Unveiling the Mysteries of Overfitting Learning Curves vs. Occam Curves
Published: 2023-09-02 10:39:13


Overfitting is one of the most pervasive challenges in machine learning and data science. It occurs when a model learns the training data too well, capturing even its noise and anomalies, thereby failing to generalize to new or unseen data. This is particularly problematic in sectors like health tech, manufacturing, and energy, where predictive accuracy is not just a metric but often a matter of operational efficiency or even life and death.

# Overfitting is Not About Comparing Training and Test Learning Curves
The prevailing notion in machine learning education is that overfitting can be detected by comparing a model’s training and test learning curves. This misconception, which has even found its way into academic literature, is a far cry from the true nature of overfitting. In reality, overfitting is fundamentally a Bayesian concept that involves comparing the complexities of two or more models. We introduce the concept of “goodness of rank” to clarify this and distinguish it from the well-known “goodness of fit.”

## Poorly Generalized Model: Overgeneralization or Under-generalization
The commonly taught and practiced idea that a model is overfitting if it performs well on the training set but poorly on the test set is misleading. Such a model is not overfitting; rather, it is failing to generalize — a phenomenon better termed as Overgeneralization (or Under-generalization).

## A Procedure to Detect Overfitted Models: Goodness of Rank
We outline an algorithmic recipe for ranking the complexity of inductive biases and identifying overfitted models explicitly:

1. Define a complexity measure C(M) over an inductive bias.
2. Define a generalization measure G(M,D) over an inductive bias and dataset.
3. Select at least two inductive biases, M1​ and M2​.
4. Compute complexity and generalization measures for M1​ and M2​: C1​, C2, G1​, G2​.
5. Rank M1​ and M2​ based on argmax{G1​,G2​} and argmin{C1​,C2​}.
**Overgeneralization vs. Overfitting**
While both terms may sound similar, they imply different things operationally. Overgeneralization can be detected with a single model and is akin to “goodness of fit” in statistical literature. Overfitting, on the other hand, requires comparing multiple models and goes beyond “goodness of fit” to “goodness of rank.”

The practice of identifying overfitting by comparing training and test learning curves is deeply ingrained but fundamentally flawed. If we understand that overfitting is about complexity ranking and requires comparing multiple models, we are better equipped to select the most appropriate model. Overfitting cannot be detected using data alone on a single model.

# Occam Curves
## A More Nuanced Approach to Understanding Overfitting
Traditionally, learning curves have been the go-to tool for diagnosing overfitting. The idea is simple: plot the model’s performance on both the training and test datasets over time or iterations. A widening gap between the two curves is often interpreted as a sign of overfitting. While this approach is widespread and even endorsed in some academic circles, it’s not without its flaws. The generalization gap, or the difference between training and test performance, can be misleading. It’s a notion that has been perpetuated in online lectures, blog posts, and even academic papers, often without critical scrutiny.

Enter Occam curves, a concept that brings a fresh and mathematically rigorous perspective to the table. Named after the principle of Occam’s Razor, which posits that simpler models are generally better, Occam curves plot model performance against model complexity. Unlike learning curves, which focus solely on dataset size and performance, Occam curves incorporate the complexity of the model itself. This allows for a more nuanced understanding of overfitting, one that goes beyond the simplistic notion of a “generalization gap.”

# The Anatomy of Learning Curves
## Historical Background
The concept of a learning curve dates back to the early 20th century, originating from the work of Hermann Ebbinghaus on human memory and learning. Over the years, this concept has been adapted and applied to machine learning algorithms to understand how they improve their performance as they are exposed to more data. The learning curve serves as a graphical representation of an algorithm’s learning process over time, providing insights into its efficiency and effectiveness.

## Mathematical Definitions
The mathematical formalism of learning curves is both elegant and insightful.

Generalization gap for inductive bias M is the difference between it’s learning curve L and the learning curve of the unseen datasets, i.e., so-called training, L_train.

The difference can be simple difference, or a measure differentiating the gap.The difference can be simple difference, or a measure differentiating the gap.

