6 research outputs found
Extremely Deep Proofs
We further the study of supercritical tradeoffs in proof and circuit complexity, which is a type of tradeoff between complexity parameters where restricting one complexity parameter forces another to exceed its worst-case upper bound. In particular, we prove a new family of supercritical tradeoffs between depth and size for Resolution, Res(k), and Cutting Planes proofs. For each of these proof systems we construct, for each c ? n^{1-?}, a formula with n^{O(c)} clauses and n variables that has a proof of size n^{O(c)} but in which any proof of size no more than roughly exponential in n^{1-?}/c must necessarily have depth ? n^c. By setting c = o(n^{1-?}) we therefore obtain exponential lower bounds on proof depth; this far exceeds the trivial worst-case upper bound of n. In doing so we give a simplified proof of a supercritical depth/width tradeoff for tree-like Resolution from [Alexander A. Razborov, 2016]. Finally, we outline several conjectures that would imply similar supercritical tradeoffs between size and depth in circuit complexity via lifting theorems
On the Power and Limitations of Branch and Cut
The Stabbing Planes proof system [Paul Beame et al., 2018] was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas - certain unsatisfiable systems of linear equations od 2 - which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari [Daniel Dadush and Samarth Tiwari, 2020] showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes.
In this paper we make progress towards answering both of these questions. First, we show that any Stabbing Planes proof with bounded coefficients (SP*) can be translated into Cutting Planes. As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*. Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari, our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this is inherent. As a step towards this conjecture, we develop a new geometric technique for proving lower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas
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Reinforcement Learning: New Algorithms and An Application for Integer Programming
Reinforcement learning (RL) is a generic paradigm for the modeling and optimization of sequential decision making. In the recent decade, progress in RL research has brought about breakthroughs in several applications, ranging from playing video games, mastering board games, to controlling simulated robots. To bring the potential benefits of RL to other domains, two elements are critical: (1) Efficient and general-purpose RL algorithms; (2) Formulations of the original applications into RL problems. These two points are the focus of this thesis.
We start by developing more efficient RL algorithms. In Chapter 2, we propose Taylor Expansion Policy Optimization, a model-free algorithmic framework that unifies a number of important prior work as special cases. This unifying framework also allows us to develop a natural algorithmic extension to prior work, with empirical performance gains. In Chapter 3, we propose Monte-Carlo Tree Search as Regularized Policy Optimization, a model-based framework that draws close connections between policy optimization and Monte-Carlo tree search. Building on this insight, we propose Policy Optimization Zero (POZero), a novel algorithm which leverages the strengths of regularized policy search to achieve significant performance gains over MuZero.
To showcase how RL can be applied to other domains where the original applications could benefit from learning systems, we study the acceleration of integer programming (IP) solvers with RL. Due to the ubiquity of IP solvers in industrial applications, such research holds the promise of significant real life impacts and practical values. In Chapter 4, we focus on a particular formulation of Reinforcement Learning for Integer Programming: Learning to Cut. By combining cutting plane methods with selection rules learned by RL, we observe that the RL-augmented cutting plane solver achieves significant performance gains over traditional heuristics. This serves as a proof-of-concept of how RL can be combined with general IP solvers, and how learning augmented optimization systems might achieve significant acceleration in general