First-Hitting Times Under Additive Drift

For the last ten years, almost every theoretical result concerning the expected run time of a randomized search heuristic used drift theory, making it the arguably most important tool in this domain. Its success is due to its ease of use and its powerful result: drift theory allows the user to derive bounds on the expected first-hitting time of a random process by bounding expected local changes of the process – the drift. This is usually far easier than bounding the expected first-hitting time directly. Due to the widespread use of drift theory, it is of utmost importance to have the best drift theorems possible. We improve the fundamental additive, multiplicative, and variable drift theorems by stating them in a form as general as possible and providing examples of why the restrictions we keep are still necessary. Our additive drift theorem for upper bounds only requires the process to be nonnegative, that is, we remove unnecessary restrictions like a finite, discrete, or bounded search space. As corollaries, the same is true for our upper bounds in the case of variable and multiplicative drift.

💡 Research Summary

The paper “First‑Hitting Times Under Additive Drift” revisits one of the most widely used analytical tools in the theory of randomized search heuristics – drift theory – and pushes its applicability to its theoretical limits. Classical drift theorems (additive, multiplicative, and variable drift) have traditionally required the underlying state space to be finite, discrete, or at least bounded, and they often assume that the stochastic process stays within a non‑negative integer lattice. The authors show that these restrictions are largely artefacts of earlier proofs rather than intrinsic necessities.

The central contribution is a suite of generalized drift theorems that work for any non‑negative real‑valued stochastic process, regardless of whether the search space is finite, countable, or unbounded. The key technical device is a careful use of martingale theory: the optional stopping theorem (Theorem 1) together with Azuma‑Hoeffding concentration for processes with bounded one‑step differences. By constructing a super‑martingale (Y_t = X_t + \delta t) (for additive drift) or appropriate transformations for the other drift types, the authors obtain clean upper bounds on the expected first‑hitting time (E