Reliable generalization lies at the heart of safe ML and AI. However, understanding when and how neural networks generalize remains one of the most important unsolved problems in the field. In this work, we conduct an extensive empirical study (20'910 models, 15 tasks) to investigate whether insights from the theory of computation can predict the limits of neural network generalization in practice. We demonstrate that grouping tasks according to the Chomsky hierarchy allows us to forecast whether certain architectures will be able to generalize to out-of-distribution inputs. This includes negative results where even extensive amounts of data and training time never lead to any non-trivial generalization, despite models having sufficient capacity to fit the training data perfectly. Our results show that, for our subset of tasks, RNNs and Transformers fail to generalize on non-regular tasks, LSTMs can solve regular and counter-language tasks, and only networks augmented with structured memory (such as a stack or memory tape) can successfully generalize on context-free and context-sensitive tasks.

Neural Networks and the Chomsky Hierarchy

Weighted pushdown automata (WPDAs) are at the core of many natural language processing tasks, like syntax-based statistical machine translation and transition-based dependency parsing. As most existing dynamic programming algorithms are designed for context-free grammars (CFGs), algorithms for PDAs often resort to a PDA-to-CFG conversion. In this paper, we develop novel algorithms that operate directly on WPDAs. Our algorithms are inspired by Lang’s algorithm, but use a more general definition of pushdown automaton and either reduce the space requirements by a factor of |Gamma| (the size of the stack alphabet) or reduce the runtime by a factor of more than |Q| (the number of states). When run on the same class of PDAs as Lang’s algorithm, our algorithm is both more space-efficient by a factor of |Gamma| and more time-efficient by a factor of |Q| x |Gamma|.

Algorithms for Weighted Pushdown Automata

NLP is deeply intertwined with the formal study of language, both conceptually and historically. Arguably, this connection goes all the way back to Chomsky's Syntactic Structures in 1957. It also still holds true today, with a strand of recent works building formal analysis of modern neural networks methods in terms of formal languages. In this document, I aim to explain background about formal languages as they relate to this recent work. I will by necessity ignore large parts of the rich history of this field, instead focusing on concepts connecting to modern deep learning-based NLP.

Formal Language Theory Meets Modern NLP

Recursive calls over recursive data are useful for generating probability distributions, and probabilistic programming allows computations over these distributions to be expressed in a modular and intuitive way. Exact inference is also useful, but unfortunately, existing probabilistic programming languages do not perform exact inference on recursive calls over recursive data, forcing programmers to code many applications manually. We introduce a probabilistic language in which a wide variety of recursion can be expressed naturally, and inference carried out exactly. For instance, probabilistic pushdown automata and their generalizations are easy to express, and polynomial-time parsing algorithms for them are derived automatically. We eliminate recursive data types using program transformations related to defunctionalization and refunctionalization. These transformations are assured correct by a linear type system, and a successful choice of transformations, if there is one, is guaranteed to be found by a greedy algorithm.

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Exact Recursive Probabilistic Programming

Beam Tree Recursive Neural Network (BT-RvNN) was recently proposed as a simple extension of Gumbel Tree RvNN and it was shown to achieve state-of-the-art length generalization performance in ListOps while maintaining comparable performance on other tasks. However, although not the worst in its kind, BT-RvNN can be still exorbitantly expensive in memory usage. In this paper, we identify the main bottleneck in BT-RvNN's memory usage to be the entanglement of the scorer function and the recursive cell function. We propose strategies to remove this bottleneck and further simplify its memory usage. Overall, our strategies not only reduce the memory usage of BT-RvNN by $10$-$16$ times but also create a new state-of-the-art in ListOps while maintaining similar performance in other tasks. In addition, we also propose a strategy to utilize the induced latent-tree node representations produced by BT-RvNN to turn BT-RvNN from a sentence encoder of the form $f:\mathbb{R}^{n \times d} \rightarrow \mathbb{R}^{d}$ into a sequence contextualizer of the form $f:\mathbb{R}^{n \times d} \rightarrow \mathbb{R}^{n \times d}$. Thus, our proposals not only open up a path for further scalability of RvNNs but also standardize a way to use BT-RvNNs as another building block in the deep learning toolkit that can be easily stacked or interfaced with other popular models such as Transformers and Structured State Space models.

Efficient Beam Tree Recursion

We present a differentiable stack data structure that simultaneously and tractably encodes an exponential number of stack configurations, based on Lang’s algorithm for simulating nondeterministic pushdown automata. We call the combination of this data structure with a recurrent neural network (RNN) controller a Nondeterministic Stack RNN. We compare our model against existing stack RNNs on various formal languages, demonstrating that our model converges more reliably to algorithmic behavior on deterministic tasks, and achieves lower cross-entropy on inherently nondeterministic tasks.

/pdf/learning-context-free-languages-with-nondeterministic-stack-3udvfqisai.pdf

Learning Context-free Languages with Nondeterministic Stack RNNs

This paper describes the Notre Dame Natural Language Processing Group’s (NDNLP) submission to the WNGT 2019 shared task (Hayashi et al., 2019). We investigated the impact of auto-sizing (Murray and Chiang, 2015; Murray et al., 2019) to the Transformer network (Vaswani et al., 2017) with the goal of substantially reducing the number of parameters in the model. Our method was able to eliminate more than 25% of the model’s parameters while suffering a decrease of only 1.1 BLEU.

/pdf/efficiency-through-auto-sizing-notre-dame-nlp-s-submission-fjjvbbjlwe.pdf

Efficiency through Auto-Sizing: Notre Dame NLP’s Submission to the WNGT 2019 Efficiency Task

Traditional recurrent neural networks (RNNs) have a fixed, finite number of memory cells. In theory (assuming bounded range and precision), this limits their formal language recognition power to regular languages, and in practice, RNNs have been shown to be unable to learn many context-free languages (CFLs). In order to expand the class of languages RNNs recognize, prior work has augmented RNNs with a nondeterministic stack data structure, putting them on par with pushdown automata and increasing their language recognition power to CFLs. Nondeterminism is needed for recognizing all CFLs (not just deterministic CFLs), but in this paper, we show that nondeterminism and the neural controller interact to produce two more unexpected abilities. First, the nondeterministic stack RNN can recognize not only CFLs, but also many non-context-free languages. Second, it can recognize languages with much larger alphabet sizes than one might expect given the size of its stack alphabet. Finally, to increase the information capacity in the stack and allow it to solve more complicated tasks with large alphabet sizes, we propose a new version of the nondeterministic stack that simulates stacks of vectors rather than discrete symbols. We demonstrate perplexity improvements with this new model on the Penn Treebank language modeling benchmark.

The Surprising Computational Power of Nondeterministic Stack RNNs

This paper describes the Notre Dame Natural Language Processing Group's (NDNLP) submission to the WNGT 2019 shared task (Hayashi et al., 2019). We investigated the impact of auto-sizing (Murray and Chiang, 2015; Murray et al., 2019) to the Transformer network (Vaswani et al., 2017) with the goal of substantially reducing the number of parameters in the model. Our method was able to eliminate more than 25% of the model's parameters while suffering a decrease of only 1.1 BLEU.

Brian DuSell

Papers

Learning Context-free Languages with Nondeterministic Stack RNNs

Algorithms for Weighted Pushdown Automata

Efficiency through Auto-Sizing: Notre Dame NLP’s Submission to the WNGT 2019 Efficiency Task

The Surprising Computational Power of Nondeterministic Stack RNNs

Efficiency through Auto-Sizing: Notre Dame NLP's Submission to the WNGT 2019 Efficiency Task.