Enhanced Convergence in p-bit Based Simulated Annealing with Partial Deactivation for Large-Scale Combinatorial Optimization Problems

TL;DR

Two novel algorithms, time average pSA (TApSA) and stalled pSA (SpSA) are proposed, designed based on partial deactivation of p-bits and are thoroughly tested using Python simulations on maximum cut benchmarks that are typical combinatorial optimization problems.

Abstract

This article critically investigates the limitations of the simulated annealing algorithm using probabilistic bits (pSA) in solving large-scale combinatorial optimization problems. The study begins with an in-depth analysis of the pSA process, focusing on the issues resulting from unexpected oscillations among p-bits. These oscillations hinder the energy reduction of the Ising model and thus obstruct the successful execution of pSA in complex tasks. Through detailed simulations, we unravel the root cause of this energy stagnation, identifying the feedback mechanism inherent to the pSA operation as the primary contributor to these disruptive oscillations. To address this challenge, we propose two novel algorithms, time average pSA (TApSA) and stalled pSA (SpSA). These algorithms are designed based on partial deactivation of p-bits and are thoroughly tested using Python simulations on maximum cut benchmarks that are typical combinatorial optimization problems. On the 16 benchmarks from 800 to 5,000 nodes, the proposed methods improve the normalized cut value from 0.8% to 98.4% on average in comparison with the conventional pSA.

Figures (10)

• Figure 1: Simulated annealing based on p-bit (pSA). p-bits (left) probabilistically operates based on \ref{['eqn:pbits']}. A combinatorial optimization problem is represented by an Ising model that corresponds to an energy (Hamiltonian). Based on an Ising model, each p-bit is biased with $h$ and is connected with other p-bits with weights $J$ (right top). During the simulated annealing process, an pseudo inverse temperature $I_0$ is gradually increased to reach the global minimum energy ($H_{min}$). pSA attempts to lower the energy of the Ising model by changing the p-bit states $\sigma_i$. If the energy reaches the global minimum energy $\sigma_i$ are a solution of the combinatorial optimization problem (right bottom).
• Figure 2: A five-node maximum cut (MAX-CUT) problem with edge weights of $-1$ and $+1$. MAX-CUT problem is a typical combinatorial optimization problem. The line cuts the edges to divide the graph into two groups while the sum of the edge weights is maximized. The graph is divided into Group A (nodes 1, 3, and 4) and Group B (nodes 2 and 5), with a sum of edge weights equal to 4.
• Figure 3: Issue of pSA. pSA is simulated by Python with G1 that is a MAX-CUT problem of the G-set benchmark. During the simulated annealing process, the pseudo inverse temperature $I_0$ is increased to control pSA (a). A mean of all the p-bits states is changed between '-1' and '+1' at every cycle (b). Because of this oscillation, the energy starts to increase after the oscillation, although the energy is expected to be lower to the global minimum energy (c).
• Figure 4: Simulation analysis of TApSA on the G1 graph with different window size $\alpha$. The mean values of all the p-bit states are oscillated between '-1' and '+1' in case of $\alpha$ of two and three (top of a and b), resulting in the energy increase of the Ising model instead of decrease (bottom of a and b). When $\alpha$ is four, no oscillation occurs and hence the energy goes down to the global minimum energy (c). TApSA can solve the oscillation issue of pSA.
• Figure 5: Normalized cut values using the TApSA algorithm on the G1, G11, G58 and K2000 MAX-CUT problems by varying the windows size $\alpha$ from one to ten. This window size $\alpha$ plays a key role in determining the quality of the solution and can control the behavior of the TApSA algorithm. When $\alpha$ is increased to a specific value, the cut values can be closer to the best-known values because of no oscillation. The peak of the normalized mean cut value is obtained with different $\alpha$ depending on the graph.