We show that for @math and for measurable subsets of the unit square with Lebesgue measure @math there exist bi-Lipschitz maps with bounded Lipschitz constant (uniformly over all such sets) which are identity on the boundary and increases the Lebesgue measure of the set to at least @math .
The question of existence of bi-Lipschitz homeomorphisms between different subsets of @math , satisfying certain conditions, has been studied classically, notably by McMullen @cite_0 . It is shown in that paper that there exists a positive function @math on @math , @math , such that there does not exist a bi-Lipschitz homeomorphism @math with @math . It is also proved that there exists separated nets in @math ( @math ) which cannot be mapped into @math in a bi-Lipschitz way.
@cite_0
2171762499
We introduce a notion of quasi regularity for points with respect to the inclusion @math , where @math is a nonlinear Frechet differentiable function from @math to @math . When @math is the set of minimum points of a convex real-valued function @math on @math and @math satisfies the @math -average Lipschitz condition of Wang, we use the majorizing function technique to establish the semilocal linear quadratic convergence of sequences generated by the Gauss-Newton method (with quasi-regular initial points) for the convex composite function @math . Results are new even when the initial point is regular and @math is Lipschitz. We prove lower bounds on the complexity of finding @math -stationary points (points @math such that @math ) of smooth, high-dimensional, and potentially non-convex functions @math . We consider oracle-based complexity measures, where an algorithm is given access to the value and all derivatives of @math at a query point @math . We show that for any (potentially randomized) algorithm @math , there exists a function @math with Lipschitz @math th order derivatives such that @math requires at least @math queries to find an @math -stationary point. Our lower bounds are sharp to within constants, and they show that gradient descent, cubic-regularized Newton's method, and generalized @math th order regularization are worst-case optimal within their natural function classes. In stochastic convex optimization the goal is to minimize a convex function @math over a convex set @math where @math is some unknown distribution and each @math in the support of @math is convex over @math . The optimization is commonly based on i.i.d. samples @math from @math . A standard approach to such problems is empirical risk minimization (ERM) that optimizes @math . Here we consider the question of how many samples are necessary for ERM to succeed and the closely related question of uniform convergence of @math to @math over @math . We demonstrate that in the standard @math setting of Lipschitz-bounded functions over a @math of bounded radius, ERM requires sample size that scales linearly with the dimension @math . This nearly matches standard upper bounds and improves on @math dependence proved for @math setting by Shalev- (2009). In stark contrast, these problems can be solved using dimension-independent number of samples for @math setting and @math dependence for @math setting using other approaches. We further show that our lower bound applies even if the functions in the support of @math are smooth and efficiently computable and even if an @math regularization term is added. Finally, we demonstrate that for a more general class of bounded-range (but not Lipschitz-bounded) stochastic convex programs an infinite gap appears already in dimension 2. In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty games, which encompass biased games and games in which players take risk into account. Here we show that if the penalty function is Lipschitz continuous, then we can provide a quasi-polynomial time approximation scheme. Finally, we study distance biased games, where we present simple strongly polynomial time algorithms for finding best responses in @math , @math , and @math biased games, and then use these algorithms to provide strongly polynomial algorithms that find @math , @math , and @math approximations for these norms, respectively.