Premature Convergence

Convergence refers to the limit of a process and can be a useful analytical tool when evaluating the expected performance of an optimization algorithm.

It can also be a useful empirical tool when exploring the learning dynamics of an optimization algorithm, and machine learning algorithms trained using an optimization algorithm, such as deep learning neural networks. This motivates the investigation of learning curves and techniques, such as early stopping.

If optimization is a process that generates candidate solutions, then convergence represents a stable point at the end of the process when no further changes or improvements are expected. Premature convergence refers to a failure mode for an optimization algorithm where the process stops at a stable point that does not represent a globally optimal solution.

In optimization, it refers to the algorithm converging upon a stable point that has worse performance than expected.

Premature convergence typically afflicts complex optimization tasks where the objective function is non-convex, meaning that the response surface contains many different good solutions (stable points), perhaps with one (or a few) best solutions.

If we consider the response surface of an objective function under optimization as a geometrical landscape and we are seeking a minimum of the function, then premature optimization refers to finding a valley close to the starting point of the search that has less depth than the deepest valley in the problem domain.

For problems that exhibit highly multi-modal (rugged) fitness landscapes or landscapes that change over time, too much exploitation generally results in premature convergence to suboptimal peaks in the space.

— Page 60, Evolutionary Computation: A Unified Approach, 2002.

In this way, premature convergence is described as finding a locally optimal solution instead of the globally optimal solution for an optimization algorithm. It is a specific failure case for an optimization algorithm.

• Premature Convergence: Convergence of an optimization algorithm to a worse than optimal stable point that is likely close to the starting point.

Put another way, convergence signifies the end of the search process, e.g. a stable point was located and further iterations of the algorithm are not likely to improve upon the solution. Premature convergence refers to reaching this stop condition of an optimization algorithm at a less than desirable stationary point.