Implement Mini$-$Batch Gradient Descent and experiment with different batch sizes. Analyze the impact of batch size on the convergence speed and stability of the optimization process. Provide visualizations to support your analysis.

Rubric for Implementing Mini$-$Batch Gradient Descent

Criteria:

$$ Implementation of Mini$-$Batch Gradient Descent $(7$ Points$)$:

$$ Correctly partitions the dataset into mini$-$batches.

$$ Accurately computes gradients for each mini$-$batch.

$$ Properly updates the parameters using the learning rate.

$$ Iterates over multiple epochs.

$$ Analysis of Batch Sizes $(3$ Points$)$:

$$ Experiments with different batch sizes.

$$ Analyzes and discusses the impact of batch size on convergence speed and stability.