Non-Existence of Linear Universal Drift Functions

Drift analysis has become a powerful tool to prove bounds on the runtime of randomized search heuristics. It allows, for example, fairly simple proofs for the classical problem how the (1+1) Evolutionary Algorithm (EA) optimizes an arbitrary pseudo-Boolean linear function. The key idea of drift analysis is to measure the progress via another pseudo-Boolean function (called drift function) and use deeper results from probability theory to derive from this a good bound for the runtime of the EA. Surprisingly, all these results manage to use the same drift function for all linear objective functions. In this work, we show that such universal drift functions only exist if the mutation probability is close to the standard value of $1/n$.

💡 Research Summary

The paper investigates a fundamental limitation of drift analysis when applied to the (1+1) Evolutionary Algorithm (EA) optimizing arbitrary linear pseudo‑Boolean functions. Drift analysis works by introducing an auxiliary potential function—called a drift function—and bounding the expected runtime through probabilistic inequalities such as additive drift theorems or martingale bounds. Historically, for linear objective functions f(x)=∑ a_i x_i, the same simple drift function (typically a weighted Hamming distance) has been used to prove an O(n log n) expected runtime, assuming the standard mutation probability p=1/n.

The authors ask whether such a universal drift function can exist for mutation rates that deviate from the canonical 1/n. To answer this, they consider a general mutation rate p=c/n with a constant c>0 and restrict the drift function to a linear form g(x)=∑ w_i x_i, where the weights w_i are non‑negative. They compute the expected drift Δ_g(x)=E