In this work, we focus on the task of controlling the level of sparsity when performing sparse learning.

This work proposes abranch and bound (BB) scheme, inwhich atailored Lagrangian relaxation method proposed in the 1970s is used to provide alower boundateachBBnode.

By exploiting smoothness of the expected losses, these bounds replace dependence onthemaximum gradient length bythevariance of the gradients, which was previously known only for linear losses. In addition, theyweakenthei.i.d.assumptionbyallowing,forexample,adversariallypoisoned rounds, which were previously considered in the expert and bandit setting.

Proximal policy optimization (PPO) is one of the most popular deep reinforcement learning (RL) methods, achieving state-of-the-art performance across a wide range of challenging tasks. However, as a model-free RL method, the success of PPO relies heavily on the effectiveness of its exploratory policy search. In this paper, we give an in-depth analysis on the exploration behavior of PPO, and show that PPO is prone to suffer from the risk of lack of exploration especially under the case of bad initialization, which may lead to the failure of training or being trapped in bad local optima. To address these issues, we proposed a novel policy optimization method, named Trust Region-Guided PPO (TRGPPO), which adaptively adjusts the clipping range within the trust region. We formally show that this method not only improves the exploration ability within the trust region but enjoys a better performance bound compared to the original PPO as well. Extensive experiments verify the advantage of the proposed method.