We propose general non-accelerated and accelerated tensor methods under inexact information on the derivatives of the objective, analyze their convergence rate. Further, we provide conditions for the inexactness in each derivative that is sufficient for each algorithm to achieve the desired accuracy. As a corollary, we propose stochastic tensor methods for convex optimization and obtain sufficient mini-batch sizes for each derivative.
It is well-known that accelerated gradient first-order methods possess optimal complexity estimates for the class of convex smooth minimization problems. In many practical situations it makes sense to work with inexact gradient information. However, this can lead to an accumulation of corresponding inexactness in the theoretical estimates of the rate of convergence. We propose some modification of the Sequential Subspace Optimization Method (SESOP) for minimization problems with quasar-convex functions with inexact gradient. A theoretical result is obtained indicating the absence of accumulati…
In this paper, we propose a new inexact version of the projected subgradient method to solve nondifferentiable constrained convex optimization problems. The method combine $ε$-subgradient method with a procedure to obtain a feasible inexact projection onto the constraint set. Asymptotic convergence results and iteration-complexity bounds for the sequence generated by the method employing the well known exogenous stepsizes, Polyak's stepsizes, and dynamic stepsizes are established.
We introduce new global and local inexact oracle concepts for a wide class of convex functions in composite convex minimization. Such inexact oracles naturally come from primal-dual framework, barrier smoothing, inexact computations of gradients and Hessian, and many other situations. We also provide examples showing that the class of convex functions equipped with the newly inexact second-order oracles is larger than standard self-concordant as well as Lipschitz gradient function classes. Further, we investigate several properties of convex and/or self-concordant functions under the inexact s…
In this paper we introduce new methods for convex optimization problems with inexact stochastic oracle. First method is an extension of the intermediate gradient method proposed by Devolder, Glineur and Nesterov for problems with inexact oracle. Our new method can be applied to the problems with composite structure, stochastic inexact oracle and allows using non-Euclidean setup. We prove estimates for mean rate of convergence and probabilities of large deviations from this rate. Also we introduce two modifications of this method for strongly convex problems. For the first modification we prove…
In view of the minimization of a function which is the sum of a differentiable function $f$ and a convex function $g$ we introduce descent methods which can be viewed as produced by inexact auxiliary problem principleor inexact variable metric forward-backward algorithm. Assuming that the global objective function satisfies the Kurdyka-Lojasiewicz inequalitywe prove the convergence of the proposed algorithm weakening assumptions found in previous works.
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"' convex part. Informally speaking, oracle inexactness means that, for the "`hard"' part, at any point we can approximately calculate the value of the function and construct a quadratic function, which approximately bounds this function from above. We give several examples of such inexactness: smooth non-convex functions with inexact Hölder-continuous gradient, fun…
In the paper we generalize universal gradient method (Yu. Nesterov) to strongly convex case and to Intermediate gradient method (Devolder-Glineur-Nesterov). We also consider possible generalizations to stochastic and online context. We show how these results can be generalized to gradient-free method and method of random direction search. But the main ingridient of this paper is assumption about the oracle. We considered the oracle to be inexact.
In this paper we will discuss two variants of an inexact feasible interior point algorithm for convex quadratic programming. We will consider two different neighbourhoods: a (small) one induced by the use of the Euclidean norm which yields a short-step algorithm and a symmetric one induced by the use of the infinity norm which yields a (practical) long-step algorithm. Both algorithms allow for the Newton equation system to be solved inexactly. For both algorithms we will provide conditions for the level of error acceptable in the Newton equation and establish the worst-case complexity results.
An inexact Newton type method for numerical minimization of convex piecewise quadratic functions is considered and its convergence is analyzed. Earlier, a similar method was successfully applied to optimizaton problems arising in numerical grid generation. The method can be applied for computing a minimum norm nonnegative solution of underdetermined system of linear equations or for finding the distance between two convex polyhedra. The performance of the method is tested using sample data from NETLIB family of the University of Florida sparse matrix collection as well as quasirandom data.
