Subgradient algorithms for training support vector machines have been quite successful for solving large-scale and online learning problems. However, they have been restricted to linear kernels and strongly convex formulations. This paper describes efficient subgradient approaches without such limitations. Our approaches make use of randomized low-dimensional approximations to nonlinear kernels, and minimization of a reduced primal formulation using an algorithm based on robust stochastic approximation, which do not require strong convexity. Experiments illustrate that our approaches produce solutions of comparable prediction accuracy with the solutions acquired from existing SVM solvers, but often in much shorter time. We also suggest efficient prediction schemes that depend only on the dimension of kernel approximation, not on the number of support vectors.
Stochastic Subgradient Methods with Guaranteed Global Stability in Nonsmooth Nonconvex Optimization
School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）；School of Mathematical Sciences, Shenzhen University（深圳大学数学科学学院）
Cornell University（康奈尔大学）；University of Washington（华盛顿大学）；University of Southern California（南加利福尼亚大学）
This work considers the question: what convergence guarantees does the stochastic subgradient method have in the absence of smoothness and convexity? We prove that the stochastic subgradient method, on any semialgebraic locally Lipschitz function, produces limit points that are all first-order stationary. More generally, our result applies to any function with a Whitney stratifiable graph. In particular, this work endows the stochastic subgradient method, and its proximal extension, with rigorous convergence guarantees for a wide class of problems arising in data science---including all popula…
We prove that the proximal stochastic subgradient method, applied to a weakly convex problem, drives the gradient of the Moreau envelope to zero at the rate $O(k^{-1/4})$. As a consequence, we resolve an open question on the convergence rate of the proximal stochastic gradient method for minimizing the sum of a smooth nonconvex function and a convex proximable function.
In this paper, we show that simple {Stochastic} subGradient Decent methods with multiple Restarting, named {\bf RSGD}, can achieve a \textit{linear convergence rate} for a class of non-smooth and non-strongly convex optimization problems where the epigraph of the objective function is a polyhedron, to which we refer as {\bf polyhedral convex optimization}. Its applications in machine learning include $\ell_1$ constrained or regularized piecewise linear loss minimization and submodular function minimization. To the best of our knowledge, this is the first result on the linear convergence rate o…
In this paper, we introduce a stochastic projected subgradient method for weakly convex (i.e., uniformly prox-regular) nonsmooth, nonconvex functions---a wide class of functions which includes the additive and convex composite classes. At a high-level, the method is an inexact proximal point iteration in which the strongly convex proximal subproblems are quickly solved with a specialized stochastic projected subgradient method. The primary contribution of this paper is a simple proof that the proposed algorithm converges at the same rate as the stochastic gradient method for smooth nonconvex p…
Convergence of Decentralized Stochastic Subgradient-based Methods for Nonsmooth Nonconvex functions
State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences, China（国家科学工程计算重点实验室，数学系统科学学院，中国科学院，中国科学院大学）
Cornell University（康奈尔大学）；Simons Institute for the Theory of Computing（西蒙斯理论计算研究所）；DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization（DIMACS/西蒙斯连续与离散优化桥接合作项目）
We extend the classic convergence rate theory for subgradient methods to apply to non-Lipschitz functions. For the deterministic projected subgradient method, we present a global $O(1/\sqrt{T})$ convergence rate for any convex function which is locally Lipschitz around its minimizers. This approach is based on Shor's classic subgradient analysis and implies generalizations of the standard convergence rates for gradient descent on functions with Lipschitz or Hölder continuous gradients. Further, we show a $O(1/\sqrt{T})$ convergence rate for the stochastic projected subgradient method on convex…
The stochastic subgradient method is a widely-used algorithm for solving large-scale optimization problems arising in machine learning. Often these problems are neither smooth nor convex. Recently, Davis et al. [1-2] characterized the convergence of the stochastic subgradient method for the weakly convex case, which encompasses many important applications (e.g., robust phase retrieval, blind deconvolution, biconvex compressive sensing, and dictionary learning). In practice, distributed implementations of the projected stochastic subgradient method (stoDPSM) are used to speed-up risk minimizati…
We show that the Subgradient algorithm is universal for online learning on the simplex in the sense that it simultaneously achieves $O(\sqrt N)$ regret for adversarial costs and $O(1)$ pseudo-regret for i.i.d costs. To the best of our knowledge this is the first demonstration of a universal algorithm on the simplex that is not a variant of Hedge. Since Subgradient is a popular and widely used algorithm our results have immediate broad application.
