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Finite sample learning of moving targets

Nikolaus Vertovec, Kostas Margellos, Maria Prandini

TL;DR

A novel bound is derived on the number of samples that are required to construct a probably approximately correct (PAC) estimate of the target when the target is a convex polytope.

Abstract

We consider a moving target that we seek to learn from samples. Our results extend randomized techniques developed in control and optimization for a constant target to the case where the target is changing. We derive a novel bound on the number of samples that are required to construct a probably approximately correct (PAC) estimate of the target. Furthermore, when the moving target is a convex polytope, we provide a constructive method of generating the PAC estimate using a mixed integer linear program (MILP). The proposed method is demonstrated on an application to autonomous emergency braking.

Finite sample learning of moving targets

TL;DR

A novel bound is derived on the number of samples that are required to construct a probably approximately correct (PAC) estimate of the target when the target is a convex polytope.

Abstract

We consider a moving target that we seek to learn from samples. Our results extend randomized techniques developed in control and optimization for a constant target to the case where the target is changing. We derive a novel bound on the number of samples that are required to construct a probably approximately correct (PAC) estimate of the target. Furthermore, when the moving target is a convex polytope, we provide a constructive method of generating the PAC estimate using a mixed integer linear program (MILP). The proposed method is demonstrated on an application to autonomous emergency braking.
Paper Structure (12 sections, 1 theorem, 42 equations, 7 figures)

This paper contains 12 sections, 1 theorem, 42 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Learning moving targets
  3. Problem statement
  4. Mathematical preliminaries and assumptions
  5. Finite sample probabilistic certificates
  6. Main result
  7. Hypothesis computation
  8. Numerical example
  9. Problem set-up
  10. Simulation results
  11. Conclusion
  12. Recovering the constant target case

Key Result

Proposition 1

Let $p_1,\ldots,p_m \in [0,1]$, and consider independent Bernoulli random variables $Y_1,\ldots,Y_m$ such that $\mathbb{P}\{Y_i = 1\} = p_i$ and $\mathbb{P}\{Y_i = 0\} = 1-p_i$, for all $i=1,\ldots,m$. For any $\tau > 0$ we then have that

Figures (7)

  • Figure 1: At each iteration, we receive a single sample along with a $\{0,1\}$-valued label. To illustrate this, consider the labeling mechanism as an indicator function over the orange set. The orange set will change between each drawn sample (we illustrate this by depicting the orange sets across multiple iterations). The green circles indicate a $0$-label, while the blue triangles represent a $1$-label. We seek to find a hypothesis on the basis of the labeling $\{(x_1,f_1(x_1)),\ldots, (x_m,f_m(x_m)) \}$ that, with certain confidence, will agree with the subsequent (unknown) target function $f_{m+1}$ on a new sample. We depict an example of such a hypothesis with the purple rectangle.
  • Figure 2: Number of samples required according to \ref{['eqn:m0']} for different accuracy levels $\epsilon$ and $\delta = 10^{-6}$ with dimension 4. The color code corresponds to different values of $\underline{\mu}, \overline{\mu}$. Notice that the term dependent on $\underline{\mu}$ in \ref{['eqn:m0']} does not depend on $\epsilon$ and thus constitutes the constant dominant at higher levels of $\epsilon$.
  • Figure 3: The evolution of the braking performance over time. Green circles indicate samples with label 0, while blue triangles show samples with label 1. The bold red halfplane represents the true safety label at the given iteration, while the opaque halfplanes show the safety boundary at previous iterations.
  • Figure 4: Illustration of the facets of the convex polytope. Since the safety label relies in this case on a single halfplane (drawn in red), we only need to consider facet 3.
  • Figure 5: All samples in black are discarded, while the red samples are kept for the computation of the hypothesis. This results in $95\%$ of the samples being omitted, greatly improving the computational feasibility.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Remark 1
  • Proposition 1
  • Definition 1: minimal disagreement
  • proof
  • Remark 2: Effect of $\underline{\mu}, \overline{\mu}$
  • Remark 3: Constant target
  • Remark 4
  • proof