Error-penalized scoring rules ($+1$ for a correct answer, $-λ$ for a wrong one, $0$ for abstaining) are increasingly prescribed against hallucination: a rational agent facing such a rule answers exactly when its correctness probability exceeds Chow's threshold $t^\ast=λ/(1+λ)$. We prove that a KL-anchored gradient learner can do the opposite. When abstention is a discrete action, the reward gradient and the anchor's restoring force are throttled by the same gate-saturation factor and die together: under explicit conditions (among them, blanket answering loses score in expectation and prompts share a bounded readout) the model drifts toward refusing everything, its mean training reward rising to zero like $1/t$ in training time $t$, so the curve reads as improvement while coverage collapses. The advantage estimator compounds the failure: in its sparse-answer regime, group normalization silently replaces every designed penalty with an effective penalty of one, moving the learned threshold from $t^\ast$ to $1/2$. The repair is structural: train a mandatory confidence report with a strictly proper score plus a correctness reward, and abstain only at deployment by thresholding the report. The always-emitted report has no gate to saturate, so no shared factor can kill its reward gradient and its anchor together, and its calibrated optimum is attracting. Simulations confirm every prediction, and experiments on language models at two scales confirm the mechanism live: the rule silences questions the models demonstrably still solve within ten optimizer steps, an ablation isolates the cause, and report-level training raises coverage, accuracy, and calibration together.
Reinforcement learning with verifiable rewards can in principle train calibrated probabilistic forecasters, since a proper scoring rule such as the Brier score is computed from outcomes alone and is minimized in expectation by the true probability. In practice it degrades calibration, and existing remedies address epistemic uncertainty, where a model's confidence accompanies a verifiably correct or incorrect answer. We study aleatoric forecasting, where the forecast itself is the output and the label is one stochastic outcome, taking NFL in-game win probability as a testbed with the betting market as a reference. Rewarding the realized per-play outcome fails, because the single outcome is a noisy target and the policy gradient corrupts the chain of thought. We introduce a verifiable, label-free reward, a state-conditioned empirical win rate estimated from past outcomes, that removes the label noise, and we keep the gradient off the reasoning, by direct prediction or a gradient mask, so it cannot be corrupted. Trained with this reward alone, without human labels or supervised fine-tuning, a 7B model reaches the calibration of the betting market by direct prediction and is better calibrated than a zero-shot frontier model. That frontier model and a tabular estimator reach the same Brier score as this model, identifying the market's small remaining edge as live in-game information beyond their shared inputs. Masking the gradient, rather than dropping the chain of thought, preserves reasoning from which the forecast follows, which ordinary chain-of-thought training corrupts.
Reward models for Reinforcement Learning from Human Feedback (RLHF) pool preferences across thousands of annotators and fit one global affine calibrator, collapsing raters with systematically different rating-scale offsets and slopes into a single average-rater fit that does not match any individual annotator. PEBS is a per-rater empirical-Bayes shrinkage estimator: it fits per-rater affine calibrators on a held-out slice of each annotator's ratings and applies Morris-James-Stein empirical-Bayes shrinkage toward the population mean, in closed form and without retraining the reward model. On PRISM, PEBS reduces within-user held-out RMSE by 8.58% over the pooled population-slope baseline. The procedure replicates on PluriHarms harm ratings (Qwen-2.5 base, in-family) with a +9.66% RMSE reduction over the same population-slope baseline. PEBS is a closed-form post-hoc estimator for annotator-specific affine calibration in RLHF reward modeling; it leaves the reward base model unchanged and estimates only the rater-level map used at inference time for new ratings.
Jungsoo Park, Hyungjoo Chae, Ethan Mendes +4cs.LG cs.AI cs.CL
Large language models can predict real-valued quantities from heterogeneous inputs such as text, code, and molecular strings, but most training objectives score each decoded floating-point number independently, improving point estimates without ensuring calibrated predictive distributions. This limits applications requiring candidate ranking or uncertainty estimation. We introduce Distribution-Aware Reward, an on-policy reinforcement learning objective whose main contribution is to train language models to produce better predictive distributions for regression tasks, rather than only optimizing individual decoded outputs against scalar targets. Our method treats multiple decoded samples as an empirical predictive distribution, evaluates it with the Continuous Ranked Probability Score, and assigns leave-one-out credit based on each rollout's marginal contribution to distribution quality, rewarding predictions that are both accurate and appropriately dispersed. We evaluate our method on a controlled Gaussian-mixture task, code performance prediction, and molecular property prediction from SMILES strings. Across tasks, our method improves over supervised fine-tuning and pointwise reinforcement learning baselines, with strong rank-correlation gains, including a 6-point Spearman improvement on KBSS. On MoleculeNet, it uses only SMILES strings yet remains competitive with strong graph-based and 3D molecular models. Further analyses show that our method mitigates rollout diversity collapse and improves uncertainty diagnostics, suggesting that directly optimizing predictive distributions makes language model regression more robust and better calibrated.