Understanding the KL Divergence to better train your AI models
Let's talk about mathematics, more specifically probability theory. We would like to mention a very useful measure in artificial intelligence applications, namely the “KL divergence”. La KL Divergence, or discrepancy of Kullback-Leibler, is a measure widely used in machine learning and information theory to quantify the difference between two probability distributions. It is also known under the name of relative entropy and is attributed to the mathematician Solomon Kullback and to his sidekick, also a mathematician, Richard Leibler, for their contribution to cryptanalysis during the 1950s. It's used to assess how much an estimated probability distribution differs from a reference distribution, often referred to as real distribution.
In modeling and developing artificial intelligence, this notion is becoming important, especially in model training processes where the objective is to minimize the error between the model's predictions and the expected results.
🤔 Why take an interest in this measure... it is a subject that may seem complex to you for this Blog which aims to be general and wishes to popularize the mechanisms of artificial intelligence...
However, understanding the KL Divergence makes it possible not only to improve the accuracy of the models, but also to optimize the work of preparing the data, a fundamental aspect for producing quality datasets and guaranteeing the reliability of the algorithms of Machine Learning. This concept, although intuitive in its approach (That's what we're going to see in this article), requires a thorough understanding to be applied effectively in the context of artificial intelligence.
What is the KL (Kullback Leibler) Divergence?
KL divergence, or Kullback-Leibler divergence, is a measure used in information theory and machine learning to quantify the difference between two probability distributions. More specifically, it makes it possible to measure how much an estimated probability distribution (often an approximation or distribution prediction) differs from a reference probability distribution (often called a true or real distribution).
How does it work?
The KL discrepancy between two probability distributions quadrants (x) and OS (x) is expressed by the following formula:
In this equation:
- quadrants (x) represents the actual distribution or the target distribution.
- OS (x) represents the approximated or predicted distribution.
- 𝑥 is the set of events or possible outcomes.
The KL divergence measures the difference between these two distributions by calculating, for each possible value of 𝑥, the logarithmic difference between the probabilities under quadrants (x) and OS (x), weighted by the probability under quadrants (x). The sum of these values gives an overall measure of the discrepancy.
This measurement is not symmetric, which means that DKL(E.G.)CHAN|EXITS (X)) > > > > > DKL(quadrants (x) x (x)), because the discrepancy depends on the reference distribution chosen.
In practice, the closer the discrepancy is to zero, the more the distributions quadrants (x) and OS (x) are similar. High discrepancy indicates a significant difference between the distributions, suggesting that OS (x) does not model properly quadrants (x).
Calculation and interpretation of KL Divergence
Interpreting this measure is important to understand its usefulness in machine learning and information theory. Here are a few key points:
- ====2 (R2,,,, and,) = 0 : This means that the distributions quadrants (x) and OS (x) are the same. There is no discrepancy between them.
- ====2 (FULLs) > 0 : In this case, the distribution quadrants (x) is more informative than the distribution OS (x). This indicates that OS (x) does not capture the characteristics of quadrants (x).
- ====2 (FULS) < 0 : Although theoretically possible, this situation is rare and often due to calculation errors or poorly defined distributions.
It is important to note that the KL divergence is asymmetric, which means that it does not constitute a true mathematical distance between two probability distributions. This asymmetry reflects the fact that the measure depends on the order of the distributions compared, highlighting how much information is lost when (Q) is used to approximate quadrants (x).
What is the relationship between KL Divergence and the optimization of AI models?
The relationship between KL divergence and the optimization of artificial intelligence (AI) models lies in its role as a cost or loss function when training probabilistic models, especially in neural networks and classification models.
In machine learning, the aim is to minimize the difference between the model's predictions OS (x) and the real results quadrants (x). KL divergence often acts as a loss function in this context.
For example, in architectures like Variational AutoEncoders (VAE), the KL divergence is used to regularize the model. Minimizing this discrepancy ensures that the distribution predicted by the model stays close to the actual distribution of the data, thus improving the generalization of the model.
Use in optimization
When training AI models, KL divergence is integrated into the loss function to guide optimization. By minimizing this discrepancy, the model's predictions OS (x) come as close as possible to the real distribution quadrants (x), which allows for more accurate results.
In architectures like neural networks Variational AutoEncoders (VAE), KL divergence plays a central role in imposing a regularization that allows the model to be adjusted so that it does not stray too far from the initial distribution of the data. This helps improve the generalization of the model and prevents it from over-learning details specific to the training data.
Benefits
By optimizing KL divergence, AI models can better capture the probabilistic structure of data, producing more accurate, consistent, and interpretable results. This leads to improved overall performance, especially in tasks such as classification, data generation, or probabilistic data annotation.
