Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Forecasting for Swap Regret for All Downstream Agents

Aaron Roth, Mirah Shi

TL;DR

This work tackles forecasting in adversarial online settings with the aim that all downstream agents, who best-respond to forecasts, experience vanishing swap regret regardless of their utilities. It moves beyond calibration by using predictions that are unbiased over carefully chosen event collections, enabling strong, dimension-robust regret guarantees. The authors derive concrete rates: in 1D they achieve $\tilde{O}(\sqrt{T})$ swap regret; in 2D they obtain $\tilde{O}(T^{5/8})$; for $d>2$ with a bound $k$ on actions and smooth best responses, they obtain $\tilde{O}(T^{2/3})$, with an alternative discretization approach recovering $\tilde{O}(\sqrt{T})$ under snapped utilities. The results leverage a high-dimensional unbiased-prediction framework, including an $\varepsilon$-net discretization, convex-geometry arguments for best-response events, and a quantal-response (logistic) extension to arbitrary agents. Overall, the paper advances swap-regret guarantees in online forecasting, offering dimension-aware rates and clarifying trade-offs between calibration, computational complexity, and agent-model assumptions.

Abstract

We study the problem of making predictions so that downstream agents who best respond to them will be guaranteed diminishing swap regret, no matter what their utility functions are. It has been known since Foster and Vohra (1997) that agents who best-respond to calibrated forecasts have no swap regret. Unfortunately, the best known algorithms for guaranteeing calibrated forecasts in sequential adversarial environments do so at rates that degrade exponentially with the dimension of the prediction space. In this work, we show that by making predictions that are not calibrated, but are unbiased subject to a carefully selected collection of events, we can guarantee arbitrary downstream agents diminishing swap regret at rates that substantially improve over the rates that result from calibrated forecasts -- while maintaining the appealing property that our forecasts give guarantees for any downstream agent, without our forecasting algorithm needing to know their utility function. We give separate results in the ``low'' (1 or 2) dimensional setting and the ``high'' ($> 2$) dimensional setting. In the low dimensional setting, we show how to make predictions such that all agents who best respond to our predictions have diminishing swap regret -- in 1 dimension, at the optimal $O(\sqrt{T})$ rate. In the high dimensional setting we show how to make forecasts that guarantee regret scaling at a rate of $O(T^{2/3})$ (crucially, a dimension independent exponent), under the assumption that downstream agents smoothly best respond. Our results stand in contrast to rates that derive from agents who best respond to calibrated forecasts, which have an exponential dependence on the dimension of the prediction space.

Forecasting for Swap Regret for All Downstream Agents

TL;DR

This work tackles forecasting in adversarial online settings with the aim that all downstream agents, who best-respond to forecasts, experience vanishing swap regret regardless of their utilities. It moves beyond calibration by using predictions that are unbiased over carefully chosen event collections, enabling strong, dimension-robust regret guarantees. The authors derive concrete rates: in 1D they achieve swap regret; in 2D they obtain ; for with a bound on actions and smooth best responses, they obtain , with an alternative discretization approach recovering under snapped utilities. The results leverage a high-dimensional unbiased-prediction framework, including an -net discretization, convex-geometry arguments for best-response events, and a quantal-response (logistic) extension to arbitrary agents. Overall, the paper advances swap-regret guarantees in online forecasting, offering dimension-aware rates and clarifying trade-offs between calibration, computational complexity, and agent-model assumptions.

Abstract

We study the problem of making predictions so that downstream agents who best respond to them will be guaranteed diminishing swap regret, no matter what their utility functions are. It has been known since Foster and Vohra (1997) that agents who best-respond to calibrated forecasts have no swap regret. Unfortunately, the best known algorithms for guaranteeing calibrated forecasts in sequential adversarial environments do so at rates that degrade exponentially with the dimension of the prediction space. In this work, we show that by making predictions that are not calibrated, but are unbiased subject to a carefully selected collection of events, we can guarantee arbitrary downstream agents diminishing swap regret at rates that substantially improve over the rates that result from calibrated forecasts -- while maintaining the appealing property that our forecasts give guarantees for any downstream agent, without our forecasting algorithm needing to know their utility function. We give separate results in the ``low'' (1 or 2) dimensional setting and the ``high'' () dimensional setting. In the low dimensional setting, we show how to make predictions such that all agents who best respond to our predictions have diminishing swap regret -- in 1 dimension, at the optimal rate. In the high dimensional setting we show how to make forecasts that guarantee regret scaling at a rate of (crucially, a dimension independent exponent), under the assumption that downstream agents smoothly best respond. Our results stand in contrast to rates that derive from agents who best respond to calibrated forecasts, which have an exponential dependence on the dimension of the prediction space.
Paper Structure (17 sections, 19 theorems, 43 equations, 1 algorithm)

This paper contains 17 sections, 19 theorems, 43 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Model and Preliminaries
  5. Predictions and Decisions
  6. Conditionally Unbiased Predictions
  7. Predictions in Low Dimensions
  8. Predictions in Higher Dimensions
  9. Discretizing the Space of Utility Functions
  10. Predicting for Arbitrary Agents
  11. Smooth Best Response
  12. Connecting Unbiased Predictions and Agent Swap Regret
  13. Discussion and Conclusion
  14. Calibration and Swap Regret
  15. Definitions and Best-Known Bounds for Calibration
  16. ...and 2 more sections

Key Result

Theorem 1

noarov2023highdimensional For a collection of events $\mathcal{E}$ and convex prediction/outcome space $\mathcal{C}$, there is an algorithm producing predictions $p_1,...,p_T \in \Delta \mathcal{C}_\varepsilon$ such that for any sequence of outcomes $y_1,...,y_T \in \mathcal{C}$ chosen by the advers For $\varepsilon \leq 1/\sqrt{T}$, the bias is at most $O\left( \sqrt{\frac{\ln(d|\mathcal{E}|T)}{T

Theorems & Definitions (39)

  • Remark 1
  • Definition 1: Swap Regret
  • Definition 2: Conditional Bias on Events
  • Definition 3: $\varepsilon$-net
  • Theorem 1
  • Definition 4: Best Response
  • Definition 5: Best Response Events
  • Theorem 2
  • Lemma 1
  • proof
  • ...and 29 more