In the Convex Body Chasing problem, we are given an initial point @math in @math and an online sequence of @math convex bodies @math . When we receive @math , we are required to move inside @math . Our goal is to minimize the total distance travelled. This fundamental online problem was first studied by Friedman and Linial (DCG 1993). They proved an @math lower bound on the competitive ratio, and conjectured that a competitive ratio depending only on d is possible. However, despite much interest in the problem, the conjecture remains wide open. We consider the setting in which the convex bodies are nested: @math . The nested setting is closely related to extending the online LP framework of Buchbinder and Naor (ESA 2005) to arbitrary linear constraints. Moreover, this setting retains much of the difficulty of the general setting and captures an essential obstacle in resolving Friedman and Linial's conjecture. In this work, we give the first @math -competitive algorithm for chasing nested convex bodies in @math .
The only known algorithms for chasing convex bodies in @math with @math are for certain restricted families of convex bodies @math such as lines and affine subspaces. For chasing lines, Friedman and Linial @cite_9 gave an @math -competitive algorithm. For chasing lines and half-line, Sitters @cite_7 showed that the generalized work function algorithm (WFA) is also @math -competitive; this is interesting as the WFA is a generic algorithm that attains nearly-optimal competitive ratios for many MSS and is a natural candidate to be @math -competitive for convex body chasing. Recently, @cite_0 gave an elegant and simple @math -competitive algorithm for chasing lines, and a @math -competitive algorithm for chasing affine subspaces. However, all these results crucially rely on the fact that the requests @math have a lower dimension and do not seem to apply to our problem.
@cite_0 @cite_9 @cite_7
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We consider three related online problems: Online Convex Optimization, Convex Body Chasing, and Lazy Convex Body Chasing. In Online Convex Optimization the input is an online sequence of convex functions over some Euclidean space. In response to a function, the online algorithm can move to any destination point in the Euclidean space. The cost is the total distance moved plus the sum of the function costs at the destination points. Lazy Convex Body Chasing is a special case of Online Convex Optimization where the function is zero in some convex region, and grows linearly with the distance from this region. And Convex Body Chasing is a special case of Lazy Convex Body Chasing where the destination point has to be in the convex region. We show that these problems are equivalent in the sense that if any of these problems have an O(1)-competitive algorithm then all of the problems have an O(1)-competitive algorithm. By leveraging these results we then obtain the first O(1)-competitive algorithm for Online Convex Optimization in two dimensions, and give the first O(1)-competitive algorithm for chasing linear subspaces. We also give a simple algorithm and O(1)-competitiveness analysis for chasing lines. For convex minimization we introduce an algorithm based on **-space decomposition. The method uses a bundle subroutine to generate a sequence of approximate proximal points. When a primal-dual track leading to a solution and zero subgradient pair exists, these points approximate the primal track points and give the algorithm's **, or corrector, steps. The subroutine also approximates dual track points that are **-gradients needed for the method's **-Newton predictor steps. With the inclusion of a simple line search the resulting algorithm is proved to be globally convergent. The convergence is superlinear if the primal-dual track points and the objective's **-Hessian are approximated well enough. We consider the adversarial convex bandit problem and we build the first @math -time algorithm with @math -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves @math -regret, and we show that a simple variant of this algorithm can be run in @math -time per step at the cost of an additional @math factor in the regret. These results improve upon the @math -regret and @math -time result of the first two authors, and the @math -regret and @math -time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve @math -regret, and moreover that this regret is unimprovable (the current best lower bound being @math and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order @math . We propose a convex-concave programming approach for the labeled weighted graph matching problem. The convex-concave programming formulation is obtained by rewriting the weighted graph matching problem as a least-square problem on the set of permutation matrices and relaxing it to two different optimization problems: a quadratic convex and a quadratic concave optimization problem on the set of doubly stochastic matrices. The concave relaxation has the same global minimum as the initial graph matching problem, but the search for its global minimum is also a hard combinatorial problem. We therefore construct an approximation of the concave problem solution by following a solution path of a convex-concave problem obtained by linear interpolation of the convex and concave formulations, starting from the convex relaxation. This method allows to easily integrate the information on graph label similarities into the optimization problem, and therefore to perform labeled weighted graph matching. The algorithm is compared with some of the best performing graph matching methods on four datasets: simulated graphs, QAPLib, retina vessel images and handwritten chinese characters. In all cases, the results are competitive with the state-of-the-art.