Whereas,linearoptimization only requires computation of the leading eigenvector.

Deep learning (DL) models, despite their remarkable success, remain vulnerable to small input perturbations that can cause erroneous outputs, motivating the recent proposal of probabilistic robustness (PR) as a complementary alternative to adversarial robustness (AR). However, existing PR formulations assume a fixed and known perturbation distribution, an unrealistic expectation in practice. T o address this limitation, we propose non-parametric probabilistic robustness (NPPR), a more practical PR metric that does not rely on any predefined perturbation distribution. F ollowing the non-parametric paradigm in statistical modeling, NPPR learns an optimized perturbation distribution directly from data, enabling conservative PR evaluation under distributional uncertainty. W e further develop an NPPR estimator based on a Gaussian Mixture Model (GMM) with Multilayer Perceptron (MLP) heads and bicubic up-sampling, covering various input-dependent and input-independent perturbation scenarios. Theoretical analyses establish the relationships among AR, PR, and NPPR. Extensive experiments on CIF AR-10, CIF AR-100, and Tiny ImageNet across ResNet18/50, WideResNet50 and VGG16 validate NPPR as a more practical robustness metric, showing up to 40% more conservative (lower) PR estimates compared to assuming those common perturbation distributions used in state-of-the-arts.

It seems that Lemma 2.1 needs a fix, and thus subsequent results as well.

In the "classical" sense of robustness, we can consider evaluating

While our algorithm has several applications, we consider the specific application of minimax games. For solving smooth convex-concave games, our algorithm only requires access to a linear optimization oracle.

While our algorithm has several applications, we consider the specific application of minimax games. For solving smooth convex-concave games, our algorithm only requires access to a linear optimization oracle.

Understanding the behavior of deep networks is crucial to increase our confidence in their results. Despite an extensive body of work for explaining their predictions, researchers have faced reliability issues, which can be attributed to insufficient formalism. In our research, we adopt novel probabilistic and spectral perspectives to formally analyze explanation methods. Our study reveals a pervasive spectral bias stemming from the use of gradient, and sheds light on some common design choices that have been discovered experimentally, in particular, the use of squared gradient and input perturbation. We further characterize how the choice of perturbation hyperparameters in explanation methods, such as SmoothGrad, can lead to inconsistent explanations and introduce two remedies based on our proposed formalism: (i) a mechanism to determine a standard perturbation scale, and (ii) an aggregation method which we call SpectralLens. Finally, we substantiate our theoretical results through quantitative evaluations.

We consider the problem of online learning and its application to solving minimax games. For the online learning problem, Follow the Perturbed Leader (FTPL) is a widely studied algorithm which enjoys the optimal $O(T^{1/2})$ worst-case regret guarantee for both convex and nonconvex losses. In this work, we show that when the sequence of loss functions is predictable, a simple modification of FTPL which incorporates optimism can achieve better regret guarantees, while retaining the optimal worst-case regret guarantee for unpredictable sequences. A key challenge in obtaining these tighter regret bounds is the stochasticity and optimism in the algorithm, which requires different analysis techniques than those commonly used in the analysis of FTPL. The key ingredient we utilize in our analysis is the dual view of perturbation as regularization. While our algorithm has several applications, we consider the specific application of minimax games. For solving smooth convex-concave games, our algorithm only requires access to a linear optimization oracle. For Lipschitz and smooth nonconvex-nonconcave games, our algorithm requires access to an optimization oracle which computes the perturbed best response. In both these settings, our algorithm solves the game up to an accuracy of $O(T^{-1/2})$ using $T$ calls to the optimization oracle. An important feature of our algorithm is that it is highly parallelizable and requires only $O(T^{1/2})$ iterations, with each iteration making $O(T^{1/2})$ parallel calls to the optimization oracle.