605 research outputs found
Elimination-based certificates for triangular equivalence and rank profiles
International audienceIn this paper, we give novel certificates for triangular equivalence and rank profiles. These certificates enable somebody to verify the row or column rank profiles or the whole rank profile matrix faster than recomputing them, with a negligible overall overhead. We first provide quadratic time and space non-interactive certificates saving the logarithmic factors of previously known ones. Then we propose interactive certificates for the same problems whose Monte Carlo verification complexity requires a small constant number of matrix-vector multiplications, a linear space, and a linear number of extra field operations, with a linear number of interactions. As an application we also give an interactive protocol, certifying the determinant or the signature of dense matrices, faster for the Prover than the best previously known one. Finally we give linear space and constant round certificates for the row or column rank profiles
The Dynamics of Household Wealth Accumulation in Italy
We examine the dynamics of wealth accumulation distribution in Italy using data drawn from the Survey of Household Income and Wealth, a representative sample of the Italian population conducted by the Bank of Italy. We compare survey data with national accounts data and discuss sample representativeness, attrition, and measurement issues. We then look at wealth inequality (the cross-sectional dispersion of wealth), wealth mobility (individual transitions across the wealth distribution), and examine the age profile of wealth using repeated cross-sectional data. Finally, we consider various explanations for the pattern of wealth accumulation in Italy, focusing on retirement, bequests, income risk, health shocks, and credit market imperfections.Wealth accumulation, Inequality, Mobility
The Discrete Dantzig Selector: Estimating Sparse Linear Models via Mixed Integer Linear Optimization
We propose a novel high-dimensional linear regression estimator: the Discrete
Dantzig Selector, which minimizes the number of nonzero regression coefficients
subject to a budget on the maximal absolute correlation between the features
and residuals. Motivated by the significant advances in integer optimization
over the past 10-15 years, we present a Mixed Integer Linear Optimization
(MILO) approach to obtain certifiably optimal global solutions to this
nonconvex optimization problem. The current state of algorithmics in integer
optimization makes our proposal substantially more computationally attractive
than the least squares subset selection framework based on integer quadratic
optimization, recently proposed in [8] and the continuous nonconvex quadratic
optimization framework of [33]. We propose new discrete first-order methods,
which when paired with state-of-the-art MILO solvers, lead to good solutions
for the Discrete Dantzig Selector problem for a given computational budget. We
illustrate that our integrated approach provides globally optimal solutions in
significantly shorter computation times, when compared to off-the-shelf MILO
solvers. We demonstrate both theoretically and empirically that in a wide range
of regimes the statistical properties of the Discrete Dantzig Selector are
superior to those of popular -based approaches. We illustrate that
our approach can handle problem instances with p = 10,000 features with
certifiable optimality making it a highly scalable combinatorial variable
selection approach in sparse linear modeling
The Discrete Dantzig Selector: Estimating Sparse Linear Models via Mixed Integer Linear Optimization
We propose a novel high-dimensional linear regression estimator: the Discrete Dantzig Selector, which minimizes the number of nonzero regression coefficients subject to a budget on the maximal absolute correlation between the features and residuals. Motivated by the significant advances in integer optimization over the past 10-15 years, we present a mixed integer linear optimization (MILO) approach to obtain certifiably optimal global solutions to this nonconvex optimization problem. The current state of algorithmics in integer optimization makes our proposal substantially more computationally attractive than the least squares subset selection framework based on integer quadratic optimization, recently proposed by Bertsimas et al. and the continuous nonconvex quadratic optimization framework of Liu et al. We propose new discrete first-order methods, which when paired with the state-of-the-art MILO solvers, lead to good solutions for the Discrete Dantzig Selector problem for a given computational budget. We illustrate that our integrated approach provides globally optimal solutions in significantly shorter computation times, when compared to off-the-shelf MILO solvers. We demonstrate both theoretically and empirically that in a wide range of regimes the statistical properties of the Discrete Dantzig Selector are superior to those of popular ell1-based approaches. We illustrate that our approach can handle problem instances with p =10,000 features with certifiable optimality making it a highly scalable combinatorial variable selection approach in sparse linear modeling
Scalable Verification of GNN-based Job Schedulers
Recently, Graph Neural Networks (GNNs) have been applied for scheduling jobs
over clusters, achieving better performance than hand-crafted heuristics.
Despite their impressive performance, concerns remain over whether these
GNN-based job schedulers meet users' expectations about other important
properties, such as strategy-proofness, sharing incentive, and stability. In
this work, we consider formal verification of GNN-based job schedulers. We
address several domain-specific challenges such as networks that are deeper and
specifications that are richer than those encountered when verifying image and
NLP classifiers. We develop vegas, the first general framework for verifying
both single-step and multi-step properties of these schedulers based on
carefully designed algorithms that combine abstractions, refinements, solvers,
and proof transfer. Our experimental results show that vegas achieves
significant speed-up when verifying important properties of a state-of-the-art
GNN-based scheduler compared to previous methods.Comment: Condensed version published at OOPSLA'2
Symmetry reduction in convex optimization with applications in combinatorics
This dissertation explores different approaches to and applications of symmetry reduction in convex optimization. Using tools from semidefinite programming, representation theory and algebraic combinatorics, hard combinatorial problems are solved or bounded. The first chapters consider the Jordan reduction method, extend the method to optimization over the doubly nonnegative cone, and apply it to quadratic assignment problems and energy minimization on a discrete torus. The following chapter uses symmetry reduction as a proving tool, to approach a problem from queuing theory with redundancy scheduling. The final chapters propose generalizations and reductions of flag algebras, a powerful tool for problems coming from extremal combinatorics
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