Mark Herbster

Danial Dervovic, Mark Herbster, Peter Mountney et al.

The Harrow-Hassidim-Lloyd (HHL) quantum algorithm for sampling from the solution of a linear system provides an exponential speed-up over its classical counterpart. The problem of solving a system of linear equations has a wide scope of applications, and thus HHL constitutes an important algorithmic primitive. In these notes, we present the HHL algorithm and its improved versions in detail, including explanations of the constituent sub- routines. More specifically, we discuss various quantum subroutines such as quantum phase estimation and amplitude amplification, as well as the important question of loading data into a quantum computer, via quantum RAM. The improvements to the original algorithm exploit variable-time amplitude amplification as well as a method for implementing linear combinations of unitary operations (LCUs) based on a decomposition of the operators using Fourier and Chebyshev series. Finally, we discuss a linear solver based on the quantum singular value estimation (QSVE) subroutine.

Stephen Pasteris, Fabio Vitale, Mark Herbster et al.

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.

Shota Saito, Mark Herbster

This paper develops an approximation to the (effective) $p$-resistance and applies it to multi-class clustering. Spectral methods based on the graph Laplacian and its generalization to the graph $p$-Laplacian have been a backbone of non-euclidean clustering techniques. The advantage of the $p$-Laplacian is that the parameter $p$ induces a controllable bias on cluster structure. The drawback of $p$-Laplacian eigenvector based methods is that the third and higher eigenvectors are difficult to compute. Thus, instead, we are motivated to use the $p$-resistance induced by the $p$-Laplacian for clustering. For $p$-resistance, small $p$ biases towards clusters with high internal connectivity while large $p$ biases towards clusters of small "extent," that is a preference for smaller shortest-path distances between vertices in the cluster. However, the $p$-resistance is expensive to compute. We overcome this by developing an approximation to the $p$-resistance. We prove upper and lower bounds on this approximation and observe that it is exact when the graph is a tree. We also provide theoretical justification for the use of $p$-resistance for clustering. Finally, we provide experiments comparing our approximated $p$-resistance clustering to other $p$-Laplacian based methods.

Stephen Pasteris, Alberto Rumi, Maximilian Thiessen et al.

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.

James Robinson, Mark Herbster

We address the problem of sequential prediction with expert advice in a non-stationary environment with long-term memory guarantees in the sense of Bousquet and Warmuth [4]. We give a linear-time algorithm that improves on the best known regret bounds [26]. This algorithm incorporates a relative entropy projection step. This projection is advantageous over previous weight-sharing approaches in that weight updates may come with implicit costs as in for example portfolio optimization. We give an algorithm to compute this projection step in linear time, which may be of independent interest.

Mark Herbster, Stephen Pasteris, Lisa Tse

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.

Stephen Pasteris, Ting He, Fabio Vitale et al.

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.

Mark Herbster, Stephen Pasteris, Lisa Tse

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 $\tilde{O}(D/γ^2)$. The term $1/γ^2$ 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 $γ$ 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 $O(k^2 + \ell^2)$.

Stephen Pasteris, Fabio Vitale, Kevin Chan et al.

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.

Mark Herbster, James Robinson

We address the problem of predicting the labeling of a graph in an online setting when the labeling is changing over time. We present an algorithm based on a specialist approach; we develop the machinery of cluster specialists which probabilistically exploits the cluster structure in the graph. Our algorithm has two variants, one of which surprisingly only requires $\mathcal{O}(\log n)$ time on any trial $t$ on an $n$-vertex graph, an exponential speed up over existing methods. We prove switching mistake-bound guarantees for both variants of our algorithm. Furthermore these mistake bounds smoothly vary with the magnitude of the change between successive labelings. We perform experiments on Chicago Divvy Bicycle Sharing data and show that our algorithms significantly outperform an existing algorithm (a kernelized Perceptron) as well as several natural benchmarks.

Carlo Ciliberto, Mark Herbster, Alessandro Davide Ialongo et al.

Recently, increased computational power and data availability, as well as algorithmic advances, have led machine learning techniques to impressive results in regression, classification, data-generation and reinforcement learning tasks. Despite these successes, the proximity to the physical limits of chip fabrication alongside the increasing size of datasets are motivating a growing number of researchers to explore the possibility of harnessing the power of quantum computation to speed-up classical machine learning algorithms. Here we review the literature in quantum machine learning and discuss perspectives for a mixed readership of classical machine learning and quantum computation experts. Particular emphasis will be placed on clarifying the limitations of quantum algorithms, how they compare with their best classical counterparts and why quantum resources are expected to provide advantages for learning problems. Learning in the presence of noise and certain computationally hard problems in machine learning are identified as promising directions for the field. Practical questions, like how to upload classical data into quantum form, will also be addressed.

Stephen Pasteris, Fabio Vitale, Claudio Gentile et al.

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.

Mark Herbster, Paul Rubenstein, James Townsend

Given a set $X$ and a function $h:X\longrightarrow\{0,1\}$ which labels each element of $X$ with either $0$ or $1$, we may define a function $h^{(s)}$ to measure the similarity of pairs of points in $X$ according to $h$. Specifically, for $h\in \{0,1\}^X$ we define $h^{(s)}\in \{0,1\}^{X\times X}$ by $h^{(s)}(w,x):= \mathbb{1}[h(w) = h(x)]$. This idea can be extended to a set of functions, or hypothesis space $\mathcal{H} \subseteq \{0,1\}^X$ by defining a similarity hypothesis space $\mathcal{H}^{(s)}:=\{h^{(s)}:h\in\mathcal{H}\}$. We show that ${vc-dimension}(\mathcal{H}^{(s)}) \in Θ({vc-dimension}(\mathcal{H}))$.

Claudio Gentile, Mark Herbster, Stephen Pasteris

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.