# Online optimization and regret guarantees for non-additive long-term constraints

arXiv: https://arxiv.org/abs/1602.05394 · HTML: https://mlsanity.com/paper/1602.05394/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Feb 2016.

## Built on by (most cited first, 3 of 3)

- [Trading-off price for data quality to achieve fair online allocation](https://mlsanity.com/paper/2306.13440/) (2023)
- [A Theoretical Framework for Auxiliary-Loss-Free Load Balancing of Sparse Mixture-of-Experts in Large-Scale AI Models](https://mlsanity.com/paper/2512.03915/) (2025)
- [Optimally Pacing Budget Spending and Learning](https://mlsanity.com/paper/2610.11074/) (2026)

## Abstract (arXiv)

We consider online optimization in the 1-lookahead setting, where the objective does not decompose additively over the rounds of the online game. The resulting formulation enables us to deal with non-stationary and/or long-term constraints , which arise, for example, in online display advertising problems. We propose an on-line primal-dual algorithm for which we obtain dynamic cumulative regret guarantees. They depend on the convexity and the smoothness of the non-additive penalty, as well as terms capturing the smoothness with which the residuals of the non-stationary and long-term constraints vary over the rounds. We conduct experiments on synthetic data to illustrate the benefits of the non-additive penalty and show vanishing regret convergence on live traffic data collected by a display advertising platform in production.