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Generalized Probabilistic Bisection for Stochastic Root-Finding
This thesis studies the stochastic root-finding problem, which consists of estimating the point x∗ that solves the equation h(x∗) = 0, where the function h : (0,1) → R is learned via a stochastic simulator (oracle). Instead of focusing on modeling h(·), we develop statistical methodologies that directly infer x∗ following a fully Bayesian approach. To do so, we investigate procedures that generalize the Probabilistic Bisection Algorithm (PBA) first introduced in Horstein (1963). The PBA is a one-dimensional stochastic root-finding routine which builds an explicit Bayesian representation (i.e., a posterior density) for x∗ based on the history of noisy function evaluations and sampling locations. The PBA starts by assuming that x∗ is the realized value of an absolutely continuous random variable, X∗ ∼ g0, with prior density g0. Then, it recursively updates a posterior, gn, leveraging the information provided by the signs (positive/negative) of the noisy function evaluations — which inform the direction where x∗ is located with respect to a given location, x—. Due to observational noise, the oracle responses are correct only with probability p(x). Waeber et al. (2013) showed that sampling at the median of gn is an optimal sampling strategy and established exponential convergence of the posterior gn to a Dirac mass at the true x∗ under the very restrictive assumption that the probability of correct response p(x) is known and constant for all x; however, in the most general and practical settings the latter condition no longer holds and the only way to implement the PBA is to estimate p(·).In the first part of this thesis, we state the Generalized PBA (G-PBA), where the above assumption is relaxed to the case where the sampling distribution of the oracle is unknown and location-dependent. Namely, as in standard PBA, we rely on a knowledge state to approximate the posterior of the root location. To implement the corresponding Bayesian updating, we also carry out inference of p(·). To this end we utilize batched querying in combination with a variety of frequentist and Bayesian estimators based on majority vote, as well as the underlying functional responses, if available. For guiding sampling selection we propose two families of sampling policies: batched Information Di- rected Sampling and Randomized Quantile Sampling, which is a reminiscent of Thompson Sampling and a generalization of the median sampling as in classical PBA. The latter leads to the first main conclusion: the G-PBA is able to efficiently learn p(·) and X∗ simultaneously.In the second part of this thesis, we propose to leverage the spatial structure of a typical oracle by constructing a non-parametric statistical surrogate for p(·) based on binomial regression. The latter leads to the second main conclusion: surrogate modeling allows to determine the batch size for querying the oracle adaptively as a function of the estimated predictive uncertainty of p(·).In the last part of this thesis, we present extensive numerical experiments in order to evaluate our sampling strategies (information-based or randomized). In particular we demonstrate the efficiency of randomized quantile sampling for balancing the ex- ploration/exploitation component; moreover, we show that spatial surrogate modeling results in significant gains relative to the local estimators, as quantified by the improved quality of the resulting root estimates (namely lower absolute residuals, narrower credible intervals and dramatically higher probability coverage). Our work is motivated by the root-finding sub-routine in pricing of Bermudan financial derivatives, illustrated in the last section of this thesis
Thompson sampling guided stochastic searching on the line for deceptive environments with applications to root-finding problems
publishedVersio
Towards Thompson Sampling for Complex Bayesian Reasoning
Paper III, IV, and VI are not available as a part of the dissertation due to the copyright.Thompson Sampling (TS) is a state-of-art algorithm for bandit problems set in a Bayesian framework. Both the theoretical foundation and the empirical efficiency of TS is wellexplored for plain bandit problems. However, the Bayesian underpinning of TS means that TS could potentially be applied to other, more complex, problems as well, beyond the bandit problem, if suitable Bayesian structures can be found.
The objective of this thesis is the development and analysis of TS-based schemes for more complex optimization problems, founded on Bayesian reasoning. We address several complex optimization problems where the previous state-of-art relies on a relatively myopic perspective on the problem. These includes stochastic searching on the line, the Goore game, the knapsack problem, travel time estimation, and equipartitioning. Instead of employing Bayesian reasoning to obtain a solution, they rely on carefully engineered rules. In all brevity, we recast each of these optimization problems in a Bayesian framework, introducing dedicated TS based solution schemes. For all of the addressed problems, the results show that besides being more effective, the TS based approaches we introduce are also capable of solving more adverse versions of the problems, such as dealing with stochastic liars.publishedVersio
Projected Power Iteration for Network Alignment
The network alignment problem asks for the best correspondence between two
given graphs, so that the largest possible number of edges are matched. This
problem appears in many scientific problems (like the study of protein-protein
interactions) and it is very closely related to the quadratic assignment
problem which has graph isomorphism, traveling salesman and minimum bisection
problems as particular cases. The graph matching problem is NP-hard in general.
However, under some restrictive models for the graphs, algorithms can
approximate the alignment efficiently. In that spirit the recent work by Feizi
and collaborators introduce EigenAlign, a fast spectral method with convergence
guarantees for Erd\H{o}s-Reny\'i graphs. In this work we propose the algorithm
Projected Power Alignment, which is a projected power iteration version of
EigenAlign. We numerically show it improves the recovery rates of EigenAlign
and we describe the theory that may be used to provide performance guarantees
for Projected Power Alignment.Comment: 8 page
Fast projections onto mixed-norm balls with applications
Joint sparsity offers powerful structural cues for feature selection,
especially for variables that are expected to demonstrate a "grouped" behavior.
Such behavior is commonly modeled via group-lasso, multitask lasso, and related
methods where feature selection is effected via mixed-norms. Several mixed-norm
based sparse models have received substantial attention, and for some cases
efficient algorithms are also available. Surprisingly, several constrained
sparse models seem to be lacking scalable algorithms. We address this
deficiency by presenting batch and online (stochastic-gradient) optimization
methods, both of which rely on efficient projections onto mixed-norm balls. We
illustrate our methods by applying them to the multitask lasso. We conclude by
mentioning some open problems.Comment: Preprint of paper under revie
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