A Geometric Proof of Calibration

We provide yet another proof of the existence of calibrated forecasters; it has two merits. First, it is valid for an arbitrary finite number of outcomes. Second, it is short and simple and it follows from a direct application of Blackwell’s approachability theorem to carefully chosen vector-valued payoff function and convex target set. Our proof captures the essence of existing proofs based on approachability (e.g., the proof by Foster, 1999 in case of binary outcomes) and highlights the intrinsic connection between approachability and calibration.

💡 Research Summary

The paper presents a concise geometric proof of the existence of calibrated forecasters by directly applying Blackwell’s approachability theorem to a suitably defined vector‑valued payoff and a convex target set. The authors begin by formalizing the calibration problem in the standard online prediction setting: at each round t a forecaster announces a probability vector pₜ belonging to the k‑dimensional simplex Δₖ, then Nature reveals an outcome xₜ that is one of the canonical basis vectors e₁,…,eₖ. Calibration requires that, for any Borel subset B of the simplex, the empirical average of the differences pₜ–xₜ over the times when pₜ∈B converges to zero. In other words, the long‑run frequency of each outcome must match the forecaster’s announced probabilities whenever a particular forecast is used sufficiently often.

To connect this requirement with approachability, the authors define a vector‑valued payoff function
 g(p, x) = p – x.
This payoff captures precisely the instantaneous calibration error. The target set C is chosen as an L₁‑ball of radius ε>0,
 C_ε = {v∈ℝᵏ : ‖v‖₁ ≤ ε}.
Because ε can be made arbitrarily small, forcing the average payoff into C_ε for every ε implies that the average payoff converges to the zero vector, which is exactly the calibration condition.

Blackwell’s theorem states that a convex set C is approachable if, for every mixed strategy of the opponent, the player can select an action whose expected payoff moves the current average vector toward C. The authors verify this condition for the calibration game. Let vₙ = (1/n)∑{t≤n} g(pₜ, xₜ) be the cumulative calibration error after n rounds. If vₙ lies outside C_ε, pick the coordinate i* with the largest absolute component of vₙ. The forecaster then plays the pure strategy pₙ₊₁ = e{i*}, i.e., predicts with certainty that outcome i* will occur. The conditional expectation of the next payoff is
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