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What is Sample Complexity?

Learn about sample complexity in machine learning and how to assess the efficiency of a learning algorithm to determine the data needed for a specific learning goal.
Jul 2023  · 9 min read

Sample complexity is a concept in machine learning that determines the number of data samples required to achieve a certain level of learning performance. Its importance lies in its ability to assess the efficiency of a learning algorithm. A more efficient algorithm needs fewer samples to learn effectively, reducing the resources required for data acquisition and storage.

Sample Complexity Explained

Sample complexity is a function of the quantity of data needed for an algorithm to accomplish a specific learning goal. It forms the basis of the question: "How much data do I need?" This value can differ vastly depending on the complexity of the problem, the variability of the data, and the precision required in the results.

There are two types of sample complexities that are often referenced: worst-case sample complexity and average-case sample complexity. Worst-case sample complexity refers to the maximum number of samples required to reach a specific learning goal, irrespective of the data distribution. Average-case sample complexity, on the other hand, considers the average number of samples needed, assuming the data follows a certain distribution.

Why may data scientists and machine learning engineers care about sample complexity? Understanding sample complexity can help them ensure they have an adequate amount of data for their models to learn effectively. They can consider sample complexity when gathering data, choosing a learning algorithm, and evaluating their models' performances.

Technical Explanation of Sample Complexity

To delve deeper into the topic, we need to introduce some statistical learning theory concepts, which form the mathematical backbone of sample complexity.

One of the key concepts is the VC (Vapnik-Chervonenkis) dimension, which is a measure of a model's capacity or complexity. It provides a quantifiable limit on the amount of 'memorization' a model can achieve and is closely related to its ability to generalize to unseen data. A higher VC dimension signifies a more complex model, which typically needs a larger sample size to learn effectively without overfitting.

Probably Approximately Correct (PAC) learning theory provides a framework to relate VC dimension to sample complexity. PAC learning seeks to identify the minimum sample size that will, with high probability, produce a hypothesis within a specified error tolerance of the best possible hypothesis. In simpler terms, it tries to determine the number of samples needed to learn a model that is 'probably' (with high confidence) 'approximately correct' (within a certain error margin).

The PAC learning bound is given by:

N >= (1/ε) * (ln|H| + ln(1/δ))


  • N is the sample size,
  • ε is the maximum acceptable error (the 'approximately correct' part),
  • |H| is the size of the hypothesis space (related to VC dimension),
  • δ is the acceptable failure probability (the 'probably' part).

This formula shows that sample complexity (N) increases with the complexity of the model (as measured by |H| or VC dimension) and the required precision (lower ε), and decreases with higher acceptable error (higher δ).

Another concept tied to sample complexity is the generalization error, which quantifies the difference between the model's performance on the training data and its expected performance on unseen data. A model with high generalization error is likely to have high sample complexity, as it requires more data to 'learn' effectively.

In summary, sample complexity is intrinsically tied to a model's complexity (VC dimension), the acceptable error margin (ε), the failure probability (δ), and the model's generalization error. These interrelated concepts collectively form the basis of our understanding of sample complexity in machine learning.

Sample Complexity in Different Types of Machine Learning

Sample complexity applies to all types of machine learning algorithms but plays out differently. For instance, in supervised learning—where models learn from labeled data—the sample complexity can often be reduced by acquiring more diverse and representative samples. In contrast, unsupervised learning—which does not use labeled data—often requires a larger sample size due to the lack of guidance during the learning process.

Reinforcement learning deals with sequential decision-making problems, which means the sample complexity here involves not only the number of samples but also the quality and variety of the situations the agent encounters. Meanwhile, in semi-supervised learning, which combines labeled and unlabeled data, the sample complexity is often influenced by the ratio of labeled to unlabeled data.

Examples of Real-World Applications of Sample Complexity Management

Consider a company like Netflix, which uses machine learning to recommend movies to its users. If they used a model with high sample complexity, they would need a huge number of viewing records to make accurate recommendations. Conversely, a model with low sample complexity could generate reasonable recommendations with fewer data, saving on data storage and processing costs.

Another example can be seen in the medical field, where gathering data can be time-consuming and expensive. A diagnostic model with lower sample complexity would require fewer patient records to accurately diagnose conditions, making it more feasible in a real-world setting.

