arXiv subjects

Cong Fang

Publications and source records attributed to Cong Fang.

76 records · Page 5Linked to original sources

arXiv2018

SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator

In this paper, we propose a new technique named \textit{Stochastic Path-Integrated Differential EstimatoR} (SPIDER), which can be used to track many deterministic quantities of interest with significantly reduced computational cost. We apply SPIDER to two tasks, namely the stochastic first-order and zeroth-order methods. For stochastic first-order method, combining SPIDER with normalized gradient descent, we propose two new algorithms, namely SPIDER-SFO and SPIDER-SFO\textsuperscript{+}, that solve non-convex stochastic optimization problems using stochastic gradients only. We provide sharp error-bound results on their convergence rates. In special, we prove that the SPIDER-SFO and SPIDER-SFO\textsuperscript{+} algorithms achieve a record-breaking gradient computation cost of $\mathcal{O}\left( \min( n^{1/2} ε^{-2}, ε^{-3} ) \right)$ for finding an $ε$-approximate first-order and $\tilde{\mathcal{O}}\left( \min( n^{1/2} ε^{-2}+ε^{-2.5}, ε^{-3} ) \right)$ for finding an $(ε, \mathcal{O}(ε^{0.5}))$-approximate second-order stationary point, respectively. In addition, we prove that SPIDER-SFO nearly matches the algorithmic lower bound for finding approximate first-order stationary points under the gradient Lipschitz assumption in the finite-sum setting. For stochastic zeroth-order method, we prove a cost of $\mathcal{O}( d \min( n^{1/2} ε^{-2}, ε^{-3}) )$ which outperforms all existing results.

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arXiv2018

Accelerating Asynchronous Algorithms for Convex Optimization by Momentum Compensation

Asynchronous algorithms have attracted much attention recently due to the crucial demands on solving large-scale optimization problems. However, the accelerated versions of asynchronous algorithms are rarely studied. In this paper, we propose the "momentum compensation" technique to accelerate asynchronous algorithms for convex problems. Specifically, we first accelerate the plain Asynchronous Gradient Descent, which achieves a faster $O(1/\sqrtε)$ (v.s. $O(1/ε)$) convergence rate for non-strongly convex functions, and $O(\sqrtκ\log(1/ε))$ (v.s. $O(κ\log(1/ε))$) for strongly convex functions to reach an $ε$- approximate minimizer with the condition number $κ$. We further apply the technique to accelerate modern stochastic asynchronous algorithms such as Asynchronous Stochastic Coordinate Descent and Asynchronous Stochastic Gradient Descent. Both of the resultant practical algorithms are faster than existing ones by order. To the best of our knowledge, we are the first to consider accelerated algorithms that allow updating by delayed gradients and are the first to propose truly accelerated asynchronous algorithms. Finally, the experimental results on a shared memory system show that acceleration can lead to significant performance gains on ill-conditioned problems.

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arXiv2017

Convergence Rates Analysis of The Quadratic Penalty Method and Its Applications to Decentralized Distributed Optimization

In this paper, we study a variant of the quadratic penalty method for linearly constrained convex problems, which has already been widely used but actually lacks theoretical justification. Namely, the penalty parameter steadily increases and the penalized objective function is minimized inexactly rather than exactly, e.g., with only one step of the proximal gradient descent. For such a variant of the quadratic penalty method, we give counterexamples to show that it may not give a solution to the original constrained problem. By choosing special penalty parameters, we ensure the convergence and further establish the convergence rates of $O\left(\frac{1}{\sqrt{K}}\right)$ for the generally convex problems and $O\left(\frac{1}{K}\right)$ for strongly convex ones, where $K$ is the number of iterations. Furthermore, by adopting Nesterov's extrapolation we show that the convergence rates can be improved to $O\left(\frac{1}{K}\right)$ for the generally convex problems and $O\left(\frac{1}{K^2}\right)$ for strongly convex ones. When applied to the decentralized distributed optimization, the penalty methods studied in this paper become the widely used distributed gradient method and the fast distributed gradient method. However, due to the totally different analysis framework, we can improve their $O\left(\frac{\log K}{\sqrt{K}}\right)$ and $O\left(\frac{\log K}{K}\right)$ convergence rates to $O\left(\frac{1}{\sqrt{K}}\right)$ and $O\left(\frac{1}{K}\right)$ with fewer assumptions on the network topology for general convex problems. Using our analysis framework, we also extend the fast distributed gradient method to a communication efficient version, i.e., finding an $\varepsilon$ solution in $O\left(\frac{1}{\varepsilon}\right)$ communications and $O\left(\frac{1}{\varepsilon^{2+δ}}\right)$ computations for the non-smooth problems, where $δ$ is a small constant.

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arXiv2017

Faster and Non-ergodic O(1/K) Stochastic Alternating Direction Method of Multipliers

We study stochastic convex optimization subjected to linear equality constraints. Traditional Stochastic Alternating Direction Method of Multipliers and its Nesterov's acceleration scheme can only achieve ergodic O(1/\sqrt{K}) convergence rates, where K is the number of iteration. By introducing Variance Reduction (VR) techniques, the convergence rates improve to ergodic O(1/K). In this paper, we propose a new stochastic ADMM which elaborately integrates Nesterov's extrapolation and VR techniques. We prove that our algorithm can achieve a non-ergodic O(1/K) convergence rate which is optimal for separable linearly constrained non-smooth convex problems, while the convergence rates of VR based ADMM methods are actually tight O(1/\sqrt{K}) in non-ergodic sense. To the best of our knowledge, this is the first work that achieves a truly accelerated, stochastic convergence rate for constrained convex problems. The experimental results demonstrate that our algorithm is significantly faster than the existing state-of-the-art stochastic ADMM methods.

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