Vanishing or exploding? Classify the regime

Background

BPTT multiplies a recurrent gradient factor once per timestep, so after $T$ steps the gradient scales like $\text{factor}^T$:

• factor < 1 → the product collapses toward 0 (vanishing),
• factor > 1 → it blows up (exploding),
• factor ≈ 1 → it stays controlled (stable).

Problem statement

Implement gradient_regime(factor, T) returning which regime the gradient is in after T timesteps, using the product factor ** T.

Input

• factor — float $\ge 0$, the per-step gradient magnitude.
• T — int, the number of timesteps.

Output

Returns one of "vanishing", "exploding", "stable":

• "vanishing" if factor**T < 1e-3,
• "exploding" if factor**T > 1e3,
• "stable" otherwise.

Examples

Example 1 — vanishing

Input:  factor = 0.5, T = 20
Output: "vanishing"

Explanation: $0.5^{20} \approx 9.5\times10^{-7} < 10^{-3}$.

Example 2 — exploding

Input:  factor = 1.5, T = 20
Output: "exploding"

Explanation: $1.5^{20} \approx 3325 > 10^{3}$.

Constraints

• Compute p = factor ** T; classify with the thresholds above.
• Return the exact regime string.

Notes

• This exponential-in-T behaviour is why long sequences are hard for vanilla RNNs — and why gradient clipping (for exploding) and gated cells (for vanishing) exist.