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Preprint Jul 2026

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 $\eta=\Theta(1/\sqrt n)$, 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.

Guglielmo Beretta, Tommaso Cesari, Roberto Colomboni et al. · 1 citation · ⚡1
Preprint Aug 2026

A lower bound for stepsize-based acceleration of gradient descent

This work presents a new lower bound of $\Omega(T^{-1.9319})$ for the last-iterate convergence rate of gradient descent with predetermined nonnegative stepsize schedules, and provides rigorous evidence that stepsize schedules alone cannot accelerate plain GD to the optimal $O(T^{-2})$ convergence rate.

Jianhao Ma, Yuxin Chen · 1 citation
Jul 2026

Data-Dependent Regret and Polyak Corrections for Constrained Online Convex Optimization

Constrained online convex optimization requires minimizing regret against adversarial convex costs while satisfying a convex constraint at every round, as needed in safety-critical applications. A computationally efficient method combines online gradient descent with a Polyak feasibility step, using one constraint evaluation and one subgradient per round. Although this method achieves O(sqrt(T)) regret with per-round feasibility, we derive a tighter, data-dependent analysis by retaining two quantities omitted by the standard worst-case argument. First, we replace the gradient envelope G_f^2 T with the observed accumulation G_T = sum_t ||grad f_t(x_t)||^2. Second, we identify a nonnegative Polyak correction P_T that measures the cumulative squared displacement caused by feasibility projections and enters the regret bound with a negative sign. The resulting improvement, Delta_T = (eta/2)(G_f^2 T - G_T) + P_T/(2 eta), is always nonnegative. We further propose AdaOGD-PFS, an adaptive-step-size method that achieves O(sqrt(G_T)) regret while preserving per-round feasibility. Experiments on ball- and halfspace-constrained problems improve the regret bound by 38 to 43 percent, with both data-dependent gradients and Polyak corrections contributing substantially.

Wentao Zhang · 0 citations

Optimal-Dimension U-Calibration by Bayesian Bootstrap

It is shown that the classical Bayesian bootstrap closes this gap in U-calibration, which asks one online probability fore-caster to have low regret for every bounded proper loss, including losses unknown when the forecasts are made.

Pahan Dewasurendra · 0 citations
Open access Aug 2026

Polynomial-Time Algorithm for Optimal Stopping with Fixed Accuracy

This work develops a novel expansion representation for the OS value and proves that truncating this expansion yields a simulation-based algorithm that implements an optimal stopping policy with computational complexity scaling polynomially in the time horizon and the underlying dimension.

Yilun Chen, David A. Goldberg · 0 citations
Preprint Aug 2026

A Mini-Batch Counterexample to Last-Iterate Convergence in Definable Optimization

We give a counterexample to the convergence conjecture in Remark 12 of [Bolte&Pauwels, 2021] for mini-batch stochastic approximation with definable potentials. The construction uses two convex piecewise-affine, hence semialgebraic, summands on $\mathbb{R}$. We choose a deterministic nonincreasing block stepsize sequence satisfying $\alpha_k = o(1/\log k)$ and an admissible minimum-norm selection from each aggregate batch field. On successive blocks, the iterates form lazy reflected random walks on nested dyadic lattices. An explicit endpoint-cover-time estimate, Markov's inequality, and the first Borel-Cantelli lemma imply that almost surely every sufficiently late block's iterates visit their entire lattice. Consequently, the iterates remain in $[-1,1]$ but do not converge, and their accumulation set is exactly $[-1,1]$, on which the averaged objective is constant. Finally, the construction has $\sum_k \alpha_k^2 =\infty$. Both Chat-GPT 5.6 (Sol) and Gemini Pro 3.1 (DeepThink) were used in the development and drafting of this result.

Weiwei Kong · 0 citations