اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدPGDOT -- Perturbed Gradient Descent Adapted with Occupation Time
85
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
This paper develops further the idea of perturbed gradient descent (PGD), by adapting perturbation with the history of states via the notion of occupation time. The proposed algorithm, perturbed gradient descent adapted with occupation time (PGDOT), is shown to converge at least as fast as the PGD algorithm and is guaranteed to avoid getting stuck at saddle points. The analysis is corroborated by empirical studies, in which a mini-batch version of PGDOT is shown to outperform alternatives such as mini-batch gradient descent, Adam, AMSGrad, and RMSProp in training multilayer perceptrons (MLPs). In particular, the mini-batch PGDOT manages to escape saddle points whereas these alternatives fail.
قيم البحث
اقرأ أيضاً
We present a strikingly simple proof that two rules are sufficient to automate gradient descent: 1) dont increase the stepsize too fast and 2) dont overstep the local curvature. No need for functional values, no line search, no information about the
function except for the gradients. By following these rules, you get a method adaptive to the local geometry, with convergence guarantees depending only on the smoothness in a neighborhood of a solution. Given that the problem is convex, our method converges even if the global smoothness constant is infinity. As an illustration, it can minimize arbitrary continuously twice-differentiable convex function. We examine its performance on a range of convex and nonconvex problems, including logistic regression and matrix factorization.
Motivated by broad applications in reinforcement learning and machine learning, this paper considers the popular stochastic gradient descent (SGD) when the gradients of the underlying objective function are sampled from Markov processes. This Markov
sampling leads to the gradient samples being biased and not independent. The existing results for the convergence of SGD under Markov randomness are often established under the assumptions on the boundedness of either the iterates or the gradient samples. Our main focus is to study the finite-time convergence of SGD for different types of objective functions, without requiring these assumptions. We show that SGD converges nearly at the same rate with Markovian gradient samples as with independent gradient samples. The only difference is a logarithmic factor that accounts for the mixing time of the Markov chain.
Despite the strong theoretical guarantees that variance-reduced finite-sum optimization algorithms enjoy, their applicability remains limited to cases where the memory overhead they introduce (SAG/SAGA), or the periodic full gradient computation they
require (SVRG/SARAH) are manageable. A promising approach to achieving variance reduction while avoiding these drawbacks is the use of importance sampling instead of control variates. While many such methods have been proposed in the literature, directly proving that they improve the convergence of the resulting optimization algorithm has remained elusive. In this work, we propose an importance-sampling-based algorithm we call SRG (stochastic reweighted gradient). We analyze the convergence of SRG in the strongly-convex case and show that, while it does not recover the linear rate of control variates methods, it provably outperforms SGD. We pay particular attention to the time and memory overhead of our proposed method, and design a specialized red-black tree allowing its efficient implementation. Finally, we present empirical results to support our findings.
This paper considers the problem of understanding the exit time for trajectories of gradient-related first-order methods from saddle neighborhoods under some initial boundary conditions. Given the `flat geometry around saddle points, first-order meth
ods can struggle in escaping these regions in a fast manner due to the small magnitudes of gradients encountered. In particular, while it is known that gradient-related first-order methods escape strict-saddle neighborhoods, existing literature does not explicitly leverage the local geometry around saddle points in order to control behavior of gradient trajectories. It is in this context that this paper puts forth a rigorous geometric analysis of the gradient-descent method around strict-saddle neighborhoods using matrix perturbation theory. In doing so, it provides a key result that can be used to generate an approximate gradient trajectory for any given initial conditions. In addition, the analysis leads to a linear exit-time solution for gradient-descent method under certain necessary initial conditions for a class of strict-saddle functions.
Distributed descent-based methods are an essential toolset to solving optimization problems in multi-agent system scenarios. Here the agents seek to optimize a global objective function through mutual cooperation. Oftentimes, cooperation is achieved
over a wireless communication network that is prone to delays and errors. There are many scenarios wherein the objective function is either non-differentiable or merely observable. In this paper, we present a cross-entropy based distributed stochastic approximation algorithm (SA) that finds a minimum of the objective, using only samples. We call this algorithm Decentralized Simultaneous Perturbation Stochastic Gradient, with Constant Sensitivity Parameters (DSPG). This algorithm is a two fold improvement over the classic Simultaneous Perturbation Stochastic Approximations (SPSA) algorithm. Specifically, DSPG allows for (i) the use of old information from other agents and (ii) easy implementation through the use simple hyper-parameter choices. We analyze the biases and variances that arise due to these two allowances. We show that the biases due to communication delays can be countered by a careful choice of algorithm hyper-parameters. The variance of the gradient estimator and its effect on the rate of convergence is studied. We present numerical results supporting our theory. Finally, we discuss an application to the stochastic consensus problem.
الأسئلة المقترحة
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
