Selecting an effective step-size is a fundamental challenge in first-order optimization, especially for problems with non-Euclidean geometries. This paper presents a novel adaptive step-size strategy for optimization algorithms that rely on linear minimization oracles, as used in the Conditional Gradient or non-Euclidean Normalized Steepest Descent algorithms. Using a simple heuristic to estimate a local Lipschitz constant for the gradient, we can determine step-sizes that guarantee sufficient decrease at each iteration. More precisely, we establish convergence guarantees for our proposed Adaptive Conditional Gradient Descent algorithm, which covers as special cases both the classical Conditional Gradient algorithm and non-Euclidean Normalized Steepest Descent algorithms with adaptive step-sizes. Our analysis covers optimization of continuously differentiable functions in non-convex, quasar-convex, and strongly convex settings, achieving convergence rates that match state-of-the-art theoretical bounds. Comprehensive numerical experiments validate our theoretical findings and illustrate the practical effectiveness of Adaptive Conditional Gradient Descent. The results exhibit competitive performance, underscoring the potential of the adaptive step-size for applications.

翻译：在一阶优化中，选择有效的步长是一个基本挑战，尤其对于具有非欧几里得几何的问题。本文提出了一种新颖的自适应步长策略，适用于依赖线性最小化预言机的优化算法，例如条件梯度算法或非欧几里得归一化最速下降算法中所使用的预言机。通过使用一个简单的启发式方法来估计梯度的局部Lipschitz常数，我们可以确定保证每次迭代有足够下降的步长。更精确地说，我们为我们提出的自适应条件梯度下降算法建立了收敛性保证，该算法作为特例涵盖了经典条件梯度算法以及具有自适应步长的非欧几里得归一化最速下降算法。我们的分析涵盖了在非凸、拟星凸和强凸设置下连续可微函数的优化，所达到的收敛速率与最先进的理论界限相匹配。全面的数值实验验证了我们的理论发现，并说明了自适应条件梯度下降法的实际有效性。结果展现了具有竞争力的性能，突显了自适应步长在应用中的潜力。