The Hawkes model is a point process observed on the real line, which generally corresponds to the time, where any previously encountered event has a direct influence on the chances of future events occurring. This past-dependent mathematical model was introduced in [1] and its first application was to model earthquakes occurrences [2, 3]. Since then, Hawkes processes have been widely used in various fields, for instance finance [4], social media [5, 6], epidemiology [7], sociology [8] and neuroscience [9]. The main advantage of Hawkes processes is their ability to model different kinds of relationships between phenomena through an unknown kernel or transfer function. The Hawkes model was originally introduced as a self-exciting point process where the appearance of an event increases the chances of another one triggering. Several estimation procedures have been proposed for the kernel function, both in parametric [2, 10, 11] and nonparametric [9, 12] frameworks. However, the inhibition setting, where the presence of an event decreases the chance of another occurring, has drawn less attention in the literature, although it can be of great interest in several fields, in particular in neuroscience [13]. In this inhibition context, the cluster representation [14] on which is based the construction of a self-exciting Hawkes process, is no longer valid.

Time-constrained decision processes have been ubiquitous in many fundamental applications in physics, biology and computer science. Recently, restart strategies have gained significant attention for boosting the efficiency of time-constrained processes by expediting the completion times. In this work, we investigate the bandit problem with controlled restarts for time-constrained decision processes, and develop provably good learning algorithms. In particular, we consider a bandit setting where each decision takes a random completion time, and yields a random and correlated reward at the end, with unknown values at the time of decision. The goal of the decision-maker is to maximize the expected total reward subject to a time constraint $\tau$. As an additional control, we allow the decision-maker to interrupt an ongoing task and forgo its reward for a potentially more rewarding alternative. For this problem, we develop efficient online learning algorithms with $O(\log(\tau))$ and $O(\sqrt{\tau\log(\tau)})$ regret in a finite and continuous action space of restart strategies, respectively. We demonstrate an applicability of our algorithm by using it to boost the performance of SAT solvers.

This work analyses the potential of restarts for probSAT, a quite successful algorithm for k-SAT, by estimating its runtime distributions on random 3-SAT instances that are close to the phase transition. We estimate an optimal restart time from empirical data, reaching a potential speedup factor of 1.39. Calculating restart times from fitted probability distributions reduces this factor to a maximum of 1.30. A spin-off result is that the Weibull distribution approximates the runtime distribution for over 93% of the used instances well. A machine learning pipeline is presented to compute a restart time for a fixed-cutoff strategy to exploit this potential. The main components of the pipeline are a random forest for determining the distribution type and a neural network for the distribution's parameters. ProbSAT performs statistically significantly better than Luby's restart strategy and the policy without restarts when using the presented approach. The structure is particularly advantageous on hard problems.