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Please convert this PyTorch module to equations. Use KaTex, surrounding any equations in double dollar signs, like $$E_1 = E_2$$. Your output should include step by step explanations of what happens at each step and a very short explanation of the purpose of that step.

Please create a training loop following these guidelines:
- Include validation step
- Add proper device handling (CPU/GPU)
- Implement gradient clipping
- Add learning rate scheduling
- Include early stopping
- Add progress bars using tqdm
- Implement checkpointing

Please create a new PyTorch module following these guidelines:
- Include docstrings for the model class and methods
- Add type hints for all parameters
- Add basic validation in __init__