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In this paper we consider the contextual bandit problem with a finite (or infinite and clustered) context set. We consider the fully adversarial problem in which, apart from having bounded losses, there are no assumptions whatsoever on the generation of the contexts and losses. In our problem we assume that the context set is partitioned into a set of protected groups. At the start of each trial we are given a probability distribution over the context set and are required (on that trial) to be fair with respect to that distribution, in that if the context (for that trial) was drawn from the distribution then our choice of action would be unbiased towards any protected group. We develop an algorithm FexEx for this problem which has remarkable efficiency, having a space and per-trial time complexity at most linear in the dimensionality of the policy space. FexEx can handle non-stationarity, in that its regret can be bounded with respect to any sequence of policies satisfying the fairness constraints. For such a sequence the regret bound of FexEx is essentially the same as that of running Exp3.S for each context independently (an approach that does not satisfy the fairness constraints).

We consider the classic problem of online convex optimisation. Whereas the notion of static regret is relevant for stationary problems, the notion of switching regret is more appropriate for non-stationary problems. A switching regret is defined relative to any segmentation of the trial sequence, and is equal to the sum of the static regrets of each segment. In this paper we show that, perhaps surprisingly, we can achieve the asymptotically optimal switching regret on every possible segmentation simultaneously. Our algorithm for doing so is very efficient: having a space and per-trial time complexity that is logarithmic in the time-horizon. Our algorithm also obtains novel bounds on its dynamic regret: being adaptive to variations in the rate of change of the comparator sequence.

We consider the problem of learning to map large instances, such as sequences and images, to outputs. Since training one large neural network end to end with backpropagation is plagued by vanishing gradients and degradation, we develop a novel neural network architecture called Extraction propagation, which works by training, in parallel, many small neural networks which interact with one another. We note that the performance of Extraction propagation is only conjectured as we have yet to implement it. We do, however, back the algorithm with some theory. A previous version of this paper was entitled "Fusion encoder networks" and detailed a slightly different architecture.

We study the classic problem of prediction with expert advice under bandit feedback. Our model assumes that one action, corresponding to the learner's abstention from play, has no reward or loss on every trial. We propose the CBA algorithm, which exploits this assumption to obtain reward bounds that can significantly improve those of the classical Exp4 algorithm. We can view our problem as the aggregation of confidence-rated predictors when the learner has the option of abstention from play. Importantly, we are the first to achieve bounds on the expected cumulative reward for general confidence-rated predictors. In the special case of specialists we achieve a novel reward bound, significantly improving previous bounds of SpecialistExp (treating abstention as another action). As an example application, we discuss learning unions of balls in a finite metric space. In this contextual setting, we devise an efficient implementation of CBA, reducing the runtime from quadratic to almost linear in the number of contexts. Preliminary experiments show that CBA improves over existing bandit algorithms.

In this paper we consider the adversarial contextual bandit problem in metric spaces. The paper "Nearest neighbour with bandit feedback" tackled this problem but when there are many contexts near the decision boundary of the comparator policy it suffers from a high regret. In this paper we eradicate this problem, designing an algorithm in which we can hold out any set of contexts when computing our regret term. Our algorithm builds on that of "Nearest neighbour with bandit feedback" and hence inherits its extreme computational efficiency.

Many online decision-making problems correspond to maximizing a sequence of submodular functions. In this work, we introduce sum-max functions, a subclass of monotone submodular functions capturing several interesting problems, including best-of-$K$-bandits, combinatorial bandits, and the bandit versions on facility location, $M$-medians, and hitting sets. We show that all functions in this class satisfy a key property that we call pseudo-concavity. This allows us to prove $\big(1 - \frac{1}{e}\big)$-regret bounds for bandit feedback in the nonstochastic setting of the order of $\sqrt{MKT}$ (ignoring log factors), where $T$ is the time horizon and $M$ is a cardinality constraint. This bound, attained by a simple and efficient algorithm, significantly improves on the $\widetilde{O}\big(T^{2/3}\big)$ regret bound for online monotone submodular maximization with bandit feedback.

In this paper we adapt the nearest neighbour rule to the contextual bandit problem. Our algorithm handles the fully adversarial setting in which no assumptions at all are made about the data-generation process. When combined with a sufficiently fast data-structure for (perhaps approximate) adaptive nearest neighbour search, such as a navigating net, our algorithm is extremely efficient - having a per trial running time polylogarithmic in both the number of trials and actions, and taking only quasi-linear space. We give generic regret bounds for our algorithm and further analyse them when applied to the stochastic bandit problem in euclidean space. We note that our algorithm can also be applied to the online classification problem.

