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 $r^{-1}\log\frac{1}ε$ calls to the oracle suffice to find an $ε$-approximate minimizer of $f$ in an $\ell_2$ unit ball. Perhaps surprisingly, this is not optimal: we design an accelerated algorithm which attains an $ε$-approximate minimizer with roughly $r^{-2/3} \log \frac{1}ε$ oracle queries, and give a matching lower bound. Further, we implement ball optimization oracles for functions with locally stable Hessians using a variant of Newton's method. The resulting algorithm applies to a number of problems of practical and theoretical import, improving upon previous results for logistic and $\ell_\infty$ regression and achieving guarantees comparable to the state-of-the-art for $\ell_p$ regression.
We develop and analyze algorithms for distributionally robust optimization (DRO) of convex losses. In particular, we consider group-structured and bounded $f$-divergence uncertainty sets. Our approach relies on an accelerated method that queries a ball optimization oracle, i.e., a subroutine that minimizes the objective within a small ball around the query point. Our main contribution is efficient implementations of this oracle for DRO objectives. For DRO with $N$ non-smooth loss functions, the resulting algorithms find an $ε$-accurate solution with $\widetilde{O}\left(Nε^{-2/3} + ε^{-2}\right…
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 $O(Nε^{-2})$ queries to a first-order oracle to compute an $ε$-suboptimal point and $\tilde{O}(Nε^{-1})$ queries if the $f_i$ are $O(1/ε)$-smooth. We develop methods with improved complexity bounds of $\tilde{O}(Nε^{-2/3} + ε^{-8/3})$ in the non-smooth case and $\tilde{O}(Nε^{-2/3} + \sqrt{N}ε^{-1})$ in the $O(1/ε)$-smooth case. Our methods consist of a recently proposed ball optimization oracle acceleration algorithm (which …
We give a simple and natural method for computing approximately optimal solutions for minimizing a convex function $f$ over a convex set $K$ given by a separation oracle. Our method utilizes the Frank--Wolfe algorithm over the cone of valid inequalities of $K$ and subgradients of $f$. Under the assumption that $f$ is $L$-Lipschitz and that $K$ contains a ball of radius $r$ and is contained inside the origin centered ball of radius $R$, using $O(\frac{(RL)^2}{\varepsilon^2} \cdot \frac{R^2}{r^2})$ iterations and calls to the oracle, our main method outputs a point $x \in K$ satisfying $f(x) \le…
Accelerated algorithms have broad applications in large-scale optimization, due to their generality and fast convergence. However, their stability in the practical setting of noise-corrupted gradient oracles is not well-understood. This paper provides two main technical contributions: (i) a new accelerated method AGDP that generalizes Nesterov's AGD and improves on the recent method AXGD (Diakonikolas & Orecchia, 2018), and (ii) a theoretical study of accelerated algorithms under noisy and inexact gradient oracles, which is supported by numerical experiments. This study leverages the simplicit…
We develop a distributed algorithm for convex Empirical Risk Minimization, the problem of minimizing large but finite sum of convex functions over networks. The proposed algorithm is derived from directly discretizing the second-order heavy-ball differential equation and results in an accelerated convergence rate, i.e, faster than distributed gradient descent-based methods for strongly convex objectives that may not be smooth. Notably, we achieve acceleration without resorting to the well-known Nesterov's momentum approach. We provide numerical experiments and contrast the proposed method with…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Institute for Information Transmission Problems RAS（俄罗斯科学院信息传输问题研究所）
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 $Ω\left(ε^{-2/(3p+1)}\right)$ on the number of the $p$-th order oracle calls required by an algorithm to find an $ε$-accurate solution to the problem, where the $p$-th order oracle stands for the computation of the objective function value and the derivatives up to the order $p$. However, the existing state-of-the-art high-order methods of Gasnikov et al. (2019b); Bubeck et al. (2019); Jiang et al. …
In constrained convex optimization, existing methods based on the ellipsoid or cutting plane method do not scale well with the dimension of the ambient space. Alternative approaches such as Projected Gradient Descent only provide a computational benefit for simple convex sets such as Euclidean balls, where Euclidean projections can be performed efficiently. For other sets, the cost of the projections can be too high. To circumvent these issues, alternative methods based on the famous Frank-Wolfe algorithm have been studied and used. Such methods use a Linear Optimization Oracle at each iterati…