Inexact alternating direction multiplier methods (ADMMs) are developed for solving general separable convex optimization problems with a linear constraint and with an objective that is the sum of smooth and nonsmooth terms. The approach involves linearized subproblems, a back substitution step, and either gradient or accelerated gradient techniques. Global convergence is established. The methods are particularly useful when the ADMM subproblems do not have closed form solution or when the solution of the subproblems is expensive. Numerical experiments based on image reconstruction problems sho…
In this paper we analyze several inexact fast augmented Lagrangian methods for solving linearly constrained convex optimization problems. Mainly, our methods rely on the combination of excessive-gap-like smoothing technique developed in [15] and the newly introduced inexact oracle framework from [4]. We analyze several algorithmic instances with constant and adaptive smoothing parameters and derive total computational complexity results in terms of projections onto a simple primal set. For the basic inexact fast augmented Lagrangian algorithm we obtain the overall computational complexity of o…
In this paper we propose a new inexact dual decomposition algorithm for solving separable convex optimization problems. This algorithm is a combination of three techniques: dual Lagrangian decomposition, smoothing and excessive gap. The algorithm requires only one primal step and two dual steps at each iteration and allows one to solve the subproblem of each component inexactly and in parallel. Moreover, the algorithmic parameters are updated automatically without any tuning strategy as in augmented Lagrangian approaches. We analyze the convergence of the algorithm and estimate its $O(\frac{1}…
We present two approximate versions of the proximal subgradient method for minimizing the sum of two convex functions (not necessarily differentiable). The algorithms involve, at each iteration, inexact evaluations of the proximal operator and approximate subgradients of the functions (namely: the $ε$-subgradients). The methods use different error criteria for approximating the proximal operators. We provide an analysis of the convergence and rate of convergence properties of these methods, considering various stepsize rules, including both, diminishing and constant stepsizes. For the case whe…
We introduce an extension of Dual Dynamic Programming (DDP) to solve convex nonlinear dynamic programming equations. We call Inexact DDP (IDDP) this extension which applies to situations where some or all primal and dual subproblems to be solved along the iterations of the method are solved with a bounded error. We show that any accumulation point of the sequence of decisions is an approximate solution to the dynamic programming equations. When these errors tend to zero as the number of iterations goes to infinity, we show that IDDP solves the dynamic programming equations. We extend the analy…
In this paper, we use Proximal Cubic regularized Newton Methods (PCNM) to optimize the sum of a smooth convex function and a non-smooth convex function, where we use inexact gradient and Hessian, and an inexact subsolver for the cubic regularized second-order subproblem. We propose inexact variants of PCNM and accelerated PCNM respectively, and show that both variants can achieve the same convergence rate as in the exact case, provided that the errors in the inexact gradient, Hessian and subsolver decrease at appropriate rates. Meanwhile, in the online stochastic setting where data comes endle…
In this paper, we propose an inexact perturbed path-following algorithm in the framework of Lagrangian dual decomposition for solving large-scale structured convex optimization problems. Unlike the exact versions considered in literature, we allow one to solve the primal problem inexactly up to a given accuracy. The inexact perturbed algorithm allows to use both approximate Hessian matrices and approximate gradient vectors to compute Newton-type directions for the dual problem. The algorithm is divided into two phases. The first phase computes an initial point which makes use of inexact pertur…
In this note, we consider a special instance of the scaled, inexact and adaptive generalised Fast Iterative Soft-Thresholding Algorithm (SAGE-FISTA) recently proposed in (Rebegoldi, Calatroni, '21) for the efficient solution of strongly convex composite optimisation problems. In particular, we address here the sole (non-strongly) convex optimisation scenario, which is frequently encountered in many imaging applications. The proposed inexact S-FISTA algorithm shows analogies to the variable metric and inexact version of FISTA studied in (Bonettini, Rebegoldi, Ruggiero, '19), the main difference…
This paper is concerned with two-block separable convex minimization problems with linear constraints, for which it is either impossible or too expensive to obtain the exact solutions of the subproblems involved in the proximal ADMM (alternating direction method of multipliers). Such structured convex minimization problems often arise from the two-block regroup settlement of three or four-block separable convex optimization problems with linear constraints, or from the constrained total-variation superresolution image reconstruction problems in image processing. For them, we propose an inexact…
For solving large-scale non-convex problems, we propose inexact variants of trust region and adaptive cubic regularization methods, which, to increase efficiency, incorporate various approximations. In particular, in addition to approximate sub-problem solves, both the Hessian and the gradient are suitably approximated. Using rather mild conditions on such approximations, we show that our proposed inexact methods achieve similar optimal worst-case iteration complexities as the exact counterparts. Our proposed algorithms, and their respective theoretical analysis, do not require knowledge of an…
We first investigate properties of M-tensor equations. In particular, we show that if the constant term of the equation is nonnegative, then finding a nonnegative solution of the equation can be done by finding a positive solution of a lower dimensional M-tensor equation. We then propose an inexact Newton method to find a positive solution to the lower dimensional equation and establish its global convergence. We also show that the convergence rate of the method is quadratic. At last, we do numerical experiments to test the proposed Newton method. The results show that the proposed Newton meth…
We study a Newton-like method for the minimization of an objective function that is the sum of a smooth convex function and an l-1 regularization term. This method, which is sometimes referred to in the literature as a proximal Newton method, computes a step by minimizing a piecewise quadratic model of the objective function. In order to make this approach efficient in practice, it is imperative to perform this inner minimization inexactly. In this paper, we give inexactness conditions that guarantee global convergence and that can be used to control the local rate of convergence of the iterat…
In this paper, we consider the linearly constrained composite convex optimization problem, whose objective is a sum of a smooth function and a possibly nonsmooth function. We propose an inexact augmented Lagrangian (IAL) framework for solving the problem. The stopping criterion used in solving the augmented Lagrangian (AL) subproblem in the proposed IAL framework is weaker and potentially much easier to check than the one used in most of the existing IAL frameworks/methods. We analyze the global convergence and the non-ergodic convergence rate of the proposed IAL framework.
