All antibiotics have to engage bacterial amphiphilic barriers such as the lipopolysaccharide-rich outer membrane or the phospholipid-based inner membrane in some manner, either by disrupting them outright and/or permeating them and thereby allow the antibiotic to get into bacteria. There is a growing class of cyclic antibiotics, many of which are of bacterial origin, that exhibit activity against Gram-negative bacteria, which constitute an urgent problem in human health. We examine a diverse collection of these cyclic antibiotics, both natural and synthetic, which include bactenecin, polymyxin B, octapeptin, capreomycin, and Kirshenbaum peptoids, in order to identify what they have in common when they interact with bacterial lipid membranes. We find that they virtually all have the ability to induce negative Gaussian curvature (NGC) in bacterial membranes, the type of curvature geometrically required for permeation mechanisms such as pore formation, blebbing, and budding. This is interesting since permeation of membranes is a function usually ascribed to antimicrobial peptides (AMPs) from innate immunity. As prototypical test cases of cyclic antibiotics, we analyzed amino acid sequences of bactenecin, polymyxin B, and capreomycin using our recently developed machine-learning classifier trained on α-helical AMP sequences. Although the original classifier was not trained on cyclic antibiotics, a modified classifier approach correctly predicted that bactenecin and polymyxin B have the ability to induce NGC in membranes, while capreomycin does not. Moreover, the classifier was able to recapitulate empirical structure-activity relationships from alanine scans in polymyxin B surprisingly well. These results suggest that there exists some common ground in the sequence design of hybrid cyclic antibiotics and linear AMPs.