Sampling Can Be Faster Than Optimization
This provides theoretical insight into the relative efficiency of sampling vs. optimization for nonconvex problems, which is incremental but important for statistical machine learning applications.
The paper tackled the problem of comparing computational complexity between optimization and Monte Carlo sampling algorithms in nonconvex settings, such as mixture modeling, and found that sampling scales linearly with dimension while optimization scales exponentially.
Optimization algorithms and Monte Carlo sampling algorithms have provided the computational foundations for the rapid growth in applications of statistical machine learning in recent years. There is, however, limited theoretical understanding of the relationships between these two kinds of methodology, and limited understanding of relative strengths and weaknesses. Moreover, existing results have been obtained primarily in the setting of convex functions (for optimization) and log-concave functions (for sampling). In this setting, where local properties determine global properties, optimization algorithms are unsurprisingly more efficient computationally than sampling algorithms. We instead examine a class of nonconvex objective functions that arise in mixture modeling and multi-stable systems. In this nonconvex setting, we find that the computational complexity of sampling algorithms scales linearly with the model dimension while that of optimization algorithms scales exponentially.