Using Minkowski addition of sets, we study linear betweenness in the hyperspace L(X) of linearly convex nonempty subsets of a normed real vector space X, as well as in the sub-hyperspace KL(X) of compact elements of L(X). We also study the metric betweenness relation induced by the Hausdorff metric on the latter. While linear betweenness in L(X) behaves reasonably like linear betweenness at the point level, the analogy is not perfect: linear intervals in X are honest line segments; this is no longer the case for L(X), where linear intervals can have exactly two elements. However, when we restrict our focus to KL(X), the R & aring;dstr & ouml;m extension theorem allows us to view this hyperspace as a linearly convex cone in a normed vector space R(X); in particular, all linear intervals are line segments that are contained in the corresponding metric intervals. We are especially interested in the notions of convexity induced by these two kinds of betweenness relation. While all closed balls and metric intervals in KL(X) are linearly convex, metric convexity has more nuanced behaviour. For example, the metric intervals in KL(X) determined by singletons are all metrically convex if and only if X is strictly convex. When X is one-dimensional, R(X) is Cartesian 2-space equipped with the max norm and KL(X) looks like the half-plane {(x, y) : x <= y}. In particular, all metric intervals - and no closed balls of positive radius - are metrically convex. When X is multi-dimensional, though, while it is still the case that closed balls are metrically nonconvex, it is now always possible to find a metrically nonconvex metric interval that is determined by a singleton and a line segment.

In this note we expand upon our results from [1] to show that every nondegenerate hereditarily decomposable Hausdorff continuum has two or more non-block points, i.e points whose complements contain a continuum-connected dense subset. The celebrated non-cut point existence theorem states that all nondegenerate Hausdorff continua have two or more non-cut points, and the corresponding result for non-block points is known to hold for metrizable continua. It is also known that there are consistent examples of Hausdorff continua with no non-block points, but that non-block point existence holds for Hausdorff continua that are either aposyndetic, irreducible, or separable.

We consider the general problem of online convex optimization with time-varying budget constraints in the presence of predictions for the next cost and constraint functions, that arises in a plethora of network resource management problems. A novel saddle-point algorithm is designed by combining a Follow-The-Regularized-Leader iteration with prediction-adaptive dynamic steps. The algorithm achieves $\mathcal O(T^{(3-\beta)/4})$ regret and $\mathcal O(T^{(1+\beta)/2})$ constraint violation bounds that are tunable via parameter $\beta \!\in \![1/2,1$ ) and have constant factors that shrink with the predictions quality, achieving eventually $\mathcal O(1)$ regret for perfect predictions. Our work extends the seminal FTRL framework for this new OCO setting and outperforms the respective state-of-the-art greedy-based solutions which naturally cannot benefit from predictions, without imposing conditions on the (unknown) quality of predictions, the cost functions or the geometry of constraints, beyond convexity.

In online linear optimisation with stochastic losses it is common to bound the pseudo-regret of an algorithm rather than the expected regret. This is attributed to the expected fluctuations for i.i.d sums making expected regret bounds better than omega( T) impossible. In this paper we show that when there is a unique optimal action and the action set is a polytope the difference between pseudo-regret and expected regret is o(1). This means that the existing upper bounds on pseudoregret in the literature can immediately be extended to also upper bound the expected regret. Our results are independent of the algorithm used to select the actions and apply equally to the bandit and full-information settings.

We consider the general problem of online convex optimization with time-varying additive constraints in the presence of predictions for the next cost and constraint functions. A novel primal-dual algorithm is designed by combining a Follow-The-Regularized-Leader iteration with prediction-adaptive dynamic steps. The algorithm achieves $\mathcal O(T^{\frac{3-\beta}{4}})$ regret and $\mathcal O(T^{\frac{1+\beta}{2}})$ constraint violation bounds that are tunable via parameter $\beta\!\in\![1/2,1)$ and have constant factors that shrink with the predictions quality, achieving eventually $\mathcal O(1)$ regret for perfect predictions. Our work extends the FTRL framework for this constrained OCO setting and outperforms the respective state-of-the-art greedy-based solutions, without imposing conditions on the quality of predictions, the cost functions or the geometry of constraints, beyond convexity.

We consider online learning problems where the aim is to achieve regret which is efficient in the sense that it is the same order as the lowest regret amongst K experts. This is a substantially stronger requirement that achieving O( √ n) or O(log n) regret with respect to the best expert and standard algorithms are insufficient, even in easy cases where the regrets of the available actions are very different from one another. We show that a particular lazy form of the online subgradient algorithm can be used to achieve minimal regret in a number of “easy” regimes while retaining an O( √ n) worst-case regret guarantee. We also show that for certain classes of problem minimal regret strategies exist for some of the remaining “hard” regimes.

