The Statistical Benefits of Multiple Responses for Learning from Demonstrations

Learning from correct demonstrations is harder than supervised learning when many answers are correct: after predicting, the learner sees one valid answer but not whether its own answer was valid, nor any reward. Existing reward-hedging guarantees consequently assume a finite reward class. We give the first horizon-free guarantee for continuous classes. The key is to hedge in one shared vote over tolerant optimality tests at every accuracy scale. A target reward has one surviving proxy per scale, and a prediction with gap above that scale doubles the proxy. This yields the simultaneous tail bound $|\{t:\ell_t>2^{-j}\}|\leq \log_2\mathcal N(\mathcal G,2^{-j-1})+j$, where $\mathcal G$ is the class of optimality-gap functions. Integrating the tails gives cumulative hidden gap bounded by a metric-entropy integral, independently of the number of rounds. Polynomial entropy $(A/ε)^d$ gives $O(d\log A)$ total gap and a fast $O(d/m)$ statistical rate. For bounded linear contextual recommendation, the result is $O(d)$ regret for arbitrary compact menus. This is the first polynomial finite bound without structural restrictions on the menus, at the price of improper prediction. Although the general vote can be expensive, it is exactly polynomial-time for one-dimensional Lipschitz parameter curves. Fixed-radius rank-two recommendation takes $O(KT^2)$ time for menus of size $K$. We also prove an $Ω(d)$ lower bound, low-rank and bounded ReLU-network corollaries, and a robust theorem that adds only the demonstrator's cumulative suboptimality. A reproducible adaptive stress test illustrates the predicted scale adaptation. After factorization, an exact MovieLens audit runs in 1.7 CPU seconds across ten users and improves mean latent gap over both a demonstrated-rating policy and a proper online baseline. The learner uses only action demonstrations and never observes a reward or a loss.

Reinforcement Learning is a cornerstone technique for modern large reasoning models. Usually, for difficult tasks such as code generation and theorem proving, the agent is evaluated by generating $K$ responses rather than sampling a single response, and performance is then measured using a retry-aware metric such as $\max$@$k$. Despite their practical importance, the theoretical foundations of learning under such criteria remain limited. In this work, we provide a theoretical study of the $\max$@$k$ learning problem in finite-horizon reinforcement learning. We show that optimizing the $\max$@$k$ objectives is fundamentally different from standard expected-return maximization. In particular, we prove that Markovian policies are in general insufficient, identify a compact state augmentation that restores optimality, and explicitly characterize the performance gap that can arise between history-dependent and non-history-dependent policies. Moreover, we show that learning $\max$@$k$-optimal policies is statistically harder than standard reinforcement learning and provide an efficient algorithm that achieves the optimal sample complexity rate.

Inverse reinforcement learning (IRL) typically assumes demonstrations from a single optimal demonstrator, but in many applications data come from multiple imperfect demonstrators with heterogeneous suboptimality levels. We study reward learning in this setting through a feasible-reward-set framework: for each demonstrator, we encode its declared suboptimality level as a linear constraint and intersect the resulting feasible sets across demonstrators. Our theoretical analysis shows that the joint feasible set shrinks monotonically as data are added, and we give an exact characterization of when a new demonstrator strictly tightens it. We further establish two recovery guarantees for the feasible reward set of the ground-truth optimal demonstrator: one bound depends on closeness to the optimal occupancy, while the other requires only sufficient coverage and no near-optimal demonstrator. On the practical side, we introduce strategies to address the inherent reward ambiguity in the obtained reward set and provide an offline algorithm with function approximation for high-dimensional environments. Experiments in tabular grid-world and large language model (LLM) fine-tuning settings are consistent with the theoretical predictions and demonstrate the effectiveness of the proposed framework over baselines.