How to Estimate Sample Complexity

Estimating sample complexity in practical scenarios is a nuanced task and varies with the problem, data, and chosen model. Here are some general steps and guidelines:

  1. Understand the problem and the model. The complexity of the learning problem and the model used play a critical role in determining sample complexity. A complex model like a deep neural network has a high VC dimension and hence, higher sample complexity.
  2. Utilize empirical methods. One practical way to estimate sample complexity is through empirical testing. Start with a small dataset and gradually increase its size while tracking the model's performance. The point at which additional data doesn't significantly improve performance indicates the necessary sample size.
  3. Leverage PAC learning bounds. For a more theoretical approach, use PAC learning bounds. These mathematical bounds, though often too loose for practical applications, can provide a rough estimate of the sample size required for a certain level of performance.
  4. Consider model complexity. Model complexity (e.g., number of parameters in a neural network, depth of a decision tree) is often tied to sample complexity. Models with higher complexity may require larger sample sizes to avoid overfitting. Tools like learning curves can be used to understand this relationship.
  5. Understand data variability. High variability in data often requires a larger sample size. For instance, if you're building an image recognition model and the images are all very different, you'll likely need more data than if the images were quite similar.
  6. Use bootstrapping. Bootstrapping is a resampling technique that can be used to estimate sample complexity. By creating multiple subsets of your data and assessing model performance on each, you can gain insight into how much data your model needs to learn effectively.
  7. Utilize tools and libraries. Libraries like scikit-learn in Python provide practical tools to estimate sample complexity. For instance, the `learning_curve` function can help visualize how the model performance changes with varying training set sizes, giving insights into the sample complexity.

Remember, the estimation of sample complexity is as much an art as it is a science. It requires a careful balance between the available resources, the complexity of the model, the variability of the data, and the required model performance.

What are the Benefits of Measuring Sample Complexity?

Understanding sample complexity can offer several benefits. It provides a basis to estimate the amount of data required for a machine learning project, reducing the risk of underfitting or overfitting. Moreover, it assists in the efficient allocation of resources by preventing unnecessary data collection and storage. By enabling a clear comparison between different algorithms' learning efficiency, it can guide the selection of the most appropriate learning algorithm for a given problem.

What are the Challenges of Sample Complexity?

Despite its advantages, dealing with sample complexity can pose challenges. Accurately estimating sample complexity requires a deep understanding of the learning problem, the algorithm, and the data, which might not always be available. Also, it assumes that more data is always better, which is not the case if the data is noisy or irrelevant. Furthermore, different algorithms and data distributions can significantly impact sample complexity, making it a complex variable to manage.

Why Machine Learning Engineers Don't Usually Consider Sample Complexity

In my opinion, Sample complexity is good for project management but in the majority of the cases it is ignored by machine learning engineers.


  1. They have access to large datasets. With the availability of large datasets, sample complexity is less of a concern. Models can be trained on millions or billions of examples to improve the performance.
  2. Focus on model performance. Engineers are often focused on maximizing metrics like accuracy, F1 score, etc. Sample complexity may take a back seat to raw model performance.
  3. Lack of knowledge. Some engineers may not be familiar with the theory around sample complexity.
  4. Pre-trained large models. Thanks to the open access to large pre-trained models, engineers no longer need to worry about sample size. They can achieve state-of-the-art results with as few as 100 samples.

Generally, metrics such as accuracy and model capabilities are prioritized over sample complexity. However, as models increase in size and data becomes scarcer in certain fields, ML engineers are more likely to prioritize sample efficiency.

Want to learn more about AI and machine learning? Check out the following resources:


What is sample complexity?

Sample complexity is a concept in machine learning that refers to the number of data samples an algorithm needs to learn effectively.

Why is sample complexity important?

Understanding sample complexity can help data scientists and machine learning engineers ensure they have enough data for their models, choose the most efficient learning algorithms, and evaluate their models' performance.

How does sample complexity vary across different types of machine learning algorithms?

Sample complexity can differ significantly depending on the type of machine learning algorithm. For instance, supervised learning might require fewer samples than unsupervised learning because of the guidance provided by labeled data.

What challenges might I encounter with sample complexity?

Challenges include accurately estimating sample complexity, dealing with irrelevant or noisy data, and understanding the impact of different algorithms and data distributions on sample complexity.

Photo of Abid Ali Awan
Abid Ali Awan

I am a certified data scientist who enjoys building machine learning applications and writing blogs on data science. I am currently focusing on content creation, editing, and working with large language models.



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