How Much Data Is Sufficient to Learn High-Performing Algorithms?

Algorithms often have tunable parameters that impact performance metrics such as runtime and solution quality. For many algorithms used in practice, no parameter settings admit meaningful worst-case bounds, so the parameters are made available for the user to tune. Alternatively, parameters may be tuned implicitly within the proof of a worst-case approximation ratio or runtime bound. Worst-case instances, however, may be rare or nonexistent in practice. A growing body of research has demonstrated that a data-driven approach to parameter tuning can lead to significant improvements in performance. This approach uses a training set of problem instances sampled from an unknown, application-specific distribution and returns a parameter setting with strong average performance on the training set. We provide techniques for deriving generalization guarantees that bound the difference between the algorithm's average performance over the training set and its expected performance on the unknown distribution. Our results apply no matter how the parameters are tuned, be it via an automated or manual approach. The challenge is that for many types of algorithms, performance is a volatile function of the parameters: slightly perturbing the parameters can cause a large change in behavior. Prior research [e.g., 12, 16, 20, 62] has proved generalization bounds by employing case-by-case analyses of greedy algorithms, clustering al- gorithms, integer programming algorithms, and selling mechanisms. We streamline these analyses with a general theorem that applies whenever an algorithm's performance is a piecewise-constant, piecewise-linear, or-more generally- piecewise-structured function of its parameters. Our results, which are tight up to loga- rithmic factors in the worst case, also imply novel bounds for configuring dynamic programming algorithms from computational biology.