56 research outputs found
Signaling in Quasipolynomial time
Strategic interactions often take place in an environment rife with
uncertainty. As a result, the equilibrium of a game is intimately related to
the information available to its players. The \emph{signaling problem}
abstracts the task faced by an informed "market maker", who must choose how to
reveal information in order to effect a desirable equilibrium.
In this paper, we consider two fundamental signaling problems: one for
abstract normal form games, and the other for single item auctions. For the
former, we consider an abstract class of objective functions which includes the
social welfare and weighted combinations of players' utilities, and for the
latter we restrict our attention to the social welfare objective and to
signaling schemes which are constrained in the number of signals used. For both
problems, we design approximation algorithms for the signaling problem which
run in quasi-polynomial time under various conditions, extending and
complementing the results of various recent works on the topic.
Underlying each of our results is a "meshing scheme" which effectively
overcomes the "curse of dimensionality" and discretizes the space of
"essentially different" posterior beliefs -- in the sense of inducing
"essentially different" equilibria. This is combined with an algorithm for
optimally assembling a signaling scheme as a convex combination of such
beliefs. For the normal form game setting, the meshing scheme leads to a convex
partition of the space of posterior beliefs and this assembly procedure is
reduced to a linear program, and in the auction setting the assembly procedure
is reduced to submodular function maximization
Quantum de Finetti Theorems under Local Measurements with Applications
Quantum de Finetti theorems are a useful tool in the study of correlations in
quantum multipartite states. In this paper we prove two new quantum de Finetti
theorems, both showing that under tests formed by local measurements one can
get a much improved error dependence on the dimension of the subsystems. We
also obtain similar results for non-signaling probability distributions. We
give the following applications of the results:
We prove the optimality of the Chen-Drucker protocol for 3-SAT, under the
exponential time hypothesis.
We show that the maximum winning probability of free games can be estimated
in polynomial time by linear programming. We also show that 3-SAT with m
variables can be reduced to obtaining a constant error approximation of the
maximum winning probability under entangled strategies of O(m^{1/2})-player
one-round non-local games, in which the players communicate O(m^{1/2}) bits all
together.
We show that the optimization of certain polynomials over the hypersphere can
be performed in quasipolynomial time in the number of variables n by
considering O(log(n)) rounds of the Sum-of-Squares (Parrilo/Lasserre) hierarchy
of semidefinite programs. As an application to entanglement theory, we find a
quasipolynomial-time algorithm for deciding multipartite separability.
We consider a result due to Aaronson -- showing that given an unknown n qubit
state one can perform tomography that works well for most observables by
measuring only O(n) independent and identically distributed (i.i.d.) copies of
the state -- and relax the assumption of having i.i.d copies of the state to
merely the ability to select subsystems at random from a quantum multipartite
state.
The proofs of the new quantum de Finetti theorems are based on information
theory, in particular on the chain rule of mutual information.Comment: 39 pages, no figure. v2: changes to references and other minor
improvements. v3: added some explanations, mostly about Theorem 1 and
Conjecture 5. STOC version. v4, v5. small improvements and fixe
Mixture Selection, Mechanism Design, and Signaling
We pose and study a fundamental algorithmic problem which we term mixture
selection, arising as a building block in a number of game-theoretic
applications: Given a function from the -dimensional hypercube to the
bounded interval , and an matrix with bounded entries,
maximize over in the -dimensional simplex. This problem arises
naturally when one seeks to design a lottery over items for sale in an auction,
or craft the posterior beliefs for agents in a Bayesian game through the
provision of information (a.k.a. signaling).
We present an approximation algorithm for this problem when
simultaneously satisfies two smoothness properties: Lipschitz continuity with
respect to the norm, and noise stability. The latter notion, which
we define and cater to our setting, controls the degree to which
low-probability errors in the inputs of can impact its output. When is
both -Lipschitz continuous and -stable, we obtain an (additive)
PTAS for mixture selection. We also show that neither assumption suffices by
itself for an additive PTAS, and both assumptions together do not suffice for
an additive FPTAS.
