arXiv subjects

Donghwan Kim

Publications and source records attributed to Donghwan Kim.

47 records · Page 3Linked to original sources

arXiv2021

Accelerated Proximal Point Method for Maximally Monotone Operators

This paper proposes an accelerated proximal point method for maximally monotone operators. The proof is computer-assisted via the performance estimation problem approach. The proximal point method includes various well-known convex optimization methods, such as the proximal method of multipliers and the alternating direction method of multipliers, and thus the proposed acceleration has wide applications. Numerical experiments are presented to demonstrate the accelerating behaviors.

math.OC↗

arXiv2020

Optimizing the Efficiency of First-Order Methods for Decreasing the Gradient of Smooth Convex Functions

This paper optimizes the step coefficients of first-order methods for smooth convex minimization in terms of the worst-case convergence bound (i.e., efficiency) of the decrease in the gradient norm. This work is based on the performance estimation problem approach. The worst-case gradient bound of the resulting method is optimal up to a constant for large-dimensional smooth convex minimization problems, under the initial bounded condition on the cost function value. This paper then illustrates that the proposed method has a computationally efficient form that is similar to the optimized gradient method.

math.OC↗

arXiv2020

PAM:Point-wise Attention Module for 6D Object Pose Estimation

6D pose estimation refers to object recognition and estimation of 3D rotation and 3D translation. The key technology for estimating 6D pose is to estimate pose by extracting enough features to find pose in any environment. Previous methods utilized depth information in the refinement process or were designed as a heterogeneous architecture for each data space to extract feature. However, these methods are limited in that they cannot extract sufficient feature. Therefore, this paper proposes a Point Attention Module that can efficiently extract powerful feature from RGB-D. In our Module, attention map is formed through a Geometric Attention Path(GAP) and Channel Attention Path(CAP). In GAP, it is designed to pay attention to important information in geometric information, and CAP is designed to pay attention to important information in Channel information. We show that the attention module efficiently creates feature representations without significantly increasing computational complexity. Experimental results show that the proposed method outperforms the existing methods in benchmarks, YCB Video and LineMod. In addition, the attention module was applied to the classification task, and it was confirmed that the performance significantly improved compared to the existing model.

cs.CV↗

arXiv2020

Generating Diverse and Consistent QA pairs from Contexts with Information-Maximizing Hierarchical Conditional VAEs

One of the most crucial challenges in question answering (QA) is the scarcity of labeled data, since it is costly to obtain question-answer (QA) pairs for a target text domain with human annotation. An alternative approach to tackle the problem is to use automatically generated QA pairs from either the problem context or from large amount of unstructured texts (e.g. Wikipedia). In this work, we propose a hierarchical conditional variational autoencoder (HCVAE) for generating QA pairs given unstructured texts as contexts, while maximizing the mutual information between generated QA pairs to ensure their consistency. We validate our Information Maximizing Hierarchical Conditional Variational AutoEncoder (Info-HCVAE) on several benchmark datasets by evaluating the performance of the QA model (BERT-base) using only the generated QA pairs (QA-based evaluation) or by using both the generated and human-labeled pairs (semi-supervised learning) for training, against state-of-the-art baseline models. The results show that our model obtains impressive performance gains over all baselines on both tasks, using only a fraction of data for training.

cs.CL↗

arXiv2018

Schroedinger Correspondence Applied to Crystals

In 1926, E. Schroedinger published a paper solving his new time dependent wave equation for a displaced ground state in a harmonic oscillator (now called a coherent state). He showed that the parameters describing the mean position and mean momentum of the wave packet obey the equations of motion of the classical oscillator while retaining its width. This was a qualitatively new kind of correspondence principle, differing from those leading up to quantum mechanics. Schroedinger surely knew that this correspondence would extend to an N-dimensional harmonic oscillator. This Schroedinger Correspondence Principle is an extremely intuitive and powerful way to approach many aspects of harmonic solids including anharmonic corrections.

cond-mat.other↗

arXiv2018

Generalizing the optimized gradient method for smooth convex minimization

This paper generalizes the optimized gradient method (OGM) that achieves the optimal worst-case cost function bound of first-order methods for smooth convex minimization. Specifically, this paper studies a generalized formulation of OGM and analyzes its worst-case rates in terms of both the function value and the norm of the function gradient. This paper also develops a new algorithm called OGM-OG that is in the generalized family of OGM and that has the best known analytical worst-case bound with rate $O(1/N^{1.5})$ on the decrease of the gradient norm among fixed-step first-order methods. This paper also proves that Nesterov's fast gradient method has an $O(1/N^{1.5})$ worst-case gradient norm rate but with constant larger than OGM-OG. The proof is based on the worst-case analysis called Performance Estimation Problem.

