47 research outputs found
Strong direct product conjecture holds for all relations in public coin randomized one-way communication complexity
Let f subset of X x Y x Z be a relation. Let the public coin one-way
communication complexity of f, with worst case error 1/3, be denoted
R^{1,pub}_{1/3}(f). We show that if for computing f^k (k independent copies of
f), o(k R^{1,pub}_{1/3}(f)) communication is provided, then the success is
exponentially small in k. This settles the strong direct product conjecture for
all relations in public coin one-way communication complexity.
We show a new tight characterization of public coin one-way communication
complexity which strengthens on the tight characterization shown in [J.,
Klauck, Nayak 08]. We use the new characterization to show our direct product
result and this may also be of independent interest.Comment: ver 2. 11 pages, proofs simplifie
Quantum Advantage on Information Leakage for Equality
We prove a lower bound on the information leakage of any classical protocol
computing the equality function in the simultaneous message passing (SMP)
model. Our bound is valid in the finite length regime and is strong enough to
demonstrate a quantum advantage in terms of information leakage for practical
quantum protocols. We prove our bound by obtaining an improved finite size
version of the communication bound due to Babai and Kimmel, relating randomized
communication to deterministic communication in the SMP model. We then relate
information leakage to randomized communication through a series of reductions.
We first provide alternative characterizations for information leakage,
allowing us to link it to average length communication while allowing for
shared randomness (pairwise, with the referee). A Markov inequality links this
with bounded length communication, and a Newman type argument allows us to go
from shared to private randomness. The only reduction in which we incur more
than a logarithmic additive factor is in the Markov inequality; in particular,
our compression method is essentially tight for the SMP model with average
length communication.Comment: 23 pages, 2 figure
One-Shot Federated Learning: Theoretical Limits and Algorithms to Achieve Them
We consider distributed statistical optimization in one-shot setting, where
there are machines each observing i.i.d. samples. Based on its observed
samples, each machine sends a -bit-long message to a server. The server then
collects messages from all machines, and estimates a parameter that minimizes
an expected convex loss function. We investigate the impact of communication
constraint, , on the expected error and derive a tight lower bound on the
error achievable by any algorithm. We then propose an estimator, which we call
Multi-Resolution Estimator (MRE), whose expected error (when )
meets the aforementioned lower bound up to poly-logarithmic factors, and is
thereby order optimal. We also address the problem of learning under tiny
communication budget, and present lower and upper error bounds when is a
constant. The expected error of MRE, unlike existing algorithms, tends to zero
as the number of machines () goes to infinity, even when the number of
samples per machine () remains upper bounded by a constant. This property of
the MRE algorithm makes it applicable in new machine learning paradigms where
is much larger than
New Results in the Simultaneous Message Passing Model
Consider the following Simultaneous Message Passing (SMP) model for computing
a relation f subset of X x Y x Z. In this model Alice, on input x in X and Bob,
on input y in Y, send one message each to a third party Referee who then
outputs a z in Z such that (x,y,z) in f. We first show optimal 'Direct sum'
results for all relations f in this model, both in the quantum and classical
settings, in the situation where we allow shared resources (shared entanglement
in quantum protocols and public coins in classical protocols) between Alice and
Referee and Bob and Referee and no shared resource between Alice and Bob. This
implies that, in this model, the communication required to compute k
simultaneous instances of f, with constant success overall, is at least k-times
the communication required to compute one instance with constant success.
This in particular implies an earlier Direct sum result, shown by
Chakrabarti, Shi, Wirth and Yao, 2001, for the Equality function (and a class
of other so-called robust functions), in the classical smp model with no shared
resources between any parties.
