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The surprising ability of Large Language Models (LLMs) to perform well on complex reasoning with only few-shot chain-of-thought prompts is believed to emerge only in very large-scale models (100+ billion parameters). We show that such abilities can, in fact, be distilled down from GPT-3.5 ($\ge$ 175B) to T5 variants ($\le$ 11B). We propose model specialization, to specialize the model's ability towards a target task. The hypothesis is that large models (commonly viewed as larger than 100B) have strong modeling power, but are spread on a large spectrum of tasks. Small models (commonly viewed as smaller than 10B) have limited model capacity, but if we concentrate their capacity on a specific target task, the model can achieve a decent improved performance. We use multi-step math reasoning as our testbed because it is a very typical emergent ability. We show two important aspects of model abilities: (1). there exists a very complex balance/ tradeoff between language models' multi-dimensional abilities; (2). by paying the price of decreased generic ability, we can clearly lift up the scaling curve of models smaller than 10B towards a specialized multi-step math reasoning ability. We further give comprehensive discussions about important design choices for better generalization, including the tuning data format, the start model checkpoint, and a new model selection method. We hope our practice and discoveries can serve as an important attempt towards specialized smaller models in the new research paradigm set by LLMs., Preprint

When large language models (LMs) are applied in zero- or few-shot settings to discriminative tasks such as multiple-choice questions, their attentiveness (i.e., probability mass) is spread across many vocabulary tokens that are not valid choices. Such a spread across multiple surface forms with identical meaning is thought to cause an underestimation of a model's true performance, referred to as the "surface form competition" (SFC) hypothesis. This has motivated the introduction of various probability normalization methods. However, many core questions remain unanswered. How do we measure SFC or attentiveness? Are there direct ways of increasing attentiveness on valid choices? Does increasing attentiveness always improve task accuracy? We propose a mathematical formalism for studying this phenomenon, provide a metric for quantifying attentiveness, and identify a simple method for increasing it -- namely, in-context learning with even just one example containing answer choices. The formalism allows us to quantify SFC and bound its impact. Our experiments on three diverse datasets and six LMs reveal several surprising findings. For example, encouraging models to generate a valid answer choice can, in fact, be detrimental to task performance for some LMs, and prior probability normalization methods are less effective (sometimes even detrimental) to instruction-tuned LMs. We conclude with practical insights for effectively using prompted LMs for multiple-choice tasks.

One way to interpret the reasoning power of transformer-based language models is to describe the types of logical rules they can resolve over some input text. Recently, Chiang et al. (2023) showed that finite-precision transformers can be equivalently expressed in a generalization of first-order logic. However, finite-precision transformers are a weak transformer variant because, as we show, a single head can only attend to a constant number of tokens and, in particular, cannot represent uniform attention. Since attending broadly is a core capability for transformers, we ask whether a minimally more expressive model that can attend universally can also be characterized in logic. To this end, we analyze transformers whose forward pass is computed in $\log n$ precision on contexts of length $n$. We prove that any log-precision transformer can be equivalently expressed as a first-order logic sentence that, in addition to standard universal and existential quantifiers, may also contain majority-vote quantifiers. This is the tightest known upper bound and first logical characterization of log-precision transformers., May 24, 2023: Restructured version of old preprint

Language models demonstrate both quantitative improvement and new qualitative capabilities with increasing scale. Despite their potentially transformative impact, these new capabilities are as yet poorly characterized. In order to inform future research, prepare for disruptive new model capabilities, and ameliorate socially harmful effects, it is vital that we understand the present and near-future capabilities and limitations of language models. To address this challenge, we introduce the Beyond the Imitation Game benchmark (BIG-bench). BIG-bench currently consists of 204 tasks, contributed by 450 authors across 132 institutions. Task topics are diverse, drawing problems from linguistics, childhood development, math, common-sense reasoning, biology, physics, social bias, software development, and beyond. BIG-bench focuses on tasks that are believed to be beyond the capabilities of current language models. We evaluate the behavior of OpenAI's GPT models, Google-internal dense transformer architectures, and Switch-style sparse transformers on BIG-bench, across model sizes spanning millions to hundreds of billions of parameters. In addition, a team of human expert raters performed all tasks in order to provide a strong baseline. Findings include: model performance and calibration both improve with scale, but are poor in absolute terms (and when compared with rater performance); performance is remarkably similar across model classes, though with benefits from sparsity; tasks that improve gradually and predictably commonly involve a large knowledge or memorization component, whereas tasks that exhibit "breakthrough" behavior at a critical scale often involve multiple steps or components, or brittle metrics; social bias typically increases with scale in settings with ambiguous context, but this can be improved with prompting., 27 pages, 17 figures + references and appendices, repo: https://github.com/google/BIG-bench

We study the task of prompting large-scale language models to perform multi-step reasoning. Existing work shows that when prompted with a chain of thoughts (CoT), sequences of short sentences describing intermediate reasoning steps towards a final answer, large language models can generate new reasoning chains and predict answers for new inputs. A central question is which reasoning examples make the most effective prompts. In this work, we propose complexity-based prompting, a simple and effective example selection scheme for multi-step reasoning. We show that prompts with higher reasoning complexity, i.e., chains with more reasoning steps, achieve substantially better performance on multi-step reasoning tasks over strong baselines. We further extend our complexity-based criteria from prompting (selecting inputs) to decoding (selecting outputs), where we sample multiple reasoning chains from the model, then choose the majority of generated answers from complex reasoning chains (over simple chains). When used to prompt GPT-3 and Codex, our approach substantially improves multi-step reasoning accuracy and achieves new state-of-the-art (SOTA) performance on three math benchmarks (GSM8K, MultiArith, and MathQA) and two BigBenchHard tasks (Date Understanding and Penguins), with an average +5.3 and up to +18 accuracy improvements. Compared with existing example selection schemes like manual tuning or retrieval-based selection, selection based on reasoning complexity is intuitive, easy to implement, and annotation-efficient. Further results demonstrate the robustness of performance gains from complex prompts under format perturbation and distribution shift., Comment: Preprint

