37 research outputs found
Lazier Than Lazy Greedy
Is it possible to maximize a monotone submodular function faster than the
widely used lazy greedy algorithm (also known as accelerated greedy), both in
theory and practice? In this paper, we develop the first linear-time algorithm
for maximizing a general monotone submodular function subject to a cardinality
constraint. We show that our randomized algorithm, STOCHASTIC-GREEDY, can
achieve a approximation guarantee, in expectation, to the
optimum solution in time linear in the size of the data and independent of the
cardinality constraint. We empirically demonstrate the effectiveness of our
algorithm on submodular functions arising in data summarization, including
training large-scale kernel methods, exemplar-based clustering, and sensor
placement. We observe that STOCHASTIC-GREEDY practically achieves the same
utility value as lazy greedy but runs much faster. More surprisingly, we
observe that in many practical scenarios STOCHASTIC-GREEDY does not evaluate
the whole fraction of data points even once and still achieves
indistinguishable results compared to lazy greedy.Comment: In Proc. Conference on Artificial Intelligence (AAAI), 201
Weakly Submodular Functions
Submodular functions are well-studied in combinatorial optimization, game
theory and economics. The natural diminishing returns property makes them
suitable for many applications. We study an extension of monotone submodular
functions, which we call {\em weakly submodular functions}. Our extension
includes some (mildly) supermodular functions. We show that several natural
functions belong to this class and relate our class to some other recent
submodular function extensions.
We consider the optimization problem of maximizing a weakly submodular
function subject to uniform and general matroid constraints. For a uniform
matroid constraint, the "standard greedy algorithm" achieves a constant
approximation ratio where the constant (experimentally) converges to 5.95 as
the cardinality constraint increases. For a general matroid constraint, a
simple local search algorithm achieves a constant approximation ratio where the
constant (analytically) converges to 10.22 as the rank of the matroid
increases
Streaming Algorithms for Submodular Function Maximization
We consider the problem of maximizing a nonnegative submodular set function
subject to a -matchoid
constraint in the single-pass streaming setting. Previous work in this context
has considered streaming algorithms for modular functions and monotone
submodular functions. The main result is for submodular functions that are {\em
non-monotone}. We describe deterministic and randomized algorithms that obtain
a -approximation using -space, where is
an upper bound on the cardinality of the desired set. The model assumes value
oracle access to and membership oracles for the matroids defining the
-matchoid constraint.Comment: 29 pages, 7 figures, extended abstract to appear in ICALP 201