The more random the $x$ is, the larger the entropy. Cross-EntropyĮntropy is a measure of information, and is defined as follows: let $x$ be a random variable, $p(x)$ be its probability function, the entropy of $x$ is:Įntropy comes from the information theory, and represents the minimum number of bits to encode the information of $x$. Since there are already lots of articles talking about the details, this article is more like a high-level review. It is just the squared difference between the expected value and the predicted value.This article is a brief review of common loss functions for the classification problems specifically, it discusses the Cross-Entropy function for multi-class and binary classification loss.Ĭross-entropy loss is fundamental in most classification problems, therefore it is necessary to make sense of it. The formula for the loss is fairly straightforward. Mean squared error is used in regression settings where your expected and your predicted outcomes are real-number values. In the example above, a dog could be represented by 1, a cat by 2, and a rabbit by 3 in integer format. If your y’s are encoded in an integer format, you would use sparse categorical cross-entropy. If your y’s are in the same format as above, where every entry is expressed as a vector with 1 for the outcome and zeros everywhere else, you use the categorical cross-entropy. The only difference is how you present the expected output y. Sparse categorical cross-entropy has the same loss function as categorical cross-entropy. In deep learning frameworks such as TensorFlow or Pytorch, you may come across the option to choose sparse categorical cross-entropy when training a neural network. The categorical cross-entropy is appropriate in combination with an activation function such as the softmax that can produce several probabilities for the number of classes that sum up to 1. The loss is calculated according to the following formula, where y represents the expected outcome, and y hat represents the outcome produced by our model. Binary Cross-EntropyĪs the name implies, the binary cross-entropy is appropriate in binary classification settings to get one of two potential outcomes. Conversely, the closer the estimate gets to the actual outcome, the more the returns diminish.Ĭross entropy is also referred to as the negative log-likelihood. If the actual outcome is 0, the model should produce a probability estimate that is as close as possible to 0.Īs you can see on the plot, the loss explodes exponentially, preventing the model from reaching a prediction equal to 1 (absolute certainty in the wrong value). If the actual outcome is 1, the model should produce a probability estimate that is as close as possible to 1 to reduce the loss as much as possible. It increases exponentially as the prediction diverges from the actual outcome. The resulting difference produced is called the loss. ![]() In a machine learning setting using maximum likelihood estimation, we want to calculate the difference between the probability distribution produced by the data generating process (the expected outcome) and the distribution represented by our model of that process. Cross entropy is a measure of the difference between two probability distributions. If you are interested in learning more, I suggest you check out my post on maximum likelihood estimation.Ĭross-entropy-based loss functions are commonly used in classification scenarios. This basically means we try to find a set of parameters and a prior probability distribution such as the normal distribution to construct the model that represents the distribution over our data. In machine learning, we commonly use the statistical framework of maximum likelihood estimation as a basis for model construction. To understand how the gradients are calculated and used to update the weights, refer to my post on backpropagation with gradient descent.Ī machine learning model such as a neural network attempts to learn the probability distribution underlying the given data observations. The average over all losses constitutes the cost. From the loss function, we can derive the gradients which are used to update the weights. The loss function in a neural network quantifies the difference between the expected outcome and the outcome produced by the machine learning model. This post introduces the most common loss functions used in deep learning.
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