Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGeneral Feasibility Bounds for Sample Average Approximation via Vapnik-Chervonenkis Dimension
143
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We investigate the feasibility of sample average approximation (SAA) for general stochastic optimization problems, including two-stage stochastic programming without the relatively complete recourse assumption. Instead of analyzing problems with specific structures, we utilize results from the Vapnik-Chervonenkis (VC) dimension and Probably Approximately Correct learning to provide a general framework that offers explicit feasibility bounds for SAA solutions under minimal structural or distributional assumption. We show that, as long as the hypothesis class formed by the feasbible region has a finite VC dimension, the infeasibility of SAA solutions decreases exponentially with computable rates and explicitly identifiable accompanying constants. We demonstrate how our bounds apply more generally and competitively compared to existing results.
rate research
Read More
The disjointness problem - where Alice and Bob are given two subsets of ${1, dots, n}$ and they have to check if their sets intersect - is a central problem in the world of communication complexity. While both deterministic and randomized communication complexities for this problem are known to be $Theta(n)$, it is also known that if the sets are assumed to be drawn from some restricted set systems then the communication complexity can be much lower. In this work, we explore how communication complexity measures change with respect to the complexity of the underlying set system. The complexity measure for the set system that we use in this work is the Vapnik-Chervonenkis (VC) dimension. More precisely, on any set system with VC dimension bounded by $d$, we analyze how large can the deterministic and randomized communication complexities be, as a function of $d$ and $n$. In this paper, we construct two natural set systems of VC dimension $d$, motivated from geometry. Using these set systems we show that the deterministic and randomized communication complexity can be $widetilde{Theta}left(dlog left( n/d right)right)$ for set systems of VC dimension $d$ and this matches the deterministic upper bound for all set systems of VC dimension $d$. We also study the deterministic and randomized communication complexities of the set intersection problem when sets belong to a set system of bounded VC dimension. We show that there exists set systems of VC dimension $d$ such that both deterministic and randomized (one-way and multi-round) complexity for the set intersection problem can be as high as $Thetaleft( dlog left( n/d right) right)$, and this is tight among all set systems of VC dimension $d$.
In machine learning and optimization community there are two main approaches for convex risk minimization problem, namely, the Stochastic Approximation (SA) and the Sample Average Approximation (SAA). In terms of oracle complexity (required number of stochastic gradient evaluations), both approaches are considered equivalent on average (up to a logarithmic factor). The total complexity depends on the specific problem, however, starting from work cite{nemirovski2009robust} it was generally accepted that the SA is better than the SAA. Nevertheless, in case of large-scale problems SA may run out of memory as storing all data on one machine and organizing online access to it can be impossible without communications with other machines. SAA in contradistinction to SA allows parallel/distributed calculations. In this paper, we shed new light on the comparison of SA and SAA for particular problem of calculating the population (regularized) Wasserstein barycenter of discrete measures. The conclusion is valid even for non-parallel (non-decentralized) setup.
We show that sparsity constrained optimization problems over low dimensional spaces tend to have a small duality gap. We use the Shapley-Folkman theorem to derive both data-driven bounds on the duality gap, and an efficient primalization procedure to recover feasible points satisfying these bounds. These error bounds are proportional to the rate of growth of the objective with the target cardinality, which means in particular that the relaxation is nearly tight as soon as the target cardinality is large enough so that only uninformative features are added.
In this paper, we will develop a systematic approach to deriving guaranteed bounds for approximate dynamic programming (ADP) schemes in optimal control problems. Our approach is inspired by our recent results on bounding the performance of greedy strategies in optimization of string-submodular functions over a finite horizon. The approach is to derive a string-submodular optimization problem, for which the optimal strategy is the optimal control solution and the greedy strategy is the ADP solution. Using this approach, we show that any ADP solution achieves a performance that is at least a factor of $beta$ of the performance of the optimal control solution, which satisfies Bellmans optimality principle. The factor $beta$ depends on the specific ADP scheme, as we will explicitly characterize. To illustrate the applicability of our bounding technique, we present examples of ADP schemes, including the popular rollout method.
Several issues in machine learning and inverse problems require to generate discrete data, as if sampled from a model probability distribution. A common way to do so relies on the construction of a uniform probability distribution over a set of $N$ points which minimizes the Wasserstein distance to the model distribution. This minimization problem, where the unknowns are the positions of the atoms, is non-convex. Yet, in most cases, a suitably adjusted version of Lloyds algorithm -- in which Voronoi cells are replaced by Power cells -- leads to configurations with small Wasserstein error. This is surprising because, again, of the non-convex nature of the problem, as well as the existence of spurious critical points. We provide explicit upper bounds for the convergence speed of this Lloyd-type algorithm, starting from a cloud of points sufficiently far from each other. This already works after one step of the iteration procedure, and similar bounds can be deduced, for the corresponding gradient descent. These bounds naturally lead to a modified Poliak-Lojasiewicz inequality for the Wasserstein distance cost, with an error term depending on the distances between Dirac masses in the discrete distribution.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
