Optimization of Discrete Parameters Using the Adaptive Gradient Method and Directed Evolution

TL;DR

This work tackles the optimization of discrete parameters under constraints by introducing CONGA, a population-based adaptive gradient method that leverages a hot sigmoid and a stochastic sigmoid to enable differentiable updates in discrete spaces. The approach combines EMA-smoothed gradients, an adaptive penalty gamma, and a directed evolutionary dynamics to guide a population of solutions toward feasibility and optimality. Applied to the 0--1 Knapsack Problem, the method shows substantial gains when using multiple agents and even higher performance when employing directed evolution, achieving fully optimal results in the tested scenarios. The proposed framework offers a practical, gradient-based alternative for constrained discrete optimization with potential applicability to a wide range of combinatorial problems and architectures that involve discrete decision variables.

Abstract

The problem is considered of optimizing discrete parameters in the presence of constraints. We use the stochastic sigmoid with temperature and put forward the new adaptive gradient method CONGA. The search for an optimal solution is carried out by a population of individuals. Each of them varies according to gradients of the 'environment' and is characterized by two temperature parameters with different annealing schedules. Unadapted individuals die, and optimal ones interbreed, the result is directed evolutionary dynamics. The proposed method is illustrated using the well-known combinatorial problem for optimal packing of a backpack (0-1 KP).

Figures (12)

• Figure 1: Heaviside function (dashed) and sigmoid for different values of temperature $\tau$
• Figure 2: The derivative of the sigmoid with temperature at different values of $\tau$
• Figure 3: Single-agent accuracy for different task types
• Figure 4: Metric $acc$ dependence on the number of agents
• Figure 5: Dependence of the mean solution time on the number of agents