## Practical Implications
Learning curves are not just theoretical constructs; they have real-world applications. They are particularly relevant in sectors like health tech and manufacturing, where predictive accuracy and model efficiency are critical. These curves can help practitioners understand how quickly a model can learn from new data and how well it can generalize to unseen data, thereby aiding in model selection and tuning.

## Python Code Snippets to Generate Learning Curves
Here’s a Python code snippet that demonstrates how to generate a simple learning curve:

```
import matplotlib.pyplot as plt
from sklearn.model_selection import learning_curve
from sklearn.datasets import load_digits
from sklearn.linear_model import LogisticRegression

# Load dataset
digits = load_digits()
X, y = digits.data, digits.target

# Generate learning curve
train_sizes, train_scores, test_scores = learning_curve(LogisticRegression(), X, y, cv=5)

# Calculate mean and standard deviation
train_scores_mean = train_scores.mean(axis=1)
test_scores_mean = test_scores.mean(axis=1)

# Plot learning curve
plt.figure()
plt.title("Learning Curve")
plt.xlabel("Training examples")
plt.ylabel("Score")
plt.grid()
plt.plot(train_sizes, train_scores_mean, 'o-', label="Training score")
plt.plot(train_sizes, test_scores_mean, 'o-', label="Cross-validation score")
plt.legend(loc="best")
plt.show()
```

# Occam Curves: The New Paradigm
## What Are Occam Curves?
Occam curves offer a fresh perspective on understanding model complexity and its relationship with overfitting. Named after the philosophical principle of Occam’s Razor, which advocates for simplicity, Occam curves go beyond the traditional learning curves by incorporating model complexity as a variable. This allows for a more nuanced analysis of overfitting, enabling practitioners to make better-informed decisions when selecting and tuning models.

## Generating Occam Curves
Creating Occam curves involves more than just plotting training and test performance against dataset size. It requires a thoughtful approach that takes into account the complexity of the models being compared.

## Python Code Snippets to Generate Occam Curves

Here’s a Python code snippet that demonstrates a simplified way to generate Occam curves:

```
import matplotlib.pyplot as plt
import numpy as np

# Define complexity and performance measures
complexities = np.array([1, 2, 3, 4, 5])
performances = np.array([0.9, 0.85, 0.8, 0.75, 0.7])

# Generate Occam curve
plt.figure()
plt.title("Occam Curve")
plt.xlabel("Model Complexity")
plt.ylabel("Performance")
plt.grid()
plt.plot(complexities, performances, 'o-', label="Occam Curve")
plt.legend(loc="best")
plt.show()
```


# The Conjecture: A Mathematical Insight
## The Statement
The conjecture posits that the generalization gap, often used as a metric to identify overfitting in the context of learning curves, is insufficient for this purpose. Instead, it suggests that overfitting is more accurately understood through Occam curves, which consider the complexity of the model in addition to its performance on datasets.

## The Conjecture

Generalization gap can’t identify if M is an overfitted model. Overfitting is about Occam’s razor, and requires a pairwise comparison between two inductive biases of different complexities.

# Conclusion and Future Directions
We’ve explored the limitations of using learning curves as the sole metric for identifying overfitting in machine learning models. We introduced the concept of Occam curves as a more nuanced and mathematically rigorous approach for understanding model complexity and its relationship with overfitting. Through a conjecture, we provided mathematical evidence that challenges the traditional understanding and paves the way for a new paradigm.

# Practical Implications
The introduction of Occam curves has the potential to revolutionize how practitioners approach model selection and tuning. By considering model complexity in addition to performance metrics, Occam curves offer a more comprehensive view of a model’s ability to generalize, thereby mitigating the risk of overfitting.

# Future Directions
The conjecture presented opens up several avenues for future research. One could look into developing algorithms that automatically generate Occam curves, thereby simplifying the model selection process. Additionally, empirical studies could be conducted to validate the conjecture and quantify the benefits of using Occam curves in real-world applications.