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We consider the problem of low probability estimation: given a machine learning model and a formally-specified input distribution, how can we estimate the probability of a binary property of the model's output, even when that probability is too small to estimate by random sampling? This problem is motivated by the need to improve worst-case performance, which distribution shift can make much more likely. We study low probability estimation in the context of argmax sampling from small transformer language models. We compare two types of methods: importance sampling, which involves searching for inputs giving rise to the rare output, and activation extrapolation, which involves extrapolating a probability distribution fit to the model's logits. We find that importance sampling outperforms activation extrapolation, but both outperform naive sampling. Finally, we explain how minimizing the probability estimate of an undesirable behavior generalizes adversarial training, and argue that new methods for low probability estimation are needed to provide stronger guarantees about worst-case performance., Comment: 27 pages, 9 figures

Christiano et al. (2022) define a *heuristic estimator* to be a hypothetical algorithm that estimates the values of mathematical expressions from arguments. In brief, a heuristic estimator $\mathbb{G}$ takes as input a mathematical expression $Y$ and a formal "heuristic argument" $\pi$, and outputs an estimate $\mathbb{G}(Y \mid \pi)$ of $Y$. In this work, we argue for the informal principle that a heuristic estimator ought not to be able to predict its own errors, and we explore approaches to formalizing this principle. Most simply, the principle suggests that $\mathbb{G}(Y - \mathbb{G}(Y \mid \pi) \mid \pi)$ ought to equal zero for all $Y$ and $\pi$. We argue that an ideal heuristic estimator ought to satisfy two stronger properties in this vein, which we term *iterated estimation* (by analogy to the law of iterated expectations) and *error orthogonality*. Although iterated estimation and error orthogonality are intuitively appealing, it can be difficult to determine whether a given heuristic estimator satisfies the properties. As an alternative approach, we explore *accuracy*: a property that (roughly) states that $\mathbb{G}$ has zero average error over a distribution of mathematical expressions. However, in the context of two estimation problems, we demonstrate barriers to creating an accurate heuristic estimator. We finish by discussing challenges and potential paths forward for finding a heuristic estimator that accords with our intuitive understanding of how such an estimator ought to behave, as well as the potential applications of heuristic estimators to understanding the behavior of neural networks., Comment: 47 pages, 2 tables, 1 figure

We introduce a formal notion of defendability against backdoors using a game between an attacker and a defender. In this game, the attacker modifies a function to behave differently on a particular input known as the "trigger", while behaving the same almost everywhere else. The defender then attempts to detect the trigger at evaluation time. If the defender succeeds with high enough probability, then the function class is said to be defendable. The key constraint on the attacker that makes defense possible is that the attacker's strategy must work for a randomly-chosen trigger. Our definition is simple and does not explicitly mention learning, yet we demonstrate that it is closely connected to learnability. In the computationally unbounded setting, we use a voting algorithm of Hanneke et al. (2022) to show that defendability is essentially determined by the VC dimension of the function class, in much the same way as PAC learnability. In the computationally bounded setting, we use a similar argument to show that efficient PAC learnability implies efficient defendability, but not conversely. On the other hand, we use indistinguishability obfuscation to show that the class of polynomial size circuits is not efficiently defendable. Finally, we present polynomial size decision trees as a natural example for which defense is strictly easier than learning. Thus, we identify efficient defendability as a notable intermediate concept in between efficient learnability and obfuscation., Comment: 29 pages

Recent work has shown that, in generative modeling, cross-entropy loss improves smoothly with model size and training compute, following a power law plus constant scaling law. One challenge in extending these results to reinforcement learning is that the main performance objective of interest, mean episode return, need not vary smoothly. To overcome this, we introduce *intrinsic performance*, a monotonic function of the return defined as the minimum compute required to achieve the given return across a family of models of different sizes. We find that, across a range of environments, intrinsic performance scales as a power law in model size and environment interactions. Consequently, as in generative modeling, the optimal model size scales as a power law in the training compute budget. Furthermore, we study how this relationship varies with the environment and with other properties of the training setup. In particular, using a toy MNIST-based environment, we show that varying the "horizon length" of the task mostly changes the coefficient but not the exponent of this relationship., Comment: 33 pages

In reinforcement learning from human feedback, it is common to optimize against a reward model trained to predict human preferences. Because the reward model is an imperfect proxy, optimizing its value too much can hinder ground truth performance, in accordance with Goodhart's law. This effect has been frequently observed, but not carefully measured due to the expense of collecting human preference data. In this work, we use a synthetic setup in which a fixed "gold-standard" reward model plays the role of humans, providing labels used to train a proxy reward model. We study how the gold reward model score changes as we optimize against the proxy reward model using either reinforcement learning or best-of-$n$ sampling. We find that this relationship follows a different functional form depending on the method of optimization, and that in both cases its coefficients scale smoothly with the number of reward model parameters. We also study the effect on this relationship of the size of the reward model dataset, the number of reward model and policy parameters, and the coefficient of the KL penalty added to the reward in the reinforcement learning setup. We explore the implications of these empirical results for theoretical considerations in AI alignment.

