In recent years, Bi-Level Optimization (BLO) techniques have received extensive attentions from both learning and vision communities. A variety of BLO models in complex and practical tasks are of non-convex follower structure in nature (a.k.a., without Lower-Level Convexity, LLC for short). However, this challenging class of BLOs is lack of developments on both efficient solution strategies and solid theoretical guarantees. In this work, we propose a new algorithmic framework, named Initialization Auxiliary and Pessimistic Trajectory Truncated Gradient Method (IAPTT-GM), to partially address the above issues. In particular, by introducing an auxiliary as initialization to guide the optimization dynamics and designing a pessimistic trajectory truncation operation, we construct a reliable approximate version of the original BLO in the absence of LLC hypothesis. Our theoretical investigations establish the convergence of solutions returned by IAPTT-GM towards those of the original BLO without LLC. As an additional bonus, we also theoretically justify the quality of our IAPTT-GM embedded with Nesterov's accelerated dynamics under LLC. The experimental results confirm both the convergence of our algorithm without LLC, and the theoretical findings under LLC.

Escaping saddle points is a central research topic in nonconvex optimization. In this paper, we propose a simple gradient-based algorithm such that for a smooth function f: Rn!R, it outputs an -approximate second-order stationary point in O(logn/ 1.75)iterations. Compared to the previous state-of-the-art algorithms by Jin et al. with O(log4 n/ 2) or O(log6 n/ 1.75) iterations, our algorithm is polynomially better in terms of logn and matches their complexities in terms of 1/ .

Let ϑ Rd1 and µ Rd2 be the dual variables corresponding to the d1 equality constraints and the d2 inequality constraints respectively. Let X? be an optimal solution to (SDP) and let X?FW be an optimal solution to (SDP-LSE). For ease of notation, let u= A(1)(X) b(1) andv = b(2) A(2)(X), (1) and define (bu,bv), (uFW,vFW) and (u?,v?) by substituting bX, XFW and X? respectively in (1). Upper bound on the objective. Rearranging the terms, using the duality of the `1 and ` norms, and the fact that µ? 0, gives hC, bX i hC,X?i+

This is the supplementary material of paper "Towards Enabling Meta-Learning from Target Models". We give implementation details, more discussions, and more experiment results in this material.

Given a point set P M from a metric space (M,d)and numbers k,z N, the metric k-center problem with z outliers is to find a set C P of k points such that the maximum distance of all but at most z outlier points of P to their nearest center in C is minimized. We consider this problem in the fully dynamic model, i.e., under insertions and deletions of points, for the case that the metric space has a bounded doubling dimension dim. We utilize a hierarchical data structure to maintain the points and their neighborhoods, which enables us to efficiently find the clusters. In particular, our data structure can be queried at any time to generate a (3 + ε)-approximate solution for input values of k and z in worst-case query time ε O(dim)klognloglog, where is the ratio between the maximum and minimum distance between two points in P. Moreover, it allows insertion/deletion of a point in worst-case update time ε O(dim) lognlog . Our result achieves a significantly faster query time with respect to k and z than the current state-of-theart by Pellizzoni, Pietracaprina, and Pucci [18], which uses ε O(dim)(k+z)2 log query time to obtain a (3+ε)-approximate solution.