Furthermore we investigate the gap between the smp model and the one-way
model in communication complexity and exhibit a partial function that is
exponentially more expensive in the former if quantum communication with
entanglement is allowed, compared to the latter even in the deterministic case.Comment: 16 pages, version
On Communication Cost of Distributed Statistical Estimation and Dimensionality
We explore the connection between dimensionality and communication cost in
distributed learning problems. Specifically we study the problem of estimating
the mean of an unknown dimensional gaussian distribution in
the distributed setting. In this problem, the samples from the unknown
distribution are distributed among different machines. The goal is to
estimate the mean at the optimal minimax rate while
communicating as few bits as possible. We show that in this setting, the
communication cost scales linearly in the number of dimensions i.e. one needs
to deal with different dimensions individually. Applying this result to
previous lower bounds for one dimension in the interactive setting
\cite{ZDJW13} and to our improved bounds for the simultaneous setting, we prove
new lower bounds of and for the bits of
communication needed to achieve the minimax squared loss, in the interactive
and simultaneous settings respectively. To complement, we also demonstrate an
interactive protocol achieving the minimax squared loss with bits of
communication, which improves upon the simple simultaneous protocol by a
logarithmic factor. Given the strong lower bounds in the general setting, we
initiate the study of the distributed parameter estimation problems with
structured parameters. Specifically, when the parameter is promised to be
-sparse, we show a simple thresholding based protocol that achieves the same
squared loss while saving a factor of communication. We conjecture that
the tradeoff between communication and squared loss demonstrated by this
protocol is essentially optimal up to logarithmic factor.Comment: to appear at NIPS'14 with oral presentatio
A lower bound for bounded round quantum communication complexity of set disjointness
We consider the class of functions whose value depends only on the
intersection of the input X_1,X_2, ..., X_t; that is, for each F in this class
there is an f_F: 2^{[n]} \to {0,1}, such that F(X_1,X_2, ..., X_t) = f_F(X_1
\cap X_2 \cap ... \cap X_t). We show that the t-party k-round communication
complexity of F is Omega(s_m(f_F)/(k^2)), where s_m(f_F) stands for the
`monotone sensitivity of f_F' and is defined by s_m(f_F) \defeq max_{S\subseteq
[n]} |{i: f_F(S \cup {i}) \neq f_F(S)|. For two-party quantum communication
protocols for the set disjointness problem, this implies that the two parties
must exchange Omega(n/k^2) qubits. For k=1, our lower bound matches the
Omega(n) lower bound observed by Buhrman and de Wolf (based on a result of
Nayak, and for 2 <= k <= n^{1/4}, improves the lower bound of Omega(sqrt{n})
shown by Razborov. (For protocols with no restrictions on the number of rounds,
we can conclude that the two parties must exchange Omega(n^{1/3}) qubits. This,
however, falls short of the optimal Omega(sqrt{n}) lower bound shown by
Razborov.)Comment: 15 pages, content added and modified, references adde
R\'enyi Information Complexity and an Information Theoretic Characterization of the Partition Bound
We introduce a new information-theoretic complexity measure for
2-party functions which is a lower-bound on communication complexity, and has
the two leading lower-bounds on communication complexity as its natural
relaxations: (external) information complexity () and logarithm of
partition complexity (), which have so far appeared conceptually
quite different from each other. is an external information
complexity measure based on R\'enyi mutual information of order infinity. In
the definition of , relaxing the order of R\'enyi mutual information
from infinity to 1 yields , while is obtained by
replacing protocol transcripts with what we term "pseudotranscripts," which
omits the interactive nature of a protocol, but only requires that the
probability of any transcript given the inputs and to the two parties,
factorizes into two terms which depend on and separately. Further
understanding might have consequences for important direct-sum
problems in communication complexity, as it lies between communication
complexity and information complexity.
