Multicalibration (Hébert-Johnson et al., 2018) is an algorithmic fairness perspective which demands that the predictions of a predictor are correct conditional on themselves and membership in a collection of potentially overlapping subgroups of a population. The work of Noarov and Roth (2023) established a surprising connection between multicalibration for an arbitrary property $\Gamma$ (e.g., mean or median) and property elicitation: a property $\Gamma$ can be multicalibrated if and only if it is elicitable, where elicitability is the notion that the true property value of a distribution can be obtained by solving a regression problem over the distribution. In the adversarial (online) setting, Noarov and Roth (2023) proposed an inefficient algorithm that achieves $\tilde{\mathcal{O}}(\sqrt{T})$ $\ell_2$-multicalibration error for a hypothesis class of group membership functions and an elicitable property $\Gamma$, after $T$ rounds of interaction between a forecaster and adversary. In this paper, we generalize multicalibration for an elicitable property $\Gamma$ from group membership functions to arbitrary bounded hypothesis classes and introduce a stronger notion—swap multicalibration, following Gopalan et al. (2023b). Subsequently, we propose an oracle-efficient algorithm which when given access to an online agnostic learner, achieves $\tilde{\mathcal{O}}(T^{\frac{1}{r+1}})$ $\ell_r$-swap multicalibration error with high probability ($r \ge 2$) for a hypothesis class with bounded sequential Rademacher complexity and an elicitable property $\Gamma$. For the special case of $r = 2$, this implies an oracle-efficient algorithm that achieves $\tilde{\mathcal{O}}(T^{\frac{1}{3}})$ $\ell_2$-swap multicalibration error, which significantly improves on the previously established bounds for the problem (Noarov and Roth, 2023; Ghuge et al., 2025; Luo et al., 2025a), and completely resolves an open question raised in Garg et al. (2024) on the possibility of an oracle-efficient algorithm that achieves $\tilde{\mathcal{O}}(\sqrt{T})$ $\ell_2$-mean multicalibration error by answering it in a strongly affirmative sense.

Calibration is a fundamental concept that aims at ensuring the reliability of probabilistic predictions by aligning them with real-world outcomes. There is a surge of studies on new calibration measures that are easier to optimize compared to the classical $\ell_1$-Calibration while still having strong implications for downstream applications. One recent such example is the work by Fishelson et al. (2025) who show that it is possible to achieve $\tilde{\mathcal{O}}(T^{1/3})$ pseudo $\ell_{2}$-Calibration error via minimizing pseudo swap regret of the squared loss, which in fact implies the same bound for all bounded proper losses with a smooth univariate form. In this work, we significantly generalize their result in the following ways: (a) in addition to smooth univariate forms, our algorithm also simultaneously achieves $\tilde{\mathcal{O}}(T^{1/3})$ swap regret for any proper loss with a twice continuously differentiable univariate form (such as Tsallis entropy); (b) our bounds hold not only for pseudo swap regret that measures losses using the forecaster's distributions on predictions, but also hold for the actual swap regret that measures losses using the forecaster's actual realized predictions. We achieve so by introducing a new stronger notion of calibration called (pseudo) KL-Calibration, which we show is equivalent to the (pseudo) swap regret with respect to log loss. We prove that there exists an algorithm that achieves $\tilde{\mathcal{O}}(T^{1/3})$ KL-Calibration error and provide an explicit algorithm that achieves $\tilde{\mathcal{O}}(T^{1/3})$ pseudo KL-Calibration error. Moreover, we show that the same algorithm achieves ${\mathcal{O}}(T^{1/3} (\log T) ^ {-\frac{1}{3}}\log (T/{\delta}))$ swap regret with probability at least $1-\delta$ for any proper loss with a smooth univariate form, which implies $\tilde{\mathcal{O}}(T^{1/3})$ $\ell_2$-Calibration error. A technical contribution of our work is a new randomized rounding procedure and a non-uniform discretization scheme to minimize the swap regret for log loss.

