_:n21f71838e43942f38d198d09f388ebdexb38
<http://schema.org/description>
"Vector norms play a fundamental role in computer science and optimization, so\nthere is an ongoing effort to generalize existing algorithms to settings beyond\nℓ_∞ and ℓ_p norms. We show that many online and bandit\napplications for general norms admit good algorithms as long as the norm can be\napproximated by a function that is “gradient-stable”, a notion that we\nintroduce. Roughly it says that the gradient of the function should not\ndrastically decrease (multiplicatively) in any component as we increase the\ninput vector. We prove that several families of norms, including all monotone\nsymmetric norms, admit a gradient-stable approximation, giving us the first\nonline and bandit algorithms for these norm families.\n  In particular, our notion of gradient-stability gives O(log^2\n(dimension))-competitive algorithms for the symmetric norm\ngeneralizations of Online Generalized Load Balancing and Bandits with\nKnapsacks. Our techniques extend to applications beyond symmetric norms as\nwell, e.g., to Online Vector Scheduling and to Online Generalized Assignment\nwith Convex Costs. Some key properties underlying our applications that are\nimplied by gradient-stable approximations are a “smooth game inequality” and\nan approximate converse to Jensen's inequality."
<https://api.deepai.org/profile/thomas-kesselheim>