In this article a family of second order ODEs associated to inertial gradient descend is studied. These ODEs are widely used to build trajectories converging to a minimizer $x^*$ of a function $F$, possibly convex. This family includes the continuous version of the Nesterov inertial scheme and the continuous heavy ball method. Several damping parameters, not necessarily vanishing, and a perturbation term $g$ are thus considered. The damping parameter is linked to the inertia of the associated inertial scheme and the perturbation term $g$ is linked to the error that can be done on the gradient …
We study distributed optimization to minimize a global objective that is a sum of smooth and strongly-convex local cost functions. Recently, several algorithms over undirected and directed graphs have been proposed that use a gradient tracking method to achieve linear convergence to the global minimizer. However, a connection between these different approaches has been unclear. In this paper, we first show that many of the existing first-order algorithms are in fact related with a simple state transformation, at the heart of which lies the $\mathcal{AB}$ algorithm. We then describe \textit{dis…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Rice University（莱斯大学）
We study the acceleration of the Local Polynomial Interpolation-based Gradient Descent method (LPI-GD) recently proposed for the approximate solution of empirical risk minimization problems (ERM). We focus on loss functions that are strongly convex and smooth with condition number $σ$. We additionally assume the loss function is $η$-Hölder continuous with respect to the data. The oracle complexity of LPI-GD is $\tilde{O}\left(σm^d \log(1/\varepsilon)\right)$ for a desired accuracy $\varepsilon$, where $d$ is the dimension of the parameter space, and $m$ is the cardinality of an approximation g…
In 1964, Polyak showed that the Heavy-ball method, the simplest momentum technique, accelerates convergence of strongly-convex problems in the vicinity of the solution. While Nesterov later developed a globally accelerated version, Polyak's original algorithm remains simpler and more widely used in applications such as deep learning. Despite this popularity, the question of whether Heavy-ball is also globally accelerated or not has not been fully answered yet, and no convincing counterexample has been provided. This is largely due to the difficulty in finding an effective Lyapunov function: in…
In this paper, a new variant of accelerated gradient descent is proposed. The pro-posed method does not require any information about the objective function, usesexact line search for the practical accelerations of convergence, converges accordingto the well-known lower bounds for both convex and non-convex objective functions,possesses primal-dual properties and can be applied in the non-euclidian set-up. Asfar as we know this is the rst such method possessing all of the above properties atthe same time. We also present a universal version of the method which is applicableto non-smooth proble…
We consider stochastic convex optimization problems with affine constraints and develop several methods using either primal or dual approach to solve it. In the primal case, we use a special penalization technique to make the initial problem more convenient for using optimization methods. We propose algorithms to solve it based on Similar Triangles Method with Inexact Proximal Step for the convex smooth and strongly convex smooth objective functions and methods based on Gradient Sliding algorithm to solve the same problems in the non-smooth case. We prove the convergence guarantees in the smoo…
We study gradient-based optimization methods obtained by direct Runge-Kutta discretization of the ordinary differential equation (ODE) describing the movement of a heavy-ball under constant friction coefficient. When the function is high order smooth and strongly convex, we show that directly simulating the ODE with known numerical integrators achieve acceleration in a nontrivial neighborhood of the optimal solution. In particular, the neighborhood can grow larger as the condition number of the function increases. Furthermore, our results also hold for nonconvex but quasi-strongly convex objec…
