We study the effects of center initialization on the performance of a family of distributed gradient-based clustering algorithms introduced in [1], that work over connected networks of users. In the considered scenario, each user contains a local dataset and communicates only with its immediate neighbours, with the aim of finding a global clustering of the joint data. We perform extensive numerical experiments, evaluating the effects of center initialization on the performance of our family of methods, demonstrating that our methods are more resilient to the effects of initialization, compared to centralized gradient clustering [2]. Next, inspired by the $K$-means++ initialization [3], we propose a novel distributed center initialization scheme, which is shown to improve the performance of our methods, compared to the baseline random initialization.

Existing convergence of distributed optimization methods in non-Euclidean geometries typically rely on kernel assumptions: (i) global Lipschitz smoothness and (ii) bi-convexity of the associated Bregman divergence function. Unfortunately, these conditions are violated by nearly all kernels used in practice, leaving a huge theory-practice gap. This work closes this gap by developing a unified analytical tool that guarantees convergence under mild conditions. Specifically, we introduce Hessian relative uniform continuity (HRUC), a regularity satisfied by nearly all standard kernels. Importantly, HRUC is closed under concatenation, positive scaling, composition, and various kernel combinations. Leveraging the geometric structure induced by HRUC, we derive convergence guarantees for mirror descent-based gradient tracking without imposing any restrictive assumptions. More broadly, our analysis techniques extend seamlessly to other decentralized optimization methods in genuinely non-Euclidean and non-Lipschitz settings.

Wegiveasmoothed analysis, showing that evenwhen contexts may be chosen by an adversary, small perturbations of the adversary's choices suffice for the algorithm to achieve "no regret", perhaps (depending on the specifics of the setting) with a constant amount of initial training data.

We therefore assume that the algorithm has access to an oracle that knows intuitively what it means to be fair, but cannot explicitly enunciate the fairness metric.

Thus,γ(ω) (0,1] for ω (0,1], which meets the algorithm design requirement. Algorithm 2 actually performs the gradient descent scheme on the function ˆfti(x) = Eu B[fti(x+ϵu)] restricted to the convex set(1 ζ)K.

Associatedwitheach segment is a hypothesis from some hypothesis class. We give algorithms that are designed to exploit the scenario where there are many such segments but significantly fewer associated hypotheses.

Similarly the ISOMAP face database consists ofimages (256levels ofgray)ofsize64 64,i.e.,vectors in R4096, whereas the correct intrinsic dimension is only3 (for the vertical, horizontal pause and lightingdirection). The second approach, is anaverage caseapproach (in the spirit of thestatistical mechanics treatment ofhighdimensional systems), thatmodelsfeaturevectorsby arandom ensemble,taken as aset ofrandom vectors with independently identically distributed (i.i.d.) components, and a small but xed fraction of non-zero components.

We first provide sufficient conditions for the feasibility ofthe attackand an efficient algorithmtoperformanattack.

Step decay step-size schedules (constant and then cut) are widely used in practice because of their excellent convergence and generalization qualities, but their theoretical properties are not yet well understood. Weprovide convergence results for step decay in the non-convexregime, ensuring that the gradient norm vanishes at an O(lnT/ T)rate.

However, in this paper, we reveal that one-to-one mappings may overlook the complex relationship between pretrained and downstream labels.