arXiv subjects

Jelena Diakonikolas

Publications and source records attributed to Jelena Diakonikolas.

54 records · Page 3Linked to original sources

arXiv2021

Parameter-free Locally Accelerated Conditional Gradients

Projection-free conditional gradient (CG) methods are the algorithms of choice for constrained optimization setups in which projections are often computationally prohibitive but linear optimization over the constraint set remains computationally feasible. Unlike in projection-based methods, globally accelerated convergence rates are in general unattainable for CG. However, a very recent work on Locally accelerated CG (LaCG) has demonstrated that local acceleration for CG is possible for many settings of interest. The main downside of LaCG is that it requires knowledge of the smoothness and strong convexity parameters of the objective function. We remove this limitation by introducing a novel, Parameter-Free Locally accelerated CG (PF-LaCG) algorithm, for which we provide rigorous convergence guarantees. Our theoretical results are complemented by numerical experiments, which demonstrate local acceleration and showcase the practical improvements of PF-LaCG over non-accelerated algorithms, both in terms of iteration count and wall-clock time.

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arXiv2021

Variance Reduction via Primal-Dual Accelerated Dual Averaging for Nonsmooth Convex Finite-Sums

We study structured nonsmooth convex finite-sum optimization that appears widely in machine learning applications, including support vector machines and least absolute deviation. For the primal-dual formulation of this problem, we propose a novel algorithm called \emph{Variance Reduction via Primal-Dual Accelerated Dual Averaging (\vrpda)}. In the nonsmooth and general convex setting, \vrpda~has the overall complexity $O(nd\log\min \{1/ε, n\} + d/ε)$ in terms of the primal-dual gap, where $n$ denotes the number of samples, $d$ the dimension of the primal variables, and $ε$ the desired accuracy. In the nonsmooth and strongly convex setting, the overall complexity of \vrpda~becomes $O(nd\log\min\{1/ε, n\} + d/\sqrtε)$ in terms of both the primal-dual gap and the distance between iterate and optimal solution. Both these results for \vrpda~improve significantly on state-of-the-art complexity estimates, which are $O(nd\log \min\{1/ε, n\} + \sqrt{n}d/ε)$ for the nonsmooth and general convex setting and $O(nd\log \min\{1/ε, n\} + \sqrt{n}d/\sqrtε)$ for the nonsmooth and strongly convex setting, in a much more simple and straightforward way. Moreover, both complexities are better than \emph{lower} bounds for general convex finite sums that lack the particular (common) structure that we consider. Our theoretical results are supported by numerical experiments, which confirm the competitive performance of \vrpda~compared to state-of-the-art.

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arXiv2021

Efficient Methods for Structured Nonconvex-Nonconcave Min-Max Optimization

The use of min-max optimization in adversarial training of deep neural network classifiers and training of generative adversarial networks has motivated the study of nonconvex-nonconcave optimization objectives, which frequently arise in these applications. Unfortunately, recent results have established that even approximate first-order stationary points of such objectives are intractable, even under smoothness conditions, motivating the study of min-max objectives with additional structure. We introduce a new class of structured nonconvex-nonconcave min-max optimization problems, proposing a generalization of the extragradient algorithm which provably converges to a stationary point. The algorithm applies not only to Euclidean spaces, but also to general $\ell_p$-normed finite-dimensional real vector spaces. We also discuss its stability under stochastic oracles and provide bounds on its sample complexity. Our iteration complexity and sample complexity bounds either match or improve the best known bounds for the same or less general nonconvex-nonconcave settings, such as those that satisfy variational coherence or in which a weak solution to the associated variational inequality problem is assumed to exist.

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arXiv2021

Potential Function-based Framework for Making the Gradients Small in Convex and Min-Max Optimization

Making the gradients small is a fundamental optimization problem that has eluded unifying and simple convergence arguments in first-order optimization, so far primarily reserved for other convergence criteria, such as reducing the optimality gap. We introduce a novel potential function-based framework to study the convergence of standard methods for making the gradients small in smooth convex optimization and convex-concave min-max optimization. Our framework is intuitive and it provides a lens for viewing algorithms that make the gradients small as being driven by a trade-off between reducing either the gradient norm or a certain notion of an optimality gap. On the lower bounds side, we discuss tightness of the obtained convergence results for the convex setup and provide a new lower bound for minimizing norm of cocoercive operators that allows us to argue about optimality of methods in the min-max setup.

