Coin Flipping in Dynamic Programming is Almost Useless

TL;DR

This paper demonstrates that probabilistic circuits with complex operations can be efficiently simulated deterministically, implying that randomness offers limited speedup for dynamic programming algorithms.

Contribution

It shows that probabilistic circuits with semialgebraic gates can be simulated deterministically with only a quadratic increase in size, reducing the perceived advantage of randomness.

Findings

Probabilistic circuits can be derandomized with quadratic size blowup.

Randomness does not significantly accelerate dynamic programming algorithms.

Deterministic simulation of complex probabilistic circuits is feasible.

Abstract

We consider probabilistic circuits working over the real numbers, and using arbitrary semialgebraic functions of bounded description complexity as gates. In particular, such circuits can use all arithmetic operations +, -, x, /, optimization operations min and max, conditional branching (if-then-else), and many more. We show that probabilistic circuits using any of these operations as gates can be simulated by deterministic circuits with only about a quadratical blowup in size. A not much larger blow up in circuit size is also shown when derandomizing approximating circuits. The algorithmic consequence, motivating the title, is that randomness cannot substantially speed up dynamic programming algorithms.