An explicit one-dimensional convex bi-objective family is constructed showing that no uniform merit estimate of order $\mathcal O(k^{-(2+\delta)})$ can hold for any fixed $\delta>0$.
Abstract
We investigate a regularized Newton method for unconstrained convex multi-objective optimization with twice continuously differentiable objectives whose Hessians are Lipschitz continuous. At each iteration, the method minimizes the quadratically regularized max-envelope of the local quadratic models. Using a Tanabe-type merit function, we prove that this merit decays at the global asymptotic rate $o(1/k^2)$ under the compactness assumption on the initial component-wise lower level set. This result also covers the single-objective case as a special case. Finally, we construct an explicit one-dimensional convex bi-objective family showing that no uniform merit estimate of order $\mathcal O(k^{-(2+\delta)})$ can hold for any fixed $\delta>0$. Thus the exponent $2$ is essentially sharp in the uniform polynomial sense, despite the $o(1/k^2)$ decay on each fixed trajectory.
A variant of the cubic-regularized Newton method for nonconvex optimization that is parameter-free in that it requires no prior knowledge of problem-dependent parameters is analyzed, and an oracle complexity bound is derived for finding an $(varepsilon, \delta)-second-order stationary point.
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.
In this paper, we present a novel second-order method called Accelerated Inexact Newton Extragradient (AINE) for convex optimization using $\delta$-inexact Hessians. We show that AINE can find an $\epsilon$-solution in the inexact second-order oracle (ISO) complexity of $\mathcal{O}( (\delta/\epsilon)^{1/2} + (L_2/\epsilon)^{2/7} )$ when the Hessian is $L_2$-Lipschitz continuous, and a better complexity of $\mathcal{O}( (\delta/\epsilon)^{1/2} + (L_3/\epsilon)^{1/5} )$ when the third-order derivative is $L_3$-Lipschitz continuous. Notably, each iteration of our method can be conducted in the same running time as matrix multiplication up to logarithmic factors. In addition, we also establish matching oracle complexity lower bounds for both setups, demonstrating the optimality of our methods.
Lesi Chen, Chengchang Liu, Luo Luo et al.· 1 citation
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.
Jacob M. Aguirre, Dmitrii M. Ostrovskii· 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.
We study the BFGS method with an Armijo-Wolfe line search for minimizing convex functions with Lipschitz-continuous gradients, without assuming strong convexity. We establish a global iteration complexity bound of $\mathcal{O}(k^{-1/2})$ for the smallest gradient norm among the first $k$ iterates. Moreover, when the initial sublevel set is bounded, we show that the function value gap converges at a rate of $\mathcal{O}(k^{-1})$. Our analysis leverages the classical trace-log-determinant potential function and reveals that a key inequality underlying this potential function remains valid without strong convexity.