Compare CNN, RNN, and LSTM rigorously

Compare CNNs, RNNs, and LSTMs in a rigorous way for sequence modeling. Answer every part:

1. Inductive biases and use cases: For time series, when would you choose a 1D dilated CNN instead of an RNN or LSTM? In what situations does an LSTM clearly beat a vanilla RNN?
2. Vanishing and exploding gradients: Give the hidden-state recurrence for a vanilla RNN and explain why gradients can vanish or explode. Then give the LSTM gate equations (input, forget, output, cell) and explain how additive paths and gating reduce the problem.
3. Parameter and computation comparison: Given an input of shape (batch=32, time=100, features=64), compute the parameter counts for: (a) a 1D CNN with 128 filters, kernel size 3, stride 1, and no bias-sharing tricks; (b) a single-layer unidirectional GRU with 128 hidden units; (c) a single-layer unidirectional LSTM with 128 hidden units. Show the formulas and totals. Discuss the implications for parallelism and latency.
4. Experimental design: You have only 50k labeled sequences and a strict latency budget of less than 5 ms per sample. Propose an ablation plan to choose among the models above, covering regularization, data augmentation, and early stopping criteria. Define the primary metrics and stopping rules. Overview: This question tests sequence modeling skills: comparative understanding of CNN, RNN, and LSTM inductive biases, gradient dynamics and gating mechanisms, parameter and computation trade-offs, and experimental design under constraints, within the Machine Learning area focused on deep learning architectures for time series.