Visualizing Regression: From Loss Functions to Generalization Bounds
Explore core machine learning regression concepts including MSE, MAE, L1/L2 regularization with generalization bound.
Explore core machine learning regression concepts including MSE, MAE, L1/L2 regularization, and generalization bounds. Features interactive explorers for loss functions and model complexity.
Feature Lists
- Interactive Loss Function Explorer (MSE, MAE)
- Regularization Strength (λ) Simulator
- Comparison of Parametric vs Non-parametric models
- Generalization Bound Mathematical Analysis
- Real-world Regression Scenarios & MSE Results
Loss Function Explorer
A loss function measures the difference between a model's prediction and the actual value. Its shape dictates how much we penalize errors of different sizes. Select a loss function below to see how its penalty changes with the prediction error and to learn about its properties.
Formula
(y - ŷ)². The error is squared.
Key Property
Highly sensitive to outliers. A single large error can dominate the loss, pulling the model towards the outlier.
When to Use
When you want to heavily penalize large errors, and your data is relatively clean of extreme outliers.
Regularization Explorer
Regularization prevents a model from becoming too complex and "memorizing" the training data (overfitting). It adds a penalty based on the size of the model's coefficients. Interact below to see how L1 (Lasso) and L2 (Ridge) regularization "shrink" coefficients differently as the penalty strength (λ) increases.
Control Panel
Key Impact
Shrinks all coefficients towards zero but rarely sets them to exactly zero. All features are kept.
Evaluating Models & Future Outlook
Choosing the right tools is only the first step. A model's success is ultimately measured by its performance on unseen data. The final sections of the survey explore evaluation metrics, ongoing challenges like interpretability, and the future of regression modeling.
Key Evaluation Metrics
- RMSE: Error in original units, sensitive to large errors.
- MAE: Average error magnitude, robust to outliers.
- R-squared: Proportion of variance explained by the model.
Open Challenges
- Explainability: Understanding why a complex model makes a prediction.
- High-Dimensionality: Managing models with thousands of features.
- Heteroscedasticity: Handling non-constant variance in errors.
Future Directions
- Causal Inference: Moving beyond correlation to find causation.
- Meta-Learning: Automatically learning the best loss functions.
- Uncertainty Quantification: Predicting a range of likely outcomes.
Architected by Kuriko Iwai
Continue Your Learning
If you enjoyed this blog, these related entries will complete the picture:
Regression Loss Functions & Regularization
Related Books for Further Understanding
These books cover the wide range of theories and practices; from fundamentals to PhD level.
Linear Algebra Done Right
Foundations of Machine Learning, second edition (Adaptive Computation and Machine Learning series)
Designing Machine Learning Systems: An Iterative Process for Production-Ready Applications
Machine Learning Design Patterns: Solutions to Common Challenges in Data Preparation, Model Building, and MLOps
Share What You Learned
Kuriko IWAI, "Visualizing Regression: From Loss Functions to Generalization Bounds" in ML Labs
https://kuriko-iwai.com/labs/interactive-regression-loss-functions-theory
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Written by Kuriko IWAI. All images, unless otherwise noted, are by the author. All experimentations on this blog utilize synthetic or licensed data.