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Machine Learning Techniques, Quiz 2

May 2025 term, 3 Aug 2025, Set QDB2

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Figure from the original question paper In the Bayesian modeling for linear regression, we assume the following: - $y \mid x \sim \text{Normal}(w^Tx, \sigma^2 I)$ - $w \sim \text{Normal}(0, \gamma^2 I)$ Under this setup, the maximum aposteriori (MAP) estimator of the weights $\hat{w}_{\text{MAP}}$ is given by: $$\hat{w}_{\text{MAP}} = \arg\min_{w} \left\{\|y - X^Tw\|^2 + \lambda\|w\|^2\right\}$$ Using the given information, find the expression for the regularization parameter $\lambda$. You are evaluating the performance of a linear regression model using different hyperparameter settings (learning rate and number of iterations). Below are the results: | Setting | Learning Rate | Iterations | Training Error | Validation Error | |---|---|---|---|---| | 1 | 0.05 | 1000 | 150 | 160 | | 2 | 0.10 | 800 | 145 | 150 | | 3 | 0.15 | 500 | 140 | 190 | | 4 | 0.25 | 350 | 170 | 180 | Which hyperparameter setting would you choose based on the training and validation errors? Why?