اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدLimitations on Variance-Reduction and Acceleration Schemes for Finite Sum Optimization
113
0
0.0
(
0
)
تأليف
Yossi Arjevani
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We study the conditions under which one is able to efficiently apply variance-reduction and acceleration schemes on finite sum optimization problems. First, we show that, perhaps surprisingly, the finite sum structure by itself, is not sufficient for obtaining a complexity bound of $tilde{cO}((n+L/mu)ln(1/epsilon))$ for $L$-smooth and $mu$-strongly convex individual functions - one must also know which individual function is being referred to by the oracle at each iteration. Next, we show that for a broad class of first-order and coordinate-descent finite sum algorithms (including, e.g., SDCA, SVRG, SAG), it is not possible to get an `accelerated complexity bound of $tilde{cO}((n+sqrt{n L/mu})ln(1/epsilon))$, unless the strong convexity parameter is given explicitly. Lastly, we show that when this class of algorithms is used for minimizing $L$-smooth and convex finite sums, the optimal complexity bound is $tilde{cO}(n+L/epsilon)$, assuming that (on average) the same update rule is used in every iteration, and $tilde{cO}(n+sqrt{nL/epsilon})$, otherwise.
قيم البحث
اقرأ أيضاً
This paper considers decentralized minimization of $N:=nm$ smooth non-convex cost functions equally divided over a directed network of $n$ nodes. Specifically, we describe a stochastic first-order gradient method, called GT-SARAH, that employs a SARA
H-type variance reduction technique and gradient tracking (GT) to address the stochastic and decentralized nature of the problem. We show that GT-SARAH, with appropriate algorithmic parameters, finds an $epsilon$-accurate first-order stationary point with $Obig(maxbig{N^{frac{1}{2}},n(1-lambda)^{-2},n^{frac{2}{3}}m^{frac{1}{3}}(1-lambda)^{-1}big}Lepsilon^{-2}big)$ gradient complexity, where ${(1-lambda)in(0,1]}$ is the spectral gap of the network weight matrix and $L$ is the smoothness parameter of the cost functions. This gradient complexity outperforms that of the existing decentralized stochastic gradient methods. In particular, in a big-data regime such that ${n = O(N^{frac{1}{2}}(1-lambda)^{3})}$, this gradient complexity furthers reduces to ${O(N^{frac{1}{2}}Lepsilon^{-2})}$, independent of the network topology, and matches that of the centralized near-optimal variance-reduced methods. Moreover, in this regime GT-SARAH achieves a non-asymptotic linear speedup, in that, the total number of gradient computations at each node is reduced by a factor of $1/n$ compared to the centralized near-optimal algorithms that perform all gradient computations at a single node. To the best of our knowledge, GT-SARAH is the first algorithm that achieves this property. In addition, we show that appropriate choices of local minibatch size balance the trade-offs between the gradient and communication complexity of GT-SARAH. Over infinite time horizon, we establish that all nodes in GT-SARAH asymptotically achieve consensus and converge to a first-order stationary point in the almost sure and mean-squared sense.
Structured problems arise in many applications. To solve these problems, it is important to leverage the structure information. This paper focuses on convex problems with a finite-sum compositional structure. Finite-sum problems appear as the sample
average approximation of a stochastic optimization problem and also arise in machine learning with a huge amount of training data. One popularly used numerical approach for finite-sum problems is the stochastic gradient method (SGM). However, the additional compositional structure prohibits easy access to unbiased stochastic approximation of the gradient, so directly applying the SGM to a finite-sum compositional optimization problem (COP) is often inefficient. We design new algorithms for solving strongly-convex and also convex two-level finite-sum COPs. Our design incorporates the Katyusha acceleration technique and adopts the mini-batch sampling from both outer-level and inner-level finite-sum. We first analyze the algorithm for strongly-convex finite-sum COPs. Similar to a few existing works, we obtain linear convergence rate in terms of the expected objective error, and from the convergence rate result, we then establish complexity results of the algorithm to produce an $varepsilon$-solution. Our complexity results have the same dependence on the number of component functions as existing works. However, due to the use of Katyusha acceleration, our results have better dependence on the condition number $kappa$ and improve to $kappa^{2.5}$ from the best-known $kappa^3$. Finally, we analyze the algorithm for convex finite-sum COPs, which uses as a subroutine the algorithm for strongly-convex finite-sum COPs. Again, we obtain better complexity results than existing works in terms of the dependence on $varepsilon$, improving to $varepsilon^{-2.5}$ from the best-known $varepsilon^{-3}$.
We propose stochastic variance reduced algorithms for solving convex-concave saddle point problems, monotone variational inequalities, and monotone inclusions. Our framework applies to extragradient, forward-backward-forward, and forward-reflected-ba
ckward methods both in Euclidean and Bregman setups. All proposed methods converge in exactly the same setting as their deterministic counterparts and they either match or improve the best-known complexities for solving structured min-max problems. Our results reinforce the correspondence between variance reduction in variational inequalities and minimization. We also illustrate the improvements of our approach with numerical evaluations on matrix games.
In this paper, we propose a unified view of gradient-based algorithms for stochastic convex composite optimization by extending the concept of estimate sequence introduced by Nesterov. More precisely, we interpret a large class of stochastic optimiza
tion methods as procedures that iteratively minimize a surrogate of the objective, which covers the stochastic gradient descent method and variants of the incremental approaches SAGA, SVRG, and MISO/Finito/SDCA. This point of view has several advantages: (i) we provide a simple generic proof of convergence for all of the aforementioned methods; (ii) we naturally obtain new algorithms with the same guarantees; (iii) we derive generic strategies to make these algorithms robust to stochastic noise, which is useful when data is corrupted by small random perturbations. Finally, we propose a new accelerated stochastic gradient descent algorithm and an accelerated SVRG algorithm with optimal complexity that is robust to stochastic noise.
In this paper, we introduce various mechanisms to obtain accelerated first-order stochastic optimization algorithms when the objective function is convex or strongly convex. Specifically, we extend the Catalyst approach originally designed for determ
inistic objectives to the stochastic setting. Given an optimization method with mild convergence guarantees for strongly convex problems, the challenge is to accelerate convergence to a noise-dominated region, and then achieve convergence with an optimal worst-case complexity depending on the noise variance of the gradients. A side contribution of our work is also a generic analysis that can handle inexact proximal operators, providing new insights about the robustness of stochastic algorithms when the proximal operator cannot be exactly computed.
الأسئلة المقترحة
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
