2,999 research outputs found
Efficient Algorithms for the Data Exchange Problem
In this paper we study the data exchange problem where a set of users is
interested in gaining access to a common file, but where each has only partial
knowledge about it as side-information. Assuming that the file is broken into
packets, the side-information considered is in the form of linear combinations
of the file packets. Given that the collective information of all the users is
sufficient to allow recovery of the entire file, the goal is for each user to
gain access to the file while minimizing some communication cost. We assume
that users can communicate over a noiseless broadcast channel, and that the
communication cost is a sum of each user's cost function over the number of
bits it transmits. For instance, the communication cost could simply be the
total number of bits that needs to be transmitted. In the most general case
studied in this paper, each user can have any arbitrary convex cost function.
We provide deterministic, polynomial-time algorithms (in the number of users
and packets) which find an optimal communication scheme that minimizes the
communication cost. To further lower the complexity, we also propose a simple
randomized algorithm inspired by our deterministic algorithm which is based on
a random linear network coding scheme.Comment: submitted to Transactions on Information Theor
Minimum Cost Multicast with Decentralized Sources
In this paper we study the multisource multicast problem where every sink in
a given directed acyclic graph is a client and is interested in a common file.
We consider the case where each node can have partial knowledge about the file
as a side information. Assuming that nodes can communicate over the capacity
constrained links of the graph, the goal is for each client to gain access to
the file, while minimizing some linear cost function of number of bits
transmitted in the network. We consider three types of side-information
settings:(ii) side information in the form of linearly correlated packets; and
(iii) the general setting where the side information at the nodes have an
arbitrary (i.i.d.) correlation structure. In this work we 1) provide a
polynomial time feasibility test, i.e., whether or not all the clients can
recover the file, and 2) we provide a polynomial-time algorithm that finds the
optimal rate allocation among the links of the graph, and then determines an
explicit transmission scheme for cases (i) and (ii)
The Lov\'asz Hinge: A Novel Convex Surrogate for Submodular Losses
Learning with non-modular losses is an important problem when sets of
predictions are made simultaneously. The main tools for constructing convex
surrogate loss functions for set prediction are margin rescaling and slack
rescaling. In this work, we show that these strategies lead to tight convex
surrogates iff the underlying loss function is increasing in the number of
incorrect predictions. However, gradient or cutting-plane computation for these
functions is NP-hard for non-supermodular loss functions. We propose instead a
novel surrogate loss function for submodular losses, the Lov\'asz hinge, which
leads to O(p log p) complexity with O(p) oracle accesses to the loss function
to compute a gradient or cutting-plane. We prove that the Lov\'asz hinge is
convex and yields an extension. As a result, we have developed the first
tractable convex surrogates in the literature for submodular losses. We
demonstrate the utility of this novel convex surrogate through several set
prediction tasks, including on the PASCAL VOC and Microsoft COCO datasets
- …