For mirror descent generated by a Legendre kernel, perhaps one of the most basic question in optimization is this: must every accumulation point of a bounded mirror descent sequence be Karush--Kuhn--Tucker (KKT) stationary under proper stepsizes? We show that the answer is no. A longstanding obstacle to resolving this question is the boundary blow-up of the Legendre gradient: it keeps every mirror step in the interior, while at a boundary limit, the inverse entropy metric vanishes on active coordinates and can erase the dual-feasibility in the KKT system. We construct $C^\infty$ objectives and bounded sequences generated by the Shannon-entropic mirror descent on the nonnegative orthant $\R_+^n$, for every $n\geq 3$, and on the probability simplex $\Delta_n$, for every $n\geq 4$, such that, in each case, the set of accumulation points is a smooth boundary circle containing a nonempty relatively open arc of non-KKT points. The steps satisfy $\alpha_k\asymp k^{-\beta}$ with $\beta\in(1/2,1)$, the objective values are nonincreasing, and the objectives are entropy-relatively smooth. Hence the pathology stems from the degeneracy of the Bregman geometry at the boundary, rather than from failure of descent, or improper stepsizes. To the best of our knowledge, these provide the first counterexamples to KKT accumulation for bounded mirror descent sequences with nonincreasing objective values.
Sequence convergence to a boundary Karush--Kuhn--Tucker (KKT) point has long remained unclear for nonconvex mirror descent with Legendre kernels. The difficulty arises from the blow-up of the gradient of the Legendre kernel at the boundary. Recent work~\cite{dingtoh2026nonkkt} shows that mirror descent can accumulate at non-KKT boundary points despite decreasing objective values, precluding a convergence guarantee to KKT points in general. Despite this negative result, mirror descent remains effective in many real applications. Motivated by this contrast, we address the boundary difficulty directly and establish KKT convergence of mirror descent for a broad class of structured nonconvex problems. We analyze mirror descent in reparameterized variables, where the Hessian metric is flattened and remains nondegenerate as the boundary is approached. Under extension and definability conditions jointly coupling the objective, the Legendre kernel, and the feasible region, the reparameterized sequence has finite length and converges, thereby recovering convergence to a KKT point of the original sequence. Our general framework applies to some concrete instances: Shannon entropy, Fermi--Dirac entropy, and power kernels on polyhedron.
Barzilai--Borwein (BB) method has shown strong practical performance in continuous optimization, yet its convergence dynamics remains poorly understood. In particular, a central unresolved question is whether BB converges superlinearly for almost every strictly convex quadratic problem and initialization. We provide a negative answer to this question. Specifically, for every finite dimension $n\geq4$, we construct a nonempty open, hence positive-Lebesgue-measure, family of strictly convex quadratic problems and initial points for which the long Barzilai--Borwein method (BB1) converges but cannot converge root-superlinearly. More precisely, with the explicit constants $\rho_{\min}=10^{-6},\rho_{\max}=0.61$, every spectral component of the gradient is bounded above and below by the corresponding geometric sequence. Consequently, the gradient norm and the energy norm of the error satisfy two-sided geometric estimates with the same rates, while the objective gap satisfies the corresponding estimates with squared rates. In particular, all three quantities are bounded below by geometric sequences, ruling out superlinear convergence. The construction is highly nontrivial, based on a computer-assisted proof of a nonresonant, attracting seven-cycle of the projectivized BB dynamics in dimension four.
We study extremizers for a trilinear Stein-Weiss inequality on $\mathbb{R}^n$. Within the known boundedness region, we prove attainment under two additional assumptions: all six weight exponents are nonnegative, and at least one pair of Lebesgue exponents is admissible. The proof combines symmetric decreasing rearrangement with a logarithmic radial reduction to a translation-invariant bilinear operator on $\mathbb{R}$ whose kernel belongs to $L^1\left(\mathbb{R}^2\right)$. A common-scale compactness argument rules out relative separation of the two arguments and yields norm attainment. We then derive the Euler-Lagrange system. In the fully symmetric case, every normalized nonnegative extremizing triple is diagonal. Finally, we establish the origin-centered Kelvin invariance of the resulting scalar equation at the scaling exponent and record the unweighted conformal example.
Given a strongly convex function $u$, equip $R^d$ with a Riemannian metric given by the Hessian $\nabla^2 u$. This is a so-called Hessian manifold. Given a probability density $\mu$ one may run a Langevin diffusion intrinsic to the manifold with stationary distribution $\mu$. Such (Hessian) manifold-valued Langevin diffusions are called Mirror Langevin diffusions (MLD) which have recently become popular. One of the questions we explore is whether, given $\mu$, one can choose $u$ to get an exponential convergence to equilibrium for the MLD, especially if $\mu$ is not strongly log-concave. Our results are based on Lyapunov function methods and give sufficient conditions for a Poincar\'e or a log-Sobolev inequality to hold for the MLD. These, in turn, imply exponential convergence. We also introduce a Markov chain approximation to the MLD given by a two step Gibbs sampler with stationary distribution $\mu$. This Markov chain is a variant of the Sinkhorn Markov chain introduced in arXiv:2307.16421 that is conjectured to converge to a time-inhomogeneous generalization of the MLD. Under suitable assumptions, we prove that the Markov chain has a guaranteed convergence rate in $\chi^2$ that is consistent with the diffusion time scale. Our proofs are based on ideas from entropic optimal transport and strong data processing inequalities.
Benjamin Capdeville, Young-Heon Kim, Soumik Pal· 0 citations
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.
Let $D\subset\mathbb{R}^n$, $n\ge2$, be a bounded convex domain, and let $u_D$ be the torsion function for the restricted half-Laplacian. We prove that $D^2u_D$ is negative definite at every point of $D$. The argument is based on the reflected harmonic extension in a slit domain. Quantitative Schauder estimates in slit domains yield parameter-uniform estimates for the first and second derivatives of the edge remainder; a Schur-complement calculation then determines the inertia of the extended Hessian near the slit edge. Superharmonicity of the logarithmic Hessian determinant and the Gleason--Wolff zero-set theorem exclude interior degeneracy. A method of continuity starting from the unit ball proves the result for smooth uniformly convex domains, and an exhaustion argument treats arbitrary bounded convex domains.
Jiahuan Li, Shujun Shi· 0 citations
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