University of Arizona（亚利桑那大学）；Boston University（波士顿大学）
We consider the problem of minimizing a convex risk with stochastic subgradients guaranteeing $ε$-locally differentially private ($ε$-LDP). While it has been shown that stochastic optimization is possible with $ε$-LDP via the standard SGD (Song et al., 2013), its convergence rate largely depends on the learning rate, which must be tuned via repeated runs. Further, tuning is detrimental to privacy loss since it significantly increases the number of gradient requests. In this work, we propose BANCO (Betting Algorithm for Noisy COins), the first $ε$-LDP SGD algorithm that essentially matches the …
We consider a distributionally robust formulation of stochastic optimization problems arising in statistical learning, where robustness is with respect to uncertainty in the underlying data distribution. Our formulation builds on risk-averse optimization techniques and the theory of coherent risk measures. It uses semi-deviation risk for quantifying uncertainty, allowing us to compute solutions that are robust against perturbations in the population data distribution. We consider a large family of loss functions that can be non-convex and non-smooth and develop an efficient stochastic subgradi…
We propose graph-dependent implicit regularisation strategies for distributed stochastic subgradient descent (Distributed SGD) for convex problems in multi-agent learning. Under the standard assumptions of convexity, Lipschitz continuity, and smoothness, we establish statistical learning rates that retain, up to logarithmic terms, centralised statistical guarantees through implicit regularisation (step size tuning and early stopping) with appropriate dependence on the graph topology. Our approach avoids the need for explicit regularisation in decentralised learning problems, such as adding con…
New Bounds for the Last Iterate of the Stochastic subGradient Method
Department of Computer Science, Aarhus University（阿arhus大学计算机科学系）；School of Electrical Engineering and Computer Science, University of Ottawa（渥太华大学电气工程与计算机科学学院）；School of Mathematics, University of Bristol（布里斯托大学数学学院）
Bayesian max-margin models have shown superiority in various practical applications, such as text categorization, collaborative prediction, social network link prediction and crowdsourcing, and they conjoin the flexibility of Bayesian modeling and predictive strengths of max-margin learning. However, Monte Carlo sampling for these models still remains challenging, especially for applications that involve large-scale datasets. In this paper, we present the stochastic subgradient Hamiltonian Monte Carlo (HMC) methods, which are easy to implement and computationally efficient. We show the approxi…
Data explosion and an increase in model size drive the remarkable advances in large-scale machine learning, but also make model training time-consuming and model storage difficult. To address the above issues in the distributed model training setting which has high computation efficiency and less device limitation, there are still two main difficulties. On one hand, the communication costs for exchanging information, e.g., stochastic gradients among different workers, is a key bottleneck for distributed training efficiency. On the other hand, less parameter model is easy for storage and commun…
Spectral risk objectives - also called $L$-risks - allow for learning systems to interpolate between optimizing average-case performance (as in empirical risk minimization) and worst-case performance on a task. We develop stochastic algorithms to optimize these quantities by characterizing their subdifferential and addressing challenges such as biasedness of subgradient estimates and non-smoothness of the objective. We show theoretically and experimentally that out-of-the-box approaches such as stochastic subgradient and dual averaging are hindered by bias and that our approach outperforms the…
Cornell University（康奈尔大学）；University of Washington（华盛顿大学）
We investigate the stochastic optimization problem of minimizing population risk, where the loss defining the risk is assumed to be weakly convex. Compositions of Lipschitz convex functions with smooth maps are the primary examples of such losses. We analyze the estimation quality of such nonsmooth and nonconvex problems by their sample average approximations. Our main results establish dimension-dependent rates on subgradient estimation in full generality and dimension-independent rates when the loss is a generalized linear model. As an application of the developed techniques, we analyze the …
We present a new class of decentralized first-order methods for nonsmooth and stochastic optimization problems defined over multiagent networks. Considering that communication is a major bottleneck in decentralized optimization, our main goal in this paper is to develop algorithmic frameworks which can significantly reduce the number of inter-node communications. We first propose a decentralized primal-dual method which can find an $ε$-solution both in terms of functional optimality gap and feasibility residual in $O(1/ε)$ inter-node communication rounds when the objective functions are convex…
We study a variant of the median problem for a collection of point sets in high dimensions. This generalizes the geometric median as well as the (probabilistic) smallest enclosing ball (pSEB) problems. Our main objective and motivation is to improve the previously best algorithm for the pSEB problem by reducing its exponential dependence on the dimension to linear. This is achieved via a novel combination of sampling techniques for clustering problems in metric spaces with the framework of stochastic subgradient descent. As a result, the algorithm becomes applicable to shape fitting problems i…
University of Science and Technology of China（中国科学技术大学）；University of Minnesota, Twin Cities（明尼苏达大学双城分校）；Sun Yat-Sen University（中山大学）
In this paper, we propose a class of robust stochastic subgradient methods for distributed learning from heterogeneous datasets at presence of an unknown number of Byzantine workers. The Byzantine workers, during the learning process, may send arbitrary incorrect messages to the master due to data corruptions, communication failures or malicious attacks, and consequently bias the learned model. The key to the proposed methods is a regularization term incorporated with the objective function so as to robustify the learning task and mitigate the negative effects of Byzantine attacks. The resulta…
Cornell University（康奈尔大学）；University of Washington（华盛顿大学）
We consider a family of algorithms that successively sample and minimize simple stochastic models of the objective function. We show that under reasonable conditions on approximation quality and regularity of the models, any such algorithm drives a natural stationarity measure to zero at the rate $O(k^{-1/4})$. As a consequence, we obtain the first complexity guarantees for the stochastic proximal point, proximal subgradient, and regularized Gauss-Newton methods for minimizing compositions of convex functions with smooth maps. The guiding principle, underlying the complexity guarantees, is tha…
Stochastic gradient algorithms are often unstable when applied to functions that do not have Lipschitz-continuous and/or bounded gradients. Gradient clipping is a simple and effective technique to stabilize the training process for problems that are prone to the exploding gradient problem. Despite its widespread popularity, the convergence properties of the gradient clipping heuristic are poorly understood, especially for stochastic problems. This paper establishes both qualitative and quantitative convergence results of the clipped stochastic (sub)gradient method (SGD) for non-smooth convex f…
