REVIEW: Portfolio Allocation for Bayesian Optimization by Siobhán Cronin

Portfolio Allocation for Bayesian Optimization (2011). Mathew Hoffman, Eric Brochu, and Nando de Freitas. 

Most of the literature seems to suggest UCB (or LCB) as the optimal acquisition function in many cases, yet this research proposes the use of a portfolio of acquisition functions governed by a multi-armed bandit strategy, as opposed to only using a single acquisition function. 

Benefits

  • The authors suggest that there may be no single acquisition function that will perform best over an entire optimization. 
  • "This can be treated as a hierarchical multi-armed bandit problem, in which each of the N arms is itself an infinite-armed bandit problem". 
  • Three strategies are suggested, but Hedge is recommended. "Hedge is an algorithm which at each time step t selects an action i with probability p_t(i) based on the cumulative rewards (gain) for that action." A gain vector is then updated from these rewards. 

Challenges

  • You can't necessarily compare convergence rates of the portfolio method directly to the single acquisition functions, since "decisions made at iteration t affect the state of the problem and the resulting rewards at all future iterations". The authors suggest an approach (Theorem 1) for setting bounds on the cumulative regret. These bounds are generated in relation to points proposed by UCB, and the authors suggest possible refinements to this theorem to take into account bounds of other acquisition functions in the portfolio. 
  • Are the improvements cited enough for us to notice a difference in applied cases?

 

REVIEW: Taking the human out of the loop - a review of Bayesian Optimization by Siobhán Cronin

Taking the Human Out of the Loop: A Review of Bayesian Optimization (2016). Shahriari et al. Proceedings of the IEEE

Introduction

  • "Mathematically we are considering the problem of finding a global maximizer (or minimizer) of an unknown objective function f, where X is some design space of interest; ..."
  • "...in global optimization, X is often a compact subset of R^d but the Bayesian optimization framework can be applied to more unusual search spaces that involve categorical or conditional inputs."
  • "The Bayesian posterior represents our updates beliefs - given data - on the likely objective function we are optimizing. Equipped with this probabilistic model, we can sequentially induce acquisition functions that leverage the uncertainty in the posterior to guide exploration."
  • "Intuitively, the acquisition function evaluates the utility of candidate points for the next evaluation of f; therefore x_n+1 is selected by maximizing \alpha_n"
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Nonparametric models

  • "The kernel trick allows us to specify an intuitive similarity between pairs of points, rather than a feature map, which in practice can be hard to define."
  • Common kernels (Matérn)
  • "The marginal likelihood is very useful in learning the hyperparameters. As long as the kernel is differentiable with respect to its hyperparameters, the marginal likelihood can be differentiated and can therefore be optimized."

 

REVIEW: Introduction to Gaussian Processes by Siobhán Cronin

Introduction to Gaussian Processes (1998) - David Mackay

Overview

  • "From a  Bayesian perspective, a choice of a neural network model can be viewed as defining a prior probability distribution over non-linear functions, and the neural network's learning process can be interpreted in terms of the posterior probability distribution over the unknown function. (Some learning algorithms search for the function with maximum posterior probability, and other Monte Carlo methods draw samples from this posterior probability)."
  • "The idea of Gaussian process modeling is, without parameterizing y(x), to place a prior P(y(x)) directly on the space of functions. The simplest type of prior over functions is called a Gaussian process. It can be thought of as the generalization of the Gaussian distribution over a finite vector space to a function of infinite dimension."
  • "Just as a Gaussian distribution is fully specified by its mean and covariance matrix, a Gaussian process is specified by a mean [often taken to be the zero function of x] and a covariance function [which expresses the expected covariance between the value of the function y at the points x and x']."
  • "The actual function y(x) in any one data modeling problem is assumed to be a single sample from this Gaussian distribution."
  • "...by concentrating on the joint probability distribution of the observed data and the quantities we wish to predict, it is possible to make predictions with resources that scale as polynomial functions of N, the number of data points."

Nonlinear Regression

  • In nonparametric methods, predictions are obtained without giving the unknown function y(x) an explicit parameterization. 
  • An example of a nonparametric approach to regression is the spline smoothing method. In this case, the spline priors are Gaussian processes. 

Multilayer Neural Networks and Gaussian Processes

  • Neal showed that the properties of a neural network with one hidden layer converges to those of a Gaussian process as the number of hidden neurons tends to infinity if the standard 'weight decay' priors are assumed. 
  • The covariance function of this Gaussian process depends on the details of the priors assumed for the weights in the network and the activation functions of the hidden units. 

Implementation

  • DIRECT: "The most obvious implementation fo these equations is to evaluate the inverse of the covariance matrix exactly. This can be done using a variety of methods such as Cholesky decomposition, LU decomposition or Gaussian-Jordan. Having obtained the explicit inverse, we then apply it directly to the appropriate vectors." 
  • APPROXIMATION METHODS 

Slow Research: Disk and Sphere Projections by Siobhán Cronin

I have begun studying differential geometry, and have been looking for points of departure in everyday life to motivate my studies. Tonight while stirring a pot of mushroom soup, I noticed something interesting about the light's reflection in the beads of fat on the surface as I moved my head from side to side. I found that if I focussed on the surrounding surface, and allowed my eyes to take in the beads less directly, I could perceive each disk as a sphere resting on the surface. 

