Related papers: Larger Greedy Sums for Reverse Partially Greedy Ba…

We analyze the performance of the greedy algorithm, and also a discrete semi-gradient based algorithm, for maximizing the sum of a suBmodular and suPermodular (BP) function (both of which are non-negative monotone non-decreasing) under two…

A base for a permutation group $G$ acting on a set $\Omega$ is a sequence $\mathcal{B}$ of points of $\Omega$ such that the pointwise stabiliser $G_{\mathcal{B}}$ is trivial. Denote the minimum size of a base for $G$ by $b(G)$. There is a…

Expansion of real numbers is a basic research topic in number theory. Usually we expand real numbers in one given base. In this paper, we begin to systematically study expansions in multiple given bases in a reasonable way, which is a…

The rate of convergence of the classical Thresholding Greedy Algorithm with respect to bases is studied in this paper. We bound the error of approximation by the product of both norms -- the norm of $f$ and the $A_1$-norm of $f$. We obtain…

Recent work on exploration in reinforcement learning (RL) has led to a series of increasingly complex solutions to the problem. This increase in complexity often comes at the expense of generality. Recent empirical studies suggest that,…

We introduce and study the notion of weak weight-semi-greedy Markushevich bases - which extends the concepts of weight semi-greedy and weak semi-greedy Markushevich bases. In particular, we study conditions under which such bases are weight…

A base for a permutation group $G$ acting on a set $\Omega$ is a subset $\mathcal{B}$ of $\Omega$ whose pointwise stabiliser $G_{(\mathcal{B})}$ is trivial. There is a natural greedy algorithm for constructing a base of relatively small…

We show that a (semi-normalized) basis in a Banach space is quasi-greedy with quasi-greedy constant equal to 1 if and only if it is unconditional with suppression-unconditional constant equal to 1.

We show that, for quasi-greedy bases in Hilbert spaces, the associated conditionality constants grow at most as $O(\log N)^{1-\epsilon}$, for some $\epsilon>0$, answering a question by Temlyakov. We show the optimality of this bound with an…

We propose a new concept named adaptive submodularity ratio to study the greedy policy for sequential decision making. While the greedy policy is known to perform well for a wide variety of adaptive stochastic optimization problems in…

$\varepsilon$-greedy is a policy used to balance exploration and exploitation in many reinforcement learning setting. In cases where the agent uses some on-policy algorithm to learn optimal behaviour, it makes sense for the agent to explore…

We characterize the approximation spaces of a broad class of bases - which includes almost greedy bases - in terms of weighted Lorentz spaces. For those bases, we also find necessary and sufficient conditions under which the approximation…

The general problem addressed in this work is the development of a systematic study of the thresholding greedy algorithm for general biorthogonal systems (also known as Markushevich bases) in quasi-Banach spaces from a functional-analytic…

We give criteria for finding the greedy $\beta$-expansion for $1$ for families of Salem numbers that approach a given Pisot number. We show that these expansions are related to the greedy expansion under the Pisot base. This expands on the…

Greedy algorithms are widely used for problems in machine learning such as feature selection and set function optimization. Unfortunately, for large datasets, the running time of even greedy algorithms can be quite high. This is because for…

Submodular maximization has been widely studied over the past decades, mostly because of its numerous applications in real-world problems. It is well known that the standard greedy algorithm guarantees a worst-case approximation factor of…

We shall present a new characterization of greedy bases and 1-greedy bases in terms of certain functionals defined using distances to one dimensional subspaces generated by the basis. We also introduce a new property that unifies the…

Let $\mathcal{F}$ be a hereditary collection of finite subsets of $\mathbb{N}$. In this paper, we introduce and characterize $\mathcal{F}$-(almost) greedy bases. Given such a family $\mathcal{F}$, a basis $(e_n)_n$ for a Banach space $X$ is…

This paper defines weak-$\alpha$-supermodularity for set functions. Many optimization objectives in machine learning and data mining seek to minimize such functions under cardinality constrains. We prove that such problems benefit from a…

An increasing sequence $(x_i)_{i=1}^n$ of positive integers is an $n$-term Egyptian underapproximation of $\theta \in (0,1]$ if $\sum_{i=1}^n \frac{1}{x_i} < \theta$. A greedy algorithm constructs an $n$-term underapproximation of $\theta$.…