Single-objective black box optimization (also known as zeroth-order optimization) is the process of minimizing a scalar objective $f(x)$, given evaluations at adaptively chosen inputs $x$. In this paper, we consider multi-objective optimization, where $f(x)$ outputs a vector of possibly competing objectives and the goal is to converge to the Pareto frontier. Quantitatively, we wish to maximize the standard \emph{hypervolume indicator} metric, which measures the dominated hypervolume of the entire set of chosen inputs. In this paper, we introduce a novel scalarization function, which we term the \emph{hypervolume scalarization}, and show that drawing random scalarizations from an appropriately chosen distribution can be used to efficiently approximate the \emph{hypervolume indicator} metric. We utilize this connection to show that Bayesian optimization with our scalarization via common acquisition functions, such as Thompson Sampling or Upper Confidence Bound, provably converges to the whole Pareto frontier by deriving tight \emph{hypervolume regret} bounds on the order of $\widetilde{O}(\sqrt{T})$. Furthermore, we highlight the general utility of our scalarization framework by showing that any provably convergent single-objective optimization process can be converted to a multi-objective optimization process with provable convergence guarantees.

Any sufficiently complex system acts as a black box when it becomes easier to experiment with than to understand. Hence, black-box optimization has become increasingly important as systems have become more complex. In this paper we describe Google Vizier, a Google-internal service for performing black-box optimization that has become the de facto parameter tuning engine at Google. Google Vizier is used to optimize many of our machine learning models and other systems, and also provides core capabilities to Google's Cloud Machine Learning HyperTune subsystem. We discuss our requirements, infrastructure design, underlying algorithms, and advanced features such as transfer learning and automated early stopping that the service provides.

Machine learning offers a fantastically powerful toolkit for building useful complexprediction systems quickly. This paper argues it is dangerous to think ofthese quick wins as coming for free. Using the software engineering frameworkof technical debt, we find it is common to incur massive ongoing maintenancecosts in real-world ML systems. We explore several ML-specific risk factors toaccount for in system design. These include boundary erosion, entanglement,hidden feedback loops, undeclared consumers, data dependencies, configurationissues, changes in the external world, and a variety of system-level anti-patterns.

We reduce the memory footprint of popular large-scale online learning methods by projecting our weight vector onto a coarse discrete set using randomized rounding. Compared to standard 32-bit float encodings, this reduces RAM usage by more than 50% during training and by up 95% when making predictions from a fixed model, with almost no loss in accuracy. We also show that randomized counting can be used to implement per-coordinate learning rates, improving model quality with little additional RAM. We prove these memory-saving methods achieve regret guarantees similar to their exact variants. Empirical evaluation confirms excellent performance, dominating standard approaches across memory versus accuracy tradeoffs.

Predicting ad click-through rates (CTR) is a massive-scale learning problem that is central to the multi-billion dollar online advertising industry. We present a selection of case studies and topics drawn from recent experiments in the setting of a deployed CTR prediction system. These include improvements in the context of traditional supervised learning based on an FTRL-Proximal online learning algorithm (which has excellent sparsity and convergence properties) and the use of per-coordinate learning rates.

How should we manage a sensor network to opti-mally guard security-critical infrastructure? How should we coordinate search and rescue helicopters to best locate survivors after a major disaster? In both applications, we would like to control sensing resources in uncertain, adversarial environments. In this paper, we introduce RSENSE, an efficient algo-rithm which guarantees near-optimal randomized sensing strategies whenever the detection perfor-mance satisfies submodularity, a natural diminishing returns property, for any fixed adversarial scenario. Our approach combines techniques from game the-ory with submodular optimization. The RSENSE algorithm applies to settings where the goal is to manage a deployed sensor network or to coordinate mobile sensing resources (such as unmanned aerial vehicles). We evaluate our algorithms on two real– world sensing problems. 1

Recent results in complexity theory suggest that various economic theories require agents to solve computationally intractable problems. However, such results assume the agents are optimizing explicit utility functions, whereas the economic theories merely assume the agents behave rationally, where rational behavior is defined via some optimization problem. Might making rational choices be easier than solving the corresponding optimization problem? For at least one major economic theory, the theory of the consumer (which simply postulates that consumers are utility maximizing), we find this is indeed the case. In other words, we prove the possibly surprising result that computational constraints have no empirical consequences for consumer choice theory.