Transformers have become a standard neural network architecture for many NLP problems, motivating theoretical analysis of their power in terms of formal languages. Recent work has shown that transformers with hard attention are quite limited in power (Hahn, 2020), as they can be simulated by constant-depth AND/OR circuits (Hao et al. 2021). However, hard attention is a strong assumption, which may complicate the relevance of these results in practice. In this work, we analyze the circuit complexity of transformers with saturated attention: a generalization of hard attention that more closely captures the attention patterns learnable in practical transformers. We first show that saturated transformers transcend the known limitations of hard-attention transformers. We then prove saturated transformers with floating-point values can be simulated by constant-depth threshold circuits, giving the class $\mathsf{TC}^0$ as an upper bound on the formal languages they recognize., To appear in TACL

Large neural models have demonstrated human-level performance on language and vision benchmarks, while their performance degrades considerably on adversarial or out-of-distribution samples. This raises the question of whether these models have learned to solve a dataset rather than the underlying task by overfitting to spurious dataset biases. We investigate one recently proposed approach, AFLite, which adversarially filters such dataset biases, as a means to mitigate the prevalent overestimation of machine performance. We provide a theoretical understanding for AFLite, by situating it in the generalized framework for optimum bias reduction. We present extensive supporting evidence that AFLite is broadly applicable for reduction of measurable dataset biases, and that models trained on the filtered datasets yield better generalization to out-of-distribution tasks. Finally, filtering results in a large drop in model performance (e.g., from 92% to 62% for SNLI), while human performance still remains high. Our work thus shows that such filtered datasets can pose new research challenges for robust generalization by serving as upgraded benchmarks., Comment: Accepted to ICML 2020

Learned neural solvers have successfully been used to solve combinatorial optimization and decision problems. More general counting variants of these problems, however, are still largely solved with hand-crafted solvers. To bridge this gap, we introduce belief propagation neural networks (BPNNs), a class of parameterized operators that operate on factor graphs and generalize Belief Propagation (BP). In its strictest form, a BPNN layer (BPNN-D) is a learned iterative operator that provably maintains many of the desirable properties of BP for any choice of the parameters. Empirically, we show that by training BPNN-D learns to perform the task better than the original BP: it converges 1.7x faster on Ising models while providing tighter bounds. On challenging model counting problems, BPNNs compute estimates 100's of times faster than state-of-the-art handcrafted methods, while returning an estimate of comparable quality.

Computing the permanent of a non-negative matrix is a core problem with practical applications ranging from target tracking to statistical thermodynamics. However, this problem is also #P-complete, which leaves little hope for finding an exact solution that can be computed efficiently. While the problem admits a fully polynomial randomized approximation scheme, this method has seen little use because it is both inefficient in practice and difficult to implement. We present AdaPart, a simple and efficient method for drawing exact samples from an unnormalized distribution. Using AdaPart, we show how to construct tight bounds on the permanent which hold with high probability, with guaranteed polynomial runtime for dense matrices. We find that AdaPart can provide empirical speedups exceeding 25x over prior sampling methods on matrices that are challenging for variational based approaches. Finally, in the context of multi-target tracking, exact sampling from the distribution defined by the matrix permanent allows us to use the optimal proposal distribution during particle filtering. Using AdaPart, we show that this leads to improved tracking performance using an order of magnitude fewer samples., 19 pages

Discrete integration in a high dimensional space of n variables poses fundamental challenges. The WISH algorithm reduces the intractable discrete integration problem into n optimization queries subject to randomized constraints, obtaining a constant approximation guarantee. The optimization queries are expensive, which limits the applicability of WISH. We propose AdaWISH, which is able to obtain the same guarantee but accesses only a small subset of queries of WISH. For example, when the number of function values is bounded by a constant, AdaWISH issues only O(log n) queries. The key idea is to query adaptively, taking advantage of the shape of the weight function being integrated. In general, we prove that AdaWISH has a regret of only O(log n) relative to an idealistic oracle that issues queries at data-dependent optimal points. Experimentally, AdaWISH gives precise estimates for discrete integration problems, of the same quality as that of WISH and better than several competing approaches, on a variety of probabilistic inference benchmarks. At the same time, it saves substantially on the number of optimization queries compared to WISH. On a suite of UAI inference challenge benchmarks, it saves 81.5% of WISH queries while retaining the quality of results.

Integration is affected by the curse of dimensionality and quickly becomes intractable as the dimensionality of the problem grows. We propose a randomized algorithm that, with high probability, gives a constant-factor approximation of a general discrete integral defined over an exponentially large set. This algorithm relies on solving only a small number of instances of a discrete combinatorial optimization problem subject to randomly generated parity constraints used as a hash function. As an application, we demonstrate that with a small number of MAP queries we can efficiently approximate the partition function of discrete graphical models, which can in turn be used, for instance, for marginal computation or model selection.