After I finished eating, I served up another bowl of soup to observe this phenomena more closely. I found that if I imagined the lamps reflection were actually a floating light "projected" by each of the spheres appearing on the soup's surface, my eye could stick with the trick more faithfully. It seemed assigning the imagined spheres this agency helped my mind hold onto the mirage. 

I can imagine the optics explanation of why my mind might be able to switch from seeing light reflected off a disk as thought it were reflecting off spheres would rest somewhere in the special relationship circles and spheres share, with no visible edges to break the illusion of an aerial view observed as a askance perspective. However I can't help wonder how these two different scenarios relate - the light reflecting off individual beads of fat on the soup's surface, and the imagined equivalently plausible scenario of suspended spheres (with our without the special light-producing agency). 

Here's the soup:

How might we define the relationship from the light to the beads and compare this to the light to the spheres? What traditions in geometry or topology might helps us do that? 

On a side note, I observed this evening that the reverse optic trick could be applied to the sun's reflection off the the surface of the moon. A quick web search for a geometric rendering of that led me to I discover that there are people who strongly believe we are circumambulating a flat earth. Wowzers!

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Tentacular thinking by Siobhán Cronin

I have become quite obnoxious as of late, reminding everyone that as holobionts, the self that many of us most closely identify with (our physical forms) is in fact a multi-species agreement. Meanwhile, my research in machine learning as a catalyst for advancing potentials of inference, imagination, and society building has led me to frontiers that seem to very much align with Donna Haraway's grounded treatments of the muddy messiness of life. 

I'm brewing up a talk that links multi-armed bandits and tentacular thinking, and will be reading this in the meantime: http://www.e-flux.com/journal/75/67125/tentacular-thinking-anthropocene-capitalocene-chthulucene/

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The Selection of Facts by Siobhán Cronin

Sri Henri Ponicaré has some thoughts on the matter:

"... care of the beautiful leads us to the same selection as care for the useful"

"economy of effort ... is a source of beauty as well as a practical advantage"

"... what we must aim at is not so much to ascertain resemblances and differences, as to discover similarities hidden under apparent discrepancies" 

"... when a rule has been established, we have first to look for the cases in which the rule stands the best chance of being found in fault"

"... facts which occur frequently appear to us simple just because we are accustomed to them"

Oh, and my favorite:

"Trying to make science contain nature is like trying to make the part contain the whole"

 

 

 

Kepler preaches Copernican astronomy by Siobhán Cronin

"I admit a soul in the body of the sun as the overseer of the rotation of the sun and as the superintendent of the movement of the whole world....But just as it was not necessary to introduce a special soul into the threads of the belly; for it is sufficient for one common soul from the heart or liver to advance, through its own form or through heat, into the belly and to employ the faculties of its threads; so too in the world that form - of light, or heat, and thus too, if you will - from the soul of the sun, flowing out together with light and heat and penetrating even when light and heat are shut out, i.e., into the inner threads of bodies, seems to be sufficient; hence, just as the soul in the body has no power without the organ of the belly, so too the soul of the world has no power without these laws and without the geometrical lay-out of bodies". 

Kepler - Epitome of Copernican Astronomy, IV

by Siobhán Cronin

And just like that, it seems I've arrived at the place I had determined would be the starting point. The moment when I would begin to find words for it all. That it has arrived before I feel ready seems fitting. When has it ever felt "right" to jump into the mess of finding words for what it means to experience living?

When I was in elementary school, I played jump rope with the other girls before class and during recess. A rhythmic symbiosis. When it was your turn to turn the ropes, you had to make sure you struck the ground right below the jumper's feet. Thwack. Thwack. Thwack. Thwack. The sound became the pulse, and assurance for the jumper of the rope's lower bounds. Standing just outside the elliptical blender, you somehow would convince your body it was possible to enter, springing forward as a ball of scrunched limbs, to land in a huddled mass of hopping urgency, no longer singing the songs that accompany the ropes, but concentrating all your attention on the relationship of feet, ground, and that sharp spark of sound.

It meant something to me that you tried on my words to see if they conveyed your feelings. After all that's happened, I sometimes feel my relationship to the primacy of cells knowing things and delivering what they know to the poetic knowledge aquifers below the palace of words, is perhaps the only true relic I have of that other place I was living. That and the non-dualism. But to see you try on the words gave me a surge of energy, as I realized that how we love each other exists here too, in these attempts to write and read. And what a beautiful lineage these more formal attempts efforts belong to. A history of letters, e-mails, notes left on desks, promises scrawled in cards. 

I wrote for a year, but in hindsight I wasn't writing to you. I was writing to stabilize my mind which seemed to be reeling from the momentum of everything that was expanding in and around me. I wrote to map the terrain you and I would someday explore. The landscape that would become the catalyst for discoveries that we couldn't neatly parse into yours or mine. 

 

 

Waulking songs by Siobhán Cronin

Can we reclaim the stories that were cast aside in our fear-driven allegiance to patriarchal lineage?  How might our children build schools, civic institutions, and systems of technology if they were raised to waulk the tweed and improvise songs with their elders?