We apply our algorithm to different game-theoretic applications from
mechanism design and optimal signaling. We make progress on a number of open
problems suggested in prior work by easily reducing them to mixture selection:
we resolve an important special case of the small-menu lottery design problem
posed by Dughmi, Han, and Nisan; we resolve the problem of revenue-maximizing
signaling in Bayesian second-price auctions posed by Emek et al. and Miltersen
and Sheffet; we design a quasipolynomial-time approximation scheme for the
optimal signaling problem in normal form games suggested by Dughmi; and we
design an approximation algorithm for the optimal signaling problem in the
voting model of Alonso and C\^{a}mara
On the Hardness of Signaling
There has been a recent surge of interest in the role of information in
strategic interactions. Much of this work seeks to understand how the realized
equilibrium of a game is influenced by uncertainty in the environment and the
information available to players in the game. Lurking beneath this literature
is a fundamental, yet largely unexplored, algorithmic question: how should a
"market maker" who is privy to additional information, and equipped with a
specified objective, inform the players in the game? This is an informational
analogue of the mechanism design question, and views the information structure
of a game as a mathematical object to be designed, rather than an exogenous
variable.
We initiate a complexity-theoretic examination of the design of optimal
information structures in general Bayesian games, a task often referred to as
signaling. We focus on one of the simplest instantiations of the signaling
question: Bayesian zero-sum games, and a principal who must choose an
information structure maximizing the equilibrium payoff of one of the players.
In this setting, we show that optimal signaling is computationally intractable,
and in some cases hard to approximate, assuming that it is hard to recover a
planted clique from an Erdos-Renyi random graph. This is despite the fact that
equilibria in these games are computable in polynomial time, and therefore
suggests that the hardness of optimal signaling is a distinct phenomenon from
the hardness of equilibrium computation. Necessitated by the non-local nature
of information structures, en-route to our results we prove an "amplification
lemma" for the planted clique problem which may be of independent interest
AM with Multiple Merlins
We introduce and study a new model of interactive proofs: AM(k), or
Arthur-Merlin with k non-communicating Merlins. Unlike with the better-known
MIP, here the assumption is that each Merlin receives an independent random
challenge from Arthur. One motivation for this model (which we explore in
detail) comes from the close analogies between it and the quantum complexity
class QMA(k), but the AM(k) model is also natural in its own right.
We illustrate the power of multiple Merlins by giving an AM(2) protocol for
3SAT, in which the Merlins' challenges and responses consist of only
n^{1/2+o(1)} bits each. Our protocol has the consequence that, assuming the
Exponential Time Hypothesis (ETH), any algorithm for approximating a dense CSP
with a polynomial-size alphabet must take n^{(log n)^{1-o(1)}} time. Algorithms
nearly matching this lower bound are known, but their running times had never
been previously explained. Brandao and Harrow have also recently used our 3SAT
protocol to show quasipolynomial hardness for approximating the values of
certain entangled games.
In the other direction, we give a simple quasipolynomial-time approximation
algorithm for free games, and use it to prove that, assuming the ETH, our 3SAT
protocol is essentially optimal. More generally, we show that multiple Merlins
never provide more than a polynomial advantage over one: that is, AM(k)=AM for
all k=poly(n). The key to this result is a subsampling theorem for free games,
which follows from powerful results by Alon et al. and Barak et al. on
subsampling dense CSPs, and which says that the value of any free game can be
closely approximated by the value of a logarithmic-sized random subgame.Comment: 48 page
On palimpsests in neural memory: an information theory viewpoint
The finite capacity of neural memory and the
reconsolidation phenomenon suggest it is important to be able
to update stored information as in a palimpsest, where new
information overwrites old information. Moreover, changing
information in memory is metabolically costly. In this paper, we
suggest that information-theoretic approaches may inform the
fundamental limits in constructing such a memory system. In
particular, we define malleable coding, that considers not only
representation length but also ease of representation update,
thereby encouraging some form of recycling to convert an old
codeword into a new one. Malleability cost is the difficulty of
synchronizing compressed versions, and malleable codes are of
particular interest when representing information and modifying
the representation are both expensive. We examine the tradeoff
between compression efficiency and malleability cost, under a
malleability metric defined with respect to a string edit distance.
