1,721,143 research outputs found
Non-convex scenario optimization
Full text linkScenario optimization is an approach to data-driven decision-making that has been introduced some fifteen years ago and has ever since then grown fast. Its most remarkable feature is that it blends the heuristic nature of data-driven methods with a rigorous theory that allows one to gain factual, reliable, insight in the solution. The usability of the scenario theory, however, has been restrained thus far by the obstacle that most results are standing on the assumption of convexity. With this paper, we aim to free the theory from this limitation. Specifically, we focus on the body of results that are known under the name of “wait-and-judge” and show that its fundamental achievements maintain their validity in a non-convex setup. While optimization is a major center of attention, this paper travels beyond it and into data-driven decision making. Adopting such a broad framework opens the door to building a new theory of truly vast applicability
La conoscenza per la valorizzazione del patrimonio architettonico. Le chiese della città di Napoli
No full textThis research consists of an accurate census, cataloguing and mapping of religious architectural monuments in the historical centre of Naples, in addition to the study and representation of many churches and small chapels.
Our study is aimed at showing and documenting some Neapolitan architectural fragments of great historical and artistic significance. Some of them are very evident in the continuity of the buildings to which they belong, some others are hidden in the dense network of alleys and therefore particularly exposed to a relentless decay.
This project consisted of more phases: the collection and organization of data from documentary researches, the census and mapping of single artifacts and, at the same time, inspections and surveys revealing some not immediately evident architectural examples that, in an overall framework, can indicate new cultural itineraries of the historical city, within a process of regeneration and reuse which takes into account the features and values of each artifact
The risk of making decisions from data through the lens of the scenario approach
Full text linkIn previous contributions, it has been shown that the “complexity” is a key indicator to quantify the “risk” associated to data-driven scenario-based solutions. Depending on the context of application, risk is interpreted as probability of misprediction, or probability of underperforming or meeting shortfalls in various control endeavors, and the acquired ability to tightly evaluate the risk is a vital element in a world where data-driven methods are being increasingly used not only for decision support but also for automated decision making. The present contribution is meant to significantly expand the area of applicability of these results: all achievements so far have been based on an assumption, called “non-degeneracy”, that hardly applies e.g. to optimization problems that are not convex. Here, we show that these results maintain their integrity in a non-convex optimization setup, and beyond into a broad domain of decision making that contains non-convex optimization as a particular case
Scenario Optimization with Constraint Relaxation in a Non-Convex Setup: A Flexible and General Framework for Data-Driven Design
No full textThe scenario approach, originally developed as a computational tool for robust problems, has through the years developed into a solid, general, framework for data-driven decision making and design. One main driving force that has fostered this process has certainly been the increasing generality of the considered schemes. In this paper, we move a further step forward in this process. By leveraging some recent results in the wake of the so-called wait-and-judge paradigm, we fully develop a scheme for scenario optimization with constraint relaxation in a non-convex setup, so greatly expanding previous achievements valid under a convexity assumption. We show that a purely data-driven, and yet tight and informative, quantification of the solution robustness is possible regardless of the mechanism through which uncertainty is generated. The generality of this new non-convex setup provides an extremely versatile scheme for data-driven design that can be applied to a variety of problems ranging from mixed-integer optimization to design in abstract spaces
A theory of the risk for optimization with relaxation and its application to support vector machines
Full text linkIn this paper we consider optimization with relaxation, an ample paradigm to make data-driven designs. This approach was previously considered by the same authors of this work in Garatti and Campi (2019), a study that revealed a deep-seated connection between two concepts: risk (probability of not satisfying a new, out-of-sample, constraint) and complexity (according to a definition introduced in paper Garatti and Campi, 2019). This connection was shown to have profound implications in applications because it implied that the risk can be estimated from the complexity, a quantity that can be measured from the data without any knowledge of the data-generation mechanism. In the present work we establish new results. First, we expand the scope of Garatti and Campi (2019) so as to embrace a more general setup that covers various algorithms in machine learning. Then, we study classical support vector methods – including SVM (Support Vector Machine), SVR (Support Vector Regression) and SVDD (Support Vector Data Description) – and derive new results for the ability of these methods to generalize. All results are valid for any finite size of the data set. When the sample size tends to infinity, we establish the unprecedented result that the risk approaches the ratio between the complexity and the cardinality of the data sample, regardless of the value of the complexity
Risk and complexity in scenario optimization
Full text linkScenario optimization is a broad methodology to perform optimization based on empirical knowledge. One collects previous cases, called “scenarios”, for the set-up in which optimization is being performed, and makes a decision that is optimal for the cases that have been collected. For convex optimization, a solid theory has been developed that provides guarantees of performance, and constraint satisfaction, of the scenario solution. In this paper, we open a new direction of investigation: the risk that a performance is not achieved, or that constraints are violated, is studied jointly with the complexity (as precisely defined in the paper) of the solution. It is shown that the joint probability distribution of risk and complexity is concentrated in such a way that the complexity carries fundamental information to tightly judge the risk. This result is obtained without requiring extra knowledge on the underlying optimization problem than that carried by the scenarios; in particular, no extra knowledge on the distribution by which scenarios are generated is assumed, so that the result is broadly applicable. This deep-seated result unveils a fundamental and general structure of data-driven optimization and suggests practical approaches for risk assessment
Compression at the service of learning: a case study for the Guaranteed Error Machine
No full textThe scenario approach is a technique for data-driven decision making that has found application in a variety of fields including systems and control design. Although initially conceived in the context of worst-case optimization, the scenario approach has progressively evolved into a general methodology that allows one to keep control on the risk of solutions designed from data according to complex decision processes. In a recent contribution, the theory of compression schemes (a paradigm that plays a fundamental role in statistical learning theory) has been deeply revisited in the wake of the scenario approach, which has led to unprecedentedly sharp generalization and risk quantification results. In this paper, we build on these achievements to gain insight on a classification paradigm called Guaranteed Error Machine (GEM). First, by leveraging the theory of reproducing kernels Hilbert spaces, we introduce a new, more flexible, GEM algorithm, which allows for complex classification geometries. The proposed scheme is then shown to fit into the new compression theory, from which new sharp results for the probability of GEM misclassification are derived in a distribution-free context
Inductive knowledge under dominance
No full textInductive reasoning aims at constructing rules and models of general applicability from a restricted set of observations. Induction is a keystone in natural sciences, and it influences diverse application fields such as engineering, medicine and economics. More generally, induction plays a major role in the way humans learn and operate in their everyday life. The level of reliability that a model achieves depends on how informative the observations are relative to the flexibility of the process by which the model is constructed. When the process is articulated so that the model can incorporate descriptive details and subtleties, a large set of informative observations are required to reliably tune the model, whereas models obtained from simple procedures can be tuned with fewer observations. This article introduces the concept of "dominance", which refers to the situation in which a reduced subset of observations suffices to reconstruct the model. A mathematical framework is presented to quantify the reliability of learning procedures as a function of the size of the subset of dominant observations. Although limited in scope, we believe that our study can contribute to the understanding of some fundamental mechanisms by which knowledge is generated from observations in inductive reasoning
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