We prove an $\Omega(L/T)$ lower bound for the convergence rate of minimization in the class of functions that are convex and $L$-smooth relative to negative entropy on the standard $d$-simplex, valid for every first-order method when $d = \Omega(T^2)$. In particular, this shows that mirror descent is optimal up to a logarithmic factor in this class. This may be surprising due to the fact that accelerated methods are readily available under the assumption of smoothness in $\ell_1$-norm. While Dragomir et al. (Mathematical Programming, 2022) have already showed that acceleration might be impossible under relative smoothness, their prox-function is pathological and constructed together with the hard instance. In contrast, we show non-acceleration for a specific prox-function with particularly favorable structure. We also extend the result to the quantum setting, proving the same lower bound in the class of functions $L$-smooth relative to negative von Neumann entropy on the spectrahedron of $d \times d$ Hermitian positive-semidefinite matrices with unit trace.
We study the deterministic query complexity of minimizing a convex Lipschitz function over a $d$-dimensional Euclidean ball using only exact function values. At accuracy $\Theta(d^{-1/2})$, the previously applicable lower bound was $\Omega(d)$, inherited from the stronger full first-order oracle, while an upper bound from Protasov's value-only method requires $O(d^2\log^2 d)$ evaluations. By providing a lower bound of $\Omega(\,\frac{d^2}{\log(d+1)})$ on the oracle complexity in this setting, we thereby close this gap dating back to 1996, up to polylogarithmic factors. Furthermore, we are able to lift this result to the mixed-integer setting: Mixed-integer convex optimization with $d$ continuous and $n$ discrete variables using function values requires $\tilde{\Omega}(d^2\cdot 2^n)$ queries.
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.
Recent work has shown that, for smooth convex optimization, plain gradient descent can be accelerated from its textbook convergence rate of $O(T^{-1})$ (where $T$ denotes the number of iterations) to $O\big(T^{-\log_2(1+\sqrt{2})}\big)$ using carefully designed stepsize schedules alone, without resorting to momentum or other algorithmic modifications. Despite this progress, however, little was known about lower bounds for such methods beyond the classical $\Omega(T^{-2})$ benchmark for general first-order methods. In this work, we present a new lower bound of $\Omega(T^{-1.9319})$ for the last-iterate convergence rate of gradient descent with predetermined nonnegative stepsize schedules. This result provides rigorous evidence that stepsize schedules alone cannot accelerate plain GD to the optimal $O(T^{-2})$ convergence rate. The proof was developed by GPT-5.6 Sol Pro under the authors'guidance.
We analyze a variant of the cubic-regularized Newton method for nonconvex optimization. This variant is parameter-free in that it requires no prior knowledge of problem-dependent parameters. Under the generalized smoothness condition $\|\nabla^3 f(x)\| \leq L_0 + L_1 \|\nabla f(x)\|$, we derive an oracle complexity bound for finding an $(\varepsilon, \delta)$-second-order stationary point. This assumption is weaker than the generalized smoothness conditions used in existing analyses of second-order methods, while the complexity bound improves upon existing guarantees for parameter-free second-order methods. In particular, when $L_1 = 0$, the bound matches the optimal dependence on $L_0$ as well as on $\varepsilon$, $\delta$, and the initial function value gap, up to additive logarithmic terms. To establish this bound, we derive Taylor-type inequalities and prove their equivalence to the generalized smoothness condition.
We explore fast convergence of the Riemannian Frank--Wolfe method for smooth geodesically convex optimization over compact feasible sets. Hadamard manifolds are the main setting. On general complete manifolds, the analysis accounts for all feasible minimizing geodesics. We consider the open loop step-size $\eta_k=a/(k+a)$, which only uses the iteration index. Under a local H\"olderian error bound and local length-normalized directional scaling, every $a>2$ gives the eventual rate $O(k^{-1/(1-\theta)})$ for $\theta\in(0,1/2]$. An interior-ball condition yields $O(k^{-2})$ for strongly geodesically convex objectives. Under an exact Riemannian scaling inequality and a uniform positive lower bound on the gradient norm, every $a\geq2$ gives $O(k^{-a})$ after an explicit threshold index. The same rate holds for the smallest Frank--Wolfe gap over the most recent half of the iterates. For geodesic balls of radius \(R<\pi/2\) in the unit sphere, we establish the scaling inequality with $\alpha_R=\tfrac12\cot R$, yielding $O(k^{-a})$ primal error and recent-window gap rates. We also analyze the standard gap-feedback short step under the local error-bound conditions, obtaining $O(k^{-1/(1-2\theta)})$ for $\theta<1/2$ and a linear rate for $\theta=1/2$. Numerical experiments illustrate the predicted rates and compare iteration-only and feedback-based step selection.
We prove a sharp lower bound for smooth nonconvex stochastic optimization with uniformly bounded gradient noise. In the \(K=1\) fresh-sample model, every randomized adaptive algorithm requires $$\Omega\left( \frac{\Delta L}{\epsilon^2} + \frac{\Delta L\sigma^2}{\epsilon^4} \right)$$ queries to find a point with expected gradient norm at most \(\epsilon\). This matches the standard upper bound and, to the best of our knowledge, resolves the question raised by [Arjevani et al. 2023] of whether almost-surely bounded oracle error permits a better rate than bounded variance. The proof was independently generated with GPT-5.6 Sol in Codex's Ultra mode during a two-hour session. The human author supplied the prompt and was responsible only forchecking the proof and revising and polishing the manuscript.