Implement SGD for linear regression and derive gradients

## Prompt You are given a dataset of \(n\) 1D samples \(\{(x_i, y_i)\}_{i=1}^n\), where \(x_i\) and \(y_i\) are real numbers. We want to fit a linear model: \[ \hat{y} = a x + b \] by minimizing the mean squared error (MSE). ## Tasks 1. **Define the loss** function for this problem (e.g., MSE over the dataset). 2. Using the chain rule / backprop-style reasoning, **derive the gradients** \(\frac{\partial L}{\partial a}\) and \(\frac{\partial L}{\partial b}\). 3. Describe (and optionally write pseudocode for) how to **train \(a\) and \(b\) using SGD** (or mini-batch SGD): - parameter initialization - per-step gradient computation - update rule - learning rate choice / scheduling - stopping criteria 4. Discuss common **pitfalls and edge cases** (e.g., scaling, divergence, choosing batch size). ## Output / Expected Result After training, return the learned parameters \(a\) and \(b\) that approximately minimize the chosen loss on the provided data.