math.OC↗

arXiv2018

Another look at the fast iterative shrinkage/thresholding algorithm (FISTA)

This paper provides a new way of developing the fast iterative shrinkage/thresholding algorithm (FISTA) that is widely used for minimizing composite convex functions with a nonsmooth term such as the $\ell_1$ regularizer. In particular, this paper shows that FISTA corresponds to an optimized approach to accelerating the proximal gradient method with respect to a worst-case bound of the cost function. This paper then proposes a new algorithm that is derived by instead optimizing the step coefficients of the proximal gradient method with respect to a worst-case bound of the composite gradient mapping. The proof is based on the worst-case analysis called Performance Estimation Problem.

math.OC↗

arXiv2017

Adaptive Restart of the Optimized Gradient Method for Convex Optimization

First-order methods with momentum such as Nesterov's fast gradient method are very useful for convex optimization problems, but can exhibit undesirable oscillations yielding slow convergence rates for some applications. An adaptive restarting scheme can improve the convergence rate of the fast gradient method, when the parameter of a strongly convex cost function is unknown or when the iterates of the algorithm enter a locally strongly convex region. Recently, we introduced the optimized gradient method, a first-order algorithm that has an inexpensive per-iteration computational cost similar to that of the fast gradient method, yet has a worst-case cost function rate that is twice faster than that of the fast gradient method and that is optimal for large-dimensional smooth convex problems. Building upon the success of accelerating the fast gradient method using adaptive restart, this paper investigates similar heuristic acceleration of the optimized gradient method. We first derive a new first-order method that resembles the optimized gradient method for strongly convex quadratic problems with known function parameters, yielding a linear convergence rate that is faster than that of the analogous version of the fast gradient method. We then provide a heuristic analysis and numerical experiments that illustrate that adaptive restart can accelerate the convergence of the optimized gradient method. Numerical results also illustrate that adaptive restart is helpful for a proximal version of the optimized gradient method for nonsmooth composite convex functions.

math.OC↗

arXiv2016

Fast dual proximal gradient algorithms with rate $O(1/k^{1.5})$ for convex minimization

We consider minimizing the composite function that consists of a strongly convex function and a convex function. The fast dual proximal gradient (FDPG) method decreases the dual function with a rate $O(1/k^2)$, leading to a rate $O(1/k)$ for decreasing the primal function. We propose a generalized FDPG method that guarantees an $O(1/k^{1.5})$ rate for the dual proximal gradient norm decrease. By relating this to the primal function decrease, the proposed approach decreases the primal function with the improved $O(1/k^{1.5})$ rate.

math.OC↗

arXiv2016

On the convergence analysis of the optimized gradient method

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 showed that it has a worst-case convergence bound for the cost function decrease that is twice as small as that of Nesterov's fast gradient method (FGM), yet has a similarly efficient practical implementation. Drori showed recently that OGM has optimal complexity over the general class of first-order methods. This optimality makes it important to study fully the convergence properties of OGM. The previous worst-case convergence bound for OGM was derived for only the last iterate of a secondary sequence. This paper provides an analytic convergence bound for the primary sequence generated by OGM. We then discuss additional convergence properties of OGM, including the interesting fact that OGM has two types of worst-case functions: a piecewise affine-quadratic function and a quadratic function. These results help complete the theory of optimal first-order methods for smooth convex minimization.

math.OC↗

arXiv2016

Optimized first-order methods for smooth convex minimization

We introduce new optimized first-order methods for smooth unconstrained convex minimization. Drori and Teboulle recently described a numerical method for computing the $N$-iteration optimal step coefficients in a class of first-order algorithms that includes gradient methods, heavy-ball methods, and Nesterov's fast gradient methods. However, Drori and Teboulle's numerical method is computationally expensive for large $N$, and the corresponding numerically optimized first-order algorithm requires impractical memory and computation for large-scale optimization problems. In this paper, we propose optimized first-order algorithms that achieve a convergence bound that is two times smaller than for Nesterov's fast gradient methods; our bound is found analytically and refines the numerical bound. Furthermore, the proposed optimized first-order methods have efficient recursive forms that are remarkably similar to Nesterov's fast gradient methods.

math.OC↗