We also show that applying both the above relaxations simultaneously to
gives a complexity measure that is lower-bounded by the (log of)
relaxed partition complexity, a complexity measure introduced by Kerenidis et
al. (FOCS 2012). We obtain a sharper connection between (external) information
complexity and relaxed partition complexity than Kerenidis et al., using an
arguably more direct proof.Comment: Full version of paper appearing at ICALP 201
Direct Sum Theorem for Bounded Round Quantum Communication Complexity
We prove a direct sum theorem for bounded round entanglement-assisted quantum
communication complexity. To do so, we use the fully quantum definition for
information cost and complexity that we recently introduced, and use both the
fact that information is a lower bound on the communication, and the fact that
a direct sum property holds for quantum information complexity. We then give a
protocol for compressing a single copy of a protocol down to its quantum
information cost, up to terms depending on the number of rounds and the allowed
increase in error. Two important tools to derive this protocol are a smooth
conditional min-entropy bound for a one-shot quantum state redistribution
protocol, and the quantum substate theorem of Jain, Radhakrishnan and Sen
(FOCS'02) to transform this bound into a von Neumann conditional entropy bound.
This result further establishes the newly introduced notions of quantum
information cost and complexity as the correct quantum generalisations of the
classical ones in the standard communication complexity setting. Finding such a
quantum generalization of information complexity was one of the open problem
recently raised by Braverman (STOC'12).Comment: 25 pages, no figure, part of prelims taken from arXiv:1404.373
The information cost of quantum memoryless protocols
We consider memoryless quantum communication protocols, where the two parties
do not possess any memory besides their classical input and they take turns
performing unitary operations on a pure quantum state that they exchange
between them. Most known quantum protocols are of this type and recently a deep
connection between memoryless protocols and Bell inequality violations has been
explored recently by Buhrman et al. We study the information cost of memoryless
quantum protocols by looking at a canonical problem: bounded-round quantum
communication protocols for the one-bit AND function. We prove directly a tight
lower bound of for the information cost of AND for
-round memoryless quantum protocols and for the input distribution needed
for the Disjointness function. It is not clear if memoryless protocols allow
for a reduction between AND and Disjointness, due to the absence of private
workspaces. We enhance the model by allowing the players to keep in their
private classical workspace apart from their input also some classical private
coins. Surprisingly, we show that every quantum protocol can be transformed
into an equivalent quantum protocol with private coins that is perfectly
private, i.e. the players only learn the value of the function and nothing
more. Last, we consider the model where the players are allowed to use one-shot
coins, i.e. private coins that can be used only once during the protocol. While
in the classical case, private coins and one-shot coins are equivalent, in the
quantum case, we prove that they are not. More precisely, we show that every
quantum memoryless protocol with one-bit inputs that uses one-shot coins can be
transformed into a memoryless quantum protocol without private coins and
without increasing too much its information cost. Hence, while private coins
always allow for private quantum protocols, one-shot coins do not.Comment: Corrected typos in the abstrac
A strong direct product theorem in terms of the smooth rectangle bound
A strong direct product theorem states that, in order to solve k instances of
a problem, if we provide less than k times the resource required to compute one
instance, then the probability of overall success is exponentially small in k.
In this paper, we consider the model of two-way public-coin communication
complexity and show a strong direct product theorem for all relations in terms
of the smooth rectangle bound, introduced by Jain and Klauck as a generic lower
bound method in this model. Our result therefore uniformly implies a strong
direct product theorem for all relations for which an (asymptotically) optimal
lower bound can be provided using the smooth rectangle bound, for example Inner
Product, Greater-Than, Set-Disjointness, Gap-Hamming Distance etc. Our result
also implies near optimal direct product results for several important
functions and relations used to show exponential separations between classical
and quantum communication complexity, for which near optimal lower bounds are
provided using the rectangle bound, for example by Raz [1999], Gavinsky [2008]
and Klartag and Regev [2011]. In fact we are not aware of any relation for
which it is known that the smooth rectangle bound does not provide an optimal
lower bound. This lower bound subsumes many of the other lower bound methods,
for example the rectangle bound (a.k.a the corruption bound), the smooth
discrepancy bound (a.k.a the \gamma_2 bound) which in turn subsumes the
discrepancy bound, the subdistribution bound and the conditional min-entropy
bound.
We show our result using information theoretic arguments. A key tool we use
is a sampling protocol due to Braverman [2012], in fact a modification of it
used by Kerenidis, Laplante, Lerays, Roland and Xiao [2012].Comment: 12 pages, version 3, improved parameters in the main resul