We derive lower bounds on the black-box oracle complexity of large-scale smooth convex minimization problems, with emphasis on minimizing smooth (with Holder continuous, with a given exponent and constant, gradient) convex functions over high-dimensional ||.||_p-balls, 1<=p<=\infty. Our bounds turn out to be tight (up to logarithmic in the design dimension factors), and can be viewed as a substantial extension of the existing lower complexity bounds for large-scale convex minimization covering the nonsmooth case and the 'Euclidean' smooth case (minimization of convex functions with Lipschitz c…
In this paper, we consider the class of quasiconvex functions and its proper subclass of conic functions. The integer minimization problem of these functions is considered in the paper, assuming that an optimized function is defined by the comparison oracle. We will show that there is no a polynomial algorithm on $\log R$ to optimize quasiconvex functions in the ball of integer radius $R$ using only the comparison oracle. On the other hand, if an optimized function is conic, then we show that there is a polynomial on $\log R$ algorithm. We also present an exponential on the dimension lower bou…
On the fast convergence of minibatch heavy ball momentum
Operations Research and Industrial Engineering（运筹学与工业工程）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Mathematics（数学）；New York University（纽约大学）；Computer Science and Engineering（计算机科学与工程）
We study the conditions under which one is able to efficiently apply variance-reduction and acceleration schemes on finite sum optimization problems. First, we show that, perhaps surprisingly, the finite sum structure by itself, is not sufficient for obtaining a complexity bound of $\tilde{\cO}((n+L/μ)\ln(1/ε))$ for $L$-smooth and $μ$-strongly convex individual functions - one must also know which individual function is being referred to by the oracle at each iteration. Next, we show that for a broad class of first-order and coordinate-descent finite sum algorithms (including, e.g., SDCA, SVRG…
In this paper, we propose new accelerated methods for smooth convex optimization, called contracting proximal methods. At every step of these methods, we need to minimize a contracted version of the objective function augmented by a regularization term in the form of Bregman divergence. We provide global convergence analysis for a general scheme admitting inexactness in solving the auxiliary subproblem. In the case of using for this purpose high-order tensor methods, we demonstrate an acceleration effect for both convex and uniformly convex composite objective functions. Thus, our construction…
Understanding the Acceleration Phenomenon via High-Resolution Differential Equations
Florida International University（佛罗里达国际大学）；Carnegie Mellon University（卡内基梅隆大学）；University of California, Berkeley（加州大学伯克利分校）；University of Pennsylvania（宾夕法尼亚大学）
University of California, Berkeley（加州大学伯克利分校）；Institute for Advanced Study（普林斯顿高等研究院）；University of Pennsylvania（宾夕法尼亚大学）
We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization schemes: an explicit Euler scheme, an implicit Euler scheme, and a symplectic scheme. We show that the optimization algorithm generated by applying the symplectic scheme to a high-resolution ODE proposed by Shi et al. [2018] achieves an accelerated rate for minimizing smooth strongly convex functions. On the other hand, the resulting algorithm either fails to ach…
Institute for Mathematical and Computational Engineering（数学与计算工程研究所）；Faculty of Mathematics（理学院）；School of Engineering（工学院）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
Arguably, the two most popular accelerated or momentum-based optimization methods in machine learning are Nesterov's accelerated gradient and Polyaks's heavy ball, both corresponding to different discretizations of a particular second order differential equation with friction. Such connections with continuous-time dynamical systems have been instrumental in demystifying acceleration phenomena in optimization. Here we study structure-preserving discretizations for a certain class of dissipative (conformal) Hamiltonian systems, allowing us to analyze the symplectic structure of both Nesterov and…
Recently classes of conic and discrete conic functions were introduced. In this paper we use the term convic instead conic. The class of convic functions properly includes the classes of convex functions, strictly quasiconvex functions and the class of quasiconvex polynomials. On the other hand, the class of convic functions is properly included in the class of quasiconvex functions. The discrete convic function is a discrete analogue of the convic function. Recently the lower bound $3^{n-1}\log (2 ρ-1)$ for the number of calls to the comparison oracle needed to find the minimum of the discret…
