# Continuous Prediction with Experts' Advice

arXiv: https://arxiv.org/abs/2206.00236 · HTML: https://mlsanity.com/paper/2206.00236/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 40 references that are in the map.
- First posted on arXiv 1 Jun 2022.

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

- [Coin Betting and Parameter-Free Online Learning](https://mlsanity.com/paper/1602.04128/) (2016)
- [Second-order Quantile Methods for Experts and Combinatorial Games](https://mlsanity.com/paper/1502.08009/) (2015)
- [PDE-Based Optimal Strategy for Unconstrained Online Learning](https://mlsanity.com/paper/2201.07877/) (2022)
- [Adaptive Online Learning](https://mlsanity.com/paper/1508.05170/) (2015)
- [Prediction with Expert Advice: a PDE Perspective](https://mlsanity.com/paper/1904.11401/) (2019)
- [New Potential-Based Bounds for Prediction with Expert Advice](https://mlsanity.com/paper/1911.01641/) (2019)
- [Optimal anytime regret with two experts](https://mlsanity.com/paper/2002.08994/) (2020)
- [Finite-Time 4-Expert Prediction Problem](https://mlsanity.com/paper/1911.10936/) (2019)
- [Towards Optimal Algorithms for Prediction with Expert Advice](https://mlsanity.com/paper/1409.3040/) (2014)
- [Optimal Non-Asymptotic Lower Bound on the Minimax Regret of Learning with Expert Advice](https://mlsanity.com/paper/1511.02176/) (2015)
- [On the asymptotic optimality of the comb strategy for prediction with expert advice](https://mlsanity.com/paper/1902.02368/) (2019)
- [Minimax Optimal Quantile and Semi-Adversarial Regret via Root-Logarithmic Regularizers](https://mlsanity.com/paper/2110.14804/) (2021)
- [A Differential Equations Approach to Optimizing Regret Trade-offs](https://mlsanity.com/paper/1305.1359/) (2013)

## Built on by (most cited first, 7 of 7)

- [Unconstrained Dynamic Regret via Sparse Coding](https://mlsanity.com/paper/2301.13349/) (2023)
- [Improving Adaptive Online Learning Using Refined Discretization](https://mlsanity.com/paper/2309.16044/) (2023)
- [Prediction with Expert Advice: Anytime Regret with Many Experts Matches the Fixed-Time Constant](https://mlsanity.com/paper/2609.27206/) (2026)
- [Instance-Optimal Matrix Multiplicative Weight Update and Its Quantum Applications](https://mlsanity.com/paper/2509.08911/) (2025)
- [A note on continuous-time online learning](https://mlsanity.com/paper/2405.10399/) (2024)
- [Operationalizing Stein's Method for Online Linear Optimization: CLT-Based Optimal Tradeoffs](https://mlsanity.com/paper/2602.06545/) (2026)
- [Cogentic: Multi-Agent Orchestration for Automated Proof Discovery](https://mlsanity.com/paper/2609.40324/) (2026)

## Abstract (arXiv)

Prediction with experts' advice is one of the most fundamental problems in online learning and captures many of its technical challenges. A recent line of work has looked at online learning through the lens of differential equations and continuous-time analysis. This viewpoint has yielded optimal results for several problems in online learning. In this paper, we employ continuous-time stochastic calculus in order to study the discrete-time experts' problem. We use these tools to design a continuous-time, parameter-free algorithm with improved guarantees for the quantile regret. We then develop an analogous discrete-time algorithm with a very similar analysis and identical quantile regret bounds. Finally, we design an anytime continuous-time algorithm with regret matching the optimal fixed-time rate when the gains are independent Brownian Motions; in many settings, this is the most difficult case. This gives some evidence that, even with adversarial gains, the optimal anytime and fixed-time regrets may coincide.