🤖 AI Summary
This study addresses the minimization of convex functions that are relatively $L$-smooth with respect to the negative entropy over the standard simplex. It establishes a fundamental lower bound of $\Omega(L/T)$ on the convergence rate for any first-order optimization algorithm in high dimensions. In contrast to prior work that relied on ill-conditioned constructions, this paper presents the first such lower bound for well-structured proximal settings based on negative entropy, and extends it to the quantum setting involving von Neumann entropy. By integrating tools from convex analysis, relative smoothness theory, complexity lower-bound constructions, and spectral simplex analysis, the work demonstrates that mirror descent is nearly optimal—up to a logarithmic factor—for these problems, and confirms that the same lower bound holds in the quantum regime.
📝 Abstract
We prove an $Ω(L/T)$ 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 $d = Ω(T^2)$. 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.