Learning compositional functions via multiplicative weight updates

Compositionality is a basic structural feature of both biological and artificial neural networks. Learning compositional functions via gradient descent incurs well known problems like vanishing and exploding gradients, making careful learning rate tuning essential for real-world applications. This paper proves that multiplicative weight updates satisfy a descent lemma tailored to compositional functions. Based on this lemma, we derive Madam—a multiplicative version of the Adam optimiser—and show that it can train state of the art neural network architectures without learning rate tuning. We further show that Madam is easily adapted to train natively compressed neural networks by representing their weights in a logarithmic number system. We conclude by drawing connections between multiplicative weight updates and recent findings about synapses in biology

Learning compositional functions via multiplicative weight updates

Compositionality is a basic structural feature of both biological and artificial neural networks. Learning compositional functions via gradient descent incurs well known problems like vanishing and exploding gradients, making careful learning rate tuning essential for real-world applications. This paper proves that multiplicative weight updates satisfy a descent lemma tailored to compositional functions. Based on this lemma, we derive Madam -- a multiplicative version of the Adam optimiser -- and show that it can train state of the art neural network architectures without learning rate tuning. We further show that Madam is easily adapted to train natively compressed neural networks by representing their weights in a logarithmic number system. We conclude by drawing connections between multiplicative weight updates and recent findings about synapses in biology

Holographic Embeddings of Knowledge Graphs

Learning embeddings of entities and relations is an efficient and versatile method to perform machine learning on relational data such as knowledge graphs. In this work, we propose holographic embeddings (HolE) to learn compositional vector space representations of entire knowledge graphs. The proposed method is related to holographic models of associative memory in that it employs circular correlation to create compositional representations. By using correlation as the compositional operator HolE can capture rich interactions but simultaneously remains efficient to compute, easy to train, and scalable to very large datasets. In extensive experiments we show that holographic embeddings are able to outperform state-of-the-art methods for link prediction in knowledge graphs and relational learning benchmark datasets.Comment: To appear in AAAI-1

Differential Privacy for Relational Algebra: Improving the Sensitivity Bounds via Constraint Systems

Differential privacy is a modern approach in privacy-preserving data analysis to control the amount of information that can be inferred about an individual by querying a database. The most common techniques are based on the introduction of probabilistic noise, often defined as a Laplacian parametric on the sensitivity of the query. In order to maximize the utility of the query, it is crucial to estimate the sensitivity as precisely as possible. In this paper we consider relational algebra, the classical language for queries in relational databases, and we propose a method for computing a bound on the sensitivity of queries in an intuitive and compositional way. We use constraint-based techniques to accumulate the information on the possible values for attributes provided by the various components of the query, thus making it possible to compute tight bounds on the sensitivity.Comment: In Proceedings QAPL 2012, arXiv:1207.055