We introduce a generalized value function of a mixed-integer program, which is simultaneously parameterized by its objective and right-hand side. We describe its fundamental properties, which we ...exploit through three algorithms to calculate it. We then show how this generalized value function can be used to reformulate two classes of mixed-integer optimization problems: two-stage stochastic mixed-integer programming and multifollower bilevel mixed-integer programming. For both of these problem classes, the generalized value function approach allows the solution of instances that are significantly larger than those solved in the literature in terms of the total number of variables and number of scenarios.
At each iteration of the simplex method there are typically many possible entering columns. We use deep value-based reinforcement learning to choose dynamically between two popular pivoting rules. We ...consider LP relaxations of the MTZ formulation of non-Euclidean TSPs with five cities. We obtain a 20–50% speed up on these very small instances. Although our methods are not remotely competitive or viable on large instances, our results indicate that there may be scope to substantially accelerate current LP solvers by augmenting them with a learned pivoting strategy.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
Given a set of feasible solutions X to a linear program, we study the set of objectives that make X optimal, known as the inverse-feasible region. We show the relationship between the dimension of a ...face of a polyhedron and the dimension of the corresponding inverse-feasible region, which leads to necessary and sufficient conditions of the extreme, boundary, and inner points of a linear program. We also characterize the set of objectives that render a given solution uniquely optimal.
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Understanding the learning dynamics and inductive bias of neural networks (NNs) is hindered by the opacity of the relationship between NN parameters and the function represented. Partially, this is ...due to symmetries inherent within the NN parameterization, allowing multiple different parameter settings to result in an identical output function, resulting in both an unclear relationship and redundant degrees of freedom. The NN parameterization is invariant under two symmetries: permutation of the neurons and a continuous family of transformations of the scale of weight and bias parameters. We propose taking a quotient with respect to the second symmetry group and reparametrizing ReLU NNs as continuous piecewise linear splines. Using this spline lens, we study learning dynamics in shallow univariate ReLU NNs, finding unexpected insights and explanations for several perplexing phenomena. We develop a surprisingly simple and transparent view of the structure of the loss surface, including its critical and fixed points, Hessian, and Hessian spectrum. We also show that standard weight initializations yield very flat initial functions, and that this flatness, together with overparametrization and the initial weight scale, is responsible for the strength and type of implicit regularization, consistent with previous work. Our implicit regularization results are complementary to recent work, showing that initialization scale critically controls implicit regularization via a kernel-based argument. Overall, removing the weight scale symmetry enables us to prove these results more simply and enables us to prove new results and gain new insights while offering a far more transparent and intuitive picture. Looking forward, our quotiented spline-based approach will extend naturally to the multivariate and deep settings, and alongside the kernel-based view, we believe it will play a foundational role in efforts to understand neural networks. Videos of learning dynamics using a spline-based visualization are available at http://shorturl.at/tFWZ2.
A novel technique was used to calculate pulse pressure variation. The algorithm reliably predicted fluid responsiveness to transfusion, with a receiver operating characteristic area under the curve ...of 0.89. This technique may assist clinicians in the management of fluids and vasoactive medications for premature infants.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Operating rooms provide crucial health services and generate significant hospital revenue. It is challenging to use operating rooms efficiently due to uncertainty in surgery durations and emergency ...patient arrivals. In this dissertation, we propose an operating room scheduling model that integrates the tactical decisions of assigning operating rooms to departments and the operational decisions of scheduling both planned and emergency patients. Our goal is to minimize the expected undertime of the operating rooms while ensuring that the probability of overtime is sufficiently small. We model the operating room scheduling problem as a three-stage decision problem and formulate it as a two-stage stochastic mixed-integer program. We then present two generalized value function reformulations that convert the model to a pure binary program. We propose a novel dynamic program to calculate the value functions. We show that our approach gives optimal scheduling policies in a reasonable amount of time. We present numerical studies using operational data from a major hospital that demonstrate the value of the integration of block assignments and considering emergency surgery arrivals. Furthermore, we investigate the properties of generalized value functions and provide reformulations for stochastic mixed-integer programming and multi-follower bilevel mixed-integer programming problems. We finally present the exact set of objectives that make a given solution uniquely optimal through inverse optimization.
Understanding the learning dynamics and inductive bias of neural networks (NNs) is hindered by the opacity of the relationship between NN parameters and the function represented. We propose ...reparametrizing ReLU NNs as continuous piecewise linear splines. Using this spline lens, we study learning dynamics in shallow univariate ReLU NNs, finding unexpected insights and explanations for several perplexing phenomena. We develop a surprisingly simple and transparent view of the structure of the loss surface, including its critical and fixed points, Hessian, and Hessian spectrum. We also show that standard weight initializations yield very flat functions, and that this flatness, together with overparametrization and the initial weight scale, is responsible for the strength and type of implicit regularization, consistent with recent work arXiv:1906.05827. Our implicit regularization results are complementary to recent work arXiv:1906.07842, done independently, which showed that initialization scale critically controls implicit regularization via a kernel-based argument. Our spline-based approach reproduces their key implicit regularization results but in a far more intuitive and transparent manner. Going forward, our spline-based approach is likely to extend naturally to the multivariate and deep settings, and will play a foundational role in efforts to understand neural networks. Videos of learning dynamics using a spline-based visualization are available at http://shorturl.at/tFWZ2.
