11 research outputs found
Almost Tight Error Bounds on Differentially Private Continual Counting
The first large-scale deployment of private federated learning uses
differentially private counting in the continual release model as a subroutine
(Google AI blog titled "Federated Learning with Formal Differential Privacy
Guarantees"). In this case, a concrete bound on the error is very relevant to
reduce the privacy parameter. The standard mechanism for continual counting is
the binary mechanism. We present a novel mechanism and show that its mean
squared error is both asymptotically optimal and a factor 10 smaller than the
error of the binary mechanism. We also show that the constants in our analysis
are almost tight by giving non-asymptotic lower and upper bounds that differ
only in the constants of lower-order terms. Our algorithm is a matrix mechanism
for the counting matrix and takes constant time per release. We also use our
explicit factorization of the counting matrix to give an upper bound on the
excess risk of the private learning algorithm of Denisov et al. (NeurIPS 2022).
Our lower bound for any continual counting mechanism is the first tight lower
bound on continual counting under approximate differential privacy. It is
achieved using a new lower bound on a certain factorization norm, denoted by
, in terms of the singular values of the matrix. In
particular, we show that for any complex matrix, , where
denotes the Schatten-1 norm.
We believe this technique will be useful in proving lower bounds for a larger
class of linear queries. To illustrate the power of this technique, we show the
first lower bound on the mean squared error for answering parity queries.Comment: Updated the citations to include two papers we learned about since
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Quantum Information and Variants of Interactive Proof Systems
For nearly three decades, the model of interactive proof systems and its variants have been central to many important and exciting developments in computational complexity theory such as exact characterization of some well known complexity classes, development of probabilistically checkable proof systems and theory of hardness of approximation, and formalization of fundamental cryptographic primitives.
On the other hand, the theory of quantum information, which is primarily concerned with harnessing quantum mechanical features for algorithmic, cryptographic, and information processing tasks has found many applications. In the past three decades, quantum information has been used to develop unconditionally secure quantum cryptography protocols, efficient quantum algorithms for certain problems that are believed to be intractable in classical world, and communication efficient protocols.
In this thesis, we study the impact of quantum information on the models of interactive proof systems and their multi-prover variants. We study various quantum models and explore two questions. The first question we address pertains to the expressive power of such models with or without resource constraints. The second question is related to error reduction technique of such proof systems via parallel repetition.
The question related to the expressive power of models of quantum interactive proof systems and their variants lead us to the following results.
(1) We show that the expressive power of quantum interactive proof systems is exactly PSPACE, the class of problems that can be solved by a polynomial-space deterministic Turing machines and that also admit a classical interactive proof systems. This result shows that in terms of complexity-theoretic characterization, both the models are equivalent. The result is obtained using an algorithmic technique known as the matrix multiplicative weights update method to solve a semidefinite program that characterizes the success probability of the quantum prover.
(2) We show that polynomially many logarithmic-size unentangled quantum proofs are no more powerful than a classical proof if the verifier has the ability to process quantum information. This result follows from an observation that logarithmic-size quantum states can be efficiently represented classically and such classical representation can be used to efficiently generate the quantum state.
(3) We also establish that the model of multi-prover quantum Merlin Arthur proof system, where the verifier is only allowed to apply nonadaptive unentangled measurement on each proof and then a quantum circuit on the classical outcomes, is no more powerful than QMA under the restriction that there are only polynomial number of outcomes per proof. This result follows from showing that such proof systems also admit a QMA verification procedure.
The question related to error reduction via parallel repetition lead us to following results on a class of two-prover one-round games with quantum provers and a class of multi-prover QMA proof systems.
(1) We establish that for a certain class of two-prover one-round games known as XOR games, admit a perfect parallel repetition theorem in the following sense. When the provers play a collection of XOR games, an optimal strategy of the provers is to play each instance of the collection independently and optimally. In particular, the success probability of the quantum provers in the n-fold repetition of an XOR game G with quantum value w(G) is exactly (w(G))^n.
