Furthermore, weprovethat, under mild assumptions, a slight variation of WQL enjoys desirable theoretical properties in the tabular setting.

Furthermore, we obtain exactp-values, which are easily computed in terms of the Tulap random variable. We show that our results also apply to distribution-free hypothesis testsforcontinuous data.

Stochastic gradient methods have been a widespread practice in machine learning.

Additionally, we show generalization performance of our proposed method across differentvisualdomains. Withthegiven problemcategory(task),asubsetforlearning can be sampled (via domain episode module in Figure 4 in main text). Here, by replacingclass with task, K-shot andN-task reasoning framework can be defined. Here, we show analogical learning with the existing meta learning framework for fast adaptation fromthesourcedomain tothetargetdomain.

However, these models suffer from quadratic computational cost in the input sequence lengthn to compute pairwise attention in each layer.

Wethenseparate theoptimization process into two steps, corresponding to weight update and structure parameter update. For the former step, we use the conventional chain rule, which can be sparse via exploiting the sparse structure.

The supplementary material is organized as follows. We first discuss additional related work and provide experiment details inSection 2andAppendix Brespectively. Adversarial Defenses: Neural networks trained using standard procedures such as SGD are extremely vulnerable [23] to -bound adversarial attacks such as FGSM [23], PGD [42], CW [11], andMomentum [17];Unrestricted attacks [7,19]cansignificantly degrade model performance as well. Defense strategies based on heuristics such as feature squeezing [82], denoising [80], encoding [10], specialized nonlinearities [83] and distillation [56] have had limited success against stronger attacks [2]. Then, we introduce a noisy version of the5-slab block,whichwelateruseinAppendixD.

In appendix, we Reward OptimizationWepropose .

Recently,researchers extended 1-hop message passing to K-hop message passing by aggregating information fromK-hop neighbors of nodes simultaneously. However, there is no work on analyzing the expressive powerofK-hopmessagepassing.

Despitethis noble goal, it has been acknowledged in the literature that statistical tests based ontheEOareoblivious totheunderlying causal mechanisms thatgenerated the disparity in the first place (Hardt et al. 2016).