# Non-stationary Stochastic Optimization under Lp,q-Variation Measures

arXiv: https://arxiv.org/abs/1708.03020 · HTML: https://mlsanity.com/paper/1708.03020/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Aug 2017.

## Built on by (most cited first, 5 of 5)

- [Adaptivity and Non-stationarity: Problem-dependent Dynamic Regret for Online Convex Optimization](https://mlsanity.com/paper/2112.14368/) (2021)
- [Optimal Dynamic Regret in Exp-Concave Online Learning](https://mlsanity.com/paper/2104.11824/) (2021)
- [Optimal Dynamic Regret in Proper Online Learning with Strongly Convex Losses and Beyond](https://mlsanity.com/paper/2201.08905/) (2022)
- [Adaptive Importance Sampling for Finite-Sum Optimization and Sampling with Decreasing Step-Sizes](https://mlsanity.com/paper/2103.12243/) (2021)
- [Second Order Path Variationals in Non-Stationary Online Learning](https://mlsanity.com/paper/2205.01921/) (2022)

## Abstract (arXiv)

We consider a non-stationary sequential stochastic optimization problem, in which the underlying cost functions change over time under a variation budget constraint. We propose an Lp,q-variation functional to quantify the change, which yields less variation for dynamic function sequences whose changes are constrained to short time periods or small subsets of input domain. Under the Lp,q-variation constraint, we derive both upper and matching lower regret bounds for smooth and strongly convex function sequences, which generalize previous results in Besbes et al. (2015). Furthermore, we provide an upper bound for general convex function sequences with noisy gradient feedback, which matches the optimal rate as p→∞. Our results reveal some surprising phenomena under this general variation functional, such as the curse of dimensionality of the function domain. The key technical novelties in our analysis include affinity lemmas that characterize the distance of the minimizers of two convex functions with bounded Lp difference, and a cubic spline based construction that attains matching lower bounds.