Xavier Glorot

Proving theorems using Incremental Learning and Hindsight Experience Replay

Maxwell Crouse

Eser Aygün

Bassem Makni

Ankit Anand

Laurent Orseau

Vernon Ralph Austel

Xavier Glorot

Cristina Cornelio

Shajith Ikbal

Stephen M Mcaleer

Pavan Kapanipathi

Vlad Firoiu

Ndivhuwo Makondo

Lei M Zhang

Doina Precup

Shibl Mourad

The highest performing ATP systems (e.g., [7, 18]) in first order logic have been evolving for decades and have grown to use an increasing n… (see more)umber of manually designed heuristics mixed with some machine learning, to obtain a large number of search strategies that are tried sequentially or in parallel. Some recent works [5, 13, 19] build on top of these provers, using modern machine learning techniques to augment, select or prioritize their already existing heuristics, with some success. Other recent works do not build on top of other provers, but still require existing proof examples as input (e.g., [9, 23]). Such machine-learning-based ATP systems can struggle to solve difficult problems when the training dataset does not provide problems of sufficiently diverse difficulties. In this paper, we propose an approach which can build a strong theorem prover without relying on existing domain-specific heuristics or on prior input data (in the form of proofs) to prime the learning. We strive to design a learning methodology for ATP that allows a system to improve even when there are large gaps in the difficulty of given set of theorems. In particular, given a set of conjectures without proofs, our system trains itself, based on its own attempts and (dis)proves an increasing number of conjectures, an approach which can be viewed as a form of incremental learning. Additionally, all the previous approaches [19, 1, 13] learn exclusively on successful proof attempts. When no new theorem can be proven, the learner may not be able to improve anymore and thus the system may not be able to obtain more training data. This could in principle happen even at the very start of training, if all the theorems available are too hard. To tackle this challenge, we adapt the idea of hindsight experience replay (HER) [3] to ATP: Clauses reached during proof attempts (whether successful or not) are turned into goals in hindsight, producing a large amount of ‘auxiliary’ theorems with proofs of varied difficulties for the learner, even in principle when no theorem from the original set can be proven initially. This leads to a smoother learning regime and a constantly improving learner. We evaluate our approach on two popular benchmarks: MPTP2078 [2] and M2k [17] and compare it both with TRAIL [1], a recent machine learning prover as well as with E prover [24, 7], one of the leading heuristic provers. Our proposed approach substantially outperforms TRAIL [1] on both datasets, surpasses E in the auto configuration with a 100s time limit, and is competitive with E in the autoschedule configuration with a 7 days time limit. In addition, our approach almost always (99.5% of cases) finds shorter proofs than E.

2022-01-01

ICML (published)

proceedings.mlr.press

openreview.net

Training a First-Order Theorem Prover from Synthetic Data

Vlad Firoiu

Eser Aygün

Ankit Anand

Zafarali Ahmed

Xavier Glorot

Laurent Orseau

Lei Zhang

Doina Precup

Shibl Mourad

2021-03-05

ArXiv (preprint)

arxiv.org

SPE: Symmetrical Prompt Enhancement for Factual Knowledge Retrieval

James M. Crawford

Matthew L. Ginsberg

Jacob Devlin

Ming-Wei Chang

Kenton Lee

Xavier Glorot

Antoine Bordes

Yoshua Bengio

Alex Graves

Abdel rahman Mohamed

Adi Haviv

Jonathan Berant

Amir Globerson

Chloe Kiddon

Pedro M. Domingos

Brian Lester

Rami Al-rfou'

Noah Constant. 2021

Pengfei Liu

Weizhe Yuan … (see 6 more)

Jinlan Fu

Zhengbao Jiang

Xiao Liu

Yanan Zheng

Zhengxiao Du

Ming Ding

Pretrained language models (PLMs) have 001 been shown to accumulate factual knowledge 002 from their unsupervised pretraining proce-003 dure… (see more)s (Petroni et al., 2019). Prompting is an 004 effective way to query such knowledge from 005 PLMs. Recently, continuous prompt methods 006 have been shown to have a larger potential 007 than discrete prompt methods in generating ef-008 fective queries (Liu et al., 2021a). However, 009 these methods do not consider symmetry of 010 the task. In this work, we propose Symmet-011 rical Prompt Enhancement (SPE), a continu-012 ous prompt-based method for fact retrieval that 013 leverages the symmetry of the task. Our results 014 on LAMA, a popular fact retrieval dataset, 015 show signiﬁcant improvement of SPE over pre-016 vious prompt methods

Learning to Prove from Synthetic Theorems

Eser Aygün

Vlad Firoiu

Laurent Orseau

Shibl Mourad

A major challenge in applying machine learning to automated theorem proving is the scarcity of training data, which is a key ingredient in t… (see more)raining successful deep learning models. To tackle this problem, we propose an approach that relies on training with synthetic theorems, generated from a set of axioms. We show that such theorems can be used to train an automated prover and that the learned prover transfers successfully to human-generated theorems. We demonstrate that a prover trained exclusively on synthetic theorems can solve a substantial fraction of problems in TPTP, a benchmark dataset that is used to compare state-of-the-art heuristic provers. Our approach outperforms a model trained on human-generated problems in most axiom sets, thereby showing the promise of using synthetic data for this task.

