TLDR: We suggest the uneven licensed robustness drawback, which requires licensed robustness for just one class and displays real-world adversarial situations. This centered setting permits us to introduce feature-convex classifiers, which produce closed-form and deterministic licensed radii on the order of milliseconds.
Determine 1. Illustration of feature-convex classifiers and their certification for sensitive-class inputs. This structure composes a Lipschitz-continuous characteristic map $varphi$ with a discovered convex perform $g$. Since $g$ is convex, it’s globally underapproximated by its tangent aircraft at $varphi(x)$, yielding licensed norm balls within the characteristic area. Lipschitzness of $varphi$ then yields appropriately scaled certificates within the unique enter area.
Regardless of their widespread utilization, deep studying classifiers are acutely susceptible to adversarial examples: small, human-imperceptible picture perturbations that idiot machine studying fashions into misclassifying the modified enter. This weak point severely undermines the reliability of safety-critical processes that incorporate machine studying. Many empirical defenses in opposition to adversarial perturbations have been proposed—usually solely to be later defeated by stronger assault methods. We due to this fact concentrate on certifiably sturdy classifiers, which offer a mathematical assure that their prediction will stay fixed for an $ell_p$-norm ball round an enter.
Typical licensed robustness strategies incur a spread of drawbacks, together with nondeterminism, gradual execution, poor scaling, and certification in opposition to just one assault norm. We argue that these points may be addressed by refining the licensed robustness drawback to be extra aligned with sensible adversarial settings.
The Uneven Licensed Robustness Drawback
Present certifiably sturdy classifiers produce certificates for inputs belonging to any class. For a lot of real-world adversarial purposes, that is unnecessarily broad. Think about the illustrative case of somebody composing a phishing rip-off electronic mail whereas attempting to keep away from spam filters. This adversary will all the time try and idiot the spam filter into considering that their spam electronic mail is benign—by no means conversely. In different phrases, the attacker is solely making an attempt to induce false negatives from the classifier. Comparable settings embody malware detection, pretend information flagging, social media bot detection, medical insurance coverage claims filtering, monetary fraud detection, phishing web site detection, and lots of extra.
Determine 2. Uneven robustness in electronic mail filtering. Sensible adversarial settings usually require licensed robustness for just one class.
These purposes all contain a binary classification setting with one delicate class that an adversary is making an attempt to keep away from (e.g., the “spam electronic mail” class). This motivates the issue of uneven licensed robustness, which goals to offer certifiably sturdy predictions for inputs within the delicate class whereas sustaining a excessive clear accuracy for all different inputs. We offer a extra formal drawback assertion in the principle textual content.
Characteristic-convex classifiers
We suggest feature-convex neural networks to handle the uneven robustness drawback. This structure composes a easy Lipschitz-continuous characteristic map ${varphi: mathbb{R}^d to mathbb{R}^q}$ with a discovered Enter-Convex Neural Community (ICNN) ${g: mathbb{R}^q to mathbb{R}}$ (Determine 1). ICNNs implement convexity from the enter to the output logit by composing ReLU nonlinearities with nonnegative weight matrices. Since a binary ICNN resolution area consists of a convex set and its complement, we add the precomposed characteristic map $varphi$ to allow nonconvex resolution areas.
Characteristic-convex classifiers allow the quick computation of sensitive-class licensed radii for all $ell_p$-norms. Utilizing the truth that convex features are globally underapproximated by any tangent aircraft, we will receive a licensed radius within the intermediate characteristic area. This radius is then propagated to the enter area by Lipschitzness. The uneven setting right here is crucial, as this structure solely produces certificates for the positive-logit class $g(varphi(x)) > 0$.
The ensuing $ell_p$-norm licensed radius components is especially elegant:
[r*p(x) = frac{ color{blue}{g(varphi(x))} } { mathrm{Lip}_p(varphi) color{red}{| nabla g(varphi(x)) | *{p,*}}}.]
The non-constant phrases are simply interpretable: the radius scales proportionally to the classifier confidence and inversely to the classifier sensitivity. We consider these certificates throughout a spread of datasets, attaining aggressive $ell_1$ certificates and comparable $ell_2$ and $ell_{infty}$ certificates—regardless of different strategies typically tailoring for a particular norm and requiring orders of magnitude extra runtime.
Determine 3. Delicate class licensed radii on the CIFAR-10 cats vs canines dataset for the $ell_1$-norm. Runtimes on the proper are averaged over $ell_1$, $ell_2$, and $ell_{infty}$-radii (observe the log scaling).
Our certificates maintain for any $ell_p$-norm and are closed kind and deterministic, requiring only one forwards and backwards cross per enter. These are computable on the order of milliseconds and scale properly with community dimension. For comparability, present state-of-the-art strategies corresponding to randomized smoothing and interval sure propagation sometimes take a number of seconds to certify even small networks. Randomized smoothing strategies are additionally inherently nondeterministic, with certificates that simply maintain with excessive likelihood.
Theoretical promise
Whereas preliminary outcomes are promising, our theoretical work suggests that there’s important untapped potential in ICNNs, even with no characteristic map. Regardless of binary ICNNs being restricted to studying convex resolution areas, we show that there exists an ICNN that achieves good coaching accuracy on the CIFAR-10 cats-vs-dogs dataset.
Reality. There exists an input-convex classifier which achieves good coaching accuracy for the CIFAR-10 cats-versus-dogs dataset.
Nonetheless, our structure achieves simply $73.4%$ coaching accuracy with no characteristic map. Whereas coaching efficiency doesn’t indicate check set generalization, this outcome means that ICNNs are at the least theoretically able to attaining the trendy machine studying paradigm of overfitting to the coaching dataset. We thus pose the next open drawback for the sphere.
Open drawback. Be taught an input-convex classifier which achieves good coaching accuracy for the CIFAR-10 cats-versus-dogs dataset.
Conclusion
We hope that the uneven robustness framework will encourage novel architectures that are certifiable on this extra centered setting. Our feature-convex classifier is one such structure and supplies quick, deterministic licensed radii for any $ell_p$-norm. We additionally pose the open drawback of overfitting the CIFAR-10 cats vs canines coaching dataset with an ICNN, which we present is theoretically potential. Please see our NeurIPS paper and codebase for extra particulars.
