Optimization from Structured Samples for Coverage Functions
This work addresses the challenge of function optimization from limited data in machine learning and algorithmic theory, offering a novel model that circumvents prior impossibility results, though it is incremental in building upon existing OPS frameworks.
The authors tackled the problem of optimizing coverage functions from sample data by introducing a stronger model called optimization from structured samples (OPSS), which incorporates structural information, and they designed efficient algorithms that achieve a constant approximation for the maximum coverage problem under specific assumptions, with tight lower bounds and proofs that removing any assumption eliminates constant approximation.
We revisit the optimization from samples (OPS) model, which studies the problem of optimizing objective functions directly from the sample data. Previous results showed that we cannot obtain a constant approximation ratio for the maximum coverage problem using polynomially many independent samples of the form $\{S_i, f(S_i)\}_{i=1}^t$ (Balkanski et al., 2017), even if coverage functions are $(1 - ε)$-PMAC learnable using these samples (Badanidiyuru et al., 2012), which means most of the function values can be approximately learned very well with high probability. In this work, to circumvent the impossibility result of OPS, we propose a stronger model called optimization from structured samples (OPSS) for coverage functions, where the data samples encode the structural information of the functions. We show that under three general assumptions on the sample distributions, we can design efficient OPSS algorithms that achieve a constant approximation for the maximum coverage problem. We further prove a constant lower bound under these assumptions, which is tight when not considering computational efficiency. Moreover, we also show that if we remove any one of the three assumptions, OPSS for the maximum coverage problem has no constant approximation.