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arXiv2020

Fair Packing and Covering on a Relative Scale

Fair resource allocation is a fundamental optimization problem with applications in operations research, networking, and economic and game theory. Research in these areas has led to the general acceptance of a class of $α$-fair utility functions parameterized by $α\in [0, \infty]$. We consider $α$-fair packing -- the problem of maximizing $α$-fair utilities under positive linear constraints -- and provide a simple first-order method for solving it with relative-error guarantees. The method has a significantly lower convergence time than the state of the art, and to analyze it, we leverage the Approximate Duality Gap Technique, which provides an intuitive interpretation of the convergence argument. Finally, we introduce a natural counterpart of $α$-fairness for minimization problems and motivate its usage in the context of fair task allocation. This generalization yields $α$-fair covering problems, for which we provide the first near-linear-time solvers with relative-error guarantees.

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arXiv2020

Generalized Momentum-Based Methods: A Hamiltonian Perspective

We take a Hamiltonian-based perspective to generalize Nesterov's accelerated gradient descent and Polyak's heavy ball method to a broad class of momentum methods in the setting of (possibly) constrained minimization in Euclidean and non-Euclidean normed vector spaces. Our perspective leads to a generic and unifying nonasymptotic analysis of convergence of these methods in both the function value (in the setting of convex optimization) and in norm of the gradient (in the setting of unconstrained, possibly nonconvex, optimization). Our approach relies upon a time-varying Hamiltonian that produces generalized momentum methods as its equations of motion. The convergence analysis for these methods is intuitive and is based on the conserved quantities of the time-dependent Hamiltonian.

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arXiv2020

Halpern Iteration for Near-Optimal and Parameter-Free Monotone Inclusion and Strong Solutions to Variational Inequalities

We leverage the connections between nonexpansive maps, monotone Lipschitz operators, and proximal mappings to obtain near-optimal (i.e., optimal up to poly-log factors in terms of iteration complexity) and parameter-free methods for solving monotone inclusion problems. These results immediately translate into near-optimal guarantees for approximating strong solutions to variational inequality problems, approximating convex-concave min-max optimization problems, and minimizing the norm of the gradient in min-max optimization problems. Our analysis is based on a novel and simple potential-based proof of convergence of Halpern iteration, a classical iteration for finding fixed points of nonexpansive maps. Additionally, we provide a series of algorithmic reductions that highlight connections between different problem classes and lead to lower bounds that certify near-optimality of the studied methods.

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arXiv2020

Langevin Monte Carlo without smoothness

Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a serious limitation for applications in machine learning. In this paper, we remove this limitation, providing polynomial-time convergence guarantees for a variant of LMC in the setting of nonsmooth log-concave distributions. At a high level, our results follow by leveraging the implicit smoothing of the log-density that comes from a small Gaussian perturbation that we add to the iterates of the algorithm and controlling the bias and variance that are induced by this perturbation.

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arXiv2020

Conjugate Gradients and Accelerated Methods Unified: The Approximate Duality Gap View

This note provides a novel, simple analysis of the method of conjugate gradients for the minimization of convex quadratic functions. In contrast with standard arguments, our proof is entirely self-contained and does not rely on the existence of Chebyshev polynomials. Another advantage of our development is that it clarifies the relation between the method of conjugate gradients and general accelerated methods for smooth minimization by unifying their analyses within the framework of the Approximate Duality Gap Technique that was introduced by the authors.

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arXiv2019

The Approximate Duality Gap Technique: A Unified Theory of First-Order Methods

We present a general technique for the analysis of first-order methods. The technique relies on the construction of a duality gap for an appropriate approximation of the objective function, where the function approximation improves as the algorithm converges. We show that in continuous time enforcement of an invariant that this approximate duality gap decreases at a certain rate exactly recovers a wide range of first-order continuous-time methods. We characterize the discretization errors incurred by different discretization methods, and show how iteration-complexity-optimal methods for various classes of problems cancel out the discretization error. The techniques are illustrated on various classes of problems -- including convex minimization for Lipschitz-continuous objectives, smooth convex minimization, composite minimization, smooth and strongly convex minimization, solving variational inequalities with monotone operators, and convex-concave saddle-point optimization -- and naturally extend to other settings.