This introduces a metric topology to the compressed domain. We
characterize the exact set of achievable rates and malleability as
the solution of a subgraph isomorphism problem. This is all done
within the optimization approach to biology framework.Accepted manuscrip
Distributed Signaling Games
A recurring theme in recent computer science literature is that proper design
of signaling schemes is a crucial aspect of effective mechanisms aiming to
optimize social welfare or revenue. One of the research endeavors of this line
of work is understanding the algorithmic and computational complexity of
designing efficient signaling schemes. In reality, however, information is
typically not held by a central authority, but is distributed among multiple
sources (third-party "mediators"), a fact that dramatically changes the
strategic and combinatorial nature of the signaling problem, making it a game
between information providers, as opposed to a traditional mechanism design
problem.
In this paper we introduce {\em distributed signaling games}, while using
display advertising as a canonical example for introducing this foundational
framework. A distributed signaling game may be a pure coordination game (i.e.,
a distributed optimization task), or a non-cooperative game. In the context of
pure coordination games, we show a wide gap between the computational
complexity of the centralized and distributed signaling problems. On the other
hand, we show that if the information structure of each mediator is assumed to
be "local", then there is an efficient algorithm that finds a near-optimal
(-approximation) distributed signaling scheme.
In the context of non-cooperative games, the outcome generated by the
mediators' signals may have different value to each (due to the auctioneer's
desire to align the incentives of the mediators with his own by relative
compensations). We design a mechanism for this problem via a novel application
of Shapley's value, and show that it possesses some interesting properties, in
particular, it always admits a pure Nash equilibrium, and it never decreases
the revenue of the auctioneer
Quantum de Finetti Theorems Under Local Measurements with Applications
Quantum de Finetti theorems are a useful tool in the study of correlations in quantum multipartite states. In this paper we prove two new quantum de Finetti theorems, both showing that under tests formed by local measurements in each of the subsystems one can get an exponential improvement in the error dependence on the dimension of the subsystems. We also obtain similar results for non-signaling probability distributions. We give several applications of the results to quantum complexity theory, polynomial optimization, and quantum information theory. The proofs of the new quantum de Finetti theorems are based on information theory, in particular on the chain rule of mutual information. The results constitute improvements and generalizations of a recent de Finetti theorem due to Brandão, Christandl and Yard
A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs
A -birthday repetition of a
two-prover game is a game in which the two provers are sent
random sets of questions from of sizes and respectively.
These two sets are sampled independently uniformly among all sets of questions
of those particular sizes. We prove the following birthday repetition theorem:
when satisfies some mild conditions, decreases exponentially in where is the total number of
questions. Our result positively resolves an open question posted by Aaronson,
Impagliazzo and Moshkovitz (CCC 2014).
As an application of our birthday repetition theorem, we obtain new
fine-grained hardness of approximation results for dense CSPs. Specifically, we
establish a tight trade-off between running time and approximation ratio for
dense CSPs by showing conditional lower bounds, integrality gaps and
approximation algorithms. In particular, for any sufficiently large and for
every , we show the following results:
- We exhibit an -approximation algorithm for dense Max -CSPs
with alphabet size via -level of Sherali-Adams relaxation.
- Through our birthday repetition theorem, we obtain an integrality gap of
for -level Lasserre relaxation for fully-dense Max
-CSP.
- Assuming that there is a constant such that Max 3SAT cannot
be approximated to within of the optimal in sub-exponential
time, our birthday repetition theorem implies that any algorithm that
approximates fully-dense Max -CSP to within a factor takes
time, almost tightly matching the algorithmic
result based on Sherali-Adams relaxation.Comment: 45 page
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