Learning and Reasoning with Fast Semidefinite Programming and Mixing Methods

Semidefinite programming has long been a theoretically powerful tool for solving relaxations of challenging, often NP-hard optimization problems. However, it has

typically not been practical for most large-scale tasks, owing to the high memory and computational cost of typical solvers for solving SDPs. In this thesis, we aim

to break the barrier and bring SDP’s power back to large-scale machine learning problems. To achieve this, we introduce a series of optimization solvers, operating on

the low-rank or low-cardinality manifolds of the semidefinite variables. We find that in many domains, these methods allow SDP relaxations to exceed the state of the art

in terms of both computational cost and the relevant performance metrics. First, we proposed the Mixing method, a low-rank SDP solver aimed at the classical MAXCUT SDP relaxation. We also show that the Mixing method can

accurately estimate the mode and partition function of the pairwise Markov Random Fields, and scales to millions of variables. Further, we show how to learn the parameters inside SDPs by analytically differentiating through the optimization problem with implicit differentiation and the mixing methods, which leads to a differentiable SAT solver that can be integrated within the loop of larger deep learning

systems. For nonnegative constraints, we propose a separate variant aimed at low cardinality SDPs, and demonstrate how to apply the method to community detection on finding clusters within large-scale networks. Finally, we show that the technique can also be applied to more generic problems, such as a generic linear programming problems (with arbitrarily structured constraints), and we use this approach to develop

a scalable sparse linear programming solver that improves solution speed over existing state-of-the-art commercial solvers