We study a stochastic multi-armed bandit problem where an agent is granted a free exploration budget before regret accumulates, a setting not captured by the classic regret minimization or pure exploration paradigms. The goal is to design an adaptive policy that strategically explores the bandit instance in the initial free exploration phase and minimizes the cumulative regret in the subsequent phase. We formalize this regret minimization with free exploration problem and identify an interesting regime where the free exploration budget scales logarithmically with the time horizon. To quantify the amount of regret saved with high probability as a result of the availability of the free exploration phase, we introduce a novel set of policies known as $(α,β)$-probably saving policies. We propose a two-phase, probably saving algorithm, UFE-KLUCB-H, which consists of a principled free exploration policy, UFE, and a history-aware regret minimization policy KLUCB-H. Instance-dependent upper bounds on UFE-KLUCB-H are derived, showing that UFE-KLUCB-H accumulates strictly less regret than policies that do not have access to a free exploration phase. Complementarily, we derive instance-dependent lower bounds based on novel multi-instance perturbation arguments tailored to the free-exploration setting, demonstrating the near-optimality of UFE-KLUCB-H for two-valued bandits. Our upper and lower bounds reveal sharp phase transitions in the accumulated regret depending on the amount of available free exploration. Simulations are conducted to demonstrate that forced exploration and adaptivity in the algorithm lead to greater regret savings.

Traffic flow estimation (TFE) is crucial for urban intelligent traffic systems. While traditional on-road detectors are hindered by limited coverage and high costs, cloud computing and data mining of vehicular network data, such as driving speeds and GPS coordinates, present a promising and cost-effective alternative. Furthermore, minimizing data collection can significantly reduce overhead. However, limited data can lead to inaccuracies and instability in TFE. To address this, we introduce the spatial-temporal Mamba (ST-Mamba), a deep learning model combining a convolutional neural network (CNN) with a Mamba framework. ST-Mamba is designed to enhance TFE accuracy and stability by effectively capturing the spatial-temporal patterns within traffic flow. Our model aims to achieve results comparable to those from extensive data sets while only utilizing minimal data. Simulations using real-world datasets have validated our model's ability to deliver precise and stable TFE across an urban landscape based on limited data, establishing a cost-efficient solution for TFE.