Related papers: Minimizing Convex Functions with Rational Minimize…
For a matrix $W \in \mathbb{Z}^{m \times n}$, $m \leq n$, and a convex function $g: \mathbb{R}^m \rightarrow \mathbb{R}$, we are interested in minimizing $f(x) = g(Wx)$ over the set $\{0,1\}^n$. We will study separable convex functions and…
The problem of minimizing the maximum of $N$ convex, Lipschitz functions plays significant roles in optimization and machine learning. It has a series of results, with the most recent one requiring $O(N\epsilon^{-2/3} + \epsilon^{-8/3})$…
Many convex optimization problems have structured objective function written as a sum of functions with different types of oracles (full gradient, coordinate derivative, stochastic gradient) and different evaluation complexity of these…
We consider submodular function minimization in the oracle model: given black-box access to a submodular set function $f:2^{[n]}\rightarrow \mathbb{R}$, find an element of $\arg\min_S \{f(S)\}$ using as few queries to $f(\cdot)$ as…
We present an efficient algorithm to find non-empty minimizers of a symmetric submodular function over any family of sets closed under inclusion. This for example includes families defined by a cardinality constraint, a knapsack constraint,…
An adaptive regularization algorithm using inexact function and derivatives evaluations is proposed for the solution of composite nonsmooth nonconvex optimization. It is shown that this algorithm needs at most…
Assume that f is a strict convex function with a unique minimum in R^n. We divide the vector of n-variables to d groups of vector subvariables with d at least two. We assume that we can find the partial minimum of f with respect to each…
We propose faster methods for unconstrained optimization of \emph{structured convex quartics}, which are convex functions of the form \begin{equation*} f(x) = c^\top x + x^\top \mathbf{G} x + \mathbf{T}[x,x,x] + \frac{1}{24} \mathopen\|…
We introduce the following submodular generalization of the Shortest Cycle problem. For a nonnegative monotone submodular cost function $f$ defined on the edges (or the vertices) of an undirected graph $G$, we seek for a cycle $C$ in $G$ of…
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…
Many fundamental problems in machine learning can be formulated by the convex program \[ \min_{\theta\in R^d}\ \sum_{i=1}^{n}f_{i}(\theta), \] where each $f_i$ is a convex, Lipschitz function supported on a subset of $d_i$ coordinates of…
We study the min-max optimization problem where each function contributing to the max operation is strongly-convex and smooth with bounded gradient in the search domain. By smoothing the max operator, we show the ability to achieve an…
Submodular optimization finds applications in machine learning and data mining. In this paper, we study the problem of maximizing functions of the form $h = f-c$, where $f$ is a monotone, non-negative, weakly submodular set function and $c$…
This paper bridges discrete and continuous optimization approaches for decomposable submodular function minimization, in both the standard and parametric settings. We provide improved running times for this problem by reducing it to a…
Many convex problems in machine learning and computer science share the same form: \begin{align*} \min_{x} \sum_{i} f_i( A_i x + b_i), \end{align*} where $f_i$ are convex functions on $\mathbb{R}^{n_i}$ with constant $n_i$, $A_i \in…
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 consider box-constrained integer programs with objective $g(Wx) + c^T x$, where $g$ is a "complicated" function with an $m$ dimensional domain. Here we assume we have $n \gg m$ variables and that $W \in \mathbb Z^{m \times n}$ is an…
We study the query complexity of min-max optimization of a nonconvex-nonconcave function $f$ over $[0,1]^d \times [0,1]^d$. We show that, given oracle access to $f$ and to its gradient $\nabla f$, any algorithm that finds an…
We consider the problem of minimizing the sum of submodular set functions assuming minimization oracles of each summand function. Most existing approaches reformulate the problem as the convex minimization of the sum of the corresponding…
Submodular function minimization is a key problem in a wide variety of applications in machine learning, economics, game theory, computer vision, and many others. The general solver has a complexity of $O(n^3 \log^2 n . E +n^4 {\log}^{O(1)}…