In this paper, we consider the related problems of multicalibration --- a multigroup fairness notion and omniprediction --- a simultaneous loss minimization paradigm, both in the distributional and online settings. The recent work of Garg et al. (2024) raised the open problem of whether it is possible to efficiently achieve $\tilde{\mathcal{O}}(\sqrt{T})$ $\ell_{2}$-multicalibration error against bounded linear functions. In this paper, we answer this question in a strongly affirmative sense. We propose an efficient algorithm that achieves $\tilde{\mathcal{O}}(T^{\frac{1}{3}})$ $\ell_{2}$-swap multicalibration error (both in high probability and expectation). On propagating this bound onward, we obtain significantly improved rates for $\ell_{1}$-swap multicalibration and swap omniprediction for a loss class of convex Lipschitz functions. In particular, we show that our algorithm achieves $\tilde{\mathcal{O}}(T^{\frac{2}{3}})$ $\ell_{1}$-swap multicalibration and swap omniprediction errors, thereby improving upon the previous best-known bound of $\tilde{\mathcal{O}}(T^{\frac{7}{8}})$. As a consequence of our improved online results, we further obtain several improved sample complexity rates in the distributional setting. In particular, we establish a $\tilde{\mathcal{O}}(\varepsilon ^ {-3})$ sample complexity of efficiently learning an $\varepsilon$-swap omnipredictor for the class of convex and Lipschitz functions, $\tilde{\mathcal{O}}(\varepsilon ^{-2.5})$ sample complexity of efficiently learning an $\varepsilon$-swap agnostic learner for the squared loss, and $\tilde{\mathcal{O}}(\varepsilon ^ {-5}), \tilde{\mathcal{O}}(\varepsilon ^ {-2.5})$ sample complexities of learning $\ell_{1}, \ell_{2}$-swap multicalibrated predictors against linear functions, all of which significantly improve on the previous best-known bounds.

The problem of minimizing the sum of $n$ functions in $d$ dimensions is ubiquitous in machine learning and statistics. In many applications where the number of observations $n$ is large, it is necessary to use incremental or stochastic methods, as their per-iteration cost is independent of $n$. Of these, Quasi-Newton (QN) methods strike a balance between the per-iteration cost and the convergence rate. Specifically, they exhibit a superlinear rate with $O(d^2)$ cost in contrast to the linear rate of first-order methods with $O(d)$ cost and the quadratic rate of second-order methods with $O(d^3)$ cost. However, existing incremental methods have notable shortcomings: Incremental Quasi-Newton (IQN) only exhibits asymptotic superlinear convergence. In contrast, Incremental Greedy BFGS (IGS) offers explicit superlinear convergence but suffers from poor empirical performance and has a per-iteration cost of $O(d^3)$. To address these issues, we introduce the Sharpened Lazy Incremental Quasi-Newton Method (SLIQN) that achieves the best of both worlds: an explicit superlinear convergence rate, and superior empirical performance at a per-iteration $O(d^2)$ cost. SLIQN features two key changes: first, it incorporates a hybrid strategy of using both classic and greedy BFGS updates, allowing it to empirically outperform both IQN and IGS. Second, it employs a clever constant multiplicative factor along with a lazy propagation strategy, which enables it to have a cost of $O(d^2)$. Additionally, our experiments demonstrate the superiority of SLIQN over other incremental and stochastic Quasi-Newton variants and establish its competitiveness with second-order incremental methods.

We consider the problem of online multiclass U-calibration, where a forecaster aims to make sequential distributional predictions over $K$ classes with low U-calibration error, that is, low regret with respect to all bounded proper losses simultaneously. Kleinberg et al. (2023) developed an algorithm with U-calibration error $\mathcal{O}(K\sqrt{T})$ after $T$ rounds and raised the open question of what the optimal bound is. We resolve this question by showing that the optimal U-calibration error is $\Theta(\sqrt{KT})$ --- we start with a simple observation that the Follow-the-Perturbed-Leader algorithm of Daskalakis and Syrgkanis (2016) achieves this upper bound, followed by a matching lower bound constructed with a specific proper loss (which, as a side result, also proves the optimality of the algorithm of Daskalakis and Syrgkanis (2016) in the context of online learning against an adversary with finite choices). We also strengthen our results under natural assumptions on the loss functions, including $\Theta(\log T)$ U-calibration error for Lipschitz proper losses, $\mathcal{O}(\log T)$ U-calibration error for a certain class of decomposable proper losses, U-calibration error bounds for proper losses with a low covering number, and others.