Many problems in artificial intelligence require adaptively making a sequence of decisions with uncertain outcomes under partial observability. Solving such stochastic optimization problems is a fundamental but notoriously difficult challenge. In this paper, we introduce the concept of adaptive submodularity, generalizing submodular set functions to adaptive policies. We prove that if a problem satisfies this property, a simple adaptive greedy algorithm is guaranteed to be competitive with the optimal policy. In addition to providing performance guarantees for both stochastic maximization and coverage, adaptive submodularity can be exploited to drastically speed up the greedy algorithm by using lazy evaluations. We illustrate the usefulness of the concept by giving several examples of adaptive submodular objectives arising in diverse AI applications including management of sensing resources, viral marketing and active learning. Proving adaptive submodularity for these problems allows us to recover existing results in these applications as special cases, improve approximation guarantees and handle natural generalizations.

A key problem in sensor networks is to decide which sensors to query when, in order to obtain the most useful information (e.g., for performing accurate prediction), subject to constraints (e.g., on power and bandwidth). In many applications the utility function is not known a priori, must be learned from data, and can even change over time. Furthermore for large sensor networks solving a centralized optimization problem to select sensors is not feasible, and thus we seek a fully distributed solution. In this paper, we present Distributed Online Greedy (DOG), an efficient, distributed algorithm for repeatedly selecting sensors online, only receiving feedback about the utility of the selected sensors. We prove very strong theoretical no-regret guarantees that apply whenever the (unknown) utility function satisfies a natural diminishing returns property called submodularity. Our algorithm has extremely low communication requirements, and scales well to large sensor deployments. We extend DOG to allow observation-dependent sensor selection. We empirically demonstrate the effectiveness of our algorithm on several real-world sensing tasks.

We tackle the fundamental problem of Bayesian active learning with noise, where we need to adaptively select from a number of expensive tests in order to identify an unknown hypothesis sampled from a known prior distribution. In the case of noise-free observations, a greedy algorithm called generalized binary search (GBS) is known to perform near-optimally. We show that if the observations are noisy, perhaps surprisingly, GBS can perform very poorly. We develop EC2, a novel, greedy active learning algorithm and prove that it is competitive with the optimal policy, thus obtaining the first competitiveness guarantees for Bayesian active learning with noisy observations. Our bounds rely on a recently discovered diminishing returns property called adaptive submodularity, generalizing the classical notion of submodular set functions to adaptive policies. Our results hold even if the tests have non–uniform cost and their noise is correlated. We also propose EffECXtive, a particularly fast approximation of EC2, and evaluate it on a Bayesian experimental design problem involving human subjects, intended to tease apart competing economic theories of how people make decisions under uncertainty.

Which ads should we display in sponsored search in order to maximize our revenue? How should we dynamically rank information sources to maximize value of information? These applications exhibit strong diminishing returns: Selection of redundant ads and information sources decreases their marginal utility. We show that these and other problems can be formalized as repeatedly selecting an assignment of items to positions to maximize a sequence of monotone submodular functions that arrive one by one. We present an efficient algorithm for this general problem and analyze it in the no-regret model. Our algorithm is equipped with strong theoretical guarantees, with a performance ratio that converges to the optimal constant of 1-1/e. We empirically evaluate our algorithms on two real-world online optimization problems on the web: ad allocation with submodular utilities, and dynamically ranking blogs to detect information cascades.

We consider the problem of simultaneous source location: selecting locations for sources in a capacitated graph such that a given set of demands can be satisfied simultaneously, with the goal of minimizing the number of locations chosen. For general directed and undirected graphs we give an O (log D )-approximation algorithm, where D is the sum of demands, and prove matching Ω(log D ) hardness results assuming P ≠ NP . For undirected trees, we give an exact algorithm and show how this can be combined with a result of Räcke to give a solution that exceeds edge capacities by at most O (log 2 n log log n ), where n is the number of nodes. For undirected graphs of bounded treewidth we show that the problem is still NP -hard, but we are able to give a PTAS with at most (1 + ϵ) violation of the capacities for arbitrarily small ϵ, or a ( k +1) approximation with exact capacities, where k is the treewidth.

