Machine Learning: The Necessity of Order (is order in order ?)

In myriad of human-tailored activities, whether in the classroom or listening to a story, human learners receive selected pieces of information, presented in a chosen order and pace. This is what it takes to facilitate learning. Yet, when machine learners exhibited sequencing effects, showing that some data sampling, ordering and tempo are better than others, it almost came as a surprise. Seemingly simple questions had suddenly to be thought anew : what are good training data? How to select them? How to present them? Why is it that there are sequencing effects? How to measure them? Should we try to avoid them or take advantage of them? This chapter is intended to present ideas and directions of research that are currently studied in the machine learning field to answer these questions and others. As any other science, machine learning strives to develop models that stress fundamental aspects of the phenomenon under study. The basic concepts and models developed in machine learning are presented here, as well as some of the findings that may have significance and counterparts in related disciplines interested in learning and education

Optimality certificates for convex minimization and Helly numbers

We consider the problem of minimizing a convex function over a subset of R^n that is not necessarily convex (minimization of a convex function over the integer points in a polytope is a special case). We define a family of duals for this problem and show that, under some natural conditions, strong duality holds for a dual problem in this family that is more restrictive than previously considered duals.Comment: 5 page

A Convex-Analysis Perspective on Disjunctive Cuts

An updated version of this paper has appeared in Math. Program., Ser. A 106, pp 567-586 (2006), DOI 10.1007/s10107-005-0670-8We treat the general problem of cutting planes with tools from convex analysis. We emphasize the case of disjunctive polyhedra and the generation of facets. We conclude with some considerations on the design of disjunctive cut generators

Idealness and 2-resistant sets

A subset of the unit hypercube {0,1}n is cube-ideal if its convex hull is described by hypercube and generalized set covering inequalities. In this note, we study sets S⊆{0,1}n such that, for any subset X⊆{0,1}n of cardinality at most 2, S∪X is cube-ideal

Unique Minimal Liftings for Simplicial Polytopes

For a minimal inequality derived from a maximal lattice-free simplicial polytope in $\R^n$, we investigate the region where minimal liftings are uniquely defined, and we characterize when this region covers $\R^n$. We then use this characterization to show that a minimal inequality derived from a maximal lattice-free simplex in $\R^n$ with exactly one lattice point in the relative interior of each facet has a unique minimal lifting if and only if all the vertices of the simplex are lattice points.Comment: 15 page

Early classification of time series as a non myopic sequential decision making problem

Classification of time series as early as possible is a valuable goal. Indeed, in many application domains, the earliest the decision, the more rewarding it can be. Yet, often, gathering more information allows one to get a better decision. The optimization of this time vs. accuracy tradeoff must generally be solved online and is a complex problem. This paper presents a formal criterion that expresses this trade-off in all generality together with a generic sequential meta algorithm to solve it. This meta algorithm is interesting in two ways. First, it pinpoints where choices can (have to) be made to obtain a computable algorithm. As a result a wealth of algorithmic solutions can be found. Second, it seeks online the earliest time in the future where a minimization of the criterion can be expected. It thus goes beyond the classical approaches that myopically decide at each time step whether to make a decision or to postpone the call one more time step. After this general setting has been expounded, we study one simple declination of the meta-algorithm, and we show the results obtained on synthetic and real time series data sets chosen for their ability to test the robustness and properties of the technique. The general approach is vindicated by the experimental results, which allows us to point to promising perspectives

On packing dijoins in digraphs and weighted digraphs

In this paper, we make some progress in addressing Woodall's Conjecture, and the refuted Edmonds-Giles Conjecture on packing dijoins in unweighted and weighted digraphs. Let $D=(V,A)$ be a digraph, and let $w\in \mathbb{Z}^A_{\geq 0}$. Suppose every dicut has weight at least $\tau$, for some integer $\tau\geq 2$. Let $\rho(\tau,D,w):=\frac{1}{\tau}\sum_{v\in V} m_v$, where each $m_v$ is the integer in $\{0,1,\ldots,\tau-1\}$ equal to $w(\delta^+(v))-w(\delta^-(v))$ mod $\tau$. In this paper, we prove the following results, amongst others: (1) If $w={\bf 1}$, then $A$ can be partitioned into a dijoin and a $(\tau-1)$-dijoin. (2) If $\rho(\tau,D,w)\in \{0,1\}$, then there is an equitable $w$-weighted packing of dijoins of size $\tau$. (3) If $\rho(\tau,D,w)= 2$, then there is a $w$-weighted packing of dijoins of size $\tau$. (4) If $w={\bf 1}$, $\tau=3$, and $\rho(\tau,D,w)=3$, then $A$ can be partitioned into three dijoins. Each result is best possible: (1) and (4) do not hold for general $w$, (2) does not hold for $\rho(\tau,D,w)=2$ even if $w={\bf 1}$, and (3) does not hold for $\rho(\tau,D,w)=3$. The results are rendered possible by a \emph{Decompose, Lift, and Reduce procedure}, which turns $(D,w)$ into a set of \emph{sink-regular weighted $(\tau,\tau+1)$-bipartite digraphs}, each of which is a weighted digraph where every vertex is a sink of weighted degree $\tau$ or a source of weighted degree $\tau,\tau+1$, and every dicut has weight at least $\tau$. Our results give rise to a number of approaches for resolving Woodall's Conjecture, fixing the refuted Edmonds-Giles Conjecture, and the $\tau=2$ Conjecture for the clutter of minimal dijoins. They also show an intriguing connection to Barnette's Conjecture.Comment: 71 page

Resistant sets in the unit hypercube

Ideal matrices and clutters are prevalent in Combinatorial Optimization, ranging from balanced matrices, clutters of T-joins, to clutters of rooted arborescences. Most of the known examples of ideal clutters are combinatorial in nature. In this paper, rendered by the recently developed theory of cuboids, we provide a different class of ideal clutters, one that is geometric in nature. The advantage of this new class of ideal clutters is that it allows for infinitely many ideal minimally non-packing clutters. We characterize the densest ideal minimally non-packing clutters of the class. Using the tools developed, we then verify the Replication Conjecture for the class