Neural networks have shown great predictive power when applied to unstructured data such as images and natural languages.

In this paper, we study a class of stochastic bilevel optimization problems, also known as stochastic simple bilevel optimization, where we minimize a smooth stochastic objective function over the optimal solution set of another stochastic convex optimization problem. We introduce novel stochastic bilevel optimization methods that locally approximate the solution set of the lower-level problem via a stochastic cutting plane, and then run a conditional gradient update with variance reduction techniques to control the error induced by using stochastic gradients. For the case that the upper-level function is convex, our method requires O(max{1/ϵ2f,1/ϵ2g}) stochastic oracle queries to obtain a solution that is ϵfoptimal for the upper-level and ϵg-optimal for the lower-level. This guarantee improves the previous best-known complexity of O(max{1/ϵ4f,1/ϵ4g}). Moreover, for the case that the upper-level function is non-convex, our method requires at most O(max{1/ϵ3f,1/ϵ3g})stochastic oracle queries to find an (ϵf,ϵg)-stationary point. In the finite-sum setting, we show that the number of stochastic oracle calls required by our method are O( n/ϵ) and O( n/ϵ2) for the convex and non-convex settings, respectively, where ϵ = min{ϵf,ϵg}.

Following Cohen and Hazan [2015] we derive a general expression for the regret upper bound applicable to any linear reward function under an anytime FTPL policy. This is accomplished in the following steps. First, we extend the argument of Cohen and Hazan [2015] to the anytime setting. Then, we specialize this bound to our problem setting. Recall the notations used in the paper - the aggregate file-request sequence from all users is denoted by {xt}t 1 and the virtual cache configuration sequence is denoted by {zt}t 1. Define the cumulative requests up to time tas: Xt = Furthermore, since the max function 14 is convex, we may interchange the expectation and gradient to obtain Φηt(Xt) =E(zt) [Bertsekas, 1973, Proposition 2.2]. Plugging in the expression of the inner product from Eqn. (25) in expression (26), we obtain: Bounding the term (a): Next, to upper bound the expected regret, we control term (a) in inequality (28).

Semi-Supervised Video Salient Object Detection (SS-VSOD) is challenging because of the lack of temporal information caused by sparse annotations in video sequences. Most works address this problem by generating pseudo labels for unlabeled data. However, error-prone pseudo labels negatively affect the VOSD model. Therefore, a deeper insight into pseudo labels should be developed. In this work, we aim to explore 1) how to utilize the incorrect predictions in pseudo labels to guide the network to generate more robust pseudo labels and 2) how to further screen out the noise that still exists in the improved pseudo labels. To this end, we propose an Uncertainty-Guided Pseudo Label Generator (UGPLG), which makes full use of inter-frame information to ensure the temporal consistency of the pseudo-labels and improves the robustness of the pseudo labels by strengthening the learning of difficult scenarios. Furthermore, we also introduce adversarial learning to address the noise problems in pseudo labels, guaranteeing the positive guidance of pseudo labels during model training. Experimental results demonstrate that our methods outperform existing semi-supervised method and partial fully-supervised methods across five public benchmarks of DAVIS, FBMS, MCL, ViSal, and SegTrack-V2. Code and dataset are available at https://github.com/Lanezzz/UGPL.

We investigate the degree to which our conditional independence assumption is satisfied empirically in the datasets used in the paper. Specifically, of interest is the assumption of conditional independence of m(x) and h(x), given y. Assessing conditional independence is not straightforward given that m(x) is a K-dimensional real-valued vector and h(x) and yeach take one of K categorical values, with K = 10 for CIFAR-10H and K = 16 for ImageNet-16H. While there exist statistical tests for assessing conditional independence for categorical random variables, with real-valued variables the situation is less straightforward and there are multiple options such as different non-parametric tests involving different tradeoffs [Runge, 2018, Marx and Vreeken, 2019, Mukherjee et al., 2020, Berrett et al., 2020]. Given these issues we investigate the degree of conditional dependence using two relatively simple approaches. The first approach looks at the conditional mutual information (CMI) between the predicted label from the model and the predicted label from the human, conditioned on the true label.

The appendix is organized as follows: In Sec. A1, we provide a generalization of Claim 1 and its the complete proof. A2, we provide the complete proof for Eq. A3, we provide additional ablations and experimental results. A4, we provide additional implementation details.