A probabilistic predictor $f:X\to[0,1]$ is calibrated if $\mathbb{E}[Y\mid f(X)]=f(X)$ almost surely. We start by briefly recalling the notion of approximate calibration, and finish the proof of the simple, efficient deterministic predictor by Arunachaleswaran et al. (SODA 2025) that achieves $O(\sqrt{T})$ distance to a perfectly calibrated prediction sequence over $T$ rounds. If time permits, we will revisit our discussion on the lack of stability in approximate calibration measures like ECE and concretely define the concepts of multicalibration and multiaccuracy. We will finish by discussing (without proof) a surprising result of Casacuberta et al. (FOCS 2025) who show that adding global calibration to multiaccuracy recovers much of the learning-theoretic and complexity-theoretic power of multicalibration, at a computational cost comparable to multiaccuracy alone.
