Variance-Reduced and Projection-Free Stochastic Optimization

The Frank-Wolfe optimization algorithm has recently regained popularity for machine learning applications due to its projection-free property and its ability to handle structured constraints. However, in the stochastic learning setting, it is still relatively understudied compared to the gradient descent counterpart. In this work, leveraging a recent variance reduction technique, we propose two stochastic Frank-Wolfe variants which substantially improve previous results in terms of the number of stochastic gradient evaluations needed to achieve $1-\epsilon$ accuracy. For example, we improve from $O(\frac{1}{\epsilon})$ to $O(\ln\frac{1}{\epsilon})$ if the objective function is smooth and strongly convex, and from $O(\frac{1}{\epsilon^2})$ to $O(\frac{1}{\epsilon^{1.5}})$ if the objective function is smooth and Lipschitz. The theoretical improvement is also observed in experiments on real-world datasets for a multiclass classification application

SalsaNet: Fast Road and Vehicle Segmentation in LiDAR Point Clouds for Autonomous Driving

In this paper, we introduce a deep encoder-decoder network, named SalsaNet, for efficient semantic segmentation of 3D LiDAR point clouds. SalsaNet segments the road, i.e. drivable free-space, and vehicles in the scene by employing the Bird-Eye-View (BEV) image projection of the point cloud. To overcome the lack of annotated point cloud data, in particular for the road segments, we introduce an auto-labeling process which transfers automatically generated labels from the camera to LiDAR. We also explore the role of imagelike projection of LiDAR data in semantic segmentation by comparing BEV with spherical-front-view projection and show that SalsaNet is projection-agnostic. We perform quantitative and qualitative evaluations on the KITTI dataset, which demonstrate that the proposed SalsaNet outperforms other state-of-the-art semantic segmentation networks in terms of accuracy and computation time. Our code and data are publicly available at https://gitlab.com/aksoyeren/salsanet.git

Stochastic Frank-Wolfe Methods for Nonconvex Optimization

We study Frank-Wolfe methods for nonconvex stochastic and finite-sum optimization problems. Frank-Wolfe methods (in the convex case) have gained tremendous recent interest in machine learning and optimization communities due to their projection-free property and their ability to exploit structured constraints. However, our understanding of these algorithms in the nonconvex setting is fairly limited. In this paper, we propose nonconvex stochastic Frank-Wolfe methods and analyze their convergence properties. For objective functions that decompose into a finite-sum, we leverage ideas from variance reduction techniques for convex optimization to obtain new variance reduced nonconvex Frank-Wolfe methods that have provably faster convergence than the classical Frank-Wolfe method. Finally, we show that the faster convergence rates of our variance reduced methods also translate into improved convergence rates for the stochastic setting

Faster Rates for the Frank-Wolfe Method over Strongly-Convex Sets

The Frank-Wolfe method (a.k.a. conditional gradient algorithm) for smooth optimization has regained much interest in recent years in the context of large scale optimization and machine learning. A key advantage of the method is that it avoids projections - the computational bottleneck in many applications - replacing it by a linear optimization step. Despite this advantage, the known convergence rates of the FW method fall behind standard first order methods for most settings of interest. It is an active line of research to derive faster linear optimization-based algorithms for various settings of convex optimization. In this paper we consider the special case of optimization over strongly convex sets, for which we prove that the vanila FW method converges at a rate of $\frac{1}{t^2}$. This gives a quadratic improvement in convergence rate compared to the general case, in which convergence is of the order $\frac{1}{t}$, and known to be tight. We show that various balls induced by $\ell_p$ norms, Schatten norms and group norms are strongly convex on one hand and on the other hand, linear optimization over these sets is straightforward and admits a closed-form solution. We further show how several previous fast-rate results for the FW method follow easily from our analysis