Salah Chikhi | Searcharxiv

arXiv2026

Last-Iterate Performance of Gradient Descent and Relaxed Proximal Point via $s$-Composability

The notion of s-composability was introduced to compose optimized stepsize schedules while preserving sharp guarantees. We show that the same joint potential has a second use: it can serve directly as a certificate of sharp last-iterate performance. We develop this viewpoint for constant and silver schedules. For the constant schedule, we prove the s-composability statement posed as an open question by Grimmer et al. (2025), at every finite horizon, through an explicit nonnegative smooth-convex interpolation certificate. The result identifies the unique constant stepsize minimizing the worst-case final gradient norm of smooth convex gradient descent under an initial-distance bound, thereby proving the optimal stepsize, value, and uniqueness predicted by Conjecture 3 of Taylor et al. (2017) without resolving its full worst-case curve. Through the Moreau envelope, the same constant is also the unique constant relaxation minimizing the worst-case final residual of relaxed proximal point. More generally, for every positive s-composable schedule, we determine the exact worst-case final proximal objective-gap constant and give a matching one-dimensional example. Applying this result to the already s-composable original silver schedule yields its exact last-iterate objective-gap constant, closing a question left open by Wang et al. (2025).

math.OC↗

arXiv2026

An analytical framework for the Levine hats problem: new strategies, bounds and generalizations

We study the Levine hat problem, a cooperative puzzle introduced by Lionel Levine in 2010, in which $n \geq 2$ players must simultaneously identify a black hat on their own infinite stack, each seeing only their teammates' stacks. While the optimal winning probability $V_n$ remains unknown even for $n=2$, we make three key advances. First, we develop a geometric and integral framework representing strategies as Lebesgue-measurable functions, yielding a new integral expression for $V_n$ and a unified treatment of finite and infinite stacks. Second, we construct a recursive strategy $\mathscr{S}_5$ processing hats in blocks of five, which attains the conjectured optimal probability $7/20$ for two players. Although this bound was already achieved by the known strategy $\mathscr{S}_3$, the existence of $\mathscr{S}_5$ refutes the previously held expectation that recursive strategies with block size greater than three yield no improvement, and produces a strictly better geometric convergence rate for $V_{2,h}$ as well as a new lower bound for $V_2(p)$ which improves known results for $p < 0.312$. Building upon this, we improve the geometric convergence rate of $V_{2,h}$ up to the near-optimal $1/4^{1-\varepsilon}$ for any $\varepsilon > 0$. Third, we introduce and completely solve a generalization of the problem where players are given uncountably infinite stacks of hats, showing that the optimal winning probability in this setting equals exactly $1/2$ for all $n \geq 2$. This new formulation allows to study the original combinatorial problem using tools from analytic optimization, and provides a natural framework for computing optimal responses to fixed strategies.

math.CO↗