On the computational and statistical complexity of over-parameterized matrix sensing

We consider solving the low rank matrix sensing problem with Factorized Gradient Descend (FGD) method when the true rank is unknown and over-specified, which we refer to as over-parameterized matrix sensing. If the ground truth signal $mathbf{X}^* in mathbb{R}^{d*d}$ is of rank $r$, but we try to recover it using $mathbf{F} mathbf{F}^top$ where $mathbf{F} in mathbb{R}^{d*k}$ and $k>r$, the existing statistical analysis falls short, due to a flat local curvature of the loss function around the global maxima. By decomposing the factorized matrix $mathbf{F}$ into separate column spaces to capture the effect of extra ranks, we show that $|mathbf{F}_t mathbf{F}_t - mathbf{X}^*|_{F}^2$ converges to a statistical error of $tilde{mathcal{O}} ({k d sigma^2/n})$ after $tilde{mathcal{O}}(frac{sigma_{r}}{sigma}sqrt{frac{n}{d}})$ number of iterations where $mathbf{F}_t$ is the output of FGD after $t$ iterations, $sigma^2$ is the variance of the observation noise, $sigma_{r}$ is the $r$-th largest eigenvalue of $mathbf{X}^*$, and $n$ is the number of sample. Our results, therefore, offer a comprehensive picture of the statistical and computational complexity of FGD for the over-parameterized matrix sensing problem.

Robust Structured Statistical Estimation via Conditional Gradient Type Methods

Structured statistical estimation problems are often solved by Conditional Gradient (CG) type methods to avoid the computationally expensive projection operation. However, the existing CG type methods are not robust to data corruption. To address this, we propose to robustify CG type methods against Hubers corruption model and heavy-tailed data. First, we show that the two Pairwise CG methods are stable, i.e., do not accumulate error. Combined with robust mean gradient estimation techniques, we can therefore guarantee robustness to a wide class of problems, but now in a projection-free algorithmic framework. Next, we consider high dimensional problems. Robust mean estimation based approaches may have an unacceptably high sample complexity. When the constraint set is a $ell_0$ norm ball, Iterative-Hard-Thresholding-based methods have been developed recently. Yet extension is non-trivial even for general sets with $O(d)$ extreme points. For setting where the feasible set has $O(text{poly}(d))$ extreme points, we develop a novel robustness method, based on a new condition we call the Robust Atom Selection Condition (RASC). When RASC is satisfied, our method converges linearly with a corresponding statistical error, with sample complexity that scales correctly in the sparsity of the problem, rather than the ambient dimension as would be required by any approach based on robust mean estimation.

Communication-Efficient Asynchronous Stochastic Frank-Wolfe over Nuclear-norm Balls

Large-scale machine learning training suffers from two prior challenges, specifically for nuclear-norm constrained problems with distributed systems: the synchronization slowdown due to the straggling workers, and high communication costs. In this work, we propose an asynchronous Stochastic Frank Wolfe (SFW-asyn) method, which, for the first time, solves the two problems simultaneously, while successfully maintaining the same convergence rate as the vanilla SFW. We implement our algorithm in python (with MPI) to run on Amazon EC2, and demonstrate that SFW-asyn yields speed-ups almost linear to the number of machines compared to the vanilla SFW.

Primal-Dual Block Frank-Wolfe

We propose a variant of the Frank-Wolfe algorithm for solving a class of sparse/low-rank optimization problems. Our formulation includes Elastic Net, regularized SVMs and phase retrieval as special cases. The proposed Primal-Dual Block Frank-Wolfe algorithm reduces the per-iteration cost while maintaining linear convergence rate. The per iteration cost of our method depends on the structural complexity of the solution (i.e. sparsity/low-rank) instead of the ambient dimension. We empirically show that our algorithm outperforms the state-of-the-art methods on (multi-class) classification tasks.

Bottom-up Object Detection by Grouping Extreme and Center Points

With the advent of deep learning, object detection drifted from a bottom-up to a top-down recognition problem. State of the art algorithms enumerate a near-exhaustive list of object locations and classify each into: object or not. In this paper, we show that bottom-up approaches still perform competitively. We detect four extreme points (top-most, left-most, bottom-most, right-most) and one center point of objects using a standard keypoint estimation network. We group the five keypoints into a bounding box if they are geometrically aligned. Object detection is then a purely appearance-based keypoint estimation problem, without region classification or implicit feature learning. The proposed method performs on-par with the state-of-the-art region based detection methods, with a bounding box AP of 43.2% on COCO test-dev. In addition, our estimated extreme points directly span a coarse octagonal mask, with a COCO Mask AP of 18.9%, much better than the Mask AP of vanilla bounding boxes. Extreme point guided segmentation further improves this to 34.6% Mask AP.