(2) We show a parallel repetition theorem for two-prover one-round unique games. More specifically, we prove that if the quantum value of a unique game is 1-e, then the quantum value of n-fold repetition of the game is at most (1-e^2/49)^n. We also establish that for certain class of unique games, the quantum value of the n-fold repetition of the game is at most (1-e/4)^n. For the special case of XOR games, our proof technique gives an alternate proof of result mentioned above.
3. Our final result on parallel repetition is concerned with SepQMA(m) proof systems, where the verifier receives m unentangled quantum proofs and the measurement operator corresponding to outcome "accept" is a fully separable operator. We give an alternate proof of a result of Harrow and Montanaro [HM10] that states that perfect parallel repetition theorem holds for such proof systems.
The first two results follow from the duality of semidefinite programs and the final result follows from cone programming duality
A Unifying Framework for Differentially Private Sums under Continual Observation
We study the problem of maintaining a differentially private decaying sum
under continual observation. We give a unifying framework and an efficient
algorithm for this problem for \emph{any sufficiently smooth} function. Our
algorithm is the first differentially private algorithm that does not have a
multiplicative error for polynomially-decaying weights. Our algorithm improves
on all prior works on differentially private decaying sums under continual
observation and recovers exactly the additive error for the special case of
continual counting from Henzinger et al. (SODA 2023) as a corollary.
Our algorithm is a variant of the factorization mechanism whose error depends
on the and norm of the underlying matrix. We give a
constructive proof for an almost exact upper bound on the and
norm and an almost tight lower bound on the norm for a
large class of lower-triangular matrices. This is the first non-trivial lower
bound for lower-triangular matrices whose non-zero entries are not all the
same. It includes matrices for all continual decaying sums problems, resulting
in an upper bound on the additive error of any differentially private decaying
sums algorithm under continual observation.
We also explore some implications of our result in discrepancy theory and
operator algebra. Given the importance of the norm in computer
science and the extensive work in mathematics, we believe our result will have
further applications.Comment: 32 page
A generalized framework for quantum state discrimination, hybrid algorithms, and the quantum change point problem
Quantum state discrimination is a central task in many quantum computing
settings where one wishes to identify what quantum state they are holding. We
introduce a framework that generalizes many of its variants and present a
hybrid quantum-classical algorithm based on semidefinite programming to
calculate the maximum reward when the states are pure and have efficient
circuits. To this end, we study the (not necessarily linearly independent) pure
state case and reduce the standard SDP problem size from to where
is the number of qubits, is the number of states, and is the number
of possible guesses (typically ). As an application, we give
now-possible algorithms for the quantum change point identification problem
which asks, given a sequence of quantum states, determine the time steps when
the quantum states changed. With our reductions, we are able to solve SDPs for
problem sizes of up to qubits in about hours and we also give
heuristics which speed up the computations.Comment: 31 pages, 13 figures. Comments welcom
QIP = PSPACE
We prove that the complexity class QIP, which consists of all problems having
quantum interactive proof systems, is contained in PSPACE. This containment is
proved by applying a parallelized form of the matrix multiplicative weights
update method to a class of semidefinite programs that captures the
computational power of quantum interactive proofs. As the containment of PSPACE
in QIP follows immediately from the well-known equality IP = PSPACE, the
equality QIP = PSPACE follows.Comment: 21 pages; v2 includes corrections and minor revision
qip = pspace
ACM Special Interest Group on Algorithms and Computation Theory; SIGACTWe prove that the complexity class QIP, which consists of all problems having quantum interactive proof systems, is contained in PSPACE. This containment is proved by applying a parallelized form of the matrix multiplicative weights update method to a class of semidefinite programs that captures the computational power of quantum interactive proofs. As the containment of PSPACE in QIP follows immediately from the well-known equality IP = PSPACE, the equality QIP = PSPACE follows. © 2010 ACM