A study of the Antlion Random Walk

TL;DR

This paper introduces the Antlion Random Walk (ARW), a memory-enabled AR(1) process with $X_t = \alpha X_{t-1} + \xi_t$ and $\xi_t$ from a (generalized) Rademacher distribution, to study how partial memory shapes exploration–exploitation dynamics in fast, chaotic-time-series decision-makers. It derives closed-form moments, proves path-uniqueness for rational memory, and delineates reachability and residence-time properties across regimes of $\alpha$, complemented by a CvM-based analysis of similarity to the normal distribution. Key findings include explicit expressions for $\mathbb{E}[X_t]$ and $\operatorname{Var}(X_t)$, a phase transition in reachability at $\alpha=1/2$, binomially distributed positive-side residence times under certain $\alpha$ values, and the result that $A_t^{(\alpha)}$ does not converge to the normal distribution for any $\alpha\in(0,1)$, although larger $\alpha$ yields closer early-time Gaussian-like behavior. The work advances memory-aware stochastic modeling by clarifying how discrete, non-Gaussian noise interacts with autoregressive memory to produce bounded, non-Gaussian yet increasingly Gaussian-like dynamics, with implications for high-speed decision systems and AR(1)/discrete-OU connections.

Abstract

Random walks (RWs) are fundamental stochastic processes with applications across physics, computer science, and information processing. A recent extension, the laser chaos decision-maker, employs chaotic time series from semiconductor lasers to solve multi-armed bandit (MAB) problems at ultrafast speeds, and its threshold adjustment mechanism has been modeled as an RW. However, previous analyses assumed complete memory preservation ($α= 1$), overlooking the role of partial memory in balancing exploration and exploitation. In this paper, we introduce the Antlion Random Walk (ARW), defined by $X_t = αX_{t-1} + ξ_t$ with $α\in [0,1]$ and Rademacher-distributed increments $(ξ_t)$, which describes a walker pulled back toward the origin before each step. We show that varying $α$ significantly alters ARW dynamics, yielding distributions that range from uniform-like to normal-like. Through mathematical and numerical analyses, we investigate expectation, variance, reachability, positive-side residence time, and distributional similarity. Our results place ARWs within the framework of autoregressive (AR(1)) processes while highlighting distinct non-Gaussian features, thereby offering new theoretical insights into memory-aware stochastic modeling of decision-making systems.

Figures (18)

• Figure 1: Schematic of the time-series decision-maker for the two-armed bandit problem. For each sampling time $t$, arm either A or B is selected according to comparison between the signal value $s_t$ and the threshold value $\theta_t$. If $s_t \geq \theta_t$, arm A is selected; otherwise, arm B is selected. In this figure, arm A is selected at $t$-th decision. Based on the result of arm play, the threshold value is updated for the next decision.
• Figure 2: Schematic of the stochastic process model proposed in Ref. okada2022theory, describing the time-series decision-making process. Herein, the threshold adjuster $X_t$ varies following Eq. \ref{['intro:eq:threshold']}. Since $\xi_t$ is determined according to the result of arm play, the process can be described by two one-dimensional random walks, in each of which the transition probability depends on the winning probability of arm A or B. Which random walk is employed depends on the comparison between the signal value $s_t$ and the threshold value $\theta_t$, as illustrated in Fig. \ref{['intro:fig:mab']}.
• Figure 3: $\alpha = 0.1$.
• Figure 4: $\alpha = 0.5$.
• Figure 5: $\alpha = 0.9$.

Theorems & Definitions (7)

• Proposition 3.1
• Proposition 3.2
• Corollary 3.3
• Proposition 3.4
• Theorem 3.5
• Proposition 3.6
• Theorem 3.7