# Data-Dependent Bounds for Online Portfolio Selection Without Lipschitzness and Smoothness

arXiv: https://arxiv.org/abs/2305.13946 · HTML: https://mlsanity.com/paper/2305.13946/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 41 references that are in the map.
- First posted on arXiv 23 May 2023.

## Builds on (most cited first, 13 of 13)

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [Optimization, Learning, and Games with Predictable Sequences](https://mlsanity.com/paper/1311.1869/) (2013)
- [Relatively-Smooth Convex Optimization by First-Order Methods, and Applications](https://mlsanity.com/paper/1610.05708/) (2016)
- [Adaptivity and Non-stationarity: Problem-dependent Dynamic Regret for Online Convex Optimization](https://mlsanity.com/paper/2112.14368/) (2021)
- [A Modular Analysis of Adaptive (Non-)Convex Optimization: Optimism, Composite Objectives, and Variational Bounds](https://mlsanity.com/paper/1709.02726/) (2017)
- [Efficient Online Portfolio with Logarithmic Regret](https://mlsanity.com/paper/1805.07430/) (2018)
- [Pushing the Efficiency-Regret Pareto Frontier for Online Learning of Portfolios and Quantum States](https://mlsanity.com/paper/2202.02765/) (2022)
- [Efficient and Near-Optimal Online Portfolio Selection](https://mlsanity.com/paper/2209.13932/) (2022)
- [Soft-Bayes: Prod for Mixtures of Experts with Log-Loss](https://mlsanity.com/paper/1901.02230/) (2019)
- [Damped Online Newton Step for Portfolio Selection](https://mlsanity.com/paper/2202.07574/) (2022)
- [Online Self-Concordant and Relatively Smooth Minimization, With Applications to Online Portfolio Selection and Learning Quantum States](https://mlsanity.com/paper/2210.00997/) (2022)
- [Minimizing Dynamic Regret on Geodesic Metric Spaces](https://mlsanity.com/paper/2302.08652/) (2023)

## Built on by (most cited first, 11 of 11)

- [Gradient-Variation Online Learning under Generalized Smoothness](https://mlsanity.com/paper/2408.09074/) (2024)
- [On the Dynamic Regret of Following the Regularized Leader: Optimism with History Pruning](https://mlsanity.com/paper/2505.22899/) (2025)
- [Improved Dimension Dependence for Bandit Convex Optimization with Gradient Variations](https://mlsanity.com/paper/2602.04761/) (2026)
- [Optimistic Interior Point Methods for Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2502.07774/) (2025)
- [Online Learning with Gradient-Variation Interval Regret](https://mlsanity.com/paper/2606.03831/) (2026)
- [Fast Minimization of Expected Logarithmic Loss via Stochastic Dual Averaging](https://mlsanity.com/paper/2311.02557/) (2023)
- [Online Learning Quantum States with the Logarithmic Loss via VB-FTRL](https://mlsanity.com/paper/2311.04237/) (2023)
- [Optimistic Learning for Communication Networks](https://mlsanity.com/paper/2504.03499/) (2025)
- [Bayesian Ensembling: Insights from Online Optimization and Empirical Bayes](https://mlsanity.com/paper/2505.15638/) (2025)
- [Duality and Policy Evaluation in Distributionally Robust Bayesian Diffusion Control](https://mlsanity.com/paper/2506.19294/) (2025)
- [From Average Sensitivity to Small-Loss Regret Bounds under Random-Order Model](https://mlsanity.com/paper/2602.09457/) (2026)

## Abstract (arXiv)

This work introduces the first small-loss and gradual-variation regret bounds for online portfolio selection, marking the first instances of data-dependent bounds for online convex optimization with non-Lipschitz, non-smooth losses. The algorithms we propose exhibit sublinear regret rates in the worst cases and achieve logarithmic regrets when the data is "easy," with per-iteration time almost linear in the number of investment alternatives. The regret bounds are derived using novel smoothness characterizations of the logarithmic loss, a local norm-based analysis of following the regularized leader (FTRL) with self-concordant regularizers, which are not necessarily barriers, and an implicit variant of optimistic FTRL with the log-barrier.