87 research outputs found
On Dynamic Optimality for Binary Search Trees
Does there exist O(1)-competitive (self-adjusting) binary search tree (BST)
algorithms? This is a well-studied problem. A simple offline BST algorithm
GreedyFuture was proposed independently by Lucas and Munro, and they
conjectured it to be O(1)-competitive. Recently, Demaine et al. gave a
geometric view of the BST problem. This view allowed them to give an online
algorithm GreedyArb with the same cost as GreedyFuture. However, no
o(n)-competitive ratio was known for GreedyArb. In this paper we make progress
towards proving O(1)-competitive ratio for GreedyArb by showing that it is
O(\log n)-competitive
Further Optimal Regret Bounds for Thompson Sampling
Thompson Sampling is one of the oldest heuristics for multi-armed bandit
problems. It is a randomized algorithm based on Bayesian ideas, and has
recently generated significant interest after several studies demonstrated it
to have better empirical performance compared to the state of the art methods.
In this paper, we provide a novel regret analysis for Thompson Sampling that
simultaneously proves both the optimal problem-dependent bound of
and the
first near-optimal problem-independent bound of on the
expected regret of this algorithm. Our near-optimal problem-independent bound
solves a COLT 2012 open problem of Chapelle and Li. The optimal
problem-dependent regret bound for this problem was first proven recently by
Kaufmann et al. [ALT 2012]. Our novel martingale-based analysis techniques are
conceptually simple, easily extend to distributions other than the Beta
distribution, and also extend to the more general contextual bandits setting
[Manuscript, Agrawal and Goyal, 2012].Comment: arXiv admin note: substantial text overlap with arXiv:1111.179
Lower Bounds for the Average and Smoothed Number of Pareto Optima
Smoothed analysis of multiobjective 0-1 linear optimization has drawn
considerable attention recently. The number of Pareto-optimal solutions (i.e.,
solutions with the property that no other solution is at least as good in all
the coordinates and better in at least one) for multiobjective optimization
problems is the central object of study. In this paper, we prove several lower
bounds for the expected number of Pareto optima. Our basic result is a lower
bound of \Omega_d(n^(d-1)) for optimization problems with d objectives and n
variables under fairly general conditions on the distributions of the linear
objectives. Our proof relates the problem of lower bounding the number of
Pareto optima to results in geometry connected to arrangements of hyperplanes.
We use our basic result to derive (1) To our knowledge, the first lower bound
for natural multiobjective optimization problems. We illustrate this for the
maximum spanning tree problem with randomly chosen edge weights. Our technique
is sufficiently flexible to yield such lower bounds for other standard
objective functions studied in this setting (such as, multiobjective shortest
path, TSP tour, matching). (2) Smoothed lower bound of min {\Omega_d(n^(d-1.5)
\phi^{(d-log d) (1-\Theta(1/\phi))}), 2^{\Theta(n)}}$ for the 0-1 knapsack
problem with d profits for phi-semirandom distributions for a version of the
knapsack problem. This improves the recent lower bound of Brunsch and Roeglin
Dynamic vs Oblivious Routing in Network Design
Consider the robust network design problem of finding a minimum cost network
with enough capacity to route all traffic demand matrices in a given polytope.
We investigate the impact of different routing models in this robust setting:
in particular, we compare \emph{oblivious} routing, where the routing between
each terminal pair must be fixed in advance, to \emph{dynamic} routing, where
routings may depend arbitrarily on the current demand. Our main result is a
construction that shows that the optimal cost of such a network based on
oblivious routing (fractional or integral) may be a factor of
\BigOmega(\log{n}) more than the cost required when using dynamic routing.
This is true even in the important special case of the asymmetric hose model.
This answers a question in \cite{chekurisurvey07}, and is tight up to constant
factors. Our proof technique builds on a connection between expander graphs and
robust design for single-sink traffic patterns \cite{ChekuriHardness07}
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