arXiv subjects

K. S. Sesh Kumar

Publications and source records attributed to K. S. Sesh Kumar.

5 recordsLinked to original sources

arXiv2020

High-dimensional Bayesian optimization using low-dimensional feature spaces

Bayesian optimization (BO) is a powerful approach for seeking the global optimum of expensive black-box functions and has proven successful for fine tuning hyper-parameters of machine learning models. However, BO is practically limited to optimizing 10--20 parameters. To scale BO to high dimensions, we usually make structural assumptions on the decomposition of the objective and\slash or exploit the intrinsic lower dimensionality of the problem, e.g. by using linear projections. We could achieve a higher compression rate with nonlinear projections, but learning these nonlinear embeddings typically requires much data. This contradicts the BO objective of a relatively small evaluation budget. To address this challenge, we propose to learn a low-dimensional feature space jointly with (a) the response surface and (b) a reconstruction mapping. Our approach allows for optimization of BO's acquisition function in the lower-dimensional subspace, which significantly simplifies the optimization problem. We reconstruct the original parameter space from the lower-dimensional subspace for evaluating the black-box function. For meaningful exploration, we solve a constrained optimization problem.

stat.ML↗

arXiv2018

Active-set Methods for Submodular Minimization Problems

We consider the submodular function minimization (SFM) and the quadratic minimization problemsregularized by the Lov'asz extension of the submodular function. These optimization problemsare intimately related; for example,min-cut problems and total variation denoising problems, wherethe cut function is submodular and its Lov'asz extension is given by the associated total variation.When a quadratic loss is regularized by the total variation of a cut function, it thus becomes atotal variation denoising problem and we use the same terminology in this paper for "general" submodularfunctions. We propose a new active-set algorithm for total variation denoising with theassumption of an oracle that solves the corresponding SFM problem. This can be seen as localdescent algorithm over ordered partitions with explicit convergence guarantees. It is more flexiblethan the existing algorithms with the ability for warm-restarts using the solution of a closely relatedproblem. Further, we also consider the case when a submodular function can be decomposed intothe sum of two submodular functions F1 and F2 and assume SFM oracles for these two functions.We propose a new active-set algorithm for total variation denoising (and hence SFM by thresholdingthe solution at zero). This algorithm also performs local descent over ordered partitions and itsability to warm start considerably improves the performance of the algorithm. In the experiments,we compare the performance of the proposed algorithms with state-of-the-art algorithms, showingthat it reduces the calls to SFM oracles.

math.OC↗

arXiv2015

Convex Optimization for Parallel Energy Minimization

Energy minimization has been an intensely studied core problem in computer vision. With growing image sizes (2D and 3D), it is now highly desirable to run energy minimization algorithms in parallel. But many existing algorithms, in particular, some efficient combinatorial algorithms, are difficult to par-allelize. By exploiting results from convex and submodular theory, we reformulate the quadratic energy minimization problem as a total variation denoising problem, which, when viewed geometrically, enables the use of projection and reflection based convex methods. The resulting min-cut algorithm (and code) is conceptually very simple, and solves a sequence of TV denoising problems. We perform an extensive empirical evaluation comparing state-of-the-art combinatorial algorithms and convex optimization techniques. On small problems the iterative convex methods match the combinatorial max-flow algorithms, while on larger problems they offer other flexibility and important gains: (a) their memory footprint is small; (b) their straightforward parallelizability fits multi-core platforms; (c) they can easily be warm-started; and (d) they quickly reach approximately good solutions, thereby enabling faster "inexact" solutions. A key consequence of our approach based on submodularity and convexity is that it is allows to combine any arbitrary combinatorial or convex methods as subroutines, which allows one to obtain hybrid combinatorial and convex optimization algorithms that benefit from the strengths of both.

cs.CV↗

arXiv2013

Maximizing submodular functions using probabilistic graphical models

We consider the problem of maximizing submodular functions; while this problem is known to be NP-hard, several numerically efficient local search techniques with approximation guarantees are available. In this paper, we propose a novel convex relaxation which is based on the relationship between submodular functions, entropies and probabilistic graphical models. In a graphical model, the entropy of the joint distribution decomposes as a sum of marginal entropies of subsets of variables; moreover, for any distribution, the entropy of the closest distribution factorizing in the graphical model provides an bound on the entropy. For directed graphical models, this last property turns out to be a direct consequence of the submodularity of the entropy function, and allows the generalization of graphical-model-based upper bounds to any submodular functions. These upper bounds may then be jointly maximized with respect to a set, while minimized with respect to the graph, leading to a convex variational inference scheme for maximizing submodular functions, based on outer approximations of the marginal polytope and maximum likelihood bounded treewidth structures. By considering graphs of increasing treewidths, we may then explore the trade-off between computational complexity and tightness of the relaxation. We also present extensions to constrained problems and maximizing the difference of submodular functions, which include all possible set functions.

cs.LG↗

arXiv2012

Convex Relaxations for Learning Bounded Treewidth Decomposable Graphs

We consider the problem of learning the structure of undirected graphical models with bounded treewidth, within the maximum likelihood framework. This is an NP-hard problem and most approaches consider local search techniques. In this paper, we pose it as a combinatorial optimization problem, which is then relaxed to a convex optimization problem that involves searching over the forest and hyperforest polytopes with special structures, independently. A supergradient method is used to solve the dual problem, with a run-time complexity of $O(k^3 n^{k+2} \log n)$ for each iteration, where $n$ is the number of variables and $k$ is a bound on the treewidth. We compare our approach to state-of-the-art methods on synthetic datasets and classical benchmarks, showing the gains of the novel convex approach.

cs.LG↗