Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNon-monotone Submodular Maximization with Nearly Optimal Adaptivity and Query Complexity
129
0
0.0
(
0
)
Added by
Matthew Fahrbach
Publication date
2018
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Submodular maximization is a general optimization problem with a wide range of applications in machine learning (e.g., active learning, clustering, and feature selection). In large-scale optimization, the parallel running time of an algorithm is governed by its adaptivity, which measures the number of sequential rounds needed if the algorithm can execute polynomially-many independent oracle queries in parallel. While low adaptivity is ideal, it is not sufficient for an algorithm to be efficient in practice---there are many applications of distributed submodular optimization where the number of function evaluations becomes prohibitively expensive. Motivated by these applications, we study the adaptivity and query complexity of submodular maximization. In this paper, we give the first constant-factor approximation algorithm for maximizing a non-monotone submodular function subject to a cardinality constraint $k$ that runs in $O(log(n))$ adaptive rounds and makes $O(n log(k))$ oracle queries in expectation. In our empirical study, we use three real-world applications to compare our algorithm with several benchmarks for non-monotone submodular maximization. The results demonstrate that our algorithm finds competitive solutions using significantly fewer rounds and queries.
rate research
Read More
Submodular optimization generalizes many classic problems in combinatorial optimization and has recently found a wide range of applications in machine learning (e.g., feature engineering and active learning). For many large-scale optimization problems, we are often concerned with the adaptivity complexity of an algorithm, which quantifies the number of sequential rounds where polynomially-many independent function evaluations can be executed in parallel. While low adaptivity is ideal, it is not sufficient for a distributed algorithm to be efficient, since in many practical applications of submodular optimization the number of function evaluations becomes prohibitively expensive. Motivated by these applications, we study the adaptivity and query complexity of adaptive submodular optimization. Our main result is a distributed algorithm for maximizing a monotone submodular function with cardinality constraint $k$ that achieves a $(1-1/e-varepsilon)$-approximation in expectation. This algorithm runs in $O(log(n))$ adaptive rounds and makes $O(n)$ calls to the function evaluation oracle in expectation. The approximation guarantee and query complexity are optimal, and the adaptivity is nearly optimal. Moreover, the number of queries is substantially less than in previous works. Last, we extend our results to the submodular cover problem to demonstrate the generality of our algorithm and techniques.
In this paper, we study the tradeoff between the approximation guarantee and adaptivity for the problem of maximizing a monotone submodular function subject to a cardinality constraint. The adaptivity of an algorithm is the number of sequential rounds of queries it makes to the evaluation oracle of the function, where in every round the algorithm is allowed to make polynomially-many parallel queries. Adaptivity is an important consideration in settings where the objective function is estimated using samples and in applications where adaptivity is the main running time bottleneck. Previous algorithms achieving a nearly-optimal $1 - 1/e - epsilon$ approximation require $Omega(n)$ rounds of adaptivity. In this work, we give the first algorithm that achieves a $1 - 1/e - epsilon$ approximation using $O(ln{n} / epsilon^2)$ rounds of adaptivity. The number of function evaluations and additional running time of the algorithm are $O(n mathrm{poly}(log{n}, 1/epsilon))$.
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 $frac{1}{2}$-approximation algorithm for continuous submodular function maximization; this approximation factor of $frac{1}{2}$ 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 $frac{1}{2}$-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.
In this work, we give a new parallel algorithm for the problem of maximizing a non-monotone diminishing returns submodular function subject to a cardinality constraint. For any desired accuracy $epsilon$, our algorithm achieves a $1/e - epsilon$ approximation using $O(log{n} log(1/epsilon) / epsilon^3)$ parallel rounds of function evaluations. The approximation guarantee nearly matches the best approximation guarantee known for the problem in the sequential setting and the number of parallel rounds is nearly-optimal for any constant $epsilon$. Previous algorithms achieve worse approximation guarantees using $Omega(log^2{n})$ parallel rounds. Our experimental evaluation suggests that our algorithm obtains solutions whose objective value nearly matches the value obtained by the state of the art sequential algorithms, and it outperforms previous parallel algorithms in number of parallel rounds, iterations, and solution quality.
The need for real time analysis of rapidly producing data streams (e.g., video and image streams) motivated the design of streaming algorithms that can efficiently extract and summarize useful information from massive data on the fly. Such problems can often be reduced to maximizing a submodular set function subject to various constraints. While efficient streaming methods have been recently developed for monotone submodular maximization, in a wide range of applications, such as video summarization, the underlying utility function is non-monotone, and there are often various constraints imposed on the optimization problem to consider privacy or personalization. We develop the first efficient single pass streaming algorithm, Streaming Local Search, that for any streaming monotone submodular maximization algorithm with approximation guarantee $alpha$ under a collection of independence systems ${cal I}$, provides a constant $1/big(1+2/sqrt{alpha}+1/alpha +2d(1+sqrt{alpha})big)$ approximation guarantee for maximizing a non-monotone submodular function under the intersection of ${cal I}$ and $d$ knapsack constraints. Our experiments show that for video summarization, our method runs more than 1700 times faster than previous work, while maintaining practically the same performance.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