We study Online Lazy Gradient Descent for optimisation on a strongly convex domain. The algorithm is known to achieve O( √ N) regret against adversarial opponents; here we show it is universal in the sense that it also achieves O(logN) expected regret against i.i.d opponents. This improves upon the more complex metaalgorithm of Huang et al [20] that only gets O( √ N logN) and O(logN) bounds. In addition we show that, unlike for the simplex, order bounds for pseudo-regret and expected regret are equivalent for strongly convex domains.

We study Online Lazy Gradient Descent for optimisation on a strongly convex domain. The algorithm is known to achieve $O(\sqrt N)$ regret against adversarial opponents; here we show it is universal in the sense that it also achieves $O(\log N)$ expected regret against i.i.d opponents. This improves upon the more complex meta-algorithm of Huang et al \cite{FTLBall} that only gets $O(\sqrt {N \log N})$ and $ O(\log N)$ bounds. In addition we show that, unlike for the simplex, order bounds for pseudo-regret and expected regret are equivalent for strongly convex domains.

How to quickly and reliably learn the preferences of new users remains a key challenge in the design of recommender systems. In this paper we introduce a new type of online learning algorithm, cluster-based bandits, to address this challenge. This exploits the fact that users can often be grouped into clusters based on the similarity of their preferences, and this allows accelerated learning of new user preferences since the task becomes one of identifying which cluster a user belongs to and typically there are far fewer clusters than there are items to be rated. Clustering by itself is not enough however. Intra-cluster variability between users can be thought of as adding noise to user ratings. Deterministic methods such as decision-trees perform poorly in the presence of such noise. We identify so-called distinguisher items that are particularly informative for deciding which cluster a new user belongs to despite the rating noise. Using these items the cluster-based bandit algorithm is able to efficiently adapt to user responses and rapidly learn the correct cluster to assign to a new user.

A betweenness structure on a set X is a ternary relation [⋅,⋅,⋅]⊆X3 that captures a rudimentary notion of one point of X lying between two others. The interval [a,b] is the set of all points lying between a and b, and a subset C of X is convex if [a,b]⊆C whenever a,b∈C. The span of a set A is the union of all intervals [a,b], where a,b∈A; by iterating the span operator countably many times, we obtain the convex hull of A. The betweenness structure is topological if X carries a topology that satisfies certain compatibility conditions with respect to betweenness; in particular, intervals are closed subsets. We are guided by questions involving how the span and convex hull operators interact with the topological closure and interior operators, especially in the domains of metric spaces and of continua. With a metric space 〈X,ϱ〉, [a,c,b] holds exactly when ϱ(a,b)=ϱ(a,c)+ϱ(c,b); and one result about this betweenness structure is that the span of any compact subset is both closed and bounded. With a continuum X, [a,c,b] holds exactly when c belongs to every subcontinuum of X that contains a and b; and one result about this betweenness structure is that when the continuum is either aposyndetic or hereditarily unicoherent, the closure of a convex subset is always convex.

Indecomposable continua with one composant are $\textit{large}$ in the sense of being non-metrisable. We adapt the method of Smith $[18]$ to construct an example which is $\textit{small}$ in the sense of being separable.

The bulk of universal algorithms in the online convex optimisation literature are variants of the Hedge (exponential weights) algorithm on the simplex. While these algorithms extend to polytope domains by assigning weights to the vertices, this process is computationally unfeasible for many important classes of polytopes where the number $V$ of vertices depends exponentially on the dimension $d$. In this paper we show the Subgradient algorithm is universal, meaning it has $O(\sqrt N)$ regret in the antagonistic setting and $O(1)$ pseudo-regret in the i.i.d setting, with two main advantages over Hedge: (1) The update step is more efficient as the action vectors have length only $d$ rather than $V$; and (2) Subgradient gives better performance if the cost vectors satisfy Euclidean rather than sup-norm bounds. This paper extends the authors' recent results for Subgradient on the simplex. We also prove the same $O(\sqrt N)$ and $O(1)$ bounds when the domain is the unit ball. To the authors' knowledge this is the first instance of these bounds on a domain other than a polytope.

