Breaking the $T^{2/3}$ Barrier for Sequential Calibration

Abstract: A set of probabilistic forecasts is calibrated if each prediction of the forecaster closely approximates the empirical distribution of outcomes on the subset of timesteps where that prediction was made. We study the fundamental problem of online calibrated forecasting of binary sequences, which was initially studied by Foster & Vohra (1998). They derived an algorithm with $O(T^{2/3})$ calibration error after $T$ time steps, and showed a lower bound of $\Omega(T^{1/2})$. These bounds remained stagnant for two decades, until Qiao & Valiant (2021) improved the lower bound to $\Omega(T^{0.528})$ by introducing a combinatorial game called sign preservation and showing that lower bounds for this game imply lower bounds for calibration. In this paper, we give the first improvement to the $O(T^{2/3})$ upper bound on calibration error of Foster & Vohra. We do this by introducing a variant of Qiao & Valiant's game that we call sign preservation with reuse (SPR). We prove that the relationship between SPR and calibrated forecasting is bidirectional: not only do lower bounds for SPR translate into lower bounds for calibration, but algorithms for SPR also translate into new algorithms for calibrated forecasting. We then give an improved \emph{upper bound} for the SPR game, which implies, via our equivalence, a forecasting algorithm with calibration error $O(T^{2/3 - \varepsilon})$ for some $\varepsilon > 0$, improving Foster & Vohra's upper bound for the first time. Using similar ideas, we then prove a slightly stronger lower bound than that of Qiao & Valiant, namely $\Omega(T^{0.54389})$. Our lower bound is obtained by an oblivious adversary, marking the first $\omega(T^{1/2})$ calibration lower bound for oblivious adversaries.

• The paper introduces the sign preservation with reuse (SPR) game to derive a new, stronger lower bound of "\u03a9(T^{0.54389})" for sequential calibration error, improving upon previous results.
• The study establishes a crucial theoretical equivalence showing that improvements in the SPR game directly translate into reducing the upper bound of calibration error below the long-standing "T^{2/3}" barrier.
• The authors construct an oblivious adversary approach using the SPR framework, proving substantial lower bounds and demonstrating robustness against both adaptive and non-adaptive scenarios.

Improved Bounds for Calibration via Stronger Sign Preservation Games

The paper "Improved bounds for calibration via stronger sign preservation games" addresses an important challenge in the domain of online learning and forecasting: achieving optimal error bounds for online calibration. Calibration refers to the process by which a sequence of probabilistic forecasts aligns closely with the actual outcomes, specifically for binary sequences. Accurate calibration has far-reaching implications, particularly in machine learning applications such as algorithmic fairness and predictive modeling in healthcare.

Background and Motivation

Historically, the calibration error for forecasts after $T$ time steps has been bounded by an upper limit of $O(T^{2/3})$ and a lower bound of $\Omega(T^{1/2})$, as established by Foster and Vohra (1998). These bounds had remained unchanged for over two decades until a recent breakthrough by Qiao and Valiant (2021) slightly improved the lower bound to $\Omega(T^{0.528})$ using a combinatorial game called the sign-preservation game.

Key Contributions

This paper introduces a novel strengthening of Qiao and Valiant's sign-preservation game, termed "sign preservation with reuse" (SPR). The authors establish a bidirectional relationship between the SPR game and the calibration forecasting problem. Key contributions include:

1. Enhanced Lower Bounds: By leveraging the SPR framework, the authors derive a new lower bound of $\Omega(T^{0.54389})$, marking a significant advancement over previous results. Importantly, this enhancement is proven even against oblivious adversaries, setting a new precedent.
2. Equivalence and Upper Bound Reduction: The paper demonstrates that improvements in the SPR game can directly translate into reducing the upper bound of calibration error below $T^{2/3}$. This equivalence forms a crucial theoretical underpinning for further exploration into more efficient forecasting algorithms.
3. Oblivious Strategy: The authors also address the limitation of adaptivity in adversaries. They construct an oblivious adversary approach, which proves substantial $\omega(T^{1/2})$ calibration error lower bounds, a distinction that positions the framework as robust against both adaptive and non-adaptive scenarios.

Methodology

The methodological core lies in transforming the calibration forecast problem into a combinatorial SPR game. The study meticulously constructs non-trivial strategies for both players in the SPR setup, encapsulating the essence of sign placement and preservation in forecasting.

• Optimization of Sign Preservation: The first main result shows that by achieving optimality in the SPR game, one can effectively bound the calibration error’s upper limit. This is showcased using theoretical constructs that entail dynamic programming across an augmented grid of choices (cells) and rounds.
• Analysis of Adversarial Outcomes: The second main result extends the understanding of oblivious adversary implications in calibrated forecasting. An analytical framework, grounded in game theory and probability, supports this outcome, highlighting techniques to maintain error resilience irrespective of adversarial dependence on forecasts.

Implications and Future Directions

The results point towards potential improvements in real-time forecasting algorithms, underlining both theoretical and practical advances in the field. Calibration, as a measure independent of accuracy, gains a nuanced representation through SPR, emphasizing fairness and predictiveness across various domains.

Future developments could harness these insights to explore omniprediction and multicalibration frameworks in greater detail. Additionally, the intersection of game theory, machine learning, and probability theory, as illustrated, presents a fertile ground for devising more robust adaptive learning models with improved calibration guarantees in dynamic environments.

In conclusion, this work significantly enhances the understanding and effectiveness of creating calibrated forecasting algorithms by introducing a stronger theoretical foundation through SPR games, aiming at pushing the frontiers in statistical learning theory and practice.