In this paper we present a complete iteration complexity analysis of inexact first order Lagrangian and penalty methods for solving cone constrained convex problems that have or may not have optimal Lagrange multipliers that close the duality gap. We first assume the existence of optimal Lagrange multipliers and study primal-dual first order methods based on inexact information and augmented Lagrangian smoothing or Nesterov type smoothing. For inexact (fast) gradient augmented Lagrangian methods we derive a total computational complexity of $\mathcal{O}\left(\frac{1}ε\right)$ projections onto …
In the paper we show that euclidian randomization in some situations (i.e. for gradient-free method on a simplex) can be as good as the randomization on the unit sphere in 1-norm. That is on the simplex example we show that for gradient-free methods the choise of the prox-structure and the choise of a way of randomization have to be connected to each other. We demonstrate how it can be done in an optimal way. It is important that we consider inexact oracle.
In this chapter we derive computational complexity certifications of first order inexact dual methods for solving general smooth constrained convex problems which can arise in real-time applications, such as model predictive control. When it is difficult to project on the primal constraint set described by a collection of general convex functions, we use the Lagrangian relaxation to handle the complicated constraints and then, we apply dual (fast) gradient algorithms based on inexact dual gradient information for solving the corresponding dual problem. The iteration complexity analysis is base…
A unified framework for inexact adaptive stepsizes in the gradient methods, the conjugate gradient methods and the quasi-Newton methods for strictly convex quadratic optimization
Locating proximal points is a component of numerous minimization algorithms. This work focuses on developing a method to find the proximal point of a convex function at a point, given an inexact oracle. Our method assumes that exact function values are at hand, but exact subgradients are either not available or not useful. We use approximate subgradients to build a model of the objective function, and prove that the method converges to the true prox-point within acceptable tolerance. The subgradient $g_k$ used at each step $k$ is such that the distance from $g_k$ to the true subdifferential of…
We study the convergence issue for inexact descent algorithm (employing general step sizes) for multiobjective optimizations on general Riemannian manifolds (without curvature constraints). Under the assumption of the local convexity/quasi-convexity, local/global convergence results are established. On the other hand, without the assumption of the local convexity/quasi-convexity, but under a Kurdyka-Łojasiewicz-like condition, local/global linear convergence results are presented, which seem new even in Euclidean spaces setting and improve sharply the corresponding results in [24] in the case …
In this paper, we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems, and variational inequalities. This framework allows obtaining many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, Bregman proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing kno…
On Some Versions of Subspace Optimization Methods with Inexact Gradient Information
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Research Center for Artificial Intelligence, Innopolis University（因诺波利斯大学人工智能研究中心）；National Research University Higher School of Economics（国立高等经济大学）；V. Vernadsky Crimean Federal University（V.维尔纳茨基克里米亚联邦大学）
We consider optimization methods for convex minimization problems under inexact information on the objective function. We introduce inexact model of the objective, which as a particular cases includes $(δ,L)$ inexact oracle and relative smoothness condition. We analyze gradient method which uses this inexact model and obtain convergence rates for convex and strongly convex problems. To show potential applications of our general framework we consider three particular problems. The first one is clustering by electorial model introduced in [Nesterov, 2018]. The second one is approximating optimal…
In this paper, we propose and analyze an inexact version of the symmetric proximal alternating direction method of multipliers (ADMM) for solving linearly constrained optimization problems. Basically, the method allows its first subproblem to be solved inexactly in such way that a relative approximate criterion is satisfied. In terms of the iteration number $k$, we establish global $\mathcal{O} (1/ \sqrt{k})$ pointwise and $\mathcal{O} (1/ {k})$ ergodic convergence rates of the method for a domain of the acceleration parameters, which is consistent with the largest known one in the exact case.…
We consider a variable metric and inexact version of the FISTA-type algorithm considered in (Chambolle, Pock, 2016, Calatroni, Chambolle, 2019) for the minimization of the sum of two (possibly strongly) convex functions. The proposed algorithm is combined with an adaptive (non-monotone) backtracking strategy, which allows for the adjustment of the algorithmic step-size along the iterations in order to improve the convergence speed. We prove a linear convergence result for the function values, which depends on both the strong convexity moduli of the two functions and the upper and lower bounds …