In many optimization problems, a solution can be viewed as ascribing a ``cost\'\' to each client, and the goal is to optimize some aggregation of the per-client costs. We often optimize some $L_p$-norm (or some other symmetric convex function or norm) of the vector of costs---though different applications may suggest different norms to use. Ideally, we could obtain a solution that optimizes several norms simultaneously. In this paper, we examine approximation algorithms that simultaneously perform well on all norms, or on all $L_p$ norms. A natural problem in this framework is the $L_p$ Set Cover problem, which generalizes \textsc{Set Cover} and \textsc{Min-Sum Set Cover}. We show that the greedy algorithm \emph{simultaneously gives a $(p + \ln p + O(1))$-approximation for all $p$, and show that this approximation ratio is optimal up to constants} under reasonable complexity-theoretic assumptions. We additionally show how to use our analysis techniques to give similar results for the more general \emph{submodular set cover}, and prove some results for the so-called \emph{pipelined set cover} problem. We then go on to examine approximation algorithms in the ``all-norms\'\' and the ``all-$L_p$-norms\'\' frameworks more broadly, and present algorithms and structural results for other problems such as $k$-facility-location, TSP, and average flow-time minimization, extending and unifying previously known results.

We present an algorithm for solving a broad class of online resource allocation problems. Our online algorithm can be applied in environments where abstract jobs arrive one at a time, and one can complete the jobs by investing time in a number of abstract activities, according to some schedule. We assume that the fraction of jobs completed by a schedule is a monotone, submodular function of a set of pairs (v,t), where t is the time invested in activity v. Under this assumption, our online algorithm performs near-optimally according to two natural metrics: (i) the fraction of jobs completed within time T, for some fixed deadline T > 0, and (ii) the average time required to complete each job. We evaluate our algorithm experimentally by using it to learn, online, a schedule for allocating CPU time among solvers entered in the 2007 SAT solver competition.

We present a strongly history independent (SHI) hash table that supports search in O(l) worst-case time, and insert and delete in O(l) expected time using O(n) data space. This matches the bounds for dynamic perfect hashing, and improves on the best previous results by Naor and league on history independent hashing, which were either weakly history independent, or only supported insertion and search (no delete) each in O(l) expected time. The results can be used to construct many other SHI data structures. We show straightforward constructions for SHI ordered dictionaries: for n keys from {l,..., nk} searches take O(log log n) worst-case time and updates (insertions and deletions) O(log log n) expected time, and for keys in the comparison model searches take O(log n) worst-case time and updates O(log n) expected time. We also describe a SHI data structure for the order-maintenance problem. It supports comparisons in O(l) worst-case time, and updates in 0(1) expected time. All structures use O(n) data space.

Motivated by the problem of centralized market clearing in a market with probabilistic supply and demand, we introduce the Stochastic Packing-Market Planning problem (SPMP), which is a stochastic generalization of the Maximum k-Set Packing problem. We provide an O(k) approximation algorithm for SPMP, as well as a O(k) approximation mechanism that is truthful in expectation. This matches up to constants the best approximation ratio known for Maximum k-Set Packing. Along the way, we develop techniques for obtaining sparse subhypergraphs of an input hypergraph that preserves EP(·) up to an O(k) factor, where EP(G) measures the expected weight of a maximum weight set packing in a random subhypergraph of G. We also give a linear programming based approximation for EP(G). These techniques may be of independent interest.

A quorum system over a universe of logical elements is a collection of subsets (quorums) of elements, any two of which intersect. In numerous distributed algorithms, the elements of the universe reside on the nodes of a physical network and the participating nodes access the system by contacting every element in some quorum, potentially causing the added network congestion induced by these quorum accesses to play a limiting factor in the performance of the algorithm.In this paper we initiate the study of algorithms to place universe elements on the nodes of a physical network so as to minimize the network congestion that results from quorum accesses, while also ensuring that no physical node is overloaded by access requests from clients. We consider two models, one in which communication routes can be chosen arbitrarily and one in which they are fixed in advance. We show that in either model, the optimal congestion (with respect to the load constraints) cannot be approximated to any factor (unless P=NP). However, we show that at most doubling the load on nodes allows us to achieve a congestion that is close to this optimal value. We also shed some light on the extent to which element migration can reduce congestion in this context.