Optimal SQ Lower Bounds for Robustly Learning Discrete Product Distributions and Ising Models
This work addresses fundamental limitations in robust machine learning for high-dimensional data, providing evidence that current algorithms are optimal, which is incremental but important for theoretical understanding.
The paper tackles the problem of robustly learning discrete high-dimensional distributions, such as binary product distributions and Ising models, from corrupted data, establishing optimal Statistical Query (SQ) lower bounds that match known algorithm error rates, specifically showing mean estimation errors of o(ε√log(1/ε)) and total variation distances of o(εlog(1/ε)).
We establish optimal Statistical Query (SQ) lower bounds for robustly learning certain families of discrete high-dimensional distributions. In particular, we show that no efficient SQ algorithm with access to an $ε$-corrupted binary product distribution can learn its mean within $\ell_2$-error $o(ε\sqrt{\log(1/ε)})$. Similarly, we show that no efficient SQ algorithm with access to an $ε$-corrupted ferromagnetic high-temperature Ising model can learn the model to total variation distance $o(ε\log(1/ε))$. Our SQ lower bounds match the error guarantees of known algorithms for these problems, providing evidence that current upper bounds for these tasks are best possible. At the technical level, we develop a generic SQ lower bound for discrete high-dimensional distributions starting from low dimensional moment matching constructions that we believe will find other applications. Additionally, we introduce new ideas to analyze these moment-matching constructions for discrete univariate distributions.