Reward-Tilted Distributions in RL

• Reward-tilted distributions are defined as reward mechanisms that shift focus from mean-based evaluation to considering the full distribution, especially under heavy-tailed or noisy conditions.
• They employ robust estimation techniques and transformation strategies like truncated means and gradient clipping to stabilize learning and manage risk.
• These methods underpin advanced RL applications, including RLHF and quantile-constrained optimization, unifying risk measures and uncertainty modeling for safer decision-making.

Reward-tilted distributions denote any scenario where the reward mechanism, its modeling, or optimization shifts the focus away from the prototypical mean-based or expectation-based approach in sequential decision problems. This shift often arises due to either intrinsic properties of the reward process (such as heavy-tailedness or perturbations), design goals related to risk, robustness, or distributional alignment, or algorithmic strategies that incorporate full distributional information rather than simple averages. The following sections survey foundational concepts, algorithmic strategies, mathematical frameworks, principal theoretical results, and major empirical observations underpinning reward-tilted distributions in reinforcement learning (RL) and related areas.

1. Fundamental Concepts and Motivations

A reward-tilted distribution arises when the distributional properties of rewards cannot be accurately summarized by the expectation alone. This tilting may result from:

• Heavy-tailed reward processes, where high-magnitude events are more probable than in light-tailed or sub-Gaussian settings, thus biasing standard estimators and confidence intervals (Zhuang et al., 2021, Cayci et al., 2023, Horii et al., 2022);
• The necessity to optimize functionals over the entire reward distribution (such as quantiles, risk measures, or distributional distances) rather than means, as in risk-sensitive control, quantile-constrained RL, or pure-exploration problems with full-distribution reward functions (Wang et al., 2021, Li et al., 17 Dec 2024, Bäuerle et al., 27 May 2025);
• Scenarios where the reward signal is itself subject to perturbation, corruption, or noise, so recovering the mode or shape of the true reward distribution becomes central rather than direct regression (Chen et al., 11 Jan 2024, Xiao et al., 20 Mar 2025);
• The need in RLHF (reinforcement learning from human feedback) or LLM alignment to model multimodal, uncertain, or preference-based rewards, or to ensure robustness under distribution shift (Liu et al., 5 Oct 2024, Sun et al., 28 Mar 2025, Hong et al., 12 May 2025, Dorka, 16 Sep 2024).

In these regimes, empirical averages or naive confidence intervals are systematically unreliable, and algorithmic advances require robust estimation, non-standard optimization objectives, or adaptive, distributionally-aware strategies.

2. Robust Estimation for Heavy-Tailed and Noisy Rewards

Heavy-tailed rewards (i.e., those with only finite low-order moments) pose profound statistical and computational challenges. Standard estimators (empirical means) or analyses (Hoeffding/Bernstein inequalities) fail due to infinite variance or high-probability violation of concentration. This is rigorously established in the context of minimax regret for MDPs with heavy-tailed rewards: the learning difficulty is fundamentally controlled by the heavy-tail index, and the regret lower bound scales as

$$\Omega\big((SA)^{\frac{\epsilon}{1+\epsilon}} T^{\frac{\epsilon}{1+\epsilon}}\big)$$

(Zhuang et al., 2021). Appropriate algorithms replace empirical means by robust (truncated mean or median-of-means) estimators. For instance, truncated empirical means,

$$\hat{\mu}_T = \frac{1}{n} \sum_{t=1}^n X_t \cdot \mathbb{1}\left\{|X_t| \leq \left(\frac{ut}{\log(1/\delta)}\right)^{\frac{1}{1+\epsilon}}\right\}$$

yield reward confidence radii scaled for only a finite (1+ε)-th moment.

In temporal difference learning, dynamic gradient clipping provides analogous robustness: a clipping threshold

$b_t = (u t)^{1/(1+p)}$

removes the influence of statistical outliers, allowing control of both bias and variance, and sample complexity results explicitly depend on the tail index p (Cayci et al., 2023). Deep RL generalizations—such as Heavy-DQN—adopt adaptive truncation based on state-action visitations, thus stabilizing value estimates under heavy-tailed noise.

Noisy or perturbed reward regimes are best handled by distributional modeling of the discrete (or binned) reward, with learning performed via classification (cross-entropy), and the estimated reward reconstructed via the predicted mode—a process that robustly tilts the observed empirical distribution back toward the "true" (unperturbed) reward mode (Chen et al., 11 Jan 2024).

3. Distributional and Functional Lifting of Reward Criteria

A paradigmatic form of reward-tilting is explicit optimization for functionals of the reward distribution, not just its expectation. The general framework introduced in (Bäuerle et al., 27 May 2025) constructs lifted Markov decision processes whose state is the probability distribution over (terminal state, accumulated reward) pairs. The “distributional BeLLMan equation”

$$J_n(F) = \sup_{\pi \in \Pi} J_{n+1}(T^\pi(F)), \quad J_N(F) = H(F)$$

recurses over joint distributions F, and the objective H can represent (for example) quantile constraints, risk measures, or distances to target distributions. Standard MDPs, quantile MDPs, and optimal transport tasks are unified under this umbrella.

In quantile-constrained RL, the safety criterion is not in expectation but on a quantile, e.g., ensuring

$q_{1-\epsilon}(\pi_\theta) \leq d,$

where the (1–ε) quantile is estimated (and differentiated) via sampled policy rollouts. Algorithmic advances include sampling-based quantile gradient estimation and adaptive, "tilted" dual update rates for Lagrange multipliers, which mitigate asymmetric distributions around safety thresholds and improve both constraint satisfaction and return (Li et al., 17 Dec 2024).

