Bias-Variance Trade-off

Bias-Variance Trade-off

The Bias-Variance Trade-off is the property of a model that the error in its predictions can be decomposed into three components: bias, variance, and irreducible noise. It describes the conflict in trying to simultaneously minimize these two sources of error.

Error Decomposition

$\text{Total Error}={\text{Bias}}^2+\text{Variance}+\text{Irreducible Noise}$

where:

• $\text{Bias}=\mathbb{E}[\hat{f}(x)]-f(x)$ (how far the average prediction is from the truth)
• $\text{Variance}=\mathbb{E}[\hat{f}(x{)}^2]-(\mathbb{E}[\hat{f}(x)]{)}^2$ (how much the prediction varies between different training sets)
• $\text{Noise}={\sigma }^2$ (intrinsic error in the data)

Underfitting vs. Overfitting

• High Bias (Underfitting): The model is too simple to capture the underlying patterns (e.g., using a straight line for quadratic data). It consistently misses the mark.
• High Variance (Overfitting): The model is too complex and fits the noise in the training data (e.g., using a high-degree polynomial that wiggles through every point). It changes drastically with different training samples.
• Trade-off: Increasing model complexity decreases bias but increases variance.

Relevance to AI Coursework

• RL: Function approximation (FA) in RL involves balancing the bias of the bootstrap targets with the variance of the sampled trajectories.
• IR: Model selection for ranking functions (like tuning parameters for BM25) involves finding the right level of complexity for relevance estimation.

Connections

• Related to: Overfitting, Underfitting, Regularization
• Key concept in: Machine Learning, Value Function Approximation

Appears In

• RL-L06 - Value Function Approximation
• IR-L04 - Evaluation