Stress-Aware Learning under KL Drift via Trust-Decayed Mirror Descent
Gabriel Nixon Raj
TL;DR
This work tackles sequential decision-making under distributional drift by introducing entropy-regularized trust-decay (TD-MD), which tilts both beliefs and decisions in response to a stress signal. A Fenchel-dual bridge shows belief tilting and stress-penalized mirror descent are equivalent, enabling unified, drift-aware dynamic-regret guarantees of $\tilde{O}(\sqrt{T})$ under KL drift path length $S_T=\sum_{t=2}^T \sqrt{\mathrm{KL}(D_t\|D_{t-1})/2}$. TD-MD achieves $O(1)$ regret per switch and supports parameter-free adaptivity, calibrated-stress, second-order and bandit variants, and distributed extensions, while a lower bound $\Omega(\lambda^2 T)$ highlights the cost of excessive tilt in stationary environments. The framework connects dynamic-regret analysis, distributionally robust objectives, and KL-regularized control in a single stress-adaptive update, with broad applicability across finance, HPC, and cognitive settings. Overall, the paper provides a principled method to modulate trust in past information as the environment drifts, delivering both formal guarantees and practical adaptability.
Abstract
We study sequential decision-making under distribution drift. We propose entropy-regularized trust-decay, which injects stress-aware exponential tilting into both belief updates and mirror-descent decisions. On the simplex, a Fenchel-dual equivalence shows that belief tilt and decision tilt coincide. We formalize robustness via fragility (worst-case excess risk in a KL ball), belief bandwidth (radius sustaining a target excess), and a decision-space Fragility Index (drift tolerated at $O(\sqrt{T})$ regret). We prove high-probability sensitivity bounds and establish dynamic-regret guarantees of $\tilde{O}(\sqrt{T})$ under KL-drift path length $S_T = \sum_{t\ge2}\sqrt{{\rm KL}(D_t|D_{t-1})/2}$. In particular, trust-decay achieves $O(1)$ per-switch regret, while stress-free updates incur $Ω(1)$ tails. A parameter-free hedge adapts the tilt to unknown drift, whereas persistent over-tilting yields an $Ω(λ^2 T)$ stationary penalty. We further obtain calibrated-stress bounds and extensions to second-order updates, bandit feedback, outliers, stress variation, distributed optimization, and plug-in KL-drift estimation. The framework unifies dynamic-regret analysis, distributionally robust objectives, and KL-regularized control within a single stress-adaptive update.