We study the Lipschitz bandit problem, where a learner sequentially maximizes an unknown Lipschitz function $f$ over a domain $\mathcal{X} \subset [0,1]^d$ using noisy pointwise evaluations. Existing regret bounds are either worst-case, scaling as $\tildeΘ \left ( T^{d+1/d+2}\right )$, or adaptive via the zooming dimension $d_z$, yielding $\tildeΘ \left ( T^{d_z+1/d_z+2}\right )$. However, such zooming-based guarantees are only partially instance-dependent, as they depend solely on the asymptotic growth of near-optimal level sets and fail to capture finer structural properties of $f$. We provide an analysis and an algorithm that characterizes the regret through integrals of the suboptimality gap of $f$ over its level sets. This yields regret bounds that adapt to the local growth of level sets, rather than only their asymptotic behavior. As a corollary, when the set of maximizers has dimension $d^\star>0$, we obtain improved adaptive rates of order $\tilde{\mathcal{O}} \left ( T^{d_z+1 / \max(d_z,d^\star)+2}\right )$ strictly improving over classical zooming bounds in this regime. Finally, we extend our analysis to the full-information setting (Lipschitz experts) and show how some of the regularity assumptions can be relaxed.

This work introduces a general framework for calibeating based on regret minimization. As compared to Foster and Hart's seminal calibeating work which had specialized treatments of Brier score (squared loss) and log loss, we consider a large family of proper losses that includes $α$-Tsallis losses (for $α\in [1, 2]$) and Lipschitz losses. Our results for Tsallis losses also hold for an unscaled version of Tsallis loss that recovers log loss. Our analysis is oriented around the Bregman divergence view of a proper loss. Technically, our results for the family of Tsallis losses that we consider are U-calibration results, simultaneously obtaining logarithmic regret for all losses in this family while having a weaker dependence on the dimension compared to previous results. Of potential independent interest, we also show a new regret equality for the regret of Be The Regularized Leader. This regret equality holds for general proper losses and itself is based on two results related to online updating formulas for the generalized variance, the latter being a previously introduced generalization of variance based on Bregman divergences.

A.1 CIFAR-10 dataset Figure 6 displays test accuracy curves for all six backbone algorithms under three distinct imbalance parameters: 2{ 0.3,1,10}. The results clearly demonstrate that FedNAR outperforms the baselines, particularly in scenarios with imbalanced data. A.2 Shakespeare dataset The experimental results presented in Figure 7 and 8 showcase the outcomes of experiments performed on the Shakespeare dataset. Six backbone algorithms were utilized, with initial weight decay values selected from {10 3,10 4}. These findings serve as evidence that FedNAR, as an adaptive weight decay scheduling algorithm, exhibits effectiveness across various initial weight decay values.

Distributional shifts pose a significant challenge to achieving robustness in contemporary machine learning. To overcome this challenge, robust satisficing (RS) seeks a robust solution to an unspecified distributional shift while achieving a utility above a desired threshold. This paper focuses on the problem of RS in contextual Bayesian optimization when there is a discrepancy between the true and reference distributions of the context. We propose a novel robust Bayesian satisficing algorithm called RoBOS for noisy black-box optimization.

The McKean-Vlasov equation (MVE) describes the collective behavior of particles subject to drift, diffusion, and mean-field interaction. In physical systems, the interaction term can be singular, i.e. it diverges when two particles collide. Notable examples of such interactions include the Coulomb interaction, fundamental in plasma physics, and the Biot-Savart interaction, present in the vorticity formulation of the 2DNavier-Stokes equation (NSE) in fluid dynamics. Solving MVEs that involve singular interaction kernels presents a significant challenge, especially when aiming to provide rigorous theoretical guarantees. In this work, we propose a novel approach based on the concept of entropy dissipation in the underlying system.

We consider online learning problems under a partial observability model capturing situations where the information conveyed to the learner is between full information and bandit feedback. In the simplest variant, we assume that in addition to its own loss, the learner also gets to observe losses of some other actions. The revealed losses depend on the learner's action and a directed observation system chosen by the environment. For this setting, we propose the first algorithm that enjoys near-optimal regret guarantees without having to know the observation system before selecting its actions. Along similar lines, we also define a new partial information setting that models online combinatorial optimization problems where the feedback received by the learner is between semi-bandit and full feedback. As the predictions of our first algorithm cannot be always computed efficiently in this setting, we propose another algorithm with similar properties and with the benefit of always being computationally efficient, at the price of a slightly more complicated tuning mechanism. Both algorithms rely on a novel exploration strategy called implicit exploration, which is shown to be more efficient both computationally and information-theoretically than previously studied exploration strategies for the problem.

Organization: This supplementary material is presented in a format parallel to the main paper. The section numbers and titles are consistent with the main paper. But, here we also add one new section: Section 10 where we describe the societal impacts and possible negative impacts of the paper. Similarly, the Theorem numbers are consistent with the main paper, but we also have several additional theorems and lemmas which were not included in the main paper. GAN-type Objective for KLEstimation Let f be a discriminator, f: X IR. Let p(x) and q(x) be two probability density functions defined over the space X.