Forecasting for Swap Regret for All Downstream Agents (2402.08753v2)

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 thehigh'' ($> 2$) 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.

• The paper presents a novel forecasting algorithm that eliminates the need for prior utility function knowledge while ensuring diminishing swap regret in adversarial settings.
• It achieves optimal O(√T) rates in one-dimensional spaces and near-optimal O(T^2/3) performance in multi-dimensional scenarios by leveraging unbiased event predictions.
• The research underscores a trade-off between improved regret guarantees and computational efficiency, pointing to the need for scalable algorithms in higher dimensions.

Addressing Swap Regret in Multi-Dimensional Prediction Spaces

Overview

In an effort to address the limitations posed by calibration approaches in guaranteeing downstream swap regret rates in sequential adversarial environments, we present a paper focused on leveraging unbiased predictions based on a set of carefully chosen events. Our work is aimed at bridging the gap between the slower regret rates associated with calibrated forecasts and the optimal rates achieved through dedicated swap-regret-minimization algorithms tailored to individual agents. However, these dedicated algorithms necessitate prior knowledge of agents' utility functions, which may not always be practical in dynamic forecasting scenarios. Our research circumvents this requirement, presenting a more versatile approach that guarantees diminishing swap regret for all downstream agents, irrespective of their utility functions.

In-Depth Analysis

The Core Strategy

Our primary contribution lies in the formulation of a forecasting algorithm that operates without calibration yet guarantees diminishing swap regret rates for arbitrary downstream agents. By defining a collection of events tied to the agents’ best response correspondences and ensuring predictions are unbiased with respect to these events, we demonstrate significant improvements over traditional calibration-based forecasts.

Dimension-Specific Results

The efficacy of our approach varies with the prediction space dimensionality:

• One-Dimensional: For the one-dimensional case, we achieve the optimal rate of O(√T), matching known swap regret rates but without the prerequisite knowledge of agents' utility functions—a substantial improvement over calibration methods.
• Two-Dimensional and Higher: When extending the technique to dimensions beyond one, specifically for cases where agents respond based on smoothed approximations of their utility (quantal response), we present guarantees of diminishing swap regret at a rate of O(T^2/3) across all dimensions. Although this does not reach the optimal O(√T) rate, it eliminates the dimension dependence that hampers calibration approaches.

Computational Considerations

While our methodology yields promising regret rates, especially in lower dimensions, it does so at the cost of computational efficiency in higher-dimensional settings. This trade-off delineates a critical area for future exploration: developing algorithms that maintain the achieved regret rates while also offering scalable computational performance across dimensions.

Conclusion and Future Directions

The research presented here marks a significant advancement in the pursuit of effective forecasting in adversarial environments, particularly for applications requiring guarantors of diminished swap regret independent of agents' specific utility functions. The delineation of our approach through dimension-specific analysis highlights its potential while also underscoring the necessity for further innovation to overcome its computational limitations in higher-dimensional spaces. Moving forward, the focal point of subsequent research should encompass the development of computationally efficient algorithms capable of sustaining the regret rates demonstrated herein, thereby broadening the applicability and accessibility of robust forecasting methodologies in diverse and dynamic prediction landscapes.