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Scenario Approach with Post-Design Certification of User-Specified Properties

Algo Carè, Marco C. Campi, Simone Garatti

TL;DR

A two-level framework of appropriateness is introduced: baseline appropriateness, which guides the design process, and post-design appropriateness, which serves as a criterion for a posteriori evaluation, and a method is provided to infer comprehensive distributional knowledge of relevant performance indexes from the available dataset.

Abstract

The scenario approach is an established data-driven design framework that comes equipped with a powerful theory linking design complexity to generalization properties. In this approach, data are simultaneously used both for design and for certifying the design's reliability, without resorting to a separate test dataset. This paper takes a step further by guaranteeing additional properties, useful in post-design usage but not considered during the design phase. To this end, we introduce a two-level framework of appropriateness: baseline appropriateness, which guides the design process, and post-design appropriateness, which serves as a criterion for a posteriori evaluation. We provide distribution-free upper bounds on the risk of failing to meet the post-design appropriateness; these bounds are computable without using any additional test data. Under additional assumptions, lower bounds are also derived. As part of an effort to demonstrate the usefulness of the proposed methodology, the paper presents two practical examples in H2 and pole-placement problems. Moreover, a method is provided to infer comprehensive distributional knowledge of relevant performance indexes from the available dataset.

Scenario Approach with Post-Design Certification of User-Specified Properties

TL;DR

A two-level framework of appropriateness is introduced: baseline appropriateness, which guides the design process, and post-design appropriateness, which serves as a criterion for a posteriori evaluation, and a method is provided to infer comprehensive distributional knowledge of relevant performance indexes from the available dataset.

Abstract

The scenario approach is an established data-driven design framework that comes equipped with a powerful theory linking design complexity to generalization properties. In this approach, data are simultaneously used both for design and for certifying the design's reliability, without resorting to a separate test dataset. This paper takes a step further by guaranteeing additional properties, useful in post-design usage but not considered during the design phase. To this end, we introduce a two-level framework of appropriateness: baseline appropriateness, which guides the design process, and post-design appropriateness, which serves as a criterion for a posteriori evaluation. We provide distribution-free upper bounds on the risk of failing to meet the post-design appropriateness; these bounds are computable without using any additional test data. Under additional assumptions, lower bounds are also derived. As part of an effort to demonstrate the usefulness of the proposed methodology, the paper presents two practical examples in H2 and pole-placement problems. Moreover, a method is provided to infer comprehensive distributional knowledge of relevant performance indexes from the available dataset.
Paper Structure (20 sections, 4 theorems, 59 equations, 4 figures)

This paper contains 20 sections, 4 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem statement and technical results
  3. A scenario decision framework with two notions of appropriateness
  4. The risk of inappropriateness - review and new goal
  5. Novel results: post-design risk certification
  6. Tight post-design risk certification
  7. Examples
  8. Probabilistically robust $H_2$ control
  9. Review of the $H_2$ control problem with no uncertainty
  10. The scenario problem
  11. Baseline and post-design appropriateness
  12. Equivalent formulation as an LMI for easy implementation
  13. Numerical results
  14. Pole placement
  15. Equivalent formulation as a linear problem
  16. ...and 5 more sections

Key Result

Lemma 1

The maps $M_m^+$ satisfy the consistency requirements with respect to the instrumental appropriateness (i.e., the same conditions in Assumption ass:Consistency_of_appr1 hold with $M_m$ replaced by $M_m^+$ and ${\cal Z}'_{\delta_{m+i}}$ replaced by ${\cal Z}^+_{\delta_{m+i}}$).

Figures (4)

  • Figure 1: Feedback configuration for the pole placement problem.
  • Figure 2: A classic pendulum
  • Figure 3: Locations in the complex plane of the closed-loop system poles corresponding to the $N=2000$ scenarios in problem \ref{['scenario_prog_polyn']}. The dotted line is the border of the desired conic sector ${\cal S}_{-0.7,0.5}$. The black dots in $-1\pm j$ and $-3\pm 3 j$ are the poles of the reference polynomial.
  • Figure 4: $\overline{F_{\epsilon}}(\ell-h^\ast_N)$ and $\underline{F_{\epsilon}}(\ell-h^\ast_N)$ (solid lines) vs. actual cumulative distribution function of $\| \eta(T) \|_\infty$ (dashed lines) for $\rho = 1$ (top) and $\rho = 0.05$ (bottom).

Theorems & Definitions (11)

  • Definition 1: sets of baseline and post-design appropriate decisions
  • Definition 2: baseline and post-design risk
  • Definition 3: baseline support list and baseline complexity
  • Definition 4: instrumental decision map
  • Definition 5: instrumentally appropriate decisions
  • Definition 6: instrumental risk
  • Lemma 1
  • Remark 1: on computing $s^{+,\ast}_N$
  • Theorem 1: upper bound for post-design risk
  • Theorem 2: upper and lower bounds under baseline non-degeneracy in the nested case
  • ...and 1 more