Asymptotic Resource Usage Bounds

When describing the resource usage of a program, it is usual to talk in asymptotic terms, such as the well-known “big O” notation, whereby we focus on the behaviour of the program for large input data and make a rough approximation by considering as equivalent programs whose resource usage grows at the same rate. Motivated by the existence of non-asymptotic resource usage analyzers, in this paper, we develop a novel transformation from a non-asymptotic cost function (which can be produced by multiple resource analyzers) into its asymptotic form. Our transformation aims at producing tight asymptotic forms which do not contain redundant subexpressions (i.e., expressions asymptotically subsumed by others). Interestingly, we integrate our transformation at the heart of a cost analyzer to generate asymptotic upper bounds without having to first compute their non-asymptotic counterparts. Our experimental results show that, while non-asymptotic cost functions become very complex, their asymptotic forms are much more compact and manageable. This is essential to improve scalability and to enable the application of cost analysis in resource-aware verification/certification

An Approach to Static Performance Guarantees for Programs with Run-time Checks

Instrumenting programs for performing run-time checking of properties, such as regular shapes, is a common and useful technique that helps programmers detect incorrect program behaviors. This is specially true in dynamic languages such as Prolog. However, such run-time checks inevitably introduce run-time overhead (in execution time, memory, energy, etc.). Several approaches have been proposed for reducing such overhead, such as eliminating the checks that can statically be proved to always succeed, and/or optimizing the way in which the (remaining) checks are performed. However, there are cases in which it is not possible to remove all checks statically (e.g., open libraries which must check their interfaces, complex properties, unknown code, etc.) and in which, even after optimizations, these remaining checks still may introduce an unacceptable level of overhead. It is thus important for programmers to be able to determine the additional cost due to the run-time checks and compare it to some notion of admissible cost. The common practice used for estimating run-time checking overhead is profiling, which is not exhaustive by nature. Instead, we propose a method that uses static analysis to estimate such overhead, with the advantage that the estimations are functions parameterized by input data sizes. Unlike profiling, this approach can provide guarantees for all possible execution traces, and allows assessing how the overhead grows as the size of the input grows. Our method also extends an existing assertion verification framework to express "admissible" overheads, and statically and automatically checks whether the instrumented program conforms with such specifications. Finally, we present an experimental evaluation of our approach that suggests that our method is feasible and promising.Comment: 15 pages, 3 tables; submitted to ICLP'18, accepted as technical communicatio

A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the supremum norm of the variance term is asymptotically distributed as a Gumbel random variable. In the following, we prove a version of this result for estimators using compactly-supported wavelets, a popular tool in nonparametric statistics. Our result relies on an assumption on the nature of the wavelet, which must be verified by provably-good numerical approximations. We verify our assumption for Daubechies wavelets and symlets, with N = 6, ..., 20 vanishing moments; larger values of N, and other wavelet bases, are easily checked, and we conjecture that our assumption holds also in those cases

Towards an Abstract Domain for Resource Analysis of Logic Programs Using Sized Types

We present a novel general resource analysis for logic programs based on sized types.Sized types are representations that incorporate structural (shape) information and allow expressing both lower and upper bounds on the size of a set of terms and their subterms at any position and depth. They also allow relating the sizes of terms and subterms occurring at different argument positions in logic predicates. Using these sized types, the resource analysis can infer both lower and upper bounds on the resources used by all the procedures in a program as functions on input term (and subterm) sizes, overcoming limitations of existing analyses and enhancing their precision. Our new resource analysis has been developed within the abstract interpretation framework, as an extension of the sized types abstract domain, and has been integrated into the Ciao preprocessor, CiaoPP. The abstract domain operations are integrated with the setting up and solving of recurrence equations for both, inferring size and resource usage functions. We show that the analysis is an improvement over the previous resource analysis present in CiaoPP and compares well in power to state of the art systems.Comment: Part of WLPE 2013 proceedings (arXiv:1308.2055

Overcommitment in Cloud Services -- Bin packing with Chance Constraints

This paper considers a traditional problem of resource allocation, scheduling jobs on machines. One such recent application is cloud computing, where jobs arrive in an online fashion with capacity requirements and need to be immediately scheduled on physical machines in data centers. It is often observed that the requested capacities are not fully utilized, hence offering an opportunity to employ an overcommitment policy, i.e., selling resources beyond capacity. Setting the right overcommitment level can induce a significant cost reduction for the cloud provider, while only inducing a very low risk of violating capacity constraints. We introduce and study a model that quantifies the value of overcommitment by modeling the problem as a bin packing with chance constraints. We then propose an alternative formulation that transforms each chance constraint into a submodular function. We show that our model captures the risk pooling effect and can guide scheduling and overcommitment decisions. We also develop a family of online algorithms that are intuitive, easy to implement and provide a constant factor guarantee from optimal. Finally, we calibrate our model using realistic workload data, and test our approach in a practical setting. Our analysis and experiments illustrate the benefit of overcommitment in cloud services, and suggest a cost reduction of 1.5% to 17% depending on the provider's risk tolerance