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Shai Ben-David, Dávid Pál, Shai Shalev-Shwartz
Agnostic Online Learning
Summary: We generalize the Littlestone's online learning model to the
agnostic setting where no hypothesis makes zero error. Littlestone defined a
combinatorial parameter of the hypothesis class, which we call Littlestone's
dimension and which determines the worst-case number of prediction mistakes
made by an online learning algorithm in the realizable setting. Point of our paper is that
Littlestone's dimension characterizes learnability in the agnostic case as
well. Namely, we give upper and lower bounds on the regret in terms of the Littlestone's
dimension.
This is a similar story to what happened to the Valiant's PAC model. The key
quantity there is Vapnik-Chervonekis dimension. Valiant's PAC model is the
``realizable case''. Our work can be paralleled to what Haussler and others
did in 1992 when they generalized the PAC model to the agnostic setting and
showed that Vapnik-Chervonekis dimension remains the key quantity
characterizing learnability there as well.
Bib info: COLT 2009
Download:
[PDF],
[slides],
[slides sources]
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PhD thesis.
Contributions to Unsupervised and Semi-Supervised Learning
Download:
[PDF],
[sources]
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Tyler Lu, Dávid Pál, Martin Pál
Showing Relevant Ads via Lipschitz Context Multi-Armed Bandits
Summary:
We study context multi-armed bandit problems where the context comes
from a metric space and the payoff satisfies a Lipschitz condition with respect
to the metric. Abstractly, a context multi-armed bandit problem models a
situation in which, in a sequence of independent trials, an online algorithm
chooses an action based on a given context (side information) from a set
of possible actions so as to maximize the total payoff of the chosen actions.
The payoff depends on both the action chosen and the context. In contrast,
context-free multi-armed bandit problems, studied previously, model
situations where no side information is available and the payoff depends only
on the action chosen.
For concreteness we focus in this paper on an application to Internet search
advertisement where the task is to display ads to a user of a Internet search
engine based on his search query so as to maximize the click-trough rate of the
ads displayed. We cast this problem as a context multi-armed bandit problem
where queries and ads form metric spaces and the payoff function is Lipschitz
with respect to both the metrics. We present an algorithm, give upper bound on its regret
and show an (almost) matching lower bound on the regret of any algorithm.
Bib info: manuscript
Download:
[PDF]
[slides]
[slides sources]
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Shai Ben-David, Tyler Lu, Dávid Pál, Miroslava Sotáková
Learning Low-Density Separators
Summary:
We define a novel, basic, unsupervised learning problem -
learning the lowest density homogeneous hyperplane separator of an
unknown probability distribution. This task is relevant to several problems
in machine learning, such as semi-supervised learning and clustering
stability. We investigate the question of existence of a universally
consistent algorithm for this problem. We propose two natural learning
paradigms and prove that, on input unlabeled random samples generated
by any member of a rich family of distributions, they are guaranteed to
converge to the optimal separator for that distribution. We complement
this result by showing that no learning algorithm for our task can achieve
uniform learning rates (that are independent of the data generating distribution).
Bib info: accepted to AISTATS 2009
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[PDF]
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Shai Ben-David, Tyler Lu, Dávid Pál
Does Unlabeled Data Provably Help? Worst-case Analysis of the Sample Complexity of Semi-Supervised Learning
Summary:
We mathematically study the potential benefits of the use of unlabeled data to
classification prediction to the learner. We propose a simple model of
semi-supervised learning (SSL) in which the unlabeled data distribution is
perfectly known to the learner. We compare this model with the standard PAC
model (and its agnostic vesrion) for supervised learning, in which the unlabeled distribution is uknown
to the learner. Can it be that there exists a supervised algorithm which
for any unlabeled distribution needs in the worst case (over possible targets)
at most by a constant factor more labeled samples than any SSL learner? It
seems that the ERM algorithm is such a candidate. We prove, for some special
cases, that this indeed the case.
Bib info: Proceedings of COLT 2008
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[PDF] [sources]
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Gagan Aggarwal, S. Muthukrishnan, Dávid Pál, Martin Pál
General Auction Mechanism for Search Advertising
Summary:
We propose a new auction mechanims for search advertising that generalizes both Generelized Second Price
auction (which is currently, in 2008, used by Google, Yahoo! and MSN) and the famous Vickrey-Clarke-Groves mechanism
adapted to the search auctions. Our mechanism allows each bidder to specify a subset of slots in which (s)he is interested,
and the value and the maximum price of each slot. For the auctioneer (the search engine) it allows to specify for every slot-bidder
pair a reserve (minimum) price. Our auction computes the bidder-optimal stable matching. The running time is O(nk^3), where
n is the number of bidders and k is the number of slots. We also prove that our auction mechanism
is truth-revealing, that is, dominant strategy of each bidder is not to lie.
Bib info: accepted to WWW 2009
Download:
[PDF]
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Shai Ben-David, Dávid Pál, Hans Ulrich Simon
Stability of K-means Clusterings
Summary:
This is follow-up of the previous paper. We consider the clustering algorithm A which minimizes the k-means cost precisely.
(Such algorithm is not realistic, since the optimization problem is known to be NP-hard.
Nevertheless, the classical Lloyd's hueristic a.k.a. "the k-mean algorithm" tries to do this.)
We analyze how stable is the clustering outputted by A with respect to a random draw of the sample points.
Given a probability distribution, draw two i.i.d. samples of the same size. If with high
probability, the clustering of the first sample does not differ from the
clustering of the second sample too much, we say that A is stable on
the probability distribution. For discrete probability distributions we show that
instability happens if and only if the probability distribution has two or more clusterings
with optimal k-means cost.
Bib info: Proceedings of COLT 2007
Download:
[PDF] (extended version), [PDF] (short conference version)
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Shai Ben-David, Dávid Pál, Ulrike von Luxburg
A Sober look at Clustering Stability
Summary:
We investigate how stable is the clustering outputted by a clustering algorithm
with respect to a random draw of the sample points. Given a probability
distribution, draw two i.i.d. samples of the same size. If with high
probability, the clustering of the first sample does not differ from the
clustering of the second sample too much, we say that algorithm is stable on
the probability distribution. We show that for that for algorithms that optimize some cost function stability depends
on the uniqueness of the optimal solution of the optimization problem for the "infinite" sample represented by the
probability distribution.
Bib info: Proceedings of COLT 2006, Best Student Paper Award
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[PDF],
[Slides in PDF],
[Slides source files]
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Dávid Pál, Martin Škoviera
Colouring Cubic Graphs by Small Steiner Triple Systems
Summary: Steiner triple system is a collection of 3-element subsets (called triples) of
a finite set of points, such that every two points appear together in exactly one triple.
Given Steiner triple system S, one can ask whether it is possible to color the edges of a cubic graph with points of S,
in such way that colors of the three edges at a vertex form a triple of S.
We construct a small Steiner triple system (of order 21) which colors all simple cubic graphs.
Note: This is the main result of my "Magister" Thesis at Comenius University under my advisor Martin Škoviera, 2004.
Bib info: Graphs and Combinatorics, 2007
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[PDF],
[Postscript],
[Source files]
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co-authors:
Gagan Aggarwal
Shai Ben-David
Tyler Lu
Ulrike von Luxburg
S. Muthukrisnan
Martin Pál
Hans Ulrich Simon
Miroslava Sotáková
Martin Škoviera
Shai Shalev-Shwartz
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