High-dimensional online calibration from harmonic weights

Key points

• For forecasting $d$ binary outcomes simultaneously ($Y=[0,1]^d$), we give the first algorithm that achieves $\varepsilon$-calibration in a number of rounds that is polynomial in $d$ for every fixed accuracy.1
• For multi-class forecasting ($Y=Δd$), we obtain the same $d^{O(1/\varepsilon)}$ rate, improving the $d^{\widetilde{O}(1/\varepsilon^2)}$ bounds of Peng and Fishelson et al.1
• More generally, it achieves $\varepsilon$-calibration after $\exp(O(γ(Y,L)/\varepsilon))$ rounds, where $γ(Y,L)$ is a geometric parameter defined by a matrix discrepancy problem.1
• The harmonic weights are motivated by the fact that the discrete Hilbert transform matrix achieves the optimal discrepancy up to a universal constant, simultaneously for every $L$.1

Sources (1)

• [1]High-dimensional online calibration from harmonic weights
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 6, 04:36 AM

  We study the online calibration of multidimensional forecasts over an arbitrary convex set $Y\subseteq\mathbb{R}^d$ relative to an arbitrary error norm $\|\cdot\|_{L}$.

  For forecasting $d$ binary outcomes simultaneously ($Y=[0,1]^d$), we give the first algorithm that achieves $\varepsilon$-calibration in a number of rounds that is polynomial in $d$ for every fixed accuracy.

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