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Title: A FRAMEWORK  FOR ROBUSTNESS CERTIFICATION  OF SMOOTHED CLASSIFIERS USING  F-DIVERGENCES

Paper content:
Formal verification techniques that compute provable guarantees on properties of machine learning models, like robustness to norm-bounded adversarial perturbations, have yielded impressive results.
Although most techniques developed so far requires knowledge of the architecture of the machine learning model and remains hard to scale to complex prediction pipelines, the method of randomized smoothing has been shown to overcome many of these obstacles.
By requiring only black-box access to the underlying model, randomized smoothing scales to large architectures and is agnostic to the internals of the network.
However, past work on randomized smoothing has focused on restricted classes of smoothing measures or perturbations (like Gaussian or discrete) and has only been able to prove robustness with respect to simple norm bounds.
In this paper we introduce a general framework for proving robustness properties of smoothed machine learning models in the black-box setting.
Specifically, we extend randomized smoothing procedures to handle arbitrary smoothing measures and prove robustness of the smoothed classifier by using $f$-divergences.
Our methodology achieves state-of-the-art}certified robustness on MNIST, CIFAR-10 and ImageNet and also audio classification task, Librispeech, with respect to several classes of adversarial perturbations.
Predictors obtained from machine learning algorithms have been shown to be vulnerable to making errors when the inputs are perturbed by carefully chosen small but imperceptible amounts (Szegedy et al., 2014; Biggio et al., 2013) .
This has motivated significant amount of research in improving adversarial robustness of a machine learning model (see, e.g. Goodfellow et al., 2015; Madry et al., 2018) .
While significant advances have been made, it has been shown that models that were estimated to be robust have later been broken by stronger attacks (Athalye et al., 2018; Uesato et al., 2018) .
This has led to the need for methods that offer provable guarantees that the predictor cannot be forced to misclassify an example by any attack algorithm restricted to produce perturbations within a certain set (for example, within an p norm ball).
While progress has been made leading to methods that are able to compute provable guarantees for several image and text classification tasks (Wong & Kolter, 2018; Wong et al., 2018; Raghunathan et al., 2018; Dvijotham et al., 2018; Katz et al., 2017; Huang et al., 2019; Jia et al., 2019) , these methods require extensive knowledge of the architecture of the predictor and are not easy to extend to new models or architectures, requiring specialized algorithms for each new class of models.
Further, the computational complexity of these methods grows significantly with input dimension and model size.
Consequently, to deal with these obstacles, recent work has proposed the randomized smoothing strategy for verifying the robustness of classifiers.
Specifically, Lecuyer et al. (2019) ; Cohen et al. (2019) have shown that robustness properties can be more easily verified for the smoothed version of a base classifier h: h s (x) = arg max y∈Y P X∼µ(x) [h(X) = y] ,
where the labels returned by the smoothed classifier h s are obtained by taking a "majority vote" over the predictions of the original classifier h on random inputs drawn from a probability distribution µ(x), called the smoothing measure (here Y denotes the set of classes in the problem).
Lecuyer et al. (2019) showed that verifying the robustness of this smoothed classifier is significantly simpler than verifying the original classifier h and only requires estimating the distribution of outputs of the classifier under random perturbations of the input, but does not require access to the internals of the classifier h.
We refer to this as black-box verification.
In this work, we develop a general framework for black-box verification that recovers prior work as special cases, and improves upon previous results in various ways.
Contributions Our contributions are summarized as follows:
1. We formulate the general problem of black-box verification via a generalized randomized smoothing procedure, which extends existing approaches to allow for arbitrary smoothing measures.
Specifically, we show that robustness certificates for smoothed classifiers can be obtained by solving a small convex optimization problem when allowed adversarial perturbations can be characterized via divergence-based bounds on the smoothing measure.
2. We prove that our certificates generalize previous results obtained in related work (Lecuyer et al., 2019; Cohen et al., 2019; Li et al., 2019) , and vastly extend the class of perturbations and smoothing measures that can be used while still allowing certifiable guarantees.
3. We introduce the notion of full-information and information-limited settings, and show that the information-limited setting that has been the main focus of prior work leads to weaker certificates for smoothed probabilistic classifiers, and can be improved by using additional information (the distribution of label scores under randomized smoothing).
4. We evaluate our framework experimentally on image and classification tasks, obtaining robustness certificates that improve upon other black-box methods either in terms of certificate tightness or computation time on robustness to 0 , 1 or 2 perturbations on MNIST, CIFAR-10 and ImageNet.
2 perturbations result from worst-case realizations of white noise that is common in many image, speech and video processing.
0 perturbations can model missing data (missing pixels in an image, or samples in a time-domain audio signal) while 1 perturbations can be used to model convex combinations of discrete perturbations in text classification (Jia et al., 2019) .
We also obtain the first, to the best of our knowledge, certifiably robust model for an audio classification task, Librispeech (Panayotov et al., 2015) , with variable-length inputs.
We have introduced a general framework for black-box verification using f -divergence constraints.
The framework improves upon state-of-the-art results on both image classification and audio tasks by a significant margin in terms of robustness certificates or computation time.
We believe that our framework can potentially enable scalable computation of robustness verification for more complex predictors and structured perturbations that can be modeled using f-divergence constraints.

assistant: text: Develop a general framework to establish certified robustness of ML models against various classes of adversarial perturbations