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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

Indonesian

Kami menunjukkan bahwa gradient descent (GD) dengan step size $η$ dapat mencapai $\varepsilon$ gap optimity pada multi-class logistic regression loss dalam sejumlah iterasi $T$ yang independen dari $\varepsilon$. Untuk melakukan ini, step size $η$ harus dipilih cukup besar sebagai fungsi dari $\varepsilon$ (dan sifat-sifat dari dataset yang dapat dipisahkan secara linear), sementara $T$ adalah konstan yang hanya tergantung pada data. Properti yang relevan dari data meliputi jumlah poin dan margin mereka (untuk $T$), dan satu jumlah tambahan (untuk $η$) yang ternyata diperlukan dalam dimensi $\dgeq 3$. Bukti-bukti didasarkan pada fakta yang diketahui, bahwa $η\toinfifty$, stable-rescaled GD mengikuti algoritma perceptron sebelumnya, dan

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.

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

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