目录

• 1 Introduction
• 2 Bayesian Optimization with Parametric Models
• 3 Nonparametric models

• Proceedings of the IEEE 2016
• https://ieeexplore.ieee.org/abstract/document/7352306
• A review of BO, an optimization algorithm typically for "hyperparameters".

1 Introduction

• design, choice, high-dim, hyperparam
  • IBM ILOG CPLEX
• \(x^* = argmax_{x\in \mathcal X}f(x)\)
  • compact subset of \(\mathbb R^d\), or ...
  • stochastic output \(\mathbb E[y|f(x)]=f(x)\)
  • unbiased noisy point-wise observations
• data efficient, evaluations are costly
• prior, refine
• best choice? acquisition function \(\alpha_n: \mathcal X\to \mathbb R\)
• 
  • mean, confidence interval
• myopic heuristics
  • uncertainty is large (exploration), or prediction is high (exploitation)
  • acquisition function: easy to find the optimum, analytic?

2 Bayesian Optimization with Parametric Models

• parametrized by \(w\)
• \(\mathcal D\): data
• bayesian: \(p(w|D)=p(D|w)p(w)/p(D)\)
  • beliefs about \(w\) after observing data \(D\)
  • \(p(D)\) intractable, but in fact a normalizing constant
• prior: conjucacy, analytically
• \(K\) drugs, independent
  • to optimize \(f\), on \(K\) indices, fully parametrized
  • beta, conjugacy
• TS, simplest strategy, posterior prob of optimality, estimated, MC
  • \(a_{n+1}=argmax_a f_{\bar w}(a)\)
  • no more param other than the prior
• linear model, feature, vector, \(f_w(a)=x_a^T w\)
• \(X\): input vectors, \(y\): outputs
• nonlinear basis functions
  • radial
  • Fourier
  • learned from data
  • feature map, regardless, weights can be computed analytically

3 Nonparametric models

• start, observation variance \(\sigma^2\), zero-mean Gaussian prior \(V_0\), preserve Gaussianity
• basis functions, linear regression, symmetric positive-semidefinite, kernel
  • intuitive similarity between pairs of points, rather than a feature map \(\Phi\)
  • tractable, linear algebra, unnecessary to explicitly define \(\Phi\)
• GP, nonparametric model, prior mean, covariance
• \(f|X\sim \mathcal N(m,K)\)
• \(y|f, \sigma^2\sim \mathcal N(f,\sigma^2 I)\)
• posterior: use \(x\) and previous data (not "abstracted by parameters")
• kernel, structure, periodic, stationary
  • Matern, diagonal, paramtrized
    
• kernel, smoothness and amplitude
• prior, possible offset, constant, expert knowledge