We present an information-theoretic framework for solving global black-box optimization problems that also have black-box constraints. Of particular interest to us is to efficiently solve problems with decoupled constraints, in which subsets of the objective and constraint functions may be evaluated independently. For example, when the objective is evaluated on a CPU and the constraints are evaluated independently on a GPU. These problems require an acquisition function that can be separated into the contributions of the individual function evaluations. We develop one such acquisition function and call it Predictive Entropy Search with Constraints (PESC). PESC is an approximation to the expected information gain criterion and it compares favorably to alternative approaches based on improvement in several synthetic and real-world problems. In addition to this, we consider problems with a mix of functions that are fast and slow to evaluate. These problems require balancing the amount of time spent in the meta-computation of PESC and in the actual evaluation of the target objective. We take a bounded rationality approach and develop partial update for PESC which trades off accuracy against speed. We then propose a method for adaptively switching between the partial and full updates for PESC. This allows us to interpolate between versions of PESC that are efficient in terms of function evaluations and those that are efficient in terms of wall-clock time. Overall, we demonstrate that PESC is an effective algorithm that provides a promising direction towards a unified solution for constrained Bayesian optimization.
In the constrained setting, the incumbent @math can be defined as the minimum expected objective value subject to all the constraints being satisfied at the corresponding location. However, we can never guarantee that all the constraints will be satisfied when they are only observed through noisy evaluations. To circumvent this problem, @cite_10 define @math as the lowest expected objective value subject to all the constraints being satisfied with posterior probability larger than the threshold @math , where @math is a small number such as @math . However, this value for @math still cannot be computed when there is no point in the search space that satisfies the constraints with posterior probability higher than @math . For example, because of lack of data for the constraints. In this case, Gelbart2014 change the original acquisition function given by eq:background:constrained-bo:EIC and ignore the factor @math in that expression. This allows them to search only for a feasible location, ignoring the objective @math entirely and just optimizing the constraint satisfaction probability. However, this can lead to inefficient optimization in practice because the data collected for the objective @math is not used to make optimal decisions.
@cite_10
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Given @math noisy samples with @math dimensions, where @math , we show that the multi-step thresholding procedure based on the Lasso -- we call it the Thresholded Lasso , can accurately estimate a sparse vector @math in a linear model @math , where @math is a design matrix normalized to have column @math norm @math , and @math . We show that under the restricted eigenvalue (RE) condition (Bickel-Ritov-Tsybakov 09), it is possible to achieve the @math loss within a logarithmic factor of the ideal mean square error one would achieve with an oracle while selecting a sufficiently sparse model -- hence achieving sparse oracle inequalities ; the oracle would supply perfect information about which coordinates are non-zero and which are above the noise level. In some sense, the Thresholded Lasso recovers the choices that would have been made by the @math penalized least squares estimators, in that it selects a sufficiently sparse model without sacrificing the accuracy in estimating @math and in predicting @math . We also show for the Gauss-Dantzig selector (Cand e s-Tao 07), if @math obeys a uniform uncertainty principle and if the true parameter is sufficiently sparse, one will achieve the sparse oracle inequalities as above, while allowing at most @math irrelevant variables in the model in the worst case, where @math is the smallest integer such that for @math , @math . Our simulation results on the Thresholded Lasso match our theoretical analysis excellently. We consider the problem of provably optimal exploration in reinforcement learning for finite horizon MDPs. We show that an optimistic modification to value iteration achieves a regret bound of @math where @math is the time horizon, @math the number of states, @math the number of actions and @math the number of time-steps. This result improves over the best previous known bound @math achieved by the UCRL2 algorithm of , 2010. The key significance of our new results is that when @math and @math , it leads to a regret of @math that matches the established lower bound of @math up to a logarithmic factor. Our analysis contains two key insights. We use careful application of concentration inequalities to the optimal value function as a whole, rather than to the transitions probabilities (to improve scaling in @math ), and we define Bernstein-based "exploration bonuses" that use the empirical variance of the estimated values at the next states (to improve scaling in @math ). We present an algorithm for the statistical learning setting with a bounded exp-concave loss in @math dimensions that obtains excess risk @math with probability at least @math . The core technique is to boost the confidence of recent in-expectation @math excess risk bounds for empirical risk minimization (ERM), without sacrificing the rate, by leveraging a Bernstein condition which holds due to exp-concavity. We also show that with probability @math the standard ERM method obtains excess risk @math . We further show that a regret bound for any online learner in this setting translates to a high probability excess risk bound for the corresponding online-to-batch conversion of the online learner. Lastly, we present two high probability bounds for the exp-concave model selection aggregation problem that are quantile-adaptive in a certain sense. The first bound is a purely exponential weights type algorithm, obtains a nearly optimal rate, and has no explicit dependence on the Lipschitz continuity of the loss. The second bound requires Lipschitz continuity but obtains the optimal rate. Many NP-complete constraint satisfaction problems appear to undergo a “phase transition” from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment (expectation) of the number of solutions tends to zero. We show that in the case of certain symmetric constraints, considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold. Specifically, we prove that the threshold for both random hypergraph 2-colorability (Property B) and random Not-All-Equal @math -SAT is @math . As a corollary, we establish that the threshold for random @math -SAT is of order @math , resolving a long-standing open problem.