Approximation Properties of NP Minimization Classes

Stability Properties of Empirical Risk Minimization over Donsker Classes

2 ) converges to zero in probability. Hence, even in the case of multiple minimizers of expected error, as n increases it becomes less and less likely that adding a sample (or a number of samples) to the training set will result in a large jump to a new hypothesis. Moreover, under some assumptions on the entropy of the class, along with an assumption of Komlos-Major-Tusnady type, we derive a po...

Some Properties of Empirical Risk Minimization Over Donsker Classes

We study properties of algorithms which minimize (or almost-minimize) empirical error over a Donsker class of functions. We show that the L2-diameter of the set of almost-minimizers is converging to zero in probability. Therefore, as the number of samples grows, it is becoming unlikely that adding a point (or a number of points) to the training set will result in a large jump (in L2 distance) t...

Weighted NP Optimization Problems: Logical Definability and Approximation Properties

Extending a well-known property of NP optimization problems in which the value of the optimum is guaranteed to be polynomially bounded in the length of the input, it is observed that, by attaching weights to tuples over the domain of the input, all NP optimization problems admit a logical characterization. It is shown that any NP optimization problem can be stated as a problem in which the cons...

Weighted NP Optimization Problems: Logical De nability and Approximation Properties

Hamming Approximation of NP Witnesses

Given a satisfiable 3-SAT formula, how hard is it to find an assignment to the variables that has Hamming distance at most n/2 to a satisfying assignment? More generally, consider any polynomial-time verifier for any NP-complete language. A d(n)-Hammingapproximation algorithm for the verifier is one that, given any member x of the language, outputs in polynomial time a string a with Hamming dis...