Adversarial examples from computational constraints (1805.10204v1)

Abstract: Why are classifiers in high dimension vulnerable to "adversarial" perturbations? We show that it is likely not due to information theoretic limitations, but rather it could be due to computational constraints. First we prove that, for a broad set of classification tasks, the mere existence of a robust classifier implies that it can be found by a possibly exponential-time algorithm with relatively few training examples. Then we give a particular classification task where learning a robust classifier is computationally intractable. More precisely we construct a binary classification task in high dimensional space which is (i) information theoretically easy to learn robustly for large perturbations, (ii) efficiently learnable (non-robustly) by a simple linear separator, (iii) yet is not efficiently robustly learnable, even for small perturbations, by any algorithm in the statistical query (SQ) model. This example gives an exponential separation between classical learning and robust learning in the statistical query model. It suggests that adversarial examples may be an unavoidable byproduct of computational limitations of learning algorithms.

• The paper demonstrates that while robust classification is information-theoretically feasible, it becomes computationally intractable under the Statistical Query model.
• It contrasts simple linear methods enabling non-robust classification with exponential query requirements needed for robust classifiers.
• The analysis highlights the need for new algorithmic strategies to overcome inherent computational barriers in designing resilient learning systems.

Analyzing the Vulnerability of Classifiers to Adversarial Examples Under Computational Constraints

This paper presents a theoretical exploration of the susceptibility of high-dimensional classifiers to adversarial perturbations, arguing against purely informational limitations as the source of this vulnerability. Instead, it posits that computational constraints may be a significant factor. The authors, Bubeck, Price, and Razenshteyn, provide evidence supporting this hypothesis by constructing specific classification tasks to illustrate this computational intractability.

The research separates the ability to learn robust classifiers from classical learning, where the latter can be achieved efficiently without robustness. The paper introduces a binary classification task in high dimensional space that underscores this distinction. It is shown that one can construct a task where, 1) robust classification is information-theoretically feasible yet complicated computationally, 2) non-robust classification remains accessible through simple linear models, and 3) robust learning presents difficulties within the Statistical Query (SQ) model. This is underscored by an exponential gap between classical and robust learning within this model.

Key to the paper's thesis is demonstrating that adversarial examples might inherently arise from the computational limitations of learning algorithms. This notion is directly evidenced by a devised scenario where robust classifiers exist yet finding such classifiers efficiently via current methods appears infeasible. The authors employ a rigorous analysis involving the SQ model of computation, a potent framework for proving learning hardness results. The example constructed serves as a case where learning a robust classifier is achievable only through exponential queries or data when using finely grained statistical queries, highlighting the limits imposed by computationally intensive approaches.

The implications of these findings compel a re-examination of current classification strategies deployed in practical applications, such as neural networks' deployment in vision systems. As these systems are increasingly utilized in critical areas, understanding the nature and origins of their vulnerability to adversarial examples is paramount. The authors caution against hypothesis 2 (robust classification needing excessive data) by showing with standard assumptions that robust classifiers may be discovered with few examples, and affirm hypothesis 3 (computational intractability) by providing an exemplar task that is robust yet computationally challenging.

The developed theory enriches the existing literature that predominantly explores susceptibility hypotheses unrelated to explicit computational hardships. The paper’s exploration of generating advesarial robustness beyond data-driven approaches invites new methodologies in addressing system vulnerabilities. Crucially, it paves the way for further investigation into whether these computational limitations are emblematic of real distribution complexities.

Looking forward, this analysis prompts several avenues for further research: refining the model to incorporate more natural distribution complexities, extending hardness results beyond the SQ model to more general algorithms, and verifying the conjectured robustness-complexity trade-offs in practical, real-world datasets. These pursuits could provide essential insights into creating more resilient learning systems in environments susceptible to adversarial perturbations. The intersection of computational complexity and learning robustness discussed herein reinforces the need for novel algorithms that balance these constraints to fortify high-dimensional classifiers against adversarial threats effectively.