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# Meta-Optimization and Optimal Optimizers

Once we have a formal definition of optimizer performance and a prior distribution over objective functions (a function prior $\mathbb{P}_F$), then we can ask what the average performance of an optimizer is against a particular function prior. Mathematically, for a performance criterion $\phi$, this is $$\left<\mathcal{G},\mathbb{P}_F\right>_\phi = \mathbb{E}_{\mathbb{P}_F}\left[\phi\left(\mathcal{G},F\right)\right].$$ This equation itself may be optimized over some optimizer set $\mathfrak{G}$ to find the optimizer $\mathcal{G}$ that has optimal performance on $\mathbb{P}_F$, i.e. $$\mathcal{G}_{\mathrm{opt}} = \mathrm{argmax}_{\mathcal{G}\in\mathfrak{G}}\,\,\left<\mathcal{G},\mathbb{P}_F\right>_\phi.$$ The problem of finding the best optimizer under a given set of assumption about the uncertainty of the objective function is termed *meta-optimization*.

## Parameter Meta-Optimization

The optimizer set $\mathfrak{G}$ might be a space of parameterized versions of an optimization method, in which case meta-optimization seeks to discover the best parameters for a particular method. For example, Differential Evolution (DE) has three parameters: the population size $K$, the crossover rate $CR$, and the learning rate $F$. Meta-optimization over DE seeks to assign optimal values to these three parameters.

## Optimal Optimizers

Meta-optimization may also be considered in a more general setting. One might ask for the best possible trajectory-restricted optimizer, in which case $\mathfrak{G} = \mathcal{O}_{\mathrm{tr}}$. In that case, the result of meta-optimization is an optimal optimizer in a general sense, i.e. an optimizer that performs better than DE, CMA-ES, Nelder Mead, or any other trajectory-restricted method when tested on the function prior $\mathbb{P}_F$.

# About Me

I am looking for an assistant professorship to research the theory of feedback controllers for the control of complex autonomous systems, from smart homes to self-driving cars and humanoid robots. A CV and research statement can be found in the links to the left.

I have published on the theory of global optimization, humanoid robotics, neural networks for perception and control, and opponent modelling in games, and am working on a book expanding my Ph.D. thesis about the theory of global optimization under contract with Springer.

I am currently a postdoctoral fellow at the Dalle Molle Institute for Artificial Intelligence Studies on a US National Science Foundation postdoc grant working with Juergen Schmidhuber in Lugano, Switzerland. My Ph.D. is from the University of Texas where I studied with Risto Miikkulainen. See my About page for contact information and more.