Gradients Must Earn Their Influence: Unifying SFT with Generalized Entropic Objectives

Standard negative log-likelihood (NLL) for Supervised Fine-Tuning (SFT) applies uniform token-level weighting. This rigidity creates a two-fold failure mode: (i) overemphasizing low-probability targets can amplify gradients on noisy supervision and disrupt robust priors, and (ii) uniform weighting provides weak sharpening when the model is already confident. Existing methods fail to resolve the resulting plasticity–stability dilemma, often suppressing necessary learning signals alongside harmful ones. To address this issue, we unify token-level SFT objectives within a generalized deformed-log family and show that they all share a gate x error gradient structure, where the gate controls how much the model trusts its current prediction. We further propose two gate designs according to the predictive information they consume–probability-level and distribution-level. At the probability level, by employing the Cayley transform, we map the model's continuously evolving uncertainty onto a continuous focus trajectory, enabling a seamless interpolation between uncertain novel concepts and well-established knowledge. At the distribution level, we propose Dynamic Entropy Fine-Tuning (DEFT), which gates on the concentration of the predictive distribution to distinguish diffuse uncertainty from concentrated disagreement. % Extensive experiments and analyses demonstrate that both Cayley-Trans and DEFT achieve a better balance between exploration and exploitation, leading to improved overall performance, with DEFT—guided by distribution-level information—consistently delivering the strongest gains. Across seven backbones and three capability regimes, both methods improve over the standard NLL objective without per-task tuning. DEFT delivers the strongest gains, raising the regime-averaged Overall score from 30.58 to 40.59 (+10.01), suggesting that distribution-level information is a substantive and underexplored dimension of token-level SFT.