# Adapting to Continuous Covariate Shift via Online Density Ratio Estimation

arXiv: https://arxiv.org/abs/2302.02552 · HTML: https://mlsanity.com/paper/2302.02552/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 97 references that are in the map.
- First posted on arXiv 6 Feb 2023.

## Builds on (most cited first, 30 of 46)

- [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (2015)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Detecting and Correcting for Label Shift with Black Box Predictors](https://mlsanity.com/paper/1802.03916/) (2018)
- [Beyond UCB: Optimal and Efficient Contextual Bandits with Regression Oracles](https://mlsanity.com/paper/2002.04926/) (2020)
- [Non-stationary Stochastic Optimization](https://mlsanity.com/paper/1307.5449/) (2013)
- [Adaptive Online Learning in Dynamic Environments](https://mlsanity.com/paper/1810.10815/) (2018)
- [A Unified View of Label Shift Estimation](https://mlsanity.com/paper/2003.07554/) (2020)
- [Online Optimization : Competing with Dynamic Comparators](https://mlsanity.com/paper/1501.06225/) (2015)
- [Non-stationary Reinforcement Learning without Prior Knowledge: An Optimal Black-box Approach](https://mlsanity.com/paper/2102.05406/) (2021)
- [A Second-order Bound with Excess Losses](https://mlsanity.com/paper/1402.2044/) (2014)
- [Rethinking Importance Weighting for Deep Learning under Distribution Shift](https://mlsanity.com/paper/2006.04662/) (2020)
- [Dynamic Regret of Convex and Smooth Functions](https://mlsanity.com/paper/2007.03479/) (2020)
- [Wild-Time: A Benchmark of in-the-Wild Distribution Shift over Time](https://mlsanity.com/paper/2211.14238/) (2022)
- [Tracking Slowly Moving Clairvoyant: Optimal Dynamic Regret of Online Learning with True and Noisy Gradient](https://mlsanity.com/paper/1605.04638/) (2016)
- [Adaptivity and Non-stationarity: Problem-dependent Dynamic Regret for Online Convex Optimization](https://mlsanity.com/paper/2112.14368/) (2021)
- [Marginal Singularity, and the Benefits of Labels in Covariate-Shift](https://mlsanity.com/paper/1803.01833/) (2018)
- [A Century of Portraits: A Visual Historical Record of American High School Yearbooks](https://mlsanity.com/paper/1511.02575/) (2015)
- [A Simple Approach for Non-stationary Linear Bandits](https://mlsanity.com/paper/2103.05324/) (2021)
- [Improved Strongly Adaptive Online Learning using Coin Betting](https://mlsanity.com/paper/1610.04578/) (2016)
- [Open-environment Machine Learning](https://mlsanity.com/paper/2206.00423/) (2022)
- [Non-Negative Bregman Divergence Minimization for Deep Direct Density Ratio Estimation](https://mlsanity.com/paper/2006.06979/) (2020)
- [Online Adaptation to Label Distribution Shift](https://mlsanity.com/paper/2107.04520/) (2021)
- [Dynamical Models and Tracking Regret in Online Convex Programming](https://mlsanity.com/paper/1301.1254/) (2013)
- [Near-Optimal Linear Regression under Distribution Shift](https://mlsanity.com/paper/2106.12108/) (2021)
- [Active Learning for Cost-Sensitive Classification](https://mlsanity.com/paper/1703.01014/) (2017)
- [Bandit Convex Optimization in Non-stationary Environments](https://mlsanity.com/paper/1907.12340/) (2019)
- [Optimal Dynamic Regret in Exp-Concave Online Learning](https://mlsanity.com/paper/2104.11824/) (2021)
- [No-Regret Learning in Time-Varying Zero-Sum Games](https://mlsanity.com/paper/2201.12736/) (2022)

## Built on by (most cited first, 15 of 15)

- [WATCH: Adaptive Monitoring for AI Deployments via Weighted-Conformal Martingales](https://mlsanity.com/paper/2505.04608/) (2025)
- [When to retrain a machine learning model](https://mlsanity.com/paper/2505.14903/) (2025)
- [Non-stationary Online Learning for Curved Losses: Improved Dynamic Regret via Mixability](https://mlsanity.com/paper/2506.10616/) (2025)
- [Fed-ADE: Adaptive Learning Rate for Federated Post-adaptation under Distribution Shift](https://mlsanity.com/paper/2603.01040/) (2026)
- [Online Feature Updates Improve Online (Generalized) Label Shift Adaptation](https://mlsanity.com/paper/2402.03545/) (2024)
- [Harnessing the Power of Vicinity-Informed Analysis for Classification under Covariate Shift](https://mlsanity.com/paper/2405.16906/) (2024)
- [Mitigating covariate shift in non-colocated data with learned parameter priors](https://mlsanity.com/paper/2411.06499/) (2024)
- [Mitigating Negative Transfer via Reducing Environmental Disagreement](https://mlsanity.com/paper/2510.24044/) (2025)
- [One-Step Score-Based Density Ratio Estimation](https://mlsanity.com/paper/2604.10672/) (2026)
- [Adapting to Covariate Shift in Real-time by Encoding Trees with Motion Equations](https://mlsanity.com/paper/2404.05168/) (2024)
- [Efficient Learning Under Density Shift in Incremental Settings Using Cram'er-Rao-Based Regularization](https://mlsanity.com/paper/2502.12949/) (2025)
- [A Guide to Failure in Machine Learning: Reliability and Robustness from Foundations to Practice](https://mlsanity.com/paper/2503.00563/) (2025)
- [Robust and Sparse Estimation of Unbounded Density Ratio under Heavy Contamination](https://mlsanity.com/paper/2512.09266/) (2025)
- [Geometry-Anchored Transport Framework for Exemplar-Free Class-Incremental Learning](https://mlsanity.com/paper/2606.25347/) (2026)
- [PIcsC: Partitioning-Induced Covariate Shift Correction](https://mlsanity.com/paper/2607.25441/) (2026)

## Abstract (arXiv)

Dealing with distribution shifts is one of the central challenges for modern machine learning. One fundamental situation is the covariate shift, where the input distributions of data change from training to testing stages while the input-conditional output distribution remains unchanged. In this paper, we initiate the study of a more challenging scenario -- continuous covariate shift -- in which the test data appear sequentially, and their distributions can shift continuously. Our goal is to adaptively train the predictor such that its prediction risk accumulated over time can be minimized. Starting with the importance-weighted learning, we show the method works effectively if the time-varying density ratios of test and train inputs can be accurately estimated. However, existing density ratio estimation methods would fail due to data scarcity at each time step. To this end, we propose an online method that can appropriately reuse historical information. Our density ratio estimation method is proven to perform well by enjoying a dynamic regret bound, which finally leads to an excess risk guarantee for the predictor. Empirical results also validate the effectiveness.