Making Gradient Descent Optimal for Strongly Convex Stochastic Optimization

نویسندگان

  • Alexander Rakhlin
  • Ohad Shamir
  • Karthik Sridharan
چکیده

Stochastic gradient descent (SGD) is a simple and popular method to solve stochastic optimization problems which arise in machine learning. For strongly convex problems, its convergence rate was known to be O(log(T )/T ), by running SGD for T iterations and returning the average point. However, recent results showed that using a different algorithm, one can get an optimal O(1/T ) rate. This might lead one to believe that standard SGD is suboptimal, and maybe should even be replaced as a method of choice. In this paper, we investigate the optimality of SGD in a stochastic setting. We show that for smooth problems, the algorithm attains the optimal O(1/T ) rate. However, for non-smooth problems, the convergence rate with averaging might really be Ω(log(T )/T ), and this is not just an artifact of the analysis. On the flip side, we show that a simple modification of the averaging step suffices to recover the O(1/T ) rate, and no other change of the algorithm is necessary. We also present experimental results which support our findings, and point out open problems.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Optimal Stochastic Strongly Convex Optimization with a Logarithmic Number of Projections

We consider stochastic strongly convex optimization with a complex inequality constraint. This complex inequality constraint may lead to computationally expensive projections in algorithmic iterations of the stochastic gradient descent (SGD) methods. To reduce the computation costs pertaining to the projections, we propose an Epoch-Projection Stochastic Gradient Descent (Epro-SGD) method. The p...

متن کامل

On Stochastic Subgradient Mirror-Descent Algorithm with Weighted Averaging

This paper considers stochastic subgradient mirror-descent method for solving constrained convex minimization problems. In particular, a stochastic subgradient mirror-descent method with weighted iterate-averaging is investigated and its per-iterate convergence rate is analyzed. The novel part of the approach is in the choice of weights that are used to construct the averages. Through the use o...

متن کامل

Stochastic Gradient Descent with Only One Projection

Although many variants of stochastic gradient descent have been proposed for large-scale convex optimization, most of them require projecting the solution at each iteration to ensure that the obtained solution stays within the feasible domain. For complex domains (e.g., positive semidefinite cone), the projection step can be computationally expensive, making stochastic gradient descent unattrac...

متن کامل

Efficient Stochastic Gradient Descent for Strongly Convex Optimization

We motivate this study from a recent work on a stochastic gradient descent (SGD) method with only one projection (Mahdavi et al., 2012), which aims at alleviating the computational bottleneck of the standard SGD method in performing the projection at each iteration, and enjoys an O(log T/T ) convergence rate for strongly convex optimization. In this paper, we make further contributions along th...

متن کامل

Stochastic Gradient Descent for Non-smooth Optimization: Convergence Results and Optimal Averaging Schemes

Stochastic Gradient Descent (SGD) is one of the simplest and most popular stochastic optimization methods. While it has already been theoretically studied for decades, the classical analysis usually required nontrivial smoothness assumptions, which do not apply to many modern applications of SGD with non-smooth objective functions such as support vector machines. In this paper, we investigate t...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:
  • CoRR

دوره abs/1109.5647  شماره 

صفحات  -

تاریخ انتشار 2012