Jacobian-Velocity Bounds for Deployment Risk Under Covariate Drift

TL;DR

This paper introduces Jacobian-velocity bounds and a drift-aligned tangent regularization method to improve the deployment risk management of models under covariate shift, validated through synthetic and real-world experiments.

Contribution

It develops a theoretical framework linking risk volatility to Jacobian energy and proposes DTR, a novel regularization technique targeting drift directions.

Findings

DTR reduces risk volatility in low-rank drift scenarios.

DTR outperforms isotropic Jacobian regularization in controlled experiments.

Deployment gains are achieved on real datasets when drift directions are accurately estimated.

Abstract

We study long-horizon deployment of a frozen predictor under dynamic covariate shift. A time-domain Poincar\'e inequality reduces temporal risk volatility to derivative energy, and a Jacobian-velocity theorem identifies directional tangent energy along the deployment path as the governing quantity under explicit along-path regularity and domination assumptions. Under low-rank drift, that quantity reduces to directional Jacobian energy in the drift subspace, motivating drift-aligned tangent regularization (DTR) and a matched monitoring proxy. Rather than smoothing the network isotropically, DTR penalizes sensitivity only along estimated drift directions. We validate the theorem-to-method pipeline in four experiments: a synthetic benchmark for the time-domain inequality, a controlled synthetic comparison against isotropic Jacobian regularization, and two frozen-deployment studies on the UCI Air Quality and Tetouan power-consumption datasets. DTR reduces risk volatility and directional gain in the controlled low-rank regime, beats isotropic smoothing there, and gives validation-selected deployment gains on both real datasets when the Air Quality drift subspace is estimated from target-orthogonal sensor motion. Moderate drift-subspace misspecification is tolerable while orthogonal misspecification largely removes the benefit.