Neural Information Processing Systems http://nips.cc/

This paper presents new projection-free algorithms for Online Convex Optimization (OCO) over a convex domain K Rd. Classical OCO algorithms (such as Online Gradient Descent) typically need to perform Euclidean projections onto the convex set K to ensure feasibility of their iterates. Alternative algorithms, such as those based on the Frank-Wolfe method, swap potentially-expensive Euclidean projections onto Kfor linear optimization over K. However, such algorithms have a sub-optimal regret in OCO compared to projection-based algorithms. In this paper, we look at a third type of algorithms that output approximate Newton iterates using a self-concordant barrier for the set of interest. The use of a self-concordant barrier automatically ensures feasibility without the need of projections. However, the computation of the Newton iterates requires a matrix inverse, which can still be expensive. As our main contribution, we show how the stability of the Newton iterates can be leveraged to only compute the inverse Hessian a vanishing fractions of the rounds, leading to a new efficient projection-free OCO algorithm with a state-of-the-art regret bound.

Neural Information Processing Systems http://nips.cc/

Ensuring that a DNN satisfies a desired property is critical when deploying DNNs in safety-critical applications. There are efficient methods that can verify whether a DNN satisfies a property, as seen in the annual DNN verification competition (VNN-COMP). However, the problem of provably editing a DNN to satisfy a property remains challenging.

Recently, several works have shown that natural modifications of the classical conditional gradient method (aka Frank-Wolfe algorithm) for constrained convex optimization, provably converge with a linear rate when the feasible set is a polytope, and the objective is smooth and strongly-convex. However, all of these results suffer from two significant shortcomings: i) large memory requirement due to the need to store an explicit convex decomposition of the current iterate, and as a consequence, large running-time overhead per iteration ii) the worst case convergence rate depends unfavorably on the dimension In this work we present a new conditional gradient variant and a corresponding analysis that improves on both of the above shortcomings. In particular, both memory and computation overheads are only linear in the dimension, and in addition, in case the optimal solution is sparse, the new convergence rate replaces a factor which is at least linear in the dimension in previous works, with a linear dependence on the number of non-zeros in the optimal solution At the heart of our method, and corresponding analysis, is a novel way to compute decomposition-invariant away-steps. While our theoretical guarantees do not apply to any polytope, they apply to several important structured polytopes that capture central concepts such as paths in graphs, perfect matchings in bipartite graphs, marginal distributions that arise in structured prediction tasks, and more. Our theoretical findings are complemented by empirical evidence that shows that our method delivers state-of-the-art performance.

We present improved algorithm for properly learning convex polytopes in the realizable PAC setting from data with a margin. Our learning algorithm constructs a consistent polytope as an intersection of about t log t halfspaces with margins in time polynomial in t (where t is the number of halfspaces forming an optimal polytope). We also identify distinct generalizations of the notion of margin from hyperplanes to polytopes and investigate how they relate geometrically; this result may be of interest beyond the learning setting.

Continuous allocation tasks are a class of problems where an agent needs to distribute a limited amount of resources over a set of entities at each time step.

Motion planning is still an open problem for many disciplines, e.g., robotics, autonomous driving, due to their need for high computational resources that hinder