TY - JOUR
AB - We consider the monotone variational inequality problem in a Hilbert space and describe a projection-type method with inertial terms under the following properties: (a) The method generates a strongly convergent iteration sequence; (b) The method requires, at each iteration, only one projection onto the feasible set and two evaluations of the operator; (c) The method is designed for variational inequality for which the underline operator is monotone and uniformly continuous; (d) The method includes an inertial term. The latter is also shown to speed up the convergence in our numerical results. A comparison with some related methods is given and indicates that the new method is promising.
AU - Shehu, Yekini
AU - Li, Xiao-Huan
AU - Dong, Qiao-Li
ID - 6593
JF - Numerical Algorithms
SN - 1017-1398
TI - An efficient projection-type method for monotone variational inequalities in Hilbert spaces
VL - 84
ER -
TY - JOUR
AB - In this paper, we introduce an inertial projection-type method with different updating strategies for solving quasi-variational inequalities with strongly monotone and Lipschitz continuous operators in real Hilbert spaces. Under standard assumptions, we establish different strong convergence results for the proposed algorithm. Primary numerical experiments demonstrate the potential applicability of our scheme compared with some related methods in the literature.
AU - Shehu, Yekini
AU - Gibali, Aviv
AU - Sagratella, Simone
ID - 7161
JF - Journal of Optimization Theory and Applications
SN - 0022-3239
TI - Inertial projection-type methods for solving quasi-variational inequalities in real Hilbert spaces
VL - 184
ER -
TY - JOUR
AB - Weak convergence of inertial iterative method for solving variational inequalities is the focus of this paper. The cost function is assumed to be non-Lipschitz and monotone. We propose a projection-type method with inertial terms and give weak convergence analysis under appropriate conditions. Some test results are performed and compared with relevant methods in the literature to show the efficiency and advantages given by our proposed methods.
AU - Shehu, Yekini
AU - Iyiola, Olaniyi S.
ID - 7577
JF - Applicable Analysis
SN - 0003-6811
TI - Weak convergence for variational inequalities with inertial-type method
ER -
TY - JOUR
AB - We develop a framework for the rigorous analysis of focused stochastic local search algorithms. These algorithms search a state space by repeatedly selecting some constraint that is violated in the current state and moving to a random nearby state that addresses the violation, while (we hope) not introducing many new violations. An important class of focused local search algorithms with provable performance guarantees has recently arisen from algorithmizations of the Lovász local lemma (LLL), a nonconstructive tool for proving the existence of satisfying states by introducing a background measure on the state space. While powerful, the state transitions of algorithms in this class must be, in a precise sense, perfectly compatible with the background measure. In many applications this is a very restrictive requirement, and one needs to step outside the class. Here we introduce the notion of measure distortion and develop a framework for analyzing arbitrary focused stochastic local search algorithms, recovering LLL algorithmizations as the special case of no distortion. Our framework takes as input an arbitrary algorithm of such type and an arbitrary probability measure and shows how to use the measure as a yardstick of algorithmic progress, even for algorithms designed independently of the measure.
AU - Achlioptas, Dimitris
AU - Iliopoulos, Fotis
AU - Kolmogorov, Vladimir
ID - 7412
IS - 5
JF - SIAM Journal on Computing
SN - 0097-5397
TI - A local lemma for focused stochastical algorithms
VL - 48
ER -
TY - CONF
AB - We present a new proximal bundle method for Maximum-A-Posteriori (MAP) inference in structured energy minimization problems. The method optimizes a Lagrangean relaxation of the original energy minimization problem using a multi plane block-coordinate Frank-Wolfe method that takes advantage of the specific structure of the Lagrangean decomposition. We show empirically that our method outperforms state-of-the-art Lagrangean decomposition based algorithms on some challenging Markov Random Field, multi-label discrete tomography and graph matching problems.
AU - Swoboda, Paul
AU - Kolmogorov, Vladimir
ID - 7468
SN - 10636919
T2 - Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition
TI - Map inference via block-coordinate Frank-Wolfe algorithm
VL - 2019-June
ER -
TY - CONF
AB - Deep neural networks (DNNs) have become increasingly important due to their excellent empirical performance on a wide range of problems. However, regularization is generally achieved by indirect means, largely due to the complex set of functions defined by a network and the difficulty in measuring function complexity. There exists no method in the literature for additive regularization based on a norm of the function, as is classically considered in statistical learning theory. In this work, we study the tractability of function norms for deep neural networks with ReLU activations. We provide, to the best of our knowledge, the first proof in the literature of the NP-hardness of computing function norms of DNNs of 3 or more layers. We also highlight a fundamental difference between shallow and deep networks. In the light on these results, we propose a new regularization strategy based on approximate function norms, and show its efficiency on a segmentation task with a DNN.
AU - Rannen-Triki, Amal
AU - Berman, Maxim
AU - Kolmogorov, Vladimir
AU - Blaschko, Matthew B.
