# No-Regret Learning with Unbounded Losses: The Case of Logarithmic Pooling

arXiv: https://arxiv.org/abs/2202.11219 · HTML: https://mlsanity.com/paper/2202.11219/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 36 references that are in the map.
- First posted on arXiv 22 Feb 2022.

## Builds on (most cited first, 7 of 7)

- [AugMix: A Simple Data Processing Method to Improve Robustness and Uncertainty](https://mlsanity.com/paper/1912.02781/) (2019)
- [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)
- [Revisiting the Calibration of Modern Neural Networks](https://mlsanity.com/paper/2106.07998/) (2021)
- [Calibration of Pre-trained Transformers](https://mlsanity.com/paper/2003.07892/) (2020)
- [Tsallis-INF: An Optimal Algorithm for Stochastic and Adversarial Bandits](https://mlsanity.com/paper/1807.07623/) (2018)
- [From Proper Scoring Rules to Max-Min Optimal Forecast Aggregation](https://mlsanity.com/paper/2102.07081/) (2021)

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

- [ANCHOR: Abductive Network Construction with Hierarchical Orchestration for Reliable Probability Inference in Large Language Models](https://mlsanity.com/paper/2605.10328/) (2026)
- [Sample Complexity of Forecast Aggregation](https://mlsanity.com/paper/2207.13126/) (2022)
- [Enhancing Online Reinforcement Learning with Meta-Learned Objective from Offline Data](https://mlsanity.com/paper/2501.07346/) (2025)
- [Tree-Based Formalization of Multi-Agent Complementarity in Human-AI Interactions](https://mlsanity.com/paper/2606.04779/) (2026)
- [Bridging Learned Visual Perception and Symbolic Belief-Space Planning](https://mlsanity.com/paper/2609.16884/) (2026)

## Abstract (arXiv)

For each of T time steps, m experts report probability distributions over n outcomes; we wish to learn to aggregate these forecasts in a way that attains a no-regret guarantee. We focus on the fundamental and practical aggregation method known as logarithmic pooling -- a weighted average of log odds -- which is in a certain sense the optimal choice of pooling method if one is interested in minimizing log loss (as we take to be our loss function). We consider the problem of learning the best set of parameters (i.e. expert weights) in an online adversarial setting. We assume (by necessity) that the adversarial choices of outcomes and forecasts are consistent, in the sense that experts report calibrated forecasts. Imposing this constraint creates a (to our knowledge) novel semi-adversarial setting in which the adversary retains a large amount of flexibility. In this setting, we present an algorithm based on online mirror descent that learns expert weights in a way that attains O(√T log T) expected regret as compared with the best weights in hindsight.