22 research outputs found
Almost Optimal Distribution-Free Junta Testing
We consider the problem of testing whether an unknown n-variable Boolean function is a k-junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over {0,1}^n. Chen, Liu, Servedio, Sheng and Xie [Zhengyang Liu et al., 2018] showed that the distribution-free k-junta testing can be performed, with one-sided error, by an adaptive algorithm that makes O~(k^2)/epsilon queries. In this paper, we give a simple two-sided error adaptive algorithm that makes O~(k/epsilon) queries
Near-Optimal Algorithm for Distribution-Free Junta Testing
We present an adaptive algorithm with one-sided error for the problem of
junta testing for Boolean function under the challenging distribution-free
setting, the query complexity of which is . This improves
the upper bound of by \cite{liu2019distribution}. From
the lower bound for junta testing under the uniform
distribution by \cite{sauglam2018near}, our algorithm is nearly optimal. In the
standard uniform distribution, the optimal junta testing algorithm is mainly
designed by bridging between relevant variables and relevant blocks. At the
heart of the analysis is the Efron-Stein orthogonal decomposition. However, it
is not clear how to generalize this tool to the general setting. Surprisingly,
we find that junta could be tested in a very simple and efficient way even in
the distribution-free setting. It is interesting that the analysis does not
rely on Fourier tools directly which are commonly used in junta testing.
Further, we present a simpler algorithm with the same query complexity
Learning and Testing Variable Partitions
Let be a multivariate function from a product set to an
Abelian group . A -partition of with cost is a partition of
the set of variables into non-empty subsets such that is -close to
for some with
respect to a given error metric. We study algorithms for agnostically learning
partitions and testing -partitionability over various groups and error
metrics given query access to . In particular we show that
Given a function that has a -partition of cost , a partition
of cost can be learned in time
for any .
In contrast, for and learning a partition of cost is NP-hard.
When is real-valued and the error metric is the 2-norm, a
2-partition of cost can be learned in time
.
When is -valued and the error metric is Hamming
weight, -partitionability is testable with one-sided error and
non-adaptive queries. We also show that even
two-sided testers require queries when .
This work was motivated by reinforcement learning control tasks in which the
set of control variables can be partitioned. The partitioning reduces the task
into multiple lower-dimensional ones that are relatively easier to learn. Our
second algorithm empirically increases the scores attained over previous
heuristic partitioning methods applied in this context.Comment: Innovations in Theoretical Computer Science (ITCS) 202
Testing and Learning Quantum Juntas Nearly Optimally
We consider the problem of testing and learning quantum -juntas: -qubit
unitary matrices which act non-trivially on just of the qubits and as
the identity on the rest. As our main algorithmic results, we give (a) a
-query quantum algorithm that can distinguish quantum
-juntas from unitary matrices that are "far" from every quantum -junta;
and (b) a -query algorithm to learn quantum -juntas. We complement
our upper bounds for testing quantum -juntas and learning quantum -juntas
with near-matching lower bounds of and
, respectively. Our techniques are Fourier-analytic and
make use of a notion of influence of qubits on unitaries
Distribution-Free Proofs of Proximity
Motivated by the fact that input distributions are often unknown in advance,
distribution-free property testing considers a setting in which the algorithmic
task is to accept functions having a certain property
and reject functions that are -far from , where the
distance is measured according to an arbitrary and unknown input distribution
. As usual in property testing, the tester is required to do so
while making only a sublinear number of input queries, but as the distribution
is unknown, we also allow a sublinear number of samples from the distribution
.
In this work we initiate the study of distribution-free interactive proofs of
proximity (df-IPP) in which the distribution-free testing algorithm is assisted
by an all powerful but untrusted prover. Our main result is a df-IPP for any
problem , with communication, sample, query,
and verification complexities, for any proximity parameter
. For such proximity parameters, this result matches the
parameters of the best-known general purpose IPPs in the standard uniform
setting, and is optimal under reasonable cryptographic assumptions.
For general values of the proximity parameter , our
distribution-free IPP has optimal query complexity but the
communication complexity is , which
is worse than what is known for uniform IPPs when . With
the aim of improving on this gap, we further show that for IPPs over
specialised, but large distribution families, such as sufficiently smooth
distributions and product distributions, the communication complexity can be
reduced to (keeping the query
complexity roughly the same as before) to match the communication complexity of
the uniform case