Section: New Results

Continuous Optimization

• Markov Chain Analysis of Evolution Strategies The theory of Markov chains with discrete time and continuous state space turns out to be very useful to analyze the convergence of adaptive evolution strategies (including simplified versions of the state-of-the are CMA-ES). Exploiting invariance of the algorithms, we can indeed construct homogeneous Markov chains underlying the algorithms whose stability implies the linear convergence of the algorithm [65] . We have also shown how the convergence on constrained problems can be analyzed with Markov chains theory [10] .

  However the stability can be very difficult to prove; even the irreducibility can be very challenging to prove with current Markov chain theory. We have hence been developing new theoretical tools exploiting deterministic control models to prove more easily the irreducibility and T-chain property of general Markov chains [67] . Those theoretical tools can be applied to the optimization algorithms we are interested in, and trivialize some stability proofs [1] , [10] .

• Benchmarking of continuous optimizers We have been pursuing our effort towards improving the standards in benchmarking of continuous optimizers. We tackled the benchmarking of bi-objective problems and transferred and adapted standard benchmarking techniques from the single-objective optimization and classical derivative free optimization community to the field of EMO [28] . In addition, we have been rewritting part of the COCO platform to improve its modularity and make it less error prone and started its extension to multiobjective optimization.

• Concentration inequalities for sampling without replacement We studied the concentration of measure phenomenon in the case of sampling without replacement, which is directly relevant for a recent MCMC techique for large data sets, see [7] accepted to the Bernoulli journal.

• Random projections for confident MCMC In the paper [66] accepted at the NIPS "Bayesian Optimization Workshop", we study the benefit of replacing uniform subsampling by random projections in recent MCMC techniques for linear regression of tall datasets.

• Automatic step size adaptation We have derived a new, low-cost strategy for online adaptation of the step size in stochastic gradient descent and related algorithms [72] . This problem is of crucial importance in many machine learning algorithms, as current approaches often rely on exploring a grid of step sizes and performing a full optimization for each of them, a lengthy process.