In this paper, we study inexact high-order Tensor Methods for solving convex optimization problems with composite objective. At every step of such methods, we use approximate solution of the auxiliary problem, defined by the bound for the residual in function value. We propose two dynamic strategies for choosing the inner accuracy: the first one is decreasing as $1/k^{p + 1}$, where $p \geq 1$ is the order of the method and $k$ is the iteration counter, and the second approach is using for the inner accuracy the last progress in the target objective. We show that inexact Tensor Methods with th…
In this paper, we propose new first-order methods for minimization of a convex function on a simple convex set. We assume that the objective function is a composite function given as a sum of a simple convex function and a convex function with inexact Hölder-continuous subgradient. We propose Universal Intermediate Gradient Method. Our method enjoys both the universality and intermediateness properties. Following the paper by Y. Nesterov (Math.Prog., 2015) on Universal Gradient Methods, our method does not require any information about the Hölder parameter and constant and adjusts itself autom…
An inexact accelerated stochastic Alternating Direction Method of Multipliers (AS-ADMM) scheme is developed for solving structured separable convex optimization problems with linear constraints. The objective function is the sum of a possibly nonsmooth convex function and a smooth function which is an average of many component convex functions. Problems having this structure often arise in machine learning and data mining applications. AS-ADMM combines the ideas of both ADMM and the stochastic gradient methods using variance reduction techniques. One of the ADMM subproblems employs a lineariza…
In this paper, we propose an inexact multi-block ADMM-type first-order method for solving a class of high-dimensional convex composite conic optimization problems to moderate accuracy. The design of this method combines an inexact 2-block majorized semi-proximal ADMM and the recent advances in the inexact symmetric Gauss-Seidel (sGS) technique for solving a multi-block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block-variable. One distinctive feature of our proposed method (the sGS-imsPADMM) is that it only needs one cycle of an in…
We propose an inexact proximal augmented Lagrangian framework with explicit inner problem termination rule for composite convex optimization problems. We consider arbitrary linearly convergent inner solver including in particular stochastic algorithms, making the resulting framework more scalable facing the ever-increasing problem dimension. Each subproblem is solved inexactly with an explicit and self-adaptive stopping criterion, without requiring to set an a priori target accuracy. When the primal and dual domain are bounded, our method achieves $O(1/\sqrtε)$ and $O(1/ε)$ complexity bound in…
We consider a misspecified optimization problem that requires minimizing of a convex function $f(x;θ^*)$ in x over a constraint set represented by $h(x;θ^*)\leq 0$, where $θ^*$ is an unknown (or misspecified) vector of parameters. Suppose $θ^*$ can be learnt by a distinct process that generates a sequence of estimators $θ_k$, each of which is an increasingly accurate approximation of $θ^*$. We develop a first-order augmented Lagrangian scheme for computing an optimal solution $x^*$ while simultaneously learning $θ^*$.
Optimal Convex Optimization with Inexact Second-Order Oracles
IIIS, Tsinghua University（清华大学交叉信息研究院）；Department of Artificial Intelligence, Westlake University（西湖大学人工智能学院）；School of Data Science, Fudan University（复旦大学数据科学系）；Department of Computer Science and Engineering, The Chinese University of Hong Kong（香港中文大学计算机科学与工程学系）
We consider the structured stochastic convex program requiring the minimization of $\mathbb{E}[\tilde f(x,ξ)]+\mathbb{E}[\tilde g(y,ξ)]$ subject to the constraint $Ax + By = b$. Motivated by the need for decentralized schemes and structure, we propose a stochastic inexact ADMM (SI-ADMM) framework where subproblems are solved inexactly via stochastic approximation schemes. Based on this framework, we prove the following: (i) under suitable assumptions on the associated batch-size of samples utilized at each iteration, the SI-ADMM scheme produces a sequence that converges to the unique solution …
In this paper we study the auxiliary problems that appear in $p$-order tensor methods for unconstrained minimization of convex functions with $ν$-Hölder continuous $p$th derivatives. This type of auxiliary problems corresponds to the minimization of a $(p+ν)$-order regularization of the $p$th order Taylor approximation of the objective. For the case $p=3$, we consider the use of Gradient Methods with Bregman distance. When the regularization parameter is sufficiently large, we prove that the referred methods take at most $\mathcal{O}(\log(ε^{-1}))$ iterations to find either a suitable approxim…
