333 research outputs found
Optimal Algorithms for Continuous Non-monotone Submodular and DR-Submodular Maximization
In this paper we study the fundamental problems of maximizing a continuous
non-monotone submodular function over the hypercube, both with and without
coordinate-wise concavity. This family of optimization problems has several
applications in machine learning, economics, and communication systems. Our
main result is the first -approximation algorithm for continuous
submodular function maximization; this approximation factor of is
the best possible for algorithms that only query the objective function at
polynomially many points. For the special case of DR-submodular maximization,
i.e. when the submodular functions is also coordinate wise concave along all
coordinates, we provide a different -approximation algorithm that
runs in quasilinear time. Both of these results improve upon prior work [Bian
et al, 2017, Soma and Yoshida, 2017].
Our first algorithm uses novel ideas such as reducing the guaranteed
approximation problem to analyzing a zero-sum game for each coordinate, and
incorporates the geometry of this zero-sum game to fix the value at this
coordinate. Our second algorithm exploits coordinate-wise concavity to identify
a monotone equilibrium condition sufficient for getting the required
approximation guarantee, and hunts for the equilibrium point using binary
search. We further run experiments to verify the performance of our proposed
algorithms in related machine learning applications
Guaranteed Non-convex Optimization: Submodular Maximization over Continuous Domains
Submodular continuous functions are a category of (generally)
non-convex/non-concave functions with a wide spectrum of applications. We
characterize these functions and demonstrate that they can be maximized
efficiently with approximation guarantees. Specifically, i) We introduce the
weak DR property that gives a unified characterization of submodularity for all
set, integer-lattice and continuous functions; ii) for maximizing monotone
DR-submodular continuous functions under general down-closed convex
constraints, we propose a Frank-Wolfe variant with approximation
guarantee, and sub-linear convergence rate; iii) for maximizing general
non-monotone submodular continuous functions subject to box constraints, we
propose a DoubleGreedy algorithm with approximation guarantee. Submodular
continuous functions naturally find applications in various real-world
settings, including influence and revenue maximization with continuous
assignments, sensor energy management, multi-resolution data summarization,
facility location, etc. Experimental results show that the proposed algorithms
efficiently generate superior solutions compared to baseline algorithms.Comment: Appears in the 20th International Conference on Artificial
Intelligence and Statistics (AISTATS) 201
Non-monotone DR-submodular Maximization: Approximation and Regret Guarantees
Diminishing-returns (DR) submodular optimization is an important field with
many real-world applications in machine learning, economics and communication
systems. It captures a subclass of non-convex optimization that provides both
practical and theoretical guarantees. In this paper, we study the fundamental
problem of maximizing non-monotone DR-submodular functions over down-closed and
general convex sets in both offline and online settings. First, we show that
for offline maximizing non-monotone DR-submodular functions over a general
convex set, the Frank-Wolfe algorithm achieves an approximation guarantee which
depends on the convex set. Next, we show that the Stochastic Gradient Ascent
algorithm achieves a 1/4-approximation ratio with the regret of
for the problem of maximizing non-monotone DR-submodular functions over
down-closed convex sets. These are the first approximation guarantees in the
corresponding settings. Finally we benchmark these algorithms on problems
arising in machine learning domain with the real-world datasets
Online Continuous Submodular Maximization
In this paper, we consider an online optimization process, where the
objective functions are not convex (nor concave) but instead belong to a broad
class of continuous submodular functions. We first propose a variant of the
Frank-Wolfe algorithm that has access to the full gradient of the objective
functions. We show that it achieves a regret bound of (where
is the horizon of the online optimization problem) against a
-approximation to the best feasible solution in hindsight. However, in
many scenarios, only an unbiased estimate of the gradients are available. For
such settings, we then propose an online stochastic gradient ascent algorithm
that also achieves a regret bound of regret, albeit against a
weaker -approximation to the best feasible solution in hindsight. We also
generalize our results to -weakly submodular functions and prove the
same sublinear regret bounds. Finally, we demonstrate the efficiency of our
algorithms on a few problem instances, including non-convex/non-concave
quadratic programs, multilinear extensions of submodular set functions, and
D-optimal design.Comment: Accepted by AISTATS 201
Continuous DR-submodular Maximization: Structure and Algorithms
DR-submodular continuous functions are important objectives with wide
real-world applications spanning MAP inference in determinantal point processes
(DPPs), and mean-field inference for probabilistic submodular models, amongst
others. DR-submodularity captures a subclass of non-convex functions that
enables both exact minimization and approximate maximization in polynomial
time.
