The condensation phase transition in the regular kk-SAT model

Much of the recent work on random constraint satisfaction problems has been inspired by ingenious but non-rigorous approaches from physics. The physics predictions typically come in the form of distributional fixed point problems that are intended to mimic Belief Propagation, a message passing algorithm, applied to the random CSP. In this paper we propose a novel method for harnessing Belief Propagation directly to obtain a rigorous proof of such a prediction, namely the existence and location of a condensation phase transition in the random regular $k$-SAT model.Comment: Revised version based on arXiv:1504.03975, version

The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective

Among various algorithms designed to exploit the specific properties of quantum computers with respect to classical ones, the quantum adiabatic algorithm is a versatile proposition to find the minimal value of an arbitrary cost function (ground state energy). Random optimization problems provide a natural testbed to compare its efficiency with that of classical algorithms. These problems correspond to mean field spin glasses that have been extensively studied in the classical case. This paper reviews recent analytical works that extended these studies to incorporate the effect of quantum fluctuations, and presents also some original results in this direction.Comment: 151 pages, 21 figure

The condensation phase transition in random graph coloring

Based on a non-rigorous formalism called the "cavity method", physicists have put forward intriguing predictions on phase transitions in discrete structures. One of the most remarkable ones is that in problems such as random $k$-SAT or random graph $k$-coloring, very shortly before the threshold for the existence of solutions there occurs another phase transition called "condensation" [Krzakala et al., PNAS 2007]. The existence of this phase transition appears to be intimately related to the difficulty of proving precise results on, e.g., the $k$-colorability threshold as well as to the performance of message passing algorithms. In random graph $k$-coloring, there is a precise conjecture as to the location of the condensation phase transition in terms of a distributional fixed point problem. In this paper we prove this conjecture for $k$ exceeding a certain constant $k_0$

Distral: Robust Multitask Reinforcement Learning

Most deep reinforcement learning algorithms are data inefficient in complex and rich environments, limiting their applicability to many scenarios. One direction for improving data efficiency is multitask learning with shared neural network parameters, where efficiency may be improved through transfer across related tasks. In practice, however, this is not usually observed, because gradients from different tasks can interfere negatively, making learning unstable and sometimes even less data efficient. Another issue is the different reward schemes between tasks, which can easily lead to one task dominating the learning of a shared model. We propose a new approach for joint training of multiple tasks, which we refer to as Distral (Distill & transfer learning). Instead of sharing parameters between the different workers, we propose to share a "distilled" policy that captures common behaviour across tasks. Each worker is trained to solve its own task while constrained to stay close to the shared policy, while the shared policy is trained by distillation to be the centroid of all task policies. Both aspects of the learning process are derived by optimizing a joint objective function. We show that our approach supports efficient transfer on complex 3D environments, outperforming several related methods. Moreover, the proposed learning process is more robust and more stable---attributes that are critical in deep reinforcement learning