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This paper considers the problem of unconstrained minimization of smooth convex functions having Lipschitz continuous gradients with known Lipschitz constant. We recently proposed an optimized gradient method (OGM) for this problem and…
In this paper, we study the fundamental open question of finding the optimal high-order algorithm for solving smooth convex minimization problems. Arjevani et al. (2019) established the lower bound $\Omega\left(\epsilon^{-2/(3p+1)}\right)$…
We provide sharp worst-case evaluation complexity bounds for nonconvex minimization problems with general inexpensive constraints, i.e.\ problems where the cost of evaluating/enforcing of the (possibly nonconvex or even disconnected)…
We consider the problem of minimizing a function that is a sum of convex agent functions plus a convex common public function that couples them. The agent functions can only be accessed via a subgradient oracle; the public function is…
We study to what extent quantum algorithms can speed up solving convex optimization problems. Following the classical literature we assume access to a convex set via various oracles, and we examine the efficiency of reductions between the…
In this paper we propose third-order methods for composite convex optimization problems in which the smooth part is a three-times continuously differentiable function with Lipschitz continuous third-order derivatives. The methods are…
We consider the problem of minimizing a continuous function given quantum access to a stochastic gradient oracle. We provide two new methods for the special case of minimizing a Lipschitz convex function. Each method obtains a dimension…
In this paper, we develop new first-order method for composite non-convex minimization problems with simple constraints and inexact oracle. The objective function is given as a sum of "`hard"', possibly non-convex part, and "`simple"'…
The unconstrained minimization of a sufficiently smooth objective function $f(x)$ is considered, for which derivatives up to order $p$, $p\geq 2$, are assumed to be available. An adaptive regularization algorithm is proposed that uses…
Second-order methods, which utilize gradients as well as Hessians to optimize a given function, are of major importance in mathematical optimization. In this work, we prove tight bounds on the oracle complexity of such methods for smooth…
We consider the problem of minimizing a convex function over a convex set given access only to an evaluation oracle for the function and a membership oracle for the set. We give a simple algorithm which solves this problem with…
The convergence behavior of gradient methods for minimizing convex differentiable functions is one of the core questions in convex optimization. This paper shows that their well-known complexities can be achieved under conditions weaker…
We characterize the complexity of minimizing $\max_{i\in[N]} f_i(x)$ for convex, Lipschitz functions $f_1,\ldots, f_N$. For non-smooth functions, existing methods require $O(N\epsilon^{-2})$ queries to a first-order oracle to compute an…
We consider sum-type strongly convex optimization problem (first term) with smooth convex not proximal friendly composite (second term). We show that the complexity of this problem can be split into optimal number of incremental oracle…
Parallelization is a popular strategy for improving the performance of iterative algorithms. Optimization methods are no exception: design of efficient parallel optimization methods and tight analysis of their theoretical properties are…
Consider an oracle which takes a point $x$ and returns the minimizer of a convex function $f$ in an $\ell_2$ ball of radius $r$ around $x$. It is straightforward to show that roughly $r^{-1}\log\frac{1}{\epsilon}$ calls to the oracle…
Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the…
We note that known methods achieving the optimal oracle complexity for first order convex optimization require quadratic memory, and ask whether this is necessary, and more broadly seek to characterize the minimax number of first order…
Universal methods for optimization are designed to achieve theoretically optimal convergence rates without any prior knowledge of the problem's regularity parameters or the accurarcy of the gradient oracle employed by the optimizer. In this…
In this paper, we develop a new adaptive regularization method for minimizing a composite function, which is the sum of a $p$th-order ($p \ge 1$) Lipschitz continuous function and a simple, convex, and possibly nonsmooth function. We use a…