We prove the familiar Lazy Online Gradient Descent algorithm is universal on polytope domains. That means it gets $O(1)$ pseudo-regret against i.i.d opponents, while simultaneously achieving the well-known $O(\sqrt N)$ worst-case regret bound. For comparison the bulk of the literature focuses on variants of the Hedge (exponential weights) algorithm on the simplex. These can in principle be lifted to general polytopes; however the process is computationally unfeasible for many important classes where the number of vertices grows quickly with the dimension. The lifting procedure also ignores any Euclidean bounds on the cost vectors, and can create extra factors of dimension in the pseudo-regret bound. Gradient Descent is simpler than the handful of purpose-built algorithms for polytopes in the literature, and works in a broader setting. In particular existing algorithms assume the optimiser is unique, while our bound allows for several optimal vertices.

A space X is microhomogeneous if for every p,q∈X there is a homeomorphism h from a neighborhood of p onto a neighborhood of q such that h(p)=q. We show that many cardinal bounds obtained using homogeneity or power homogeneity can be obtained using microhomogeneity. Since microhomogeneous spaces need not be either homogeneous or power homogeneous, this extends those cardinal bounds to a broader class of spaces. We also explore possible connections between microhomogeneity and both homogeneity and power homogeneity, providing a simple example of a connected microhomogeneous space that is not homogeneous. We also give examples of microhomogeneous spaces in which no open set is homogeneous, and a microhomogeneous space which is not uniformly microhomogeneous in the sense that for every p∈X and every neighborhood U of p there is q∈X which does not have a neighborhood that is homeomorphic to U. In the process of establishing the properties of the last example we obtain the following result, which is of independent interest. Let X be a locally compact, homogeneous, and strongly locally homogeneous space (such as a manifold). Let A and B be countable dense subsets of X and let {Ai:i∈I} and {Bi:i∈I} be partitions of A and B respectively into dense subsets of X. For every a0,b0∈X there is a homeomorphism F:X→X such that F(a0)=b0 and F[Ai−{a0}]=Bi−{b0}.

We construct an indecomposable continuum with exactly one strong non-cut point. The method is an adaptation of Bellamy $[1]$. We start with an $ω_1$-chain of indecomposable metric continua and retractions. The inverse limit is an indecomposable continuum with exactly two composants. Our example is formed by identifying a point in each composant.

Recently Jaouad Mourtada and St\' ephane Ga\"iffas showed the anytime hedge algorithm has pseudo-regret $O(\log (d) / \Delta)$ if the cost vectors are generated by an i.i.d sequence in the cube $[0,1]^d$. Here $d$ is the dimension and $\Delta$ the suboptimality gap. This is remarkable because the Hedge algorithm was designed for the antagonistic setting. We prove a similar result for the anytime subgradient algorithm on the simplex. Given i.i.d cost vectors in the unit ball our pseudo-regret bound is $O(1/\Delta)$ and does not depend on the dimension of the problem.

We consider online learning problems where the aim is to achieve regret which is efficient in the sense that it is the same order as the lowest regret amongst K experts. This is a substantially stronger requirement that achieving $O(\sqrt{n})$ or $O(\log n)$ regret with respect to the best expert and standard algorithms are insufficient, even in easy cases where the regrets of the available actions are very different from one another. We show that a particular lazy form of the online subgradient algorithm can be used to achieve minimal regret in a number of "easy" regimes while retaining an $O(\sqrt{n})$ worst-case regret guarantee. We also show that for certain classes of problem minimal regret strategies exist for some of the remaining "hard" regimes.

We prove the three propositions are equivalent: (a) Every Hausdorff continuum has two or more shore points. (b) Every Hausdorff continuum has two or more non-block points. (c) Every Hausdorff continuum is coastal at each point. Thus it is consistent that all three properties fail. We also give the following characterization of shore points: The point p of the continuum X is a shore point if and only if there is a net of subcontinua in {K∈C(X):K⊂κ(p)−p} tending to X in the Vietoris topology. This contrasts with the standard characterization which only demands the net elements be contained in X−p. In addition we prove every point of an indecomposable continuum is a shore point.

We show that the Subgradient algorithm is universal for online learning on the simplex in the sense that it simultaneously achieves $O(\sqrt N)$ regret for adversarial costs and $O(1)$ pseudo-regret for i.i.d costs. To the best of our knowledge this is the first demonstration of a universal algorithm on the simplex that is not a variant of Hedge. Since Subgradient is a popular and widely used algorithm our results have immediate broad application.

We import into continuum theory the notion of extreme point of a convex set from the theory of topological vector spaces. We explore how extreme points relate to other established types of "edge point" of a continuum; for example we prove that extreme points are always shore points, and that any extreme point is also non-block if the continuum is either decomposable or irreducible (in particular, metrizable). In addition we discuss some continuum-theoretic analogues of the celebrated Krein-Milman theorem. (C) 2019 Elsevier B.V. All rights reserved.