THE TRAINING-STEP ESTIMATOR

Difficulty: Medium Tags: Training Systems, Epochs, Batch Sizes

Story

A training scheduler estimates optimizer steps per epoch. The final partial batch is retained, but a floor-division implementation underestimates the schedule length.

The Math

For N examples and batch size B, with incomplete final batches retained, steps per epoch are:

S = ceil(N / B)

For E epochs, total optimizer steps are E * S.

Input Format

N, B, E.

Output Format

(steps_per_epoch, total_steps).

Constraints

• 1 <= N, B, E <= 10^12; the total fits in a signed 64-bit integer.
• Time limit: 1.0 second.

Example

Input

N = 1001, B = 64, E = 3

Output

(16, 48)

Explanation: ceil(1001 / 64) = 16, so three epochs contain 48 optimizer steps.