In this work we study the problem of maximizing non-monotone DR-submodular
continuous functions under general down-closed convex constraints. We start by
investigating geometric properties that underlie such objectives, e.g., a
strong relation between (approximately) stationary points and global optimum is
proved. These properties are then used to devise two optimization algorithms
with provable guarantees. Concretely, we first devise a "two-phase" algorithm
with approximation guarantee. This algorithm allows the use of existing
methods for finding (approximately) stationary points as a subroutine, thus,
harnessing recent progress in non-convex optimization. Then we present a
non-monotone Frank-Wolfe variant with approximation guarantee and
sublinear convergence rate. Finally, we extend our approach to a broader class
of generalized DR-submodular continuous functions, which captures a wider
spectrum of applications. Our theoretical findings are validated on synthetic
and real-world problem instances.Comment: Published in NIPS 201
Conditional Gradient Method for Stochastic Submodular Maximization: Closing the Gap
In this paper, we study the problem of \textit{constrained} and
\textit{stochastic} continuous submodular maximization. Even though the
objective function is not concave (nor convex) and is defined in terms of an
expectation, we develop a variant of the conditional gradient method, called
\alg, which achieves a \textit{tight} approximation guarantee. More precisely,
for a monotone and continuous DR-submodular function and subject to a
\textit{general} convex body constraint, we prove that \alg achieves a
[(1-1/e)\text{OPT} -\eps] guarantee (in expectation) with
\mathcal{O}{(1/\eps^3)} stochastic gradient computations. This guarantee
matches the known hardness results and closes the gap between deterministic and
stochastic continuous submodular maximization. By using stochastic continuous
optimization as an interface, we also provide the first tight
approximation guarantee for maximizing a \textit{monotone but stochastic}
submodular \textit{set} function subject to a general matroid constraint
Maximizing Monotone DR-submodular Continuous Functions by Derivative-free Optimization
In this paper, we study the problem of monotone (weakly) DR-submodular
continuous maximization. While previous methods require the gradient
information of the objective function, we propose a derivative-free algorithm
LDGM for the first time. We define and to characterize how
close a function is to continuous DR-submodulr and submodular, respectively.
Under a convex polytope constraint, we prove that LDGM can achieve a
-approximation guarantee after
iterations, which is the same as the best previous gradient-based algorithm.
Moreover, in some special cases, a variant of LDGM can achieve a
-approximation guarantee for (weakly)
submodular functions. We also compare LDGM with the gradient-based algorithm
Frank-Wolfe under noise, and show that LDGM can be more robust. Empirical
results on budget allocation verify the effectiveness of LDGM
Optimal approximation for unconstrained non-submodular minimization
Submodular function minimization is a well studied problem; existing
algorithms solve it exactly or up to arbitrary accuracy. However, in many
applications, the objective function is not exactly submodular. No theoretical
guarantees exist in this case. While submodular minimization algorithms rely on
intricate connections between submodularity and convexity, we show that these
relations can be extended sufficiently to obtain approximation guarantees for
non-submodular minimization. In particular, we prove how a projected
subgradient method can perform well even for certain non-submodular functions.
This includes important examples, such as objectives for structured sparse
learning and variance reduction in Bayesian optimization. We also extend this
result to noisy function evaluations. Our algorithm works in the value oracle
model. We prove that in this model, the approximation result we obtain is the
best possible with a subexponential number of queries
Stochastic Conditional Gradient++
In this paper, we consider the general non-oblivious stochastic optimization
where the underlying stochasticity may change during the optimization procedure
and depends on the point at which the function is evaluated. We develop
Stochastic Frank-Wolfe++ (), an efficient variant of the
conditional gradient method for minimizing a smooth non-convex function subject
to a convex body constraint. We show that converges to an
-first order stationary point by using stochastic
gradients. Once further structures are present, 's theoretical
guarantees, in terms of the convergence rate and quality of its solution,
improve. In particular, for minimizing a convex function,
achieves an -approximate optimum while using
stochastic gradients. It is known that this rate is optimal in terms of
stochastic gradient evaluations. Similarly, for maximizing a monotone
continuous DR-submodular function, a slightly different form of , called Stochastic Continuous Greedy++ (), achieves a tight
solution while using
stochastic gradients. Through an information theoretic argument, we also prove
that 's convergence rate is optimal. Finally, for maximizing a
non-monotone continuous DR-submodular function, we can achieve a
solution by using stochastic
gradients. We should highlight that our results and our novel variance
reduction technique trivially extend to the standard and easier oblivious
stochastic optimization settings for (non-)covex and continuous submodular
settings
Robust Budget Allocation via Continuous Submodular Functions
The optimal allocation of resources for maximizing influence, spread of
information or coverage, has gained attention in the past years, in particular
in machine learning and data mining. But in applications, the parameters of the
problem are rarely known exactly, and using wrong parameters can lead to
undesirable outcomes. We hence revisit a continuous version of the Budget
Allocation or Bipartite Influence Maximization problem introduced by Alon et
al. (2012) from a robust optimization perspective, where an adversary may
choose the least favorable parameters within a confidence set. The resulting
problem is a nonconvex-concave saddle point problem (or game). We show that
this nonconvex problem can be solved exactly by leveraging connections to
continuous submodular functions, and by solving a constrained submodular
minimization problem. Although constrained submodular minimization is hard in
general, here, we establish conditions under which such a problem can be solved
to arbitrary precision .Comment: ICML 201
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