319 research outputs found
On the Convergence Rate of Decomposable Submodular Function Minimization
Submodular functions describe a variety of discrete problems in machine
learning, signal processing, and computer vision. However, minimizing
submodular functions poses a number of algorithmic challenges. Recent work
introduced an easy-to-use, parallelizable algorithm for minimizing submodular
functions that decompose as the sum of "simple" submodular functions.
Empirically, this algorithm performs extremely well, but no theoretical
analysis was given. In this paper, we show that the algorithm converges
linearly, and we provide upper and lower bounds on the rate of convergence. Our
proof relies on the geometry of submodular polyhedra and draws on results from
spectral graph theory.Comment: 17 pages, 3 figure
Privately Releasing Conjunctions and the Statistical Query Barrier
Suppose we would like to know all answers to a set of statistical queries C
on a data set up to small error, but we can only access the data itself using
statistical queries. A trivial solution is to exhaustively ask all queries in
C. Can we do any better?
+ We show that the number of statistical queries necessary and sufficient for
this task is---up to polynomial factors---equal to the agnostic learning
complexity of C in Kearns' statistical query (SQ) model. This gives a complete
answer to the question when running time is not a concern.
+ We then show that the problem can be solved efficiently (allowing arbitrary
error on a small fraction of queries) whenever the answers to C can be
described by a submodular function. This includes many natural concept classes,
such as graph cuts and Boolean disjunctions and conjunctions.
While interesting from a learning theoretic point of view, our main
applications are in privacy-preserving data analysis:
Here, our second result leads to the first algorithm that efficiently
releases differentially private answers to of all Boolean conjunctions with 1%
average error. This presents significant progress on a key open problem in
privacy-preserving data analysis.
Our first result on the other hand gives unconditional lower bounds on any
differentially private algorithm that admits a (potentially
non-privacy-preserving) implementation using only statistical queries. Not only
our algorithms, but also most known private algorithms can be implemented using
only statistical queries, and hence are constrained by these lower bounds. Our
result therefore isolates the complexity of agnostic learning in the SQ-model
as a new barrier in the design of differentially private algorithms
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