This paper proposes a new easy-to-implement parameter-free gradient-based optimizer: DoWG (Distance over Weighted Gradients). We prove that DoWG is efficient--matching the convergence rate of optimally tuned gradient descent in convex optimization up to a logarithmic factor without tuning any parameters, and universal--automatically adapting to both smooth and nonsmooth problems. While popular algorithms following the AdaGrad framework compute a running average of the squared gradients to use for normalization, DoWG maintains a new distance-based weighted version of the running average, which is crucial to achieve the desired properties. To complement our theory, we also show empirically that DoWG trains at the edge of stability, and validate its effectiveness on practical machine learning tasks.

The sharpest known high probability generalization bounds for uniformly stable algorithms (Feldman, Vondrák, NeurIPS 2018, COLT, 2019), (Bousquet, Klochkov, Zhivotovskiy, COLT, 2020) contain a generally inevitable sampling error term of order Θ(1/ n). When applied to excess risk bounds, this leads to suboptimal results in several standard stochastic convex optimization problems. We show that if the so-called Bernstein condition is satisfied, the term Θ(1/ n) can be avoided, and high probability excess risk bounds of order up to O(1/n) are possible via uniform stability. Using this result, we show a high probability excess risk bound with the rate O(log n/n) for strongly convex and Lipschitz losses valid for any empirical risk minimization method.

A worker checks the radiation level inside the control room of reactor No 4, where the Chornobyl disaster happened in 1986. A worker checks the radiation level inside the control room of reactor No 4, where the Chornobyl disaster happened in 1986. In February 2025, a cheap Russian drone tore through Chornobyl's confinement shelter. Workers warn the site of the world's worst nuclear accident is not safe yet The dosimeter clipped to your chest ticks faster the moment you step off the designated path inside the Chornobyl nuclear power plant. Step back, and it slows again - an invisible line between clean ground and contamination.

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