2020-06-19

ArXiv (preprint)

arxiv.org

Theano: A Python framework for fast computation of mathematical expressions

Rami Al-rfou'

Guillaume Alain

Amjad Almahairi

Christof Angermüller

Dzmitry Bahdanau

Nicolas Ballas

Frédéric Bastien

Justin S. Bayer

A. Belikov

A. Belopolsky

J. Bergstra

Josh Bleecher Snyder

Nicolas Boulanger-Lewandowski

Xavier Bouthillier

Alexandre De Brébisson

Olivier Breuleux … (see 92 more)

Paul F. Christiano

Marc-Alexandre Côté

Myriam Côté

Julien Demouth

Sander Dieleman

M'elanie Ducoffe

Samira Ebrahimi Kahou

Dumitru Erhan

Ziye Fan

Orhan Firat

Mathieu Germain

Xavier Glorot

Ian J. Goodfellow

Matthew Graham

Balázs Hidasi

Arjun Jain

S'ebastien Jean

Kai Jia

Mikhail V. Korobov

Vivek Kulkarni

Alex Lamb

Pascal Lamblin

Eric P. Larsen

César Laurent

S. Lee

Simon-mark Lefrancois

Simon Lemieux

Nicholas Léonard

Zhouhan Lin

J. Livezey

Cory R. Lorenz

Jeremiah L. Lowin

Qianli M. Ma

Pierre-Antoine Manzagol

R. McGibbon

Mehdi Mirza

Alberto Orlandi

Chris Pal

Razvan Pascanu

Mohammad Pezeshki

Colin Raffel

Daniel Renshaw

Matthew David Rocklin

Adriana Romero Soriano

Markus Dr. Roth

Peter Sadowski

John Salvatier

François Savard

Jan Schlüter

John D. Schulman

Gabriel Schwartz

Iulian V. Serban

Dmitriy Serdyuk

Samira Shabanian

Etienne Simon

Sigurd Spieckermann

S. Subramanyam

Jakub Sygnowski

Jeremie Tanguay

Gijs van Tulder

Joseph P. Turian

Sebastian Urban

Dustin J. Webb

M. Willson

Lijun Xue

Theano is a Python library that allows to define, optimize, and evaluate mathematical expressions involving multi-dimensional arrays efficie… (see more)ntly. Since its introduction, it has been one of the most used CPU and GPU mathematical compilers - especially in the machine learning community - and has shown steady performance improvements. Theano is being actively and continuously developed since 2008, multiple frameworks have been built on top of it and it has been used to produce many state-of-the-art machine learning models. The present article is structured as follows. Section I provides an overview of the Theano software and its community. Section II presents the principal features of Theano and how to use them, and compares them with other similar projects. Section III focuses on recently-introduced functionalities and improvements. Section IV compares the performance of Theano against Torch7 and TensorFlow on several machine learning models. Section V discusses current limitations of Theano and potential ways of improving it.

2016-05-09

ArXiv (preprint)

arxiv.org

Theano: A Python framework for fast computation of mathematical expressions

Rami Al-rfou'

Guillaume Alain

Amjad Almahairi

Christof Angermüller

Dzmitry Bahdanau

Nicolas Ballas

Frédéric Bastien

Justin S. Bayer

A. Belikov

A. Belopolsky

Josh Bleecher Snyder

Nicolas Boulanger-Lewandowski

Xavier Bouthillier

Alexandre De Brébisson

Olivier Breuleux … (see 92 more)

Paul F. Christiano

Marc-Alexandre Côté

Myriam Côté

Julien Demouth

Sander Dieleman

M'elanie Ducoffe

Samira Ebrahimi Kahou

Dumitru Erhan

Ziye Fan

Orhan Firat

Mathieu Germain

Xavier Glorot

Ian G Goodfellow

Matthew Graham

Balázs Hidasi

Arjun Jain

Kai Jia

Mikhail V. Korobov

Vivek Kulkarni

Alex Lamb

Pascal Lamblin

Eric Larsen

César Laurent

S. Lee

Simon-mark Lefrancois

Simon Lemieux

Nicholas Léonard

Zhouhan Lin

J. Livezey

Cory R. Lorenz

Jeremiah L. Lowin

Qianli M. Ma

Pierre-Antoine Manzagol

R. McGibbon

Mehdi Mirza

Alberto Orlandi

Chris Pal

Razvan Pascanu

Mohammad Pezeshki

Colin Raffel

Daniel Renshaw

Matthew David Rocklin

Adriana Romero Soriano

Markus Dr. Roth

Peter Sadowski

John Salvatier

François Savard

Jan Schlüter

John D. Schulman

Gabriel Schwartz

Iulian V. Serban

Dmitriy Serdyuk

Samira Shabanian

Etienne Simon

Sigurd Spieckermann

S. Subramanyam

Gijs van Tulder

Sebastian Urban

Dustin J. Webb

M. Willson

Lijun Xue

Theano is a Python library that allows to define, optimize, and evaluate mathematical expressions involving multi-dimensional arrays efficie… (see more)ntly. Since its introduction, it has been one of the most used CPU and GPU mathematical compilers - especially in the machine learning community - and has shown steady performance improvements. Theano is being actively and continuously developed since 2008, multiple frameworks have been built on top of it and it has been used to produce many state-of-the-art machine learning models. The present article is structured as follows. Section I provides an overview of the Theano software and its community. Section II presents the principal features of Theano and how to use them, and compares them with other similar projects. Section III focuses on recently-introduced functionalities and improvements. Section IV compares the performance of Theano against Torch7 and TensorFlow on several machine learning models. Section V discusses current limitations of Theano and potential ways of improving it.

2016-05-09

ArXiv (preprint)

arxiv.org

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Xavier Glorot

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Xavier Glorot

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