6,262 research outputs found
Deterministic and Probabilistic Binary Search in Graphs
We consider the following natural generalization of Binary Search: in a given
undirected, positively weighted graph, one vertex is a target. The algorithm's
task is to identify the target by adaptively querying vertices. In response to
querying a node , the algorithm learns either that is the target, or is
given an edge out of that lies on a shortest path from to the target.
We study this problem in a general noisy model in which each query
independently receives a correct answer with probability (a
known constant), and an (adversarial) incorrect one with probability .
Our main positive result is that when (i.e., all answers are
correct), queries are always sufficient. For general , we give an
(almost information-theoretically optimal) algorithm that uses, in expectation,
no more than queries, and identifies the target correctly with probability at
leas . Here, denotes the
entropy. The first bound is achieved by the algorithm that iteratively queries
a 1-median of the nodes not ruled out yet; the second bound by careful repeated
invocations of a multiplicative weights algorithm.
Even for , we show several hardness results for the problem of
determining whether a target can be found using queries. Our upper bound of
implies a quasipolynomial-time algorithm for undirected connected
graphs; we show that this is best-possible under the Strong Exponential Time
Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs
with non-uniform node querying costs, the problem is PSPACE-complete. For a
semi-adaptive version, in which one may query nodes each in rounds, we
show membership in in the polynomial hierarchy, and hardness
for
Local Testing for Membership in Lattices
Motivated by the structural analogies between point lattices and linear error-correcting codes, and by the mature theory on locally testable codes, we initiate a systematic study of local testing for membership in lattices. Testing membership in lattices is also motivated in practice, by applications to integer programming, error detection in lattice-based communication, and cryptography. Apart from establishing the conceptual foundations of lattice testing, our results include the following: 1. We demonstrate upper and lower bounds on the query complexity of local testing for the well-known family of code formula lattices. Furthermore, we instantiate our results with code formula lattices constructed from Reed-Muller codes, and obtain nearly-tight bounds. 2. We show that in order to achieve low query complexity, it is sufficient to design one-sided non-adaptive canonical tests. This result is akin to, and based on an analogous result for error-correcting codes due to Ben-Sasson et al. (SIAM J. Computing 35(1) pp1-21)
Privacy Risk in Machine Learning: Analyzing the Connection to Overfitting
Machine learning algorithms, when applied to sensitive data, pose a distinct
threat to privacy. A growing body of prior work demonstrates that models
produced by these algorithms may leak specific private information in the
training data to an attacker, either through the models' structure or their
observable behavior. However, the underlying cause of this privacy risk is not
well understood beyond a handful of anecdotal accounts that suggest overfitting
and influence might play a role.
This paper examines the effect that overfitting and influence have on the
ability of an attacker to learn information about the training data from
machine learning models, either through training set membership inference or
attribute inference attacks. Using both formal and empirical analyses, we
illustrate a clear relationship between these factors and the privacy risk that
arises in several popular machine learning algorithms. We find that overfitting
is sufficient to allow an attacker to perform membership inference and, when
the target attribute meets certain conditions about its influence, attribute
inference attacks. Interestingly, our formal analysis also shows that
overfitting is not necessary for these attacks and begins to shed light on what
other factors may be in play. Finally, we explore the connection between
membership inference and attribute inference, showing that there are deep
connections between the two that lead to effective new attacks
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