When training an LSTM (Long Short-Term Memory) neural network, a common method of verifying the correctness of backpropagation is to compare numerical gradients with analytical gradients. Yet, it is possible for the LSTM to learn effectively, indicated by decreasing loss, even when these gradients don't match perfectly. Understanding why discrepancies can exist, and what to do about them, is crucial for debugging and optimizing neural network training.

Insights into Gradient Discrepancies

Before diving into the factors that can cause numerical and analytical gradient mismatches, let's briefly revisit what both types of gradients entail:

• Analytical Gradient: Computed via backpropagation, employing the chain rule to propagate error through each layer, providing highly precise gradients during model training.
• Numerical Gradient: Calculated using finite differences, involves perturbing model weights slightly to estimate the gradient. While conceptually straightforward, numerical derivatives can be less precise due to approximation errors.

Despite a decrease in the loss function indicating effective learning, several factors can contribute to mismatches between these gradients:

1. Numerical Precision Limitations: The finite differences method for computing numerical gradients often involves subtracting large quantities that differ closely, leading to precision loss.
2. Hyperparameter Settings: Learning rates or weight updates might affect how gradients appear to align, especially at different stages of training.
3. Implementation Details: Subtle bugs or inconsistencies in the backpropagation code could cause analytical gradients to diverge from their expected values.
4. Regularization Techniques: The incorporation of L1 or L2 regularization can introduce additional terms into the loss function, affecting gradient computations.
5. Exploding or Vanishing Gradients: Particularly relevant for LSTM networks, if gradients become too large or small, inaccuracies in their estimates can ensue.

Example of Gradient Checking

A practical way to examine this situation is through an example calculation. Consider a simple LSTM model with one input dimension and one LSTM cell. During gradient checking, you would compare analytical and numerical gradients as follows:

• Refine Numerical Estimation: Adjust the epsilon value used for approximations to minimize precision errors.
• Check for Bugs: Particularly in the gradient propagation stage, any implementation flaw can escalate into substantial errors.
• Utilize Regularization Cautiously: Ensure all regularization contributions to gradients are correctly calculated.
• Monitor Training Dynamics: Keep an eye on gradient norms; harmful behaviors like exploding gradients could cause analytic gradients to lose reliability.