ID - 7639
SN - 9781728150239
T2 - Proceedings of the 2019 International Conference on Computer Vision Workshop
TI - Function norms for neural networks
ER -
TY - JOUR
AB - It is well known that many problems in image recovery, signal processing, and machine learning can be modeled as finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. Many papers have studied forward-backward splitting methods for finding zeros of the sum of two monotone operators in Hilbert spaces. Most of the proposed splitting methods in the literature have been proposed for the sum of maximal monotone and inverse-strongly monotone operators in Hilbert spaces. In this paper, we consider splitting methods for finding zeros of the sum of maximal monotone operators and Lipschitz continuous monotone operators in Banach spaces. We obtain weak and strong convergence results for the zeros of the sum of maximal monotone and Lipschitz continuous monotone operators in Banach spaces. Many already studied problems in the literature can be considered as special cases of this paper.
AU - Shehu, Yekini
ID - 6596
IS - 4
JF - Results in Mathematics
SN - 1422-6383
TI - Convergence results of forward-backward algorithms for sum of monotone operators in Banach spaces
VL - 74
ER -
TY - CONF
AB - A Valued Constraint Satisfaction Problem (VCSP) provides a common framework that can express a wide range of discrete optimization problems. A VCSP instance is given by a finite set of variables, a finite domain of labels, and an objective function to be minimized. This function is represented as a sum of terms where each term depends on a subset of the variables. To obtain different classes of optimization problems, one can restrict all terms to come from a fixed set Γ of cost functions, called a language.
Recent breakthrough results have established a complete complexity classification of such classes with respect to language Γ: if all cost functions in Γ satisfy a certain algebraic condition then all Γ-instances can be solved in polynomial time, otherwise the problem is NP-hard. Unfortunately, testing this condition for a given language Γ is known to be NP-hard. We thus study exponential algorithms for this meta-problem. We show that the tractability condition of a finite-valued language Γ can be tested in O(3‾√3|D|⋅poly(size(Γ))) time, where D is the domain of Γ and poly(⋅) is some fixed polynomial. We also obtain a matching lower bound under the Strong Exponential Time Hypothesis (SETH). More precisely, we prove that for any constant δ<1 there is no O(3‾√3δ|D|) algorithm, assuming that SETH holds.
AU - Kolmogorov, Vladimir
ID - 6725
SN - 1868-8969
T2 - 46th International Colloquium on Automata, Languages and Programming
TI - Testing the complexity of a valued CSP language
VL - 132
ER -
TY - JOUR
AB - The main contributions of this paper are the proposition and the convergence analysis of a class of inertial projection-type algorithm for solving variational inequality problems in real Hilbert spaces where the underline operator is monotone and uniformly continuous. We carry out a unified analysis of the proposed method under very mild assumptions. In particular, weak convergence of the generated sequence is established and nonasymptotic O(1 / n) rate of convergence is established, where n denotes the iteration counter. We also present some experimental results to illustrate the profits gained by introducing the inertial extrapolation steps.
AU - Shehu, Yekini
AU - Iyiola, Olaniyi S.
AU - Li, Xiao-Huan
AU - Dong, Qiao-Li
ID - 7000
IS - 4
JF - Computational and Applied Mathematics
SN - 2238-3603
TI - Convergence analysis of projection method for variational inequalities
VL - 38
ER -
TY - CONF
AB - The accuracy of information retrieval systems is often measured using complex loss functions such as the average precision (AP) or the normalized discounted cumulative gain (NDCG). Given a set of positive and negative samples, the parameters of a retrieval system can be estimated by minimizing these loss functions. However, the non-differentiability and non-decomposability of these loss functions does not allow for simple gradient based optimization algorithms. This issue is generally circumvented by either optimizing a structured hinge-loss upper bound to the loss function or by using asymptotic methods like the direct-loss minimization framework. Yet, the high computational complexity of loss-augmented inference, which is necessary for both the frameworks, prohibits its use in large training data sets. To alleviate this deficiency, we present a novel quicksort flavored algorithm for a large class of non-decomposable loss functions. We provide a complete characterization of the loss functions that are amenable to our algorithm, and show that it includes both AP and NDCG based loss functions. Furthermore, we prove that no comparison based algorithm can improve upon the computational complexity of our approach asymptotically. We demonstrate the effectiveness of our approach in the context of optimizing the structured hinge loss upper bound of AP and NDCG loss for learning models for a variety of vision tasks. We show that our approach provides significantly better results than simpler decomposable loss functions, while requiring a comparable training time.
AU - Mohapatra, Pritish
AU - Rolinek, Michal
AU - Jawahar, C V
AU - Kolmogorov, Vladimir
AU - Kumar, M Pawan
ID - 273
SN - 9781538664209
T2 - 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition
TI - Efficient optimization for rank-based loss functions
ER -