In bandit pure exploration, the target function may be the τ-quantile, total variation to a target, or support for a particular distribution type—the so-called “reward-tilted” best-arm paradigm—requiring plug-in estimators of functionals H(D) and associated confidence/sample complexity bounds (Wang et al., 2021).

4. Algorithmic and Modeling Strategies for Reward-Tilting

Reward-tilting can be implemented:

• Directly, by mapping all reward signals (or their aggregates) through a homeomorphic transformation prior to value propagation. The “conjugated distributional operator”

$$T_{(\varphi)} \xi = \int_{\mathcal{R}\times \mathcal{S}} [\varphi \circ f_{r,\gamma} \circ \varphi^{-1} \# \xi^{s',a^*(s')}] d\rho(r,s'|s,a)$$

applies such a transformation "in distribution," preserving optimal policy invariance and enabling learning over unaltered rewards, with training guided by proper distributional metrics, e.g., the squared Cramér distance (Lindenberg et al., 2021).

• Through probabilistic reward redistribution, where the reward of each (s,a) pair is drawn from a parametrized distribution (e.g., Gaussian or Skew Normal) whose parameters are learned to maximize the likelihood of observed episodic returns via leave-one-out strategies. This approach introduces principled uncertainty regularization and is flexible enough to model skew or asymmetry as a form of reward-tilting (Xiao et al., 20 Mar 2025).
• In RLHF reward modeling, via quantile regression (yielding multimodal, uncertainty-aware reward distributions for downstream, risk-sensitive optimization) (Dorka, 16 Sep 2024), or by modeling the reward as a Gaussian and quantifying uncertainty via the overlap (Bhattacharyya coefficient) between reward distributions. This penalizes high-uncertainty predictions, discouraging reward hacking and overfitting (Sun et al., 28 Mar 2025).

Regularization strategies, such as batch-wise sum-to-zero, further "tilt" the distribution of rewards seen by an RL agent, constraining over-optimization and improving both generalization and alignment to human preferences (Hong et al., 12 May 2025).

5. Theoretical Regret, Convergence, and Large Deviations

Reward-tilted approaches introduce new minimax regret, sample complexity, and convergence characterizations:

• In heavy-tailed MDPs, the optimal regret bound is the sum of a term familiar from transition learning and a heavy-tail-dominated term scaling with tail index ε: $$\text{Regret} \leq c D S \sqrt{A T \log(T/\delta)} + C\, (SA T)^{\frac{\epsilon}{1+\epsilon}}$$ (Zhuang et al., 2021).
• For robust TD learning under heavy-tailed rewards, sample complexity bounds are shown to scale as $\mathcal{O}(\varepsilon^{-1/p})$ or $\mathcal{O}(\varepsilon^{-1-1/p})$ (Cayci et al., 2023).
• In distributional approaches, operator convergence to correct value functions is proven under generalized transformations, with risk-sensitive and quantile approaches covered as special cases (Lindenberg et al., 2021, Li et al., 17 Dec 2024, Rojas et al., 3 Jun 2025).
• Renewal-reward processes with power-law decaying waiting times (density $\sim 1/t^3$) exhibit anomalous fluctuation scaling: the variance of time-averaged observables decays as $(\ln t)/t$ rather than $1/t$ (Horii et al., 2022). The large deviation function has an “affine part,” indicating a finite probability for "tilted" reward outcomes rather than exponential suppression.

The empirical and theoretical guarantees thus jointly underscore that distributionally robust or reward-tilted algorithms not only maintain meaningful performance under structural reward uncertainty but often are strictly necessary for provable learning guarantees when the reward process itself challenges classical statistical assumptions.

6. Empirical Benchmarks and Applications

Reward-tilted algorithms are validated on both synthetic and real-world benchmarks:

• Tabular and deep RL with synthetic MDPs exhibiting heavy-tailed rewards: conventional algorithms are outperformed by approaches using robust estimation or adaptive truncation (Zhuang et al., 2021, Cayci et al., 2023).
• Deep RL on continuous control or Atari domains, with reward corruptions or adversarial/noise perturbations: distributional critics outperform baseline regression-based critics, especially when the perturbation model is complex or unknown (Chen et al., 11 Jan 2024, Xiao et al., 20 Mar 2025).
• RLHF and LLM alignment: quantile regression models yield policies with fewer extremely negative responses and better robustness to conflicting or noisy labels (Dorka, 16 Sep 2024); combining uncertainty estimation (via Bhattacharyya coefficient or regularization) systematically delays reward hacking and enhances alignment with human-preference "gold" models (Sun et al., 28 Mar 2025, Hong et al., 12 May 2025).
• Bandit and optimal transport applications: reward-tilted criteria (quantiles, risk measures, distances to target distributions) are enabled by plug-in estimation and lifted BeLLMan recursion (Wang et al., 2021, Bäuerle et al., 27 May 2025).

7. Synthesis and Outlook

Reward-tilted distributions encompass a broad suite of approaches—statistical, algorithmic, and decision-theoretic—for dealing with scenarios where reward’s full distributional structure fundamentally shapes learning or control.

• In statistical terms, robustness to tail events or noise (finite low-order moments, mode preservation) is essential for meaningful regret, sample complexity, and safety guarantees.
• From an optimization and modeling standpoint, distributional BeLLMan equations, conjugated operators, quantile gradients, and uncertainty-aware architectures generalize and unify many risk-sensitive and distributionally robust objectives.
• For applications in RLHF and sequential decision-making with feedback distribution shift, reward-tilted modeling—through multimodal regression, regularization, or lifted loss functions—is critical for aligning learning targets with true preferences, robustness, and safe policy deployment.

Further research directions include tighter integration of risk and robustness objectives with scalable distributional RL, unified frameworks for reward perturbations across discrete and continuous domains, and adaptive learning objectives that can tailor tilting dynamically in response to evolving environment or user feedback structure.