We investigate the problem of online collaborative filtering under no-repetition constraints, whereby users need to be served content in an online fashion and a given user cannot be recommended the same content item more than once. We start by designing and analyzing an algorithm that works under biclustering assumptions on the user-item preference matrix, and show that this algorithm exhibits an optimal regret guarantee, while being fully adaptive, in that it is oblivious to any prior knowledge about the sequence of users, the universe of items, as well as the biclustering parameters of the preference matrix. We then propose a more robust version of this algorithm which operates with general matrices. Also this algorithm is parameter free, and we prove regret guarantees that scale with the amount by which the preference matrix deviates from a biclustered structure. To our knowledge, these are the first results on online collaborative filtering that hold at this level of generality and adaptivity under no-repetition constraints. Finally, we complement our theoretical findings with simple experiments on real-world datasets aimed at both validating the theory and empirically comparing to standard baselines. This comparison shows the competitive advantage of our approach over these baselines.

Coresets are small, weighted summaries of larger datasets, aiming at providing provable error bounds for machine learning (ML) tasks while significantly reducing the communication and computation costs. To achieve a better trade-off between ML error bounds and costs, we propose the first framework to incorporate quantization techniques into the process of coreset construction. Specifically, we theoretically analyze the ML error bounds caused by a combination of coreset construction and quantization. Based on that, we formulate an optimization problem to minimize the ML error under a fixed budget of communication cost. To improve the scalability for large datasets, we identify two proxies of the original objective function, for which efficient algorithms are developed. For the case of data on multiple nodes, we further design a novel algorithm to allocate the communication budget to the nodes while minimizing the overall ML error. Through extensive experiments on multiple real-world datasets, we demonstrate the effectiveness and efficiency of our proposed algorithms for a variety of ML tasks. In particular, our algorithms have achieved more than 90% data reduction with less than 10% degradation in ML performance in most cases.

We consider the problem of computing the k-means centers for a large high-dimensional dataset in the context of edge-based machine learning, where data sources offload machine learning computation to nearby edge servers. k-Means computation is fundamental to many data analytics, and the capability of computing provably accurate k-means centers by leveraging the computation power of the edge servers, at a low communication and computation cost to the data sources, will greatly improve the performance of these analytics. We propose to let the data sources send small summaries, generated by joint dimensionality reduction (DR), cardinality reduction (CR), and quantization (QT), to support approximate k-means computation at reduced complexity and communication cost. By analyzing the complexity, the communication cost, and the approximation error of k-means algorithms based on carefully designed composition of DR/CR/QT methods, we show that: (i) it is possible to compute near-optimal k-means centers at a near-linear complexity and a constant or logarithmic communication cost, (ii) the order of applying DR and CR significantly affects the complexity and the communication cost, and (iii) combining DR/CR methods with a properly configured quantizer can further reduce the communication cost without compromising the other performance metrics. Our theoretical analysis has been validated through experiments based on real datasets.

We introduce a novel online multitask setting. In this setting each task is partitioned into a sequence of segments that is unknown to the learner. Associated with each segment is a hypothesis from some hypothesis class. We give algorithms that are designed to exploit the scenario where there are many such segments but significantly fewer associated hypotheses. We prove regret bounds that hold for any segmentation of the tasks and any association of hypotheses to the segments. In the single-task setting this is equivalent to switching with long-term memory in the sense of [Bousquet and Warmuth; 2003]. We provide an algorithm that predicts on each trial in time linear in the number of hypotheses when the hypothesis class is finite. We also consider infinite hypothesis classes from reproducing kernel Hilbert spaces for which we give an algorithm whose per trial time complexity is cubic in the number of cumulative trials. In the single-task special case this is the first example of an efficient regret-bounded switching algorithm with long-term memory for a non-parametric hypothesis class.

In this paper, we provide a rigorous theoretical investigation of an online learning version of the Facility Location problem which is motivated by emerging problems in real-world applications. In our formulation, we are given a set of sites and an online sequence of user requests. At each trial, the learner selects a subset of sites and then incurs a cost for each selected site and an additional cost which is the price of the user's connection to the nearest site in the selected subset. The problem may be solved by an application of the well-known Hedge algorithm. This would, however, require time and space exponential in the number of the given sites, which motivates our design of a novel quasi-linear time algorithm for this problem, with good theoretical guarantees on its performance.

Mobile edge computing (MEC) is a promising technique for providing low-latency access to services at the network edge. The services are hosted at various types of edge nodes with both computation and communication capabilities. Due to the heterogeneity of edge node characteristics and user locations, the performance of MEC varies depending on where the service is hosted. In this paper, we consider such a heterogeneous MEC system, and focus on the problem of placing multiple services in the system to maximize the total reward. We show that the problem is NP-hard via reduction from the set cover problem, and propose a deterministic approximation algorithm to solve the problem, which has an approximation ratio that is not worse than $\left(1-e^{-1}\right)/4$. The proposed algorithm is based on two sub-routines that are suitable for small and arbitrarily sized services, respectively. The algorithm is designed using a novel way of partitioning each edge node into multiple slots, where each slot contains one service. The approximation guarantee is obtained via a specialization of the method of conditional expectations, which uses a randomized procedure as an intermediate step. In addition to theoretical guarantees, simulation results also show that the proposed algorithm outperforms other state-of-the-art approaches., Comment: Extended version of paper presented at IEEE INFOCOM 2019

