arXiv subjects

Guglielmo Beretta

Publications and source records attributed to Guglielmo Beretta.

3 recordsLinked to original sources

arXiv2026

Sharp Dimension Dependence for the Last Iterate of the SubGradient Method

We study the last iterate of the projected subGradient Method (sGM) for convex Lipschitz objectives defined on $\mathbb{R}^d$. We prove that, for a finite horizon $n$ and a constant stepsize $η=Θ(1/\sqrt n)$, the last iterate achieves an optimization error of order $d/\sqrt n$, showing that the extra $\log n$ factor appearing in high dimensions is unnecessary in every fixed dimension. We complement this result with a matching linear-in-$d$ lower bound and show that the sharp worst-case dimension-horizon dependence is of order $\min\{d,\log n\}/\sqrt n$. This solves, in particular, a COLT open problem posed by Koren and Segal in 2020 and shows that the correct dependence on the dimension is linear rather than logarithmic.

math.OC↗

arXiv2026

New Bounds for the Last Iterate of the Stochastic subGradient Method

We study the last iterate of the stochastic subgradient method for one-dimensional convex Lipschitz objectives. For a fixed horizon $n$, we consider the standard fixed stepsizes $η=Θ(1/\sqrt n)$. We prove that, for such stepsize policies, under additive i.i.d. subgradient noise with uniformly bounded variance, the last iterate features an optimization error of order $1/\sqrt n$, thereby removing the extra $(\log n)$ factor present in existing generic bounds. On the other hand, we show that without the i.i.d. assumption, the optimization error can be of order $(\log n)/\sqrt n$. Thus, under the uniformly bounded variance assumption alone, the last iterate of SsGM is suboptimal even in dimension one, resolving negatively an open problem posed in Koren and Segal, COLT, 2020.

math.OC↗

arXiv2026

Generalized Pinsker Inequality for Bregman Divergences of Negative Tsallis Entropies

The Pinsker inequality lower bounds the Kullback--Leibler divergence $D_{\textrm{KL}}$ in terms of total variation and provides a canonical way to convert $D_{\textrm{KL}}$ control into $\lVert \cdot \rVert_1$-control. Motivated by applications to probabilistic prediction with Tsallis losses and online learning, we establish a generalized Pinsker inequality for the Bregman divergences $D_α$ generated by the negative $α$-Tsallis entropies -- also known as $β$-divergences. Specifically, for any $p$, $q$ in the relative interior of the probability simplex $Δ^K$, we prove the sharp bound \[ D_α(p\Vert q) \ge \frac{C_{α,K}}{2}\cdot \|p-q\|_1^2, \] and we determine the optimal constant $C_{α,K}$ explicitly for every choice of $(α,K)$.

cs.IT↗