We introduce a notion of non-black-box access to computational devices (such as circuits, formulas, decision trees, and so forth) that we call \emph{restriction access}. Restrictions are partial assignments to input variables. Each restriction simplifies the device, and yields a new device for the restricted function on the unassigned variables. On one extreme, full restrictions (assigning all variables) correspond to evaluating the device on a complete input, yielding the result of the computation on that input, which is the same as standard black-box access. On the other extreme, empty restrictions (assigning no variables) yield a full description of the original device. We explore the grey-scale of possibilities in the middle.
Focusing on learning theory, we show that restriction access provides a setting in which one can obtain positive results for problems that have resisted attack in the black-box access model. We introduce a PAC-learning version of restriction access, and show that one can efficiently learn both decision trees and DNF formulas in this model. These two classes are not known to be learnable in the PAC model with black-box access.
Our DNF learning algorithm is obtained by a reduction to a general learning problem we call {\em population recovery}, in which random samples from an unknown distribution become available only after a random part of each is obliterated. Specifically, assume that every member of an unknown population is described by a vector of values. The algorithm has access to random samples, each of which is a random member of the population, whose values are given {\em only} on a {\em random} subset of the attributes. Analyzing our efficient algorithm to fully recover the unknown population calls for understanding another basic problem of independent interest: ``robust {\em local} inversion'' of matrices. The population recovery algorithm and construction of robust local inverses for some families of matrices are the main technical contributions of the paper.
We also discuss other possible variants of restriction access, in which the values to restricted variables, as well as the subset of free (unassigned) variables, are generated deterministically or randomly, in friendly or adversarial fashions. We discuss how these models may naturally suit situations in computational learning, computational biology, automated proofs, cryptography and complexity theory.
