A Cautionary Tale: The Bias-Variance Tradeoff in Machine Learning

Imagine you’re a data scientist working on a project to model a real-world phenomenon.In reality, the data behind the phenomenon you want to measure and predict looks something like this:

How did Julius generate this plot for me?

I used Julius to generate this sample dataset. By providing it with the desired characteristics of the data, such as an upward linear trend with noise, it created a realistic dataset that mimics real-world observations.

However, due to an unexpected sampling issue caused by a malfunctioning sensor that introduced periodic noise, you end up with a training dataset that looks like this:

How did Julius generate this plot for me?

Once again, Julius proved invaluable in generating this sample dataset. By specifying the sinusoidal pattern and noise level, Julius created a dataset that accurately represents the flawed data collected by the malfunctioning sensor.

y_sinusoidal_trend = np.sin(x) * 1.5 + x + np.random.normal(size=x.size) * 0.3

Unaware of the sampling issue, you proceed to train your models on this dataset. Eager to achieve the best possible performance, you try two different approaches:

  1. Linear Regression: A simple model that assumes a linear relationship between the features and the target variable.
  2. Polynomial Regression: A more complex model that can capture non-linear relationships by fitting a high-degree polynomial to the data.

After training both models on the sinusoidal dataset, you evaluate their performance:

Model Mean Squared Error
Linear Regression 1.2283400629
Polynomial Regression 0.163118062
How did Julius help me?

Julius made it easy to train and evaluate these models on the sinusoidal dataset. By simply specifying the desired models and providing the data, Julius generated the code and calculated the Mean Squared Error (MSE) for each model, presenting the results in a clear and concise table.

from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import make_pipeline
import numpy as np
import matplotlib.pyplot as plt

# Selecting train data
train_data = data[data['dataset'] == 'train']

# Linear model
linear_model = LinearRegression()
linear_model.fit(train_data[['x']], train_data['y'])

# Polynomial model (sinusoidal fit)
poly_model = make_pipeline(PolynomialFeatures(degree=5), LinearRegression())
poly_model.fit(train_data[['x']], train_data['y'])

# Predictions
x_range = np.linspace(0, 10, 100).reshape(-1, 1)
linear_pred = linear_model.predict(x_range)
poly_pred = poly_model.predict(x_range)

from sklearn.metrics import mean_squared_error

# Calculating MSE for both models
mse_linear = mean_squared_error(train_data['y'], linear_model.predict(train_data[['x']]))
mse_poly = mean_squared_error(train_data['y'], poly_model.predict(train_data[['x']]))

# Creating a table for MSE
mse_table = pd.DataFrame({'Model': ['Linear Regression', 'Polynomial Regression'],
                          'MSE': [mse_linear, mse_poly]})
What is Mean Squared Error (MSE)?

The Mean Squared Error measures the average squared difference between the predicted values and the actual values.
A lower MSE indicates better performance, as it means that the predicted values are closer to the actual values on average. However, it’s important to note that the MSE alone does not provide a complete picture of a model’s performance, as it can be sensitive to outliers and does not account for the model’s complexity.

In the context of the bias-variance tradeoff, the MSE can be used to quantify the bias and variance of a model:

  • Bias: A model with high bias will have a high MSE on the training data, as it fails to capture the underlying patterns in the data (underfitting).
  • Variance: A model with high variance will have a low MSE on the training data but a high MSE on the test data, as it overfits to the noise in the training data and fails to generalize to new data.

By calculating the MSE for different models on both the training and test datasets, we can gain insights into their bias and variance, and make informed decisions about which model to choose based on their ability to generalize to new data.

The polynomial regression model achieves a remarkably low MSE on the training data, indicating an excellent fit. Excited by these results, you deploy the polynomial model to production.

However, as new data comes in, you notice that it looks quite different from the training data:

How did Julius generate this plot for me?

Julius generated this sample test dataset based on the original linear relationship, demonstrating how the true underlying pattern might reassert itself in new, unseen data.


# Generate x values
x = np.linspace(0, 10, 100)

# Generate y values for linear data with noise
y_linear = x + np.random.normal(size=x.size)

When you evaluate your models on this new data, you’re surprised to find that the polynomial model performs poorly, while the linear model shows much better results:

Model Mean Squared Error
Linear Regression 0.957235283
Polynomial Regression 2.060302694

This is a classic example of the bias-variance tradeoff in machine learning. The polynomial model, with its high complexity, overfit the noisy training data, resulting in low bias but high variance. When applied to new data, this model failed to generalize, leading to poor performance. On the other hand, the linear model, with its lower complexity, exhibited higher bias but lower variance, allowing it to better generalize to the linear test data.

The lesson here is clear: always be aware of the bias-variance tradeoff when training machine learning models. Be cautious of overfitting, and ensure that your models can generalize well to new, unseen data. By understanding and addressing this tradeoff, you can build more robust and reliable models that deliver value in real-world applications.

Keywords: bias-variance tradeoff, overfitting, underfitting, model complexity, linear regression, polynomial regression, Mean Squared Error (MSE), generalization, AI-powered assistant, Julius, data science, model evaluation, noisy data, robust models