In this paper, we comparatively analyze the Bures-Wasserstein (BW) geometry with the popular Affine-Invariant (AI) geometry for Riemannian optimization on the symmetric positive definite (SPD) matrix manifold. Our study begins with an observation that the BW metric has a linear dependence on SPD matrices in contrast to the quadratic dependence of the AI metric. We build on this to show that the BW metric is a more suitable and robust choice for several Riemannian optimization problems over ill-conditioned SPD matrices. We show that the BW geometry has a non-negative curvature, which further improves convergence rates of algorithms over the non-positively curved AI geometry. Finally, we verify that several popular cost functions, which are known to be geodesic convex under the AI geometry, are also geodesic convex under the BW geometry. Extensive experiments on various applications support our findings.

There is an increasing need for effective active learning algorithms that are compatible with deep neural networks. This paper motivates and revisits a classic, Fisher-based active selection objective, and proposes BAIT, a practical, tractable, and high-performing algorithm that makes it viable for use with neural models. BAIT draws inspiration from the theoretical analysis of maximum likelihood estimators (MLE) for parametric models. It selects batches of samples by optimizing a bound on the MLE error in terms of the Fisher information, which we show can be implemented efficiently at scale by exploiting linear-algebraic structure especially amenable to execution on modern hardware. Our experiments demonstrate that BAIT outperforms the previous state of the art on both classification and regression problems, and is flexible enough to be used with a variety of model architectures.

Motivated by the scenario of large-scale learning in distributed systems, this paper studies a scenario where M agents cooperate together to solve the same instance of a K-armed stochastic bandit problem. The agents have limited access to a local subset of arms and are asynchronous with different gaps between decision-making rounds. The goal is to find the global optimal arm, and agents are able to pull any arm; however, they can only observe the reward when the selected arm is local. The challenge is a tradeoff for agents between pulling a local arm with observable feedback or pulling external arms without feedback and relying on others' observations that occur at different rates. We propose AAE-LCB, a two-stage algorithm that prioritizes pulling local arms following an active arm elimination policy and switches to other arms only if all local arms are dominated by some external arms. We analyze the regret of AAE-LCBand show it matches the regret lower bound up to a small factor.

In this paper, we evaluate the quality of knowledge representations encoded in deep neural networks (DNNs) for 3D point cloud processing. We propose a method to disentangle the overall model vulnerability into the sensitivity to the rotation, the translation, the scale, and local 3D structures. Besides, we also propose metrics to evaluate the spatial smoothness of encoding 3D structures, and the representation complexity of the DNN. Based on such analysis, experiments expose representation problems with classic DNNs, and explain the utility of the adversarial training. The code will be released when this paper is accepted.

For medical image segmentation, contrastive learning is the dominant practice to improve the quality of visual representations by contrasting semantically similar and dissimilar pairs of samples. This is enabled by the observation that without accessing ground truth labels, negative examples with truly dissimilar anatomical features, if sampled, can significantly improve the performance. In reality, however, these samples may come from similar anatomical regions and the models may struggle to distinguish the minority tail-class samples, making the tail classes more prone to misclassification, both of which typically lead to model collapse. In this paper, we propose ARCO, a semi-supervised contrastive learning (CL) framework with stratified group theory for medical image segmentation. In particular, we first propose building ARCO through the concept of variance-reduced estimation and show that certain variance-reduction techniques are particularly beneficial in pixel/voxel-level segmentation tasks with extremely limited labels. Furthermore, we theoretically prove these sampling techniques are universal in variance reduction. Finally, we experimentally validate our approaches on eight benchmarks, i.e., five 2D/3D medical and three semantic segmentation datasets, with different label settings, and our methods consistently outperform state-of-the-art semi-supervised methods. Additionally, we augment the CL frameworks with these sampling techniques and demonstrate significant gains over previous methods. We believe our work is an important step towards semi-supervised medical image segmentation by quantifying the limitation of current self-supervision objectives for accomplishing such challenging safety-critical tasks. 1

One of the key drivers of complexity in the classical (stochastic) multi-armed bandit (MAB) problem is the difference between mean rewards in the top two arms, also known as the instance gap. The celebrated Upper Confidence Bound (UCB) policy is among the simplest optimism-based MAB algorithms that naturally adapts to this gap: for a horizon of play n, it achieves optimal O(log n) regret in instances with "large" gaps, and a near-optimal O nlog n minimax regret when the gap can be arbitrarily "small." This paper provides new results on the arm-sampling behavior of UCB, leading to several important insights. Among these, it is shown that arm-sampling rates under UCB are asymptotically deterministic, regardless of the problem complexity.