459 research outputs found
On Algorithmic Statistics for space-bounded algorithms
Algorithmic statistics studies explanations of observed data that are good in
the algorithmic sense: an explanation should be simple i.e. should have small
Kolmogorov complexity and capture all the algorithmically discoverable
regularities in the data. However this idea can not be used in practice because
Kolmogorov complexity is not computable.
In this paper we develop algorithmic statistics using space-bounded
Kolmogorov complexity. We prove an analogue of one of the main result of
`classic' algorithmic statistics (about the connection between optimality and
randomness deficiences). The main tool of our proof is the Nisan-Wigderson
generator.Comment: accepted to CSR 2017 conferenc
Weak Parity
We study the query complexity of Weak Parity: the problem of computing the
parity of an n-bit input string, where one only has to succeed on a 1/2+eps
fraction of input strings, but must do so with high probability on those inputs
where one does succeed. It is well-known that n randomized queries and n/2
quantum queries are needed to compute parity on all inputs. But surprisingly,
we give a randomized algorithm for Weak Parity that makes only
O(n/log^0.246(1/eps)) queries, as well as a quantum algorithm that makes only
O(n/sqrt(log(1/eps))) queries. We also prove a lower bound of
Omega(n/log(1/eps)) in both cases; and using extremal combinatorics, prove
lower bounds of Omega(log n) in the randomized case and Omega(sqrt(log n)) in
the quantum case for any eps>0. We show that improving our lower bounds is
intimately related to two longstanding open problems about Boolean functions:
the Sensitivity Conjecture, and the relationships between query complexity and
polynomial degree.Comment: 18 page
Auctions with Severely Bounded Communication
We study auctions with severe bounds on the communication allowed: each
bidder may only transmit t bits of information to the auctioneer. We consider
both welfare- and profit-maximizing auctions under this communication
restriction. For both measures, we determine the optimal auction and show that
the loss incurred relative to unconstrained auctions is mild. We prove
non-surprising properties of these kinds of auctions, e.g., that in optimal
mechanisms bidders simply report the interval in which their valuation lies in,
as well as some surprising properties, e.g., that asymmetric auctions are
better than symmetric ones and that multi-round auctions reduce the
communication complexity only by a linear factor
Single Parameter Combinatorial Auctions with Partially Public Valuations
We consider the problem of designing truthful auctions, when the bidders'
valuations have a public and a private component. In particular, we consider
combinatorial auctions where the valuation of an agent for a set of
items can be expressed as , where is a private single parameter
of the agent, and the function is publicly known. Our motivation behind
studying this problem is two-fold: (a) Such valuation functions arise naturally
in the case of ad-slots in broadcast media such as Television and Radio. For an
ad shown in a set of ad-slots, is, say, the number of {\em unique}
viewers reached by the ad, and is the valuation per-unique-viewer. (b)
From a theoretical point of view, this factorization of the valuation function
simplifies the bidding language, and renders the combinatorial auction more
amenable to better approximation factors. We present a general technique, based
on maximal-in-range mechanisms, that converts any -approximation
non-truthful algorithm () for this problem into
and -approximate truthful
mechanisms which run in polynomial time and quasi-polynomial time,
respectively
When Analysis Fails: Heuristic Mechanism Design via Self-Correcting Procedures
Computational mechanism design (CMD) seeks to understand how to design game forms that induce desirable outcomes in multi-agent systems despite private information, self-interest and limited computational resources. CMD finds application in many settings, in the public sector for wireless spectrum and airport landing rights, to Internet advertising, to expressive sourcing in the supply chain, to allocating computational resources. In meeting the demands for CMD in these rich domains, we often need to bridge from the theory of economic mechanism design to the practice of deployable, computational mechanisms. A compelling example of this need arises in dynamic combinatorial environments, where classic analytic approaches fail and heuristic, computational approaches are required. In this talk I outline the direction of self-correcting mechanisms, which dynamically modify decisions via “output ironing" to ensure truthfulness and provide a fully computational approach to mechanism design. For an application, I suggest heuristic mechanisms for dynamic auctions in which bids arrive over time and supply may also be uncertain.Engineering and Applied Science
Truthful Multi-unit Procurements with Budgets
We study procurement games where each seller supplies multiple units of his
item, with a cost per unit known only to him. The buyer can purchase any number
of units from each seller, values different combinations of the items
differently, and has a budget for his total payment.
For a special class of procurement games, the {\em bounded knapsack} problem,
we show that no universally truthful budget-feasible mechanism can approximate
the optimal value of the buyer within , where is the total number of
units of all items available. We then construct a polynomial-time mechanism
that gives a -approximation for procurement games with {\em concave
additive valuations}, which include bounded knapsack as a special case. Our
mechanism is thus optimal up to a constant factor. Moreover, for the bounded
knapsack problem, given the well-known FPTAS, our results imply there is a
provable gap between the optimization domain and the mechanism design domain.
Finally, for procurement games with {\em sub-additive valuations}, we
construct a universally truthful budget-feasible mechanism that gives an
-approximation in polynomial time with a
demand oracle.Comment: To appear at WINE 201
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