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Efficient Algorithms for k-Regret Minimizing Sets
A regret minimizing set Q is a small size representation of a much larger database P so that user queries executed on Q return answers whose scores are not much worse than those on the full dataset. In particular, a k-regret minimizing set has the property that the regret ratio between the score of the top-1 item in Q and the score of the top-k item in P is minimized, where the score of an item is the inner product of the item\u27s attributes with a user\u27s weight (preference) vector. The problem is challenging because we want to find a single representative set Q whose regret ratio is small with respect to all possible user weight vectors.
We show that k-regret minimization is NP-Complete for all dimensions d>=3, settling an open problem from Chester et al. [VLDB 2014]. Our main algorithmic contributions are two approximation algorithms, both with provable guarantees, one based on coresets and another based on hitting sets. We perform extensive experimental evaluation of our algorithms, using both real-world and synthetic data, and compare their performance against the solution proposed in [VLDB 14]. The results show that our algorithms are significantly faster and scalable to much larger sets than the greedy algorithm of Chester et al. for comparable quality answers
RRR: Rank-Regret Representative
Selecting the best items in a dataset is a common task in data exploration.
However, the concept of "best" lies in the eyes of the beholder: different
users may consider different attributes more important, and hence arrive at
different rankings. Nevertheless, one can remove "dominated" items and create a
"representative" subset of the data set, comprising the "best items" in it. A
Pareto-optimal representative is guaranteed to contain the best item of each
possible ranking, but it can be almost as big as the full data. Representative
can be found if we relax the requirement to include the best item for every
possible user, and instead just limit the users' "regret". Existing work
defines regret as the loss in score by limiting consideration to the
representative instead of the full data set, for any chosen ranking function.
However, the score is often not a meaningful number and users may not
understand its absolute value. Sometimes small ranges in score can include
large fractions of the data set. In contrast, users do understand the notion of
rank ordering. Therefore, alternatively, we consider the position of the items
in the ranked list for defining the regret and propose the {\em rank-regret
representative} as the minimal subset of the data containing at least one of
the top- of any possible ranking function. This problem is NP-complete. We
use the geometric interpretation of items to bound their ranks on ranges of
functions and to utilize combinatorial geometry notions for developing
effective and efficient approximation algorithms for the problem. Experiments
on real datasets demonstrate that we can efficiently find small subsets with
small rank-regrets
Online Isotonic Regression
We consider the online version of the isotonic regression problem. Given a
set of linearly ordered points (e.g., on the real line), the learner must
predict labels sequentially at adversarially chosen positions and is evaluated
by her total squared loss compared against the best isotonic (non-decreasing)
function in hindsight. We survey several standard online learning algorithms
and show that none of them achieve the optimal regret exponent; in fact, most
of them (including Online Gradient Descent, Follow the Leader and Exponential
Weights) incur linear regret. We then prove that the Exponential Weights
algorithm played over a covering net of isotonic functions has a regret bounded
by and present a matching
lower bound on regret. We provide a computationally efficient version of this
algorithm. We also analyze the noise-free case, in which the revealed labels
are isotonic, and show that the bound can be improved to or even to
(when the labels are revealed in isotonic order). Finally, we extend the
analysis beyond squared loss and give bounds for entropic loss and absolute
loss.Comment: 25 page
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