Jesús A. De Loera

Yue Wu, Jesús A. De Loera

Can agents be trained to answer difficult mathematical questions by playing a game? We consider the integer feasibility problem, a challenge of deciding whether a system of linear equations and inequalities has a solution with integer values. This is a famous NP-complete problem with applications in many areas of Mathematics and Computer Science. Our paper describes a novel algebraic reinforcement learning framework that allows an agent to play a game equivalent to the integer feasibility problem. We explain how to transform the integer feasibility problem into a game over a set of arrays with fixed margin sums. The game starts with an initial state (an array), and by applying a legal move that leaves the margins unchanged, we aim to eventually reach a winning state with zeros in specific positions. To win the game the player must find a path between the initial state and a final terminal winning state if one exists. Finding such a winning state is equivalent to solving the integer feasibility problem. The key algebraic ingredient is a Gröbner basis of the toric ideal for the underlying axial transportation polyhedron. The Gröbner basis can be seen as a set of connecting moves (actions) of the game. We then propose a novel RL approach that trains an agent to predict moves in continuous space to cope with the large size of action space. The continuous move is then projected onto the set of legal moves so that the path always leads to valid states. As a proof of concept we demonstrate in experiments that our agent can play well the simplest version of our game for 2-way tables. Our work highlights the potential to train agents to solve non-trivial mathematical queries through contemporary machine learning methods used to train agents to play games.

Félix Almendra-Hernández, Jesús A. De Loera, Sonja Petrović

We study a colored generalization of the famous simple-switch Markov chain for sampling the set of graphs with a fixed degree sequence. Here we consider the space of graphs with colored vertices, in which we fix the degree sequence and another statistic arising from the vertex coloring, and prove that the set can be connected with simple color-preserving switches or moves. These moves form a basis for defining an irreducible Markov chain necessary for testing statistical model fit to block-partitioned network data. Our methods further generalize well-known algebraic results from the 1990s: namely, that the corresponding moves can be used to construct a regular triangulation for a generalization of the second hypersimplex. On the other hand, in contrast to the monochromatic case, we show that for \emph{simple} graphs, the 1-norm of the moves necessary to connect the space increases with the number of colors.

Imran Adham, Jesus De Loera, Zhenyang Zhang

This paper discusses a data-driven, empirically-based framework to make algorithmic decisions or recommendations without expert knowledge. We improve the performance of two algorithmic case studies: the selection of a pivot rule for the Simplex method and the selection of an all-pair shortest paths algorithm. We train machine learning methods to select the optimal algorithm for given data without human expert opinion. We use two types of techniques, neural networks and boosted decision trees. We concluded, based on our experiments, that: 1) Our selection framework recommends various pivot rules that improve overall total performance over just using a fixed default pivot rule. Over many years experts identified steepest-edge pivot rule as a favorite pivot rule. Our data analysis corroborates that the number of iterations by steepest-edge is no more than 4 percent more than the optimal selection which corroborates human expert knowledge, but this time the knowledge was obtained using machine learning. Here our recommendation system is best when using gradient boosted trees. 2) For the all-pairs shortest path problem, the models trained made a large improvement and our selection is on average .07 percent away from the optimal choice. The conclusions do not seem to be affected by the machine learning method we used. We tried to make a parallel analysis of both algorithmic problems, but it is clear that there are intrinsic differences. For example, in the all-pairs shortest path problem the graph density is a reasonable predictor, but there is no analogous single parameter for decisions in the Simplex method.

Jesus A. De Loera, Jamie Haddock, Anna Ma et al.

Machine learning algorithms typically rely on optimization subroutines and are well-known to provide very effective outcomes for many types of problems. Here, we flip the reliance and ask the reverse question: can machine learning algorithms lead to more effective outcomes for optimization problems? Our goal is to train machine learning methods to automatically improve the performance of optimization and signal processing algorithms. As a proof of concept, we use our approach to improve two popular data processing subroutines in data science: stochastic gradient descent and greedy methods in compressed sensing. We provide experimental results that demonstrate the answer is ``yes'', machine learning algorithms do lead to more effective outcomes for optimization problems, and show the future potential for this research direction.