Probabilistic Insights for Efficient Exploration Strategies in Reinforcement Learning

TL;DR

This work tackles efficient exploration in model-free stochastic environments with unknown dynamics and sparse rewards by analyzing two core strategies: parallel exploration and restart mechanisms. Using simplified toy dynamics—random walks and Lévy processes—the paper derives a phase-transition phenomenon for the success probability as a function of the number of parallel simulations and identifies an optimal parallelization level $N^*$. It further shows that restarting trajectories from strategically chosen states, including quasi-stationary distributions, can yield exponential gains in reaching rare target states, with explicit bounds and sharp asymptotics. The results link to Fleming–Viot particle systems and quasi-stationary distributions, offering theoretically grounded guidance for designing more efficient exploration in reinforcement learning and rare-event estimation. Overall, the study provides a rigorous, model-free perspective on when and how to deploy parallel sampling and restarting to improve exploration efficiency under finite budgets, with practical implications for policy-gradient estimation and gradient quality in sparse-reward RL settings.

Abstract

We investigate efficient exploration strategies of environments with unknown stochastic dynamics and sparse rewards. Specifically, we analyze first the impact of parallel simulations on the probability of reaching rare states within a finite time budget. Using simplified models based on random walks and Lévy processes, we provide analytical results that demonstrate a phase transition in reaching probabilities as a function of the number of parallel simulations. We identify an optimal number of parallel simulations that balances exploration diversity and time allocation. Additionally, we analyze a restarting mechanism that exponentially enhances the probability of success by redirecting efforts toward more promising regions of the state space. Our findings contribute to a more qualitative and quantitative theory of some exploration schemes in reinforcement learning, offering insights into developing more efficient strategies for environments characterized by rare events.

Figures (4)

• Figure 1: Parallel exploration for random walks with negative mean $p - (1-p)=-0.1$. Estimated ratio between ${\mathbb P}(\tau^{(N)}(x)\leq B(x)/N)$ and ${\mathbb P}(\tau(x)\leq B(x))$ as a function of number $N$ with $B(x)=C\cdot x = 300\cdot x$ . The phase transition is observed at the expected threshold $N^* = \lceil C (1-2p)\rceil - 1=29$.
• Figure 2: Parallel exploration Lévy processes. Estimated ratio between ${\mathbb P}(\tau^{(N)}(x)\leq B(x)/N)$ and ${\mathbb P}(\tau(x)\leq B(x))$ as a function of the number $N$ of independent Lévy processes with exponential jumps, with parameters described in Section \ref{['subsec:simulations_LP_parallel']}.
• Figure 3: Probability of exceeding the level $x=50$ for the restarted Lévy process as a function of the time horizon. The underlying process combines a Brownian motion with drift $\mu=-1$ and volatility $\sigma=1$ with exponential jumps: positive jumps arrive at rate $\lambda_+=2$ with jump sizes of rate $4$, while negative jumps arrive at rate $\lambda_-=3$ with jump sizes of rate $1$. Whenever the process exits the interval $(0,50)$, it is restarted according to a truncated exponential distribution with mean $10$.
• Figure 4: Simulation of parallel $\mathrm{M/M/1}$ queues with $K=40$, $J=12$, $\lambda=0.7$, and $\mu=1$, comparing renewal-based stationary probability estimates across time horizons $10^5$, $10^6$, and $10^7$. The number of parallel copies to deploy in each time regime is computed using the results of our main theorems.

Theorems & Definitions (37)

• Theorem 1
• Corollary 1: Optimal number of particles
• Example 1
• Example 2
• Theorem 2
• Example 3
• Remark 1
• Definition 1
• Theorem 3
• Example 4