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arXiv2019

Locally Accelerated Conditional Gradients

Conditional gradients constitute a class of projection-free first-order algorithms for smooth convex optimization. As such, they are frequently used in solving smooth convex optimization problems over polytopes, for which the computational cost of orthogonal projections would be prohibitive. However, they do not enjoy the optimal convergence rates achieved by projection-based accelerated methods; moreover, achieving such globally-accelerated rates is information-theoretically impossible for these methods. To address this issue, we present Locally Accelerated Conditional Gradients -- an algorithmic framework that couples accelerated steps with conditional gradient steps to achieve local acceleration on smooth strongly convex problems. Our approach does not require projections onto the feasible set, but only on (typically low-dimensional) simplices, thus keeping the computational cost of projections at bay. Further, it achieves the optimal accelerated local convergence. Our theoretical results are supported by numerical experiments, which demonstrate significant speedups of our framework over state of the art methods in both per-iteration progress and wall-clock time.

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arXiv2019

Alternating Randomized Block Coordinate Descent

Block-coordinate descent algorithms and alternating minimization methods are fundamental optimization algorithms and an important primitive in large-scale optimization and machine learning. While various block-coordinate-descent-type methods have been studied extensively, only alternating minimization -- which applies to the setting of only two blocks -- is known to have convergence time that scales independently of the least smooth block. A natural question is then: is the setting of two blocks special? We show that the answer is "no" as long as the least smooth block can be optimized exactly -- an assumption that is also needed in the setting of alternating minimization. We do so by introducing a novel algorithm AR-BCD, whose convergence time scales independently of the least smooth (possibly non-smooth) block. The basic algorithm generalizes both alternating minimization and randomized block coordinate (gradient) descent, and we also provide its accelerated version -- AAR-BCD. As a special case of AAR-BCD, we obtain the first nontrivial accelerated alternating minimization algorithm.

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arXiv2019

Lower Bounds for Parallel and Randomized Convex Optimization

We study the question of whether parallelization in the exploration of the feasible set can be used to speed up convex optimization, in the local oracle model of computation. We show that the answer is negative for both deterministic and randomized algorithms applied to essentially any of the interesting geometries and nonsmooth, weakly-smooth, or smooth objective functions. In particular, we show that it is not possible to obtain a polylogarithmic (in the sequential complexity of the problem) number of parallel rounds with a polynomial (in the dimension) number of queries per round. In the majority of these settings and when the dimension of the space is polynomial in the inverse target accuracy, our lower bounds match the oracle complexity of sequential convex optimization, up to at most a logarithmic factor in the dimension, which makes them (nearly) tight. Prior to our work, lower bounds for parallel convex optimization algorithms were only known in a small fraction of the settings considered in this paper, mainly applying to Euclidean ($\ell_2$) and $\ell_\infty$ spaces. Our work provides a more general approach for proving lower bounds in the setting of parallel convex optimization.

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arXiv2019

A Hierarchical WDM-based Scalable Data Center Network Architecture

Massive data centers are at the heart of the Internet. The rapid growth of Internet traffic and the abundance of rich data-driven applications have raised the need for enormous network bandwidth. Towards meeting this growing traffic demand, optical interconnects have gained significant attention, as they can provide high throughput, low latency, and scalability. In particular, optical Wavelength Division Multiplexing (WDM) provides the possibility to build data centers comprising of millions of servers, while providing hundreds of terabits per second bandwidth. In this paper, we propose a WDM-based Reconfigurable Hierarchical Optical Data Center Architecture (RHODA) that can satisfy future Internet traffic demands. To improve scalability, our DCN architecture is hierarchical, as it groups server racks into clusters. Cluster membership is reconfigurable through the use of optical switches. Each cluster enables heavy-traffic communication among the racks within. To support varying traffic patterns, the inter-cluster network topology and link capacities are also reconfigurable, which is achieved through the use of optical space switches and Wavelength Selective Switches (WSSs). Our simulation results demonstrate that in terms of average hop distance, RHODA outperforms OSA, FatTree and WaveCube by up to 81%, 66% and 60%, respectively.

