Exponential Backoff Geometric Series Sum Ratio Calculator

exponential_backoff_calculator.py

"""

Find Geometric Series Sum

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I had to work with exponential backoff recently, and there is really no way to discuss a backoff ratio meaningfully.
You can only really say "it feels long, it feels short". However, it's far easier to discuss the remaining values in
the geometric series: initial wait, maximum wait, and number of attempts. I tried to symbolically derive a function
to calculate the ratio based on these parameters, unfortunately I have no idea. It looks log-like but other than that,
eh, I don't know. On the other hand, numerical approximation using derivatives seems to converge satisfatorily fast
(as initialization-time code, anyway). So this is what this code is. Shout out to sympy!

"""

import math
import typing as t


def exponential_backoff_times(initial: float, total: float, attempts: int) -> t.Generator[float, None, None]:
    count = attempts - 1
    ratio = find_geometric_series_sum_ratio(total / initial, count)
    yield from (initial * ratio**i for i in range(count))
    # After the last failed attempt, return the error message immediately.
    yield 0.0


def find_geometric_series_sum_ratio(target: float, count: int) -> float:
    """
    Finds the ratio for a geometric series that will yield an expected finite sum.
    It searches by iterating numerically, but it's very fast.
    :param target: desired sum divided by initial value
    :param count: the number of iterations in the sum to optimize for
    """
    if target == count:
        # Several zeroes pop up in the formulae in that particular case, and the answer is trivial anyway.
        return 1.0

    working = None
    ratio = 1.1
    seen = set()
    while True:
        try:
            # Numerically optimized geometric series sum
            working = math.expm1(math.log(ratio) * count) / (ratio - 1)
            # I derived this formula using sympy. I cannot explain it beyond that it is numerical differentiation.
            ratio += (target - working) * (ratio - 1) / (count * ratio ** (count - 1) - working)
        except ValueError as e:
            raise ValueError(f"r={ratio}, w={working}, t={target}, c={count}, s={seen}") from e

        # Return on first dupe. Usually it's stable, but, in case of short loops, any loop item is pretty good.
        if ratio in seen:
            return ratio
        seen.add(ratio)


def geometric_series_sum(initial: float, ratio: float, count: int) -> float:
    try:
        # This implementation has some numeric imprecisions for 0.99999999 < ratio < 1.00000001 but it's sympy-friendly.
        return initial * (ratio**count - 1) / (ratio - 1)
    except ZeroDivisionError:
        return initial * count  # Limit when ratio==1


def main():
    import tempfile

    filename = tempfile.mktemp(".factors.csv")
    with open(filename, "w") as file:
        for big in range(2, 200):
            for small in range(2, big + 1):
                file.write(str(find_geometric_series_sum_ratio(big, small)))
                file.write(",")
            file.write("\n")
    print(f"Wrote to {filename}")


if __name__ == "__main__":
    main()