We give an online algorithm and prove novel mistake and regret bounds for online binary matrix completion with side information. The mistake bounds we prove are of the form $\tilde{O}(D/\gamma^2)$. The term $1/\gamma^2$ is analogous to the usual margin term in SVM (perceptron) bounds. More specifically, if we assume that there is some factorization of the underlying $m \times n$ matrix into $P Q^\intercal$ where the rows of $P$ are interpreted as "classifiers" in $\mathcal{R}^d$ and the rows of $Q$ as "instances" in $\mathcal{R}^d$, then $\gamma$ is the maximum (normalized) margin over all factorizations $P Q^\intercal$ consistent with the observed matrix. The quasi-dimension term $D$ measures the quality of side information. In the presence of vacuous side information, $D= m+n$. However, if the side information is predictive of the underlying factorization of the matrix, then in an ideal case, $D \in O(k + \ell)$ where $k$ is the number of distinct row factors and $\ell$ is the number of distinct column factors. We additionally provide a generalization of our algorithm to the inductive setting. In this setting, we provide an example where the side information is not directly specified in advance. For this example, the quasi-dimension $D$ is now bounded by $O(k^2 + \ell^2)$.

We introduce a new online learning framework where, at each trial, the learner is required to select a subset of actions from a given known action set. Each action is associated with an energy value, a reward and a cost. The sum of the energies of the actions selected cannot exceed a given energy budget. The goal is to maximise the cumulative profit, where the profit obtained on a single trial is defined as the difference between the maximum reward among the selected actions and the sum of their costs. Action energy values and the budget are known and fixed. All rewards and costs associated with each action change over time and are revealed at each trial only after the learner's selection of actions. Our framework encompasses several online learning problems where the environment changes over time; and the solution trades-off between minimising the costs and maximising the maximum reward of the selected subset of actions, while being constrained to an action energy budget. The algorithm that we propose is efficient and general in that it may be specialised to multiple natural online combinatorial problems., Comment: Published in AISTATS 2019

Pairwise clustering, in general, partitions a set of items via a known similarity function. In our treatment, clustering is modeled as a transductive prediction problem. Thus rather than beginning with a known similarity function, the function instead is hidden and the learner only receives a random sample consisting of a subset of the pairwise similarities. An additional set of pairwise side-information may be given to the learner, which then determines the inductive bias of our algorithms. We measure performance not based on the recovery of the hidden similarity function, but instead on how well we classify each item. We give tight bounds on the number of misclassifications. We provide two algorithms. The first algorithm SACA is a simple agglomerative clustering algorithm which runs in near linear time, and which serves as a baseline for our analyses. Whereas the second algorithm, RGCA, enables the incorporation of side-information which may lead to improved bounds at the cost of a longer running time.

The junction tree algorithm is a way of computing marginals of boolean multivariate probability distributions that factorise over sets of random variables. The junction tree algorithm first constructs a tree called a junction tree who's vertices are sets of random variables. The algorithm then performs a generalised version of belief propagation on the junction tree. The Shafer-Shenoy and Hugin architectures are two ways to perform this belief propagation that tradeoff time and space complexities in different ways: Hugin propagation is at least as fast as Shafer-Shenoy propagation and in the cases that we have large vertices of high degree is significantly faster. However, this speed increase comes at the cost of an increased space complexity. This paper first introduces a simple novel architecture, ARCH-1, which has the best of both worlds: the speed of Hugin propagation and the low space requirements of Shafer-Shenoy propagation. A more complicated novel architecture, ARCH-2, is then introduced which has, up to a factor only linear in the maximum cardinality of any vertex, time and space complexities at least as good as ARCH-1 and in the cases that we have large vertices of high degree is significantly faster than ARCH-1.

We consider online similarity prediction problems over networked data. We begin by relating this task to the more standard class prediction problem, showing that, given an arbitrary algorithm for class prediction, we can construct an algorithm for similarity prediction with "nearly" the same mistake bound, and vice versa. After noticing that this general construction is computationally infeasible, we target our study to {\em feasible} similarity prediction algorithms on networked data. We initially assume that the network structure is {\em known} to the learner. Here we observe that Matrix Winnow \cite{w07} has a near-optimal mistake guarantee, at the price of cubic prediction time per round. This motivates our effort for an efficient implementation of a Perceptron algorithm with a weaker mistake guarantee but with only poly-logarithmic prediction time. Our focus then turns to the challenging case of networks whose structure is initially {\em unknown} to the learner. In this novel setting, where the network structure is only incrementally revealed, we obtain a mistake-bounded algorithm with a quadratic prediction time per round.