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arXiv2018

Hybrid Scheduling in Heterogeneous Half- and Full-Duplex Wireless Networks

Full-duplex (FD) wireless is an attractive communication paradigm with high potential for improving network capacity and reducing delay in wireless networks. Despite significant progress on the physical layer development, the challenges associated with developing medium access control (MAC) protocols for heterogeneous networks composed of both legacy half-duplex (HD) and emerging FD devices have not been fully addressed. Therefore, we focus on the design and performance evaluation of scheduling algorithms for infrastructure-based heterogeneous HD-FD networks (composed of HD and FD users). We first show that centralized Greedy Maximal Scheduling (GMS) is throughput-optimal in heterogeneous HD-FD networks. We propose the Hybrid-GMS (H-GMS) algorithm, a distributed implementation of GMS that combines GMS and a queue-based random-access mechanism. We prove that H-GMS is throughput-optimal. Moreover, we analyze the delay performance of H-GMS by deriving lower bounds on the average queue length. We further demonstrate the benefits of upgrading HD nodes to FD nodes in terms of throughput gains for individual nodes and the whole network. Finally, we evaluate the performance of H-GMS and its variants in terms of throughput, delay, and fairness between FD and HD users via extensive simulations. We show that in heterogeneous HD-FD networks, H-GMS achieves 16-30x better delay performance and improves fairness between HD and FD users by up to 50% compared with the fully decentralized Q-CSMA algorithm.

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arXiv2018

On Acceleration with Noise-Corrupted Gradients

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 simplicity of AGDP and its analysis to clarify the interaction between noise and acceleration and to suggest modifications to the algorithm that reduce the mean and variance of the error incurred due to the gradient noise.

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arXiv2018

Accelerated Extra-Gradient Descent: A Novel Accelerated First-Order Method

We provide a novel accelerated first-order method that achieves the asymptotically optimal convergence rate for smooth functions in the first-order oracle model. To this day, Nesterov's Accelerated Gradient Descent (AGD) and variations thereof were the only methods achieving acceleration in this standard blackbox model. In contrast, our algorithm is significantly different from AGD, as it relies on a predictor-corrector approach similar to that used by Mirror-Prox [Nemirovski, 2004] and Extra-Gradient Descent [Korpelevich, 1977] in the solution of convex-concave saddle point problems. For this reason, we dub our algorithm Accelerated Extra-Gradient Descent (AXGD). Its construction is motivated by the discretization of an accelerated continuous-time dynamics [Krichene et al., 2015] using the classical method of implicit Euler discretization. Our analysis explicitly shows the effects of discretization through a conceptually novel primal-dual viewpoint. Moreover, we show that the method is quite general: it attains optimal convergence rates for other classes of objectives (e.g., those with generalized smoothness properties or that are non-smooth and Lipschitz-continuous) using the appropriate choices of step lengths. Finally, we present experiments showing that our algorithm matches the performance of Nesterov's method, while appearing more robust to noise in some cases.

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arXiv2017

Solving Packing and Covering LPs in $\tilde{O}(\frac{1}{ε^2})$ Distributed Iterations with a Single Algorithm and Simpler Analysis

Packing and covering linear programs belong to the narrow class of linear programs that are efficiently solvable in parallel and distributed models of computation, yet are a powerful modeling tool for a wide range of fundamental problems in theoretical computer science, operations research, and many other areas. Following recent progress in obtaining faster distributed and parallel algorithms for packing and covering linear programs, we present a simple algorithm whose iteration count matches the best known $\tilde{O}(\frac{1}{ε^2})$ for this class of problems. The algorithm is similar to the algorithm of [Allen-Zhu and Orecchia, 2015], it can be interpreted as Nesterov's dual averaging, and it constructs approximate solutions to both primal (packing) and dual (covering) problems. However, the analysis relies on the construction of an approximate optimality gap and a primal-dual view, leading to a more intuitive interpretation. Moreover, our analysis suggests that all existing algorithms for solving packing and covering linear programs in parallel/distributed models of computation are, in fact, unaccelerated, and raises the question of designing accelerated algorithms for this class of problems.

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