Newest papers

Gradient descent reaches arbitrary accuracy in a fixed number of large steps on multiclass logistic regression

Andreea-Alexandra Muşat, Nicolas Boumal · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.26748 · cited by 0 works (OpenAlex) · open-access PDF

Japanese

われわれは、ステップサイズ$η$のグレディエントダウン(GD)が、複数のイテレーションから独立した$\varepsilon$のマルチクラスロジスティック回帰損失における$\varepsilon$の最適化差を達成できることを示す。これを行うには、ステップサイズ$η$が$\varepsilon$の機能として十分に大きく選択されなければならない(および線形的に分離可能なデータセットの属性)と、$T$は、データに依存する常数である。データの関連性は、ポイントの数とそのマージン($T$のための)と、1つの追加量($η$のための)が、次元$\dgeq 3$で必要であることが判明する。証拠は、 $η\toin

Machine translation (Meta M2M100); the English original follows.

Abstract

We show that gradient descent (GD) with step size $η$ can achieve $\varepsilon$ optimality gap on the multiclass logistic regression loss in a number of iterations $T$ that is independent of $\varepsilon$. To do so, the step size $η$ must be chosen large enough as a function of $\varepsilon$ (and properties of the linearly separable dataset), while $T$ is a constant that only depends on the data. The relevant properties of the data include the number of points and their margin (for $T$), and one additional quantity (for $η$) that turns out to be necessary in dimension $d \geq 3$. The proofs build on the known fact that, as $η\to \infty$, suitably rescaled GD becomes the perceptron algorithm, and the latter finds a perfect classifier in $T$ iterations. A similar claim was shown for two classes in dimension $d = 2$ using other proof techniques. This result adds to previous works that showed GD on the logistic regression loss can be accelerated by taking large step sizes. Those provoke a wild, oscillatory initial phase, followed by a stable phase. We provide results of this (qualitatively different) nature, too.

Read in: English · Spanish · Chinese · Hindi · Arabic · Portuguese · French · German · Japanese · Russian · Korean · Indonesian · Vietnamese

Topics: Mathematics, Gradient descent, Separable space, Dimension (graph theory), Mathematical proof, Logistic regression

Sources: OpenAlex (CC0), arXiv, Crossref. Agent Exchange