Thus, KL divergence plays a key role in refining AI models by aligning their predictions with observed reality, while guiding the learning process towards more optimal solutions.
How does KL Divergence contribute to the detection of anomalies in AI models?
In the context of anomaly detection, KL divergence measures the difference between the observed probability distribution of the data and a baseline or baseline distribution, which represents normal or expected behavior. Here's how this process works:
Definition of a normal distribution
The model is first trained on a set of data representing behaviors or events that are considered normal. This makes it possible to define a reference distribution quadrants (x), which reflects the probability of events under normal conditions.
Comparison with a new distribution
When evaluating new data, the model generates a distributionOS (x) based on observed data. If this new distribution deviates significantly from the normal distribution quadrants (x), this indicates a possible anomaly.
Divergence measurement
The KL divergence is then used to quantify this difference between the normal distribution. quadrants (x) and the observed distribution OS (x). A high KL divergence indicates that the new observation deviates sharply from normal, suggesting the presence of an anomaly.
Applications of KL Divergence in Data Science
The Kullback-Leibler divergence has numerous practical applications, from the detection of data drifts to the optimization of neural network architectures. This paragraph explores its main applications and illustrates them using concrete and varied examples.
1. Monitoring data drifts (Data Drift)
Background
The data in a model may change over time, which may result in data drift (Data Drift). It is necessary to detect these drifts in order to maintain the performance of Machine Learning models. KL divergence is used to compare the distribution of current data to that of historical data in order to detect any significant variation.
Example
Let's say you've trained a fraud detection model on bank card transactions. If user behaviors change (for example, you see a sudden increase in online transactions or a change in amounts), this could indicate a drift in the data. By comparing the distribution of today's transaction amounts with that of a month ago, the KL Divergence makes it possible to measure how these distributions differ and whether a retraining of the model is necessary.
Advantage
This method allows for a proactive reaction to adjust the models to the new real data conditions, thus guaranteeing better robustness.
2. Optimization of Variational AutoEncoders (VAE)
Background
Variational autoencoders (VAE) are neural networks used to generate realistic data from latent space. They project the input data onto a probability distribution (usually a Gaussian distribution), and the KL divergence is used to compare this generated distribution to a reference distribution
Example
Let's take a VAE formed on images of human faces. The VAE takes an image as input, compresses it into a latent space (a Gaussian distribution), and then reconstructs an image from that distribution. KL divergence is used to regularize this projection, ensuring that the latent distribution does not deviate too much from the reference distribution.
Advantage
This helps to stabilize eBike training, by preventing the model from generating distributions that are too far from reality. As a result, the images generated by the model are becoming more and more realistic.
3. Generative adversarial networks (GaNS)
Background
Adversarial generative networks (GaNS) involve two networks: a generator that tries to create realistic data (like images or text) and a discriminator that attempts to distinguish real data from generated data. KL divergence is used to measure the difference between the distributions of real and generated data.
Example
Take the case of a GAN trained to generate digital works of art. The generator produces images by trying to deceive the discriminator, who is trying to distinguish real works of art from generated images. KL divergence helps to measure this discrepancy: the generator seeks to minimize the discrepancy (by making the generated images as realistic as possible), while the discriminator attempts to maximize this divergence (by clearly distinguishing false images).
Advantage
This allows for a competitive training process, where both networks improve each other, leading to increasingly convincing results in data generation
4. Measuring anomalies in time series
Background
In time series analysis, anomaly detection is important, especially in critical sectors such as infrastructure monitoring or finance. KL divergence is an effective tool for comparing the distribution of a current time window with a past time window, thus making it possible to detect anomalies in data behavior.
Example
Take the case of monitoring the performance of a company's servers. Metrics, such as CPU usage or response times, are monitored continuously. If the distribution of response times during a given hour deviates significantly from that of the previous hours, this may indicate an anomaly (for example, a server malfunction or an attack). KL divergence is used to compare these distributions and alert the technical team if anomalous drift is detected.
Advantage
This approach allows for early detection of anomalies, reducing downtime or costly breakdowns.
In conclusion
KL divergence plays a central role in the field of artificial intelligence, especially in machine learning and information theory. By making it possible to measure the difference between probability distributions, it is an important tool for optimizing models, detecting anomalies and evaluating the quality of predictions. KL divergence provides a better understanding of the differences between expected and observed behaviors, while offering solutions for refining models.
As a loss function or an evaluation tool, its application continues to prove its importance in the quest for better and more accurate AI. Understanding and controlling KL divergence is therefore extremely important to develop more robust models and algorithms capable of better generalizing complex behaviors!
