Variational Neural Annealing (2101.10154v1)

Abstract: Many important challenges in science and technology can be cast as optimization problems. When viewed in a statistical physics framework, these can be tackled by simulated annealing, where a gradual cooling procedure helps search for groundstate solutions of a target Hamiltonian. While powerful, simulated annealing is known to have prohibitively slow sampling dynamics when the optimization landscape is rough or glassy. Here we show that by generalizing the target distribution with a parameterized model, an analogous annealing framework based on the variational principle can be used to search for groundstate solutions. Modern autoregressive models such as recurrent neural networks provide ideal parameterizations since they can be exactly sampled without slow dynamics even when the model encodes a rough landscape. We implement this procedure in the classical and quantum settings on several prototypical spin glass Hamiltonians, and find that it significantly outperforms traditional simulated annealing in the asymptotic limit, illustrating the potential power of this yet unexplored route to optimization.

• The paper introduces Variational Neural Annealing, a hybrid approach that combines simulated annealing with variational techniques to overcome traditional sampling limitations.
• The paper proposes two variants—Variational Classical Annealing and Variational Quantum Annealing—using RNN-based autoregressive models for efficient equilibrium sampling.
• The paper demonstrates that VNA outperforms standard simulated annealing methods in accurately solving complex spin glass Hamiltonians and combinatorial optimization problems.

Variational Neural Annealing

The paper "Variational Neural Annealing" introduces a novel optimization methodology leveraging the strengths of autoregressive models in tackling rough or glassy optimization landscapes. The method proposed combines concepts from simulated annealing with variational principles to improve the efficiency of finding ground states in complex combinatorial optimization problems embodied by spin glass Hamiltonians.

Problem Statement and Approach

Simulated annealing (SA) is a traditional approach that exploits thermal fluctuations to explore optimization landscapes by "cooling" a system until it stabilizes at its minimum energy configuration. However, SA is known to struggle with landscapes characterized by numerous local minima due to its slow sampling dynamics. The authors propose a variational alternative, dubbed Variational Neural Annealing (VNA), which employs autoregressive models to parameterize distributions. These models precisely sample from these distributions, thus bypassing the limitations of slow sampling inherent to traditional methods.

Methodology

The methodology of VNA is broken down into two variants:

1. Variational Classical Annealing (VCA): Uses the variational principle of statistical mechanics to approximate the equilibrium properties of a system by minimizing a variational free energy that includes both energy and entropy terms. This approach ensures near-equilibrium sampling during the annealing protocol, which enhances the exploration of the solution space. During VCA, a recurrent neural network (RNN) is trained to represent the probability distribution of states, and gradient descent is used to optimize the parameters of the network, conditioned on decreasing temperature.
2. Variational Quantum Annealing (VQA): Emulates quantum annealing by adding quantum fluctuations to the optimization process. VQA leverages the variational Monte Carlo approach to approximate the ground state of quantum Hamiltonians. The method uses RNNs to model quantum states by translating wavefunctions into the classical domain, where they can be optimized using variational principles.

Results

The authors apply the proposed methodologies to several classical and quantum spin glass Hamiltonians. They observe that VNA methods, particularly VCA, consistently outperform traditional SA and simulated quantum annealing (SQA) approaches in finding more accurate solutions to optimization problems, especially as the number of annealing steps increases. This advantage is attributed to the ability of VNA to effectively explore complex energy landscapes, facilitated by the autoregressive sampling of recurrent neural networks.

Implications and Future Directions

The results hold significant implications for the use of machine learning in combinatorial optimization problems. VNA showcases the potential to improve upon classical solutions by effectively addressing the pitfalls associated with sampling and convergence in SA. The methodologies presented could influence diverse fields including operations research, materials science, and quantum computing.

Future research could explore the deployment of more complex neural architectures, such as transformers, and investigate hyperparameter optimization to further enhance the efficiency of VNA. Additionally, integrating reinforcement learning to dynamically optimize the annealing schedule and exploiting domain knowledge to tailor models to specific problem types represent promising areas for further exploration.

In summary, "Variational Neural Annealing" provides an insightful contribution to the optimization landscape, illustrating a path towards leveraging the intrinsic strengths of modern machine learning techniques in overcoming the limitations of traditional simulated annealing processes.