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In many high-throughput experimental design settings, such as those common in biochemical engineering, batched queries are often more cost effective than one-by-one sequential queries. Furthermore, it is often not possible to directly choose items to query. Instead, the experimenter specifies a set of constraints that generates a library of possible items, which are then selected stochastically. Motivated by these considerations, we investigate Batched Stochastic Bayesian Optimization (BSBO), a novel Bayesian optimization scheme for choosing the constraints in order to guide exploration towards items with greater utility. We focus on site saturation mutagenesis, a prototypical setting of BSBO in biochemical engineering, and propose a natural objective function for this problem. Importantly, we show that our objective function can be efficiently decomposed as a difference of submodular functions (DS), which allows us to employ DS optimization tools to greedily identify sets of constraints that increase the likelihood of finding items with high utility. Our experimental results show that our algorithm outperforms common heuristics on both synthetic and two real protein datasets. 

How can we efficiently gather information to optimize an unknown function, when presented with multiple, mutually dependent information sources with different costs? For example, when optimizing a physical system, intelligently trading off computer simulations and real-world tests can lead to significant savings. Existing multi-fidelity Bayesian optimization methods, such as multi-fidelity GP-UCB or Entropy Search-based approaches, either make simplistic assumptions on the interaction among different fidelities or use simple heuristics that lack theoretical guarantees. In this paper, we study multifidelity Bayesian optimization with complex structural dependencies among multiple outputs, and propose MF-MI-Greedy, a principled algorithmic framework for addressing this problem. In particular, we model different fidelities using additive Gaussian processes based on shared latent relationships with the target function. Then we use cost-sensitive mutual information gain for efficient Bayesian optimization. We propose a simple notion of regret which incorporates the varying cost of different fidelities, and prove that MF-MI-Greedy achieves low regret. We demonstrate the strong empirical performance of our algorithm on both synthetic and real-world datasets. 

This paper considers the problem of measure estimation under the barycentric coding model (BCM), in which an unknown measure is assumed to belong to the set of Wasserstein-2 barycenters of a finite set of known measures. Estimating a measure under this model is equivalent to estimating the unknown barycentric coordinates. We provide novel geometrical, statistical, and computational insights for measure estimation under the BCM, consisting of three main results. Our first main result leverages the Riemannian geometry of Wasserstein-2 space to provide a procedure for recovering the barycentric coordinates as the solution to a quadratic optimization problem assuming access to the true reference measures. The essential geometric insight is that the parameters of this quadratic problem are determined by inner products between the optimal displacement maps from the given measure to the reference measures defining the BCM. Our second main result then establishes an algorithm for solving for the coordinates in the BCM when all the measures are observed empirically via i.i.d. samples. We prove precise rates of convergence for this algorithm—determined by the smoothness of the underlying measures and their dimensionality—thereby guaranteeing its statistical consistency. Finally, we demonstrate the utility of the BCM and associated estimation procedures in three application areas: (i) covariance estimation for Gaussian measures; (ii) image processing; and (iii) natural language processing.