Not all uncertainty is alike: volatility, stochasticity, and exploration

Most exploration algorithms search broadly until uncertainty is resolved. When the action space is too large to resolve within budget, practitioners default to $\varepsilon$-greedy, which bounds disruption but spends its override blindly. We introduce \textit{Delight-gated exploration} (DE), a host--override rule that spends exploratory actions only when their prospective delight (expected improvement times surprisal) exceeds a gate price. This practical heuristic recovers a classical result: Pandora's reservation-value rule for costly search, with surprisal setting the effective inspection cost. Resolved arms exit the gate, fresh arms shut off above a prior-determined threshold, and selected linear-bandit overrides consume finite information budget. Across Bernoulli bandits, linear bandits, and tabular MDPs, the same hyperparameters transfer without retuning, and DE shows much weaker regret growth than Thompson Sampling and $\varepsilon$-greedy in the tested unresolved regimes. Delight improves acting for the same reason it improves learning: it prices scarce resources by the product of upside and surprisal.

We pose uncertainty quantification and exploration in online decision-making as a problem of training and generation from an autoregressive sequence model, an area experiencing rapid innovation. Our approach rests on viewing uncertainty as arising from missing future outcomes that could be revealed through action choices, rather than from unobservable latent parameters of the environment. This reformulation aligns naturally with modern machine learning capabilities: we can i) train generative models through next-outcome prediction rather than fit explicit priors, ii) assess uncertainty through autoregressive generation rather than sampling latent parameters from posteriors, and iii) adapt to new information by extending the sequence model's context rather than explicit posterior updating. Our main theoretical result establishes a reduction from online decision-making to offline next-outcome prediction: Bayesian regret is controlled directly by the sequence model's offline prediction loss, without requiring an explicit latent-variable posterior. Experiments, including a semi-synthetic news recommendation task, show that autoregressive generation produces calibrated epistemic uncertainty and enables effective exploration by using article text as prior information to focus exploration on resolving remaining uncertainties.

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.