Improved Regret for Bandit Convex Optimization with Delayed Feedback

We investigate bandit convex optimization (BCO) with delayed feedback, where only the loss value of the action is revealed under an arbitrary delay. Previous studies have established a regret bound of $O(T^{3/4}+d^{1/3}T^{2/3})$ for this problem, where $d$ is the maximum delay, by simply feeding delayed loss values to the classical bandit gradient descent (BGD) algorithm. In this paper, we develop a novel algorithm to enhance the regret, which carefully exploits the delayed bandit feedback via a blocking update mechanism. Our analysis first reveals that the proposed algorithm can decouple the joint effect of the delays and bandit feedback on the regret, and improve the regret bound to $O(T^{3/4}+\sqrt{dT})$ for convex functions. Compared with the previous result, our regret matches the $O(T^{3/4})$ regret of BGD in the non-delayed setting for a larger amount of delay, i.e., $d=O(\sqrt{T})$, instead of $d=O(T^{1/4})$. Furthermore, we consider the case with strongly convex functions, and prove that the proposed algorithm can enjoy a better regret bound of $O(T^{2/3}\log^{1/3}T+d\log T)$. Finally, we show that in a special case with unconstrained action sets, it can be simply extended to achieve a regret bound of $O(\sqrt{T\log T}+d\log T)$ for strongly convex and smooth functions.

Improved Projection-free Online Continuous Submodular Maximization

We investigate the problem of online learning with monotone and continuous DR-submodular reward functions, which has received great attention recently. To efficiently handle this problem, especially in the case with complicated decision sets, previous studies have proposed an efficient projection-free algorithm called Mono-Frank-Wolfe (Mono-FW) using $O(T)$ gradient evaluations and linear optimization steps in total. However, it only attains a $(1-1/e)$-regret bound of $O(T^{4/5})$. In this paper, we propose an improved projection-free algorithm, namely POBGA, which reduces the regret bound to $O(T^{3/4})$ while keeping the same computational complexity as Mono-FW. Instead of modifying Mono-FW, our key idea is to make a novel combination of a projection-based algorithm called online boosting gradient ascent, an infeasible projection technique, and a blocking technique. Furthermore, we consider the decentralized setting and develop a variant of POBGA, which not only reduces the current best regret bound of efficient projection-free algorithms for this setting from $O(T^{4/5})$ to $O(T^{3/4})$, but also reduces the total communication complexity from $O(T)$ to $O(\sqrt{T})$.

Non-stationary Online Convex Optimization with Arbitrary Delays

Online convex optimization (OCO) with arbitrary delays, in which gradients or other information of functions could be arbitrarily delayed, has received increasing attention recently. Different from previous studies that focus on stationary environments, this paper investigates the delayed OCO in non-stationary environments, and aims to minimize the dynamic regret with respect to any sequence of comparators. To this end, we first propose a simple algorithm, namely DOGD, which performs a gradient descent step for each delayed gradient according to their arrival order. Despite its simplicity, our novel analysis shows that DOGD can attain an $O(\sqrt{dT}(P_T+1)$ dynamic regret bound in the worst case, where $d$ is the maximum delay, $T$ is the time horizon, and $P_T$ is the path length of comparators. More importantly, in case delays do not change the arrival order of gradients, it can automatically reduce the dynamic regret to $O(\sqrt{S}(1+P_T))$, where $S$ is the sum of delays. Furthermore, we develop an improved algorithm, which can reduce those dynamic regret bounds achieved by DOGD to $O(\sqrt{dT(P_T+1)})$ and $O(\sqrt{S(1+P_T)})$, respectively. The essential idea is to run multiple DOGD with different learning rates, and utilize a meta-algorithm to track the best one based on their delayed performance. Finally, we demonstrate that our improved algorithm is optimal in both cases by deriving a matching lower bound.

Claw U-Net: A Unet-based Network with Deep Feature Concatenation for Scleral Blood Vessel Segmentation

Sturge-Weber syndrome (SWS) is a vascular malformation disease, and it may cause blindness if the patient's condition is severe. Clinical results show that SWS can be divided into two types based on the characteristics of scleral blood vessels. Therefore, how to accurately segment scleral blood vessels has become a significant problem in computer-aided diagnosis. In this research, we propose to continuously upsample the bottom layer's feature maps to preserve image details, and design a novel Claw UNet based on UNet for scleral blood vessel segmentation. Specifically, the residual structure is used to increase the number of network layers in the feature extraction stage to learn deeper features. In the decoding stage, by fusing the features of the encoding, upsampling, and decoding parts, Claw UNet can achieve effective segmentation in the fine-grained regions of scleral blood vessels. To effectively extract small blood vessels, we use the attention mechanism to calculate the attention coefficient of each position in images. Claw UNet outperforms other UNet-based networks on scleral blood vessel image dataset.