Simultaneous Swap Regret Minimization via KL-Calibration

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 $O(T^{1/3})$ 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 $O(T^{1/3})$ 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 for log loss. We prove that there exists an algorithm that achieves $O(T^{1/3})$ KL-Calibration error and provide an explicit algorithm that achieves $O(T^{1/3})$ pseudo KL-Calibration error. Moreover, we show that the same algorithm achieves $O(T^{1/3}(\log T)^{-1/3}\log(T/\delta))$ swap regret w.p. $\ge 1-\delta$ for any proper loss with a smooth univariate form, which implies $O(T^{1/3})$ $\ell_2$-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.

Optimal Multiclass U-Calibration Error and Beyond

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 $O(K\sqrt{T})$ 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, $O(\log T)$ 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.

Online Convex Optimization with Switching Cost and Delayed Gradients

We consider the online convex optimization (OCO) problem with quadratic and linear switching cost in the limited information setting, where an online algorithm can choose its action using only gradient information about the previous objective function. For $L$-smooth and $\mu$-strongly convex objective functions, we propose an online multiple gradient descent (OMGD) algorithm and show that its competitive ratio for the OCO problem with quadratic switching cost is at most $4(L + 5) + \frac{16(L + 5)}{\mu}$. The competitive ratio upper bound for OMGD is also shown to be order-wise tight in terms of $L,\mu$. In addition, we show that the competitive ratio of any online algorithm is $\max\{\Omega(L), \Omega(\frac{L}{\sqrt{\mu}})\}$ in the limited information setting when the switching cost is quadratic. We also show that the OMGD algorithm achieves the optimal (order-wise) dynamic regret in the limited information setting. For the linear switching cost, the competitive ratio upper bound of the OMGD algorithm is shown to depend on both the path length and the squared path length of the problem instance, in addition to $L, \mu$, and is shown to be order-wise, the best competitive ratio any online algorithm can achieve. Consequently, we conclude that the optimal competitive ratio for the quadratic and linear switching costs are fundamentally different in the limited information setting.

Sharpened Lazy Incremental Quasi-Newton Method

We consider the finite sum minimization of $n$ strongly convex and smooth functions with Lipschitz continuous Hessians in $d$ dimensions. In many applications where such problems arise, including maximum likelihood estimation, empirical risk minimization, and unsupervised learning, the number of observations $n$ is large, and it becomes necessary to use incremental or stochastic algorithms whose per-iteration complexity is independent of $n$. Of these, the incremental/stochastic variants of the Newton method exhibit superlinear convergence, but incur a per-iteration complexity of $O(d^3)$, which may be prohibitive in large-scale settings. On the other hand, the incremental Quasi-Newton method incurs a per-iteration complexity of $O(d^2)$ but its superlinear convergence rate has only been characterized asymptotically. This work puts forth the Sharpened Lazy Incremental Quasi-Newton (SLIQN) method that achieves the best of both worlds: an explicit superlinear convergence rate with a per-iteration complexity of $O(d^2)$. Building upon the recently proposed Sharpened Quasi-Newton method, the proposed incremental variant incorporates a hybrid update strategy incorporating both classic and greedy BFGS updates. The proposed lazy update rule distributes the computational complexity between the iterations, so as to enable a per-iteration complexity of $O(d^2)$. Numerical tests demonstrate the superiority of SLIQN over all other incremental and stochastic Quasi-Newton variants.