7 research outputs found
Budget Feasible Mechanisms for Experimental Design
In the classical experimental design setting, an experimenter E has access to
a population of potential experiment subjects , each
associated with a vector of features . Conducting an experiment
with subject reveals an unknown value to E. E typically assumes
some hypothetical relationship between 's and 's, e.g., , and estimates from experiments, e.g., through linear
regression. As a proxy for various practical constraints, E may select only a
subset of subjects on which to conduct the experiment.
We initiate the study of budgeted mechanisms for experimental design. In this
setting, E has a budget . Each subject declares an associated cost to be part of the experiment, and must be paid at least her cost. In
particular, the Experimental Design Problem (EDP) is to find a set of
subjects for the experiment that maximizes V(S) = \log\det(I_d+\sum_{i\in
S}x_i\T{x_i}) under the constraint ; our objective
function corresponds to the information gain in parameter that is
learned through linear regression methods, and is related to the so-called
-optimality criterion. Further, the subjects are strategic and may lie about
their costs.
We present a deterministic, polynomial time, budget feasible mechanism
scheme, that is approximately truthful and yields a constant factor
approximation to EDP. In particular, for any small and , we can construct a (12.98, )-approximate mechanism that is
-truthful and runs in polynomial time in both and
. We also establish that no truthful,
budget-feasible algorithms is possible within a factor 2 approximation, and
show how to generalize our approach to a wide class of learning problems,
beyond linear regression
The Limitations of Optimization from Samples
In this paper we consider the following question: can we optimize objective
functions from the training data we use to learn them? We formalize this
question through a novel framework we call optimization from samples (OPS). In
OPS, we are given sampled values of a function drawn from some distribution and
the objective is to optimize the function under some constraint.
While there are interesting classes of functions that can be optimized from
samples, our main result is an impossibility. We show that there are classes of
functions which are statistically learnable and optimizable, but for which no
reasonable approximation for optimization from samples is achievable. In
particular, our main result shows that there is no constant factor
approximation for maximizing coverage functions under a cardinality constraint
using polynomially-many samples drawn from any distribution.
We also show tight approximation guarantees for maximization under a
cardinality constraint of several interesting classes of functions including
unit-demand, additive, and general monotone submodular functions, as well as a
constant factor approximation for monotone submodular functions with bounded
curvature
FPT-Algorithms for the l-Matchoid Problem with Linear and Submodular Objectives
We design a fixed-parameter deterministic algorithm for computing a maximum
weight feasible set under a -matchoid of rank , parameterized by
and . Unlike previous work that presumes linear representativity of
matroids, we consider the general oracle model. Our result, combined with the
lower bounds of Lovasz, and Jensen and Korte, demonstrates a separation between
the -matchoid and the matroid -parity problems in the setting of
fixed-parameter tractability.
Our algorithms are obtained by means of kernelization: we construct a small
representative set which contains an optimal solution. Such a set gives us much
flexibility in adapting to other settings, allowing us to optimize not only a
linear function, but also several important submodular functions. It also helps
to transform our algorithms into streaming algorithms.
In the streaming setting, we show that we can find a feasible solution of
value and the number of elements to be stored in memory depends only on
and but totally independent of . This shows that it is possible to
circumvent the recent space lower bound of Feldman et al., by parameterizing
the solution value. This result, combined with existing lower bounds, also
provides a new separation between the space and time complexity of maximizing
an arbitrary submodular function and a coverage function in the value oracle
model