470 research outputs found
Universal Algorithms: Beyond the Simplex
The bulk of universal algorithms in the online convex optimisation literature
are variants of the Hedge (exponential weights) algorithm on the simplex. While
these algorithms extend to polytope domains by assigning weights to the
vertices, this process is computationally unfeasible for many important classes
of polytopes where the number of vertices depends exponentially on the
dimension . In this paper we show the Subgradient algorithm is universal,
meaning it has regret in the antagonistic setting and
pseudo-regret in the i.i.d setting, with two main advantages over Hedge: (1)
The update step is more efficient as the action vectors have length only
rather than ; and (2) Subgradient gives better performance if the cost
vectors satisfy Euclidean rather than sup-norm bounds. This paper extends the
authors' recent results for Subgradient on the simplex. We also prove the same
and bounds when the domain is the unit ball. To the
authors' knowledge this is the first instance of these bounds on a domain other
than a polytope.Comment: 1 figure, 40 page
Universal Algorithms for Clustering Problems
This paper presents universal algorithms for clustering problems, including
the widely studied -median, -means, and -center objectives. The input
is a metric space containing all potential client locations. The algorithm must
select cluster centers such that they are a good solution for any subset of
clients that actually realize. Specifically, we aim for low regret, defined as
the maximum over all subsets of the difference between the cost of the
algorithm's solution and that of an optimal solution. A universal algorithm's
solution for a clustering problem is said to be an -approximation if for all subsets of clients , it satisfies , where is the cost of the
optimal solution for clients and is the minimum regret achievable by
any solution.
Our main results are universal algorithms for the standard clustering
objectives of -median, -means, and -center that achieve -approximations. These results are obtained via a novel framework for
universal algorithms using linear programming (LP) relaxations. These results
generalize to other -objectives and the setting where some subset of
the clients are fixed. We also give hardness results showing that -approximation is NP-hard if or is at most a certain
constant, even for the widely studied special case of Euclidean metric spaces.
This shows that in some sense, -approximation is the strongest
type of guarantee obtainable for universal clustering.Comment: Appeared in ICALP 2021, Track A. Fixed mismatch between paper title
and arXiv titl
Optimal Lower Bounds for Universal and Differentially Private Steiner Tree and TSP
Given a metric space on n points, an {\alpha}-approximate universal algorithm
for the Steiner tree problem outputs a distribution over rooted spanning trees
such that for any subset X of vertices containing the root, the expected cost
of the induced subtree is within an {\alpha} factor of the optimal Steiner tree
cost for X. An {\alpha}-approximate differentially private algorithm for the
Steiner tree problem takes as input a subset X of vertices, and outputs a tree
distribution that induces a solution within an {\alpha} factor of the optimal
as before, and satisfies the additional property that for any set X' that
differs in a single vertex from X, the tree distributions for X and X' are
"close" to each other. Universal and differentially private algorithms for TSP
are defined similarly. An {\alpha}-approximate universal algorithm for the
Steiner tree problem or TSP is also an {\alpha}-approximate differentially
private algorithm. It is known that both problems admit O(logn)-approximate
universal algorithms, and hence O(log n)-approximate differentially private
algorithms as well. We prove an {\Omega}(logn) lower bound on the approximation
ratio achievable for the universal Steiner tree problem and the universal TSP,
matching the known upper bounds. Our lower bound for the Steiner tree problem
holds even when the algorithm is allowed to output a more general solution of a
distribution on paths to the root.Comment: 14 page
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