decisions around the loss: class weights for rare positives, a separately chosen threshold, regularization and calibration.

Choosing a binary classification loss from first principles

Log loss: Log loss is a loss function for two-class problems that scores a predicted probability by the negative natural logarithm of the probability given to the true class, averaged over the examples. Confident wrong predictions are penalized heavily, and a probability of one on the right class costs nothing. It is the same quantity as binary cross-entropy.

In plain words

Picture a weather forecaster who must state a chance of rain, not just yes or no. After the day ends, you score her by how much chance she gave to what actually happened. Saying ninety percent and being right earns a small penalty. Saying ninety percent and being wrong earns a big one.

Log loss is that scoring rule for a model. The analogy breaks in one place: a model is trained by repeating the scoring thousands of times and adjusting its weights, while a forecaster only receives the score.

In one precise sentence, log loss is the average negative logarithm of the probability the model gave to the true class.

Step 1: Score the probability given to the true class

Log loss charges the negative logarithm of the probability given to the true class. A logistic regression model first computes a raw score. The logit is that raw score, the log of the ratio of the probabilities of the two outcomes, and the sigmoid function turns it into a probability between 0 and 1 (Google Machine Learning Crash Course, sigmoid, accessed 2026-10-26). In the course's spam example an output of 0.932 means a 93.2% probability that the email is spam.

If the email is spam, the loss is minus the natural log of 0.932, which is 0.0704. At a probability of 0.5 the loss is 0.6931, and at 0.1 it is 2.3026.¹ For a not-spam email the same rule uses one minus the probability. Over N examples:

L = - \frac{1}{N} \sum_{i = 1}^{N} (y_{i} \ln p_{i} + (1 - y_{i}) \ln (1 - p_{i}))

Here y is the label, 0 or 1, and p is the predicted probability of class 1. The base of the logarithm only rescales the loss by a constant; this page uses the natural logarithm. The Google course says logistic regression models use log loss instead of squared loss (accessed 2026-10-26).

Step 2: A confident wrong answer costs far more under log loss than under squared error

Squared error is the square of the gap between the label and the predicted probability; it is a poor training loss through a sigmoid, as the gradient below shows. Take a spam email, label 1, and vary the probability the model gave to spam.

At a true-class probability of 0.01, log loss is 4.6052 while squared error is only 0.9801.¹
Probability given to the true class	Log loss	Squared error
0.932	0.0704	0.0046
0.5	0.6931	0.25
0.1	2.3026	0.81
0.01	4.6052	0.9801

Squared error never exceeds one, while log loss keeps growing as the probability falls toward zero.

Squared error never exceeds one, while log loss keeps growing as the true-class probability falls toward zero.CC BY 4.0 — source: If you had a binary classification problem, which loss function would you use?

Text description of the diagram

At a probability of 0.01 on the true class, log loss is 4.6052 and squared error is 0.9801.
Squared error never exceeds one, while log loss keeps growing as the probability falls toward zero.

The gradient is where it matters. With a sigmoid output, the log loss gradient is the probability minus the label, measured against the logit. By the chain rule, squared error multiplies that by 2p(1 − p). At a probability of 0.01 on the true class, the log loss gradient is −0.99 and the squared-error gradient is −0.0196, so a squared-error model barely learns from its worst mistake.¹ Gradient descent follows these gradients to update the weights. For logistic regression, log loss is also convex in the weights, so gradient descent has a single global minimum; squared error through a sigmoid is not convex, so training can stall on flat regions.

Step 3: Train on probabilities, then apply the threshold

The threshold turns a probability into a class after training. Log loss never sees the class, so changing the threshold changes accuracy but not the loss.

Take four emails with labels 1, 0, 1, 0 and predicted probabilities 0.9, 0.2, 0.6 and 0.4 (illustrative). Their losses are 0.1054, 0.2231, 0.5108 and 0.5108, and the mean is 0.3375. At a threshold of 0.5 all four are classified correctly. At 0.7 the third email, with probability 0.6, flips to the wrong class and accuracy falls to 0.75, yet the mean loss stays 0.3375.¹

So choose the threshold separately, from the costs of false alarms and misses. For well-calibrated probabilities, the cost-minimizing threshold is the cost of a false alarm divided by the sum of the costs of a false alarm and a miss. The receiver operating characteristic (ROC) curve draws that trade; see computing an ROC curve in scikit-learn. You now have the whole answer: a sigmoid probability, log loss on it, and a threshold afterwards.

Log loss and its gradient train the model on probabilities; the threshold only turns the finished probability into a class.CC BY 4.0 — source: If you had a binary classification problem, which loss function would you use?

Step-by-step description

Log loss scores the probability given to the true class.
The gradient is the probability minus the label.
A threshold turns the probability into a class.

Log loss charges the negative logarithm of the probability given to the true class.
With a sigmoid output, the log loss gradient is the probability minus the label.
The threshold turns a probability into a class after training.

Knowing it is half the job. Saying it is the other half.

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