Popular inference algorithms such as belief propagation (BP) and generalized belief propagation (GBP) are intimately related to linear programming (LP) relaxation within the Sherali-Adams hierarchy. Despite the popularity of these algorithms, it is well understood that the Sum-of-Squares (SOS) hierarchy based on semidefinite programming (SDP) can provide superior guarantees. Unfortunately, SOS relaxations for a graph with $n$ vertices require solving an SDP with $n^{\Theta(d)}$ variables where $d$ is the degree in the hierarchy. In practice, for $d\ge 4$, this approach does not scale beyond a few tens of variables. In this paper, we propose binary SDP relaxations for MAP inference using the SOS hierarchy with two innovations focused on computational efficiency. Firstly, in analogy to BP and its variants, we only introduce decision variables corresponding to contiguous regions in the graphical model. Secondly, we solve the resulting SDP using a non-convex Burer-Monteiro style method, and develop a sequential rounding procedure. We demonstrate that the resulting algorithm can solve problems with tens of thousands of variables within minutes, and outperforms BP and GBP on practical problems such as image denoising and Ising spin glasses. Finally, for specific graph types, we establish a sufficient condition for the tightness of the proposed partial SOS relaxation.

The design of revenue-maximizing combinatorial auctions, i.e. multi item auctions over bundles of goods, is one of the most fundamental problems in computational economics, unsolved even for two bidders and two items for sale. In the traditional economic models, it is assumed that the bidders' valuations are drawn from an underlying distribution and that the auction designer has perfect knowledge of this distribution. Despite this strong and oftentimes unrealistic assumption, it is remarkable that the revenue-maximizing combinatorial auction remains unknown. In recent years, automated mechanism design has emerged as one of the most practical and promising approaches to designing high-revenue combinatorial auctions. The most scalable automated mechanism design algorithms take as input samples from the bidders' valuation distribution and then search for a high-revenue auction in a rich auction class. In this work, we provide the first sample complexity analysis for the standard hierarchy of deterministic combinatorial auction classes used in automated mechanism design. In particular, we provide tight sample complexity bounds on the number of samples needed to guarantee that the empirical revenue of the designed mechanism on the samples is close to its expected revenue on the underlying, unknown distribution over bidder valuations, for each of the auction classes in the hierarchy. In addition to helping set automated mechanism design on firm foundations, our results also push the boundaries of learning theory. In particular, the hypothesis functions used in our contexts are defined through multi stage combinatorial optimization procedures, rather than simple decision boundaries, as are common in machine learning.

Building systems that autonomously create temporal abstractions from data is a key challenge in scaling learning and planning in reinforcement learning. One popular approach for addressing this challenge is the options framework (Sutton et al., 1999). However, only recently in (Bacon et al., 2017) was a policy gradient theorem derived for online learning of general purpose options in an end to end fashion. In this work, we extend previous work on this topic that only focuses on learning a two-level hierarchy including options and primitive actions to enable learning simultaneously at multiple resolutions in time. We achieve this by considering an arbitrarily deep hierarchy of options where high level temporally extended options are composed of lower level options with finer resolutions in time. We extend results from (Bacon et al., 2017) and derive policy gradient theorems for a deep hierarchy of options. Our proposed hierarchical option-critic architecture is capable of learning internal policies, termination conditions, and hierarchical compositions over options without the need for any intrinsic rewards or subgoals. Our empirical results in both discrete and continuous environments demonstrate the efficiency of our framework.

The zero-shot open-vocabulary setting poses challenges for conventional image classification.