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., Comment: 27 pages, 17 figures + references and appendices, repo: https://github.com/google/BIG-bench

We show that a GPT-3 model can learn to express uncertainty about its own answers in natural language -- without use of model logits. When given a question, the model generates both an answer and a level of confidence (e.g. "90% confidence" or "high confidence"). These levels map to probabilities that are well calibrated. The model also remains moderately calibrated under distribution shift, and is sensitive to uncertainty in its own answers, rather than imitating human examples. To our knowledge, this is the first time a model has been shown to express calibrated uncertainty about its own answers in natural language. For testing calibration, we introduce the CalibratedMath suite of tasks. We compare the calibration of uncertainty expressed in words ("verbalized probability") to uncertainty extracted from model logits. Both kinds of uncertainty are capable of generalizing calibration under distribution shift. We also provide evidence that GPT-3's ability to generalize calibration depends on pre-trained latent representations that correlate with epistemic uncertainty over its answers., Comment: CalibratedMath tasks and evaluation code are available at https://github.com/sylinrl/CalibratedMath

Making language models bigger does not inherently make them better at following a user's intent. For example, large language models can generate outputs that are untruthful, toxic, or simply not helpful to the user. In other words, these models are not aligned with their users. In this paper, we show an avenue for aligning language models with user intent on a wide range of tasks by fine-tuning with human feedback. Starting with a set of labeler-written prompts and prompts submitted through the OpenAI API, we collect a dataset of labeler demonstrations of the desired model behavior, which we use to fine-tune GPT-3 using supervised learning. We then collect a dataset of rankings of model outputs, which we use to further fine-tune this supervised model using reinforcement learning from human feedback. We call the resulting models InstructGPT. In human evaluations on our prompt distribution, outputs from the 1.3B parameter InstructGPT model are preferred to outputs from the 175B GPT-3, despite having 100x fewer parameters. Moreover, InstructGPT models show improvements in truthfulness and reductions in toxic output generation while having minimal performance regressions on public NLP datasets. Even though InstructGPT still makes simple mistakes, our results show that fine-tuning with human feedback is a promising direction for aligning language models with human intent.

We fine-tune GPT-3 to answer long-form questions using a text-based web-browsing environment, which allows the model to search and navigate the web. By setting up the task so that it can be performed by humans, we are able to train models on the task using imitation learning, and then optimize answer quality with human feedback. To make human evaluation of factual accuracy easier, models must collect references while browsing in support of their answers. We train and evaluate our models on ELI5, a dataset of questions asked by Reddit users. Our best model is obtained by fine-tuning GPT-3 using behavior cloning, and then performing rejection sampling against a reward model trained to predict human preferences. This model's answers are preferred by humans 56% of the time to those of our human demonstrators, and 69% of the time to the highest-voted answer from Reddit., Comment: 32 pages

State-of-the-art language models can match human performance on many tasks, but they still struggle to robustly perform multi-step mathematical reasoning. To diagnose the failures of current models and support research, we introduce GSM8K, a dataset of 8.5K high quality linguistically diverse grade school math word problems. We find that even the largest transformer models fail to achieve high test performance, despite the conceptual simplicity of this problem distribution. To increase performance, we propose training verifiers to judge the correctness of model completions. At test time, we generate many candidate solutions and select the one ranked highest by the verifier. We demonstrate that verification significantly improves performance on GSM8K, and we provide strong empirical evidence that verification scales more effectively with increased data than a finetuning baseline.

We say an algorithm is batch size-invariant if changes to the batch size can largely be compensated for by changes to other hyperparameters. Stochastic gradient descent is well-known to have this property at small batch sizes, via the learning rate. However, some policy optimization algorithms (such as PPO) do not have this property, because of how they control the size of policy updates. In this work we show how to make these algorithms batch size-invariant. Our key insight is to decouple the proximal policy (used for controlling policy updates) from the behavior policy (used for off-policy corrections). Our experiments help explain why these algorithms work, and additionally show how they can make more efficient use of stale data., Comment: 32 pages. Code is available at https://github.com/openai/ppo-ewma

We propose a benchmark to measure whether a language model is truthful in generating answers to questions. The benchmark comprises 817 questions that span 38 categories, including health, law, finance and politics. We crafted questions that some humans would answer falsely due to a false belief or misconception. To perform well, models must avoid generating false answers learned from imitating human texts. We tested GPT-3, GPT-Neo/J, GPT-2 and a T5-based model. The best model was truthful on 58% of questions, while human performance was 94%. Models generated many false answers that mimic popular misconceptions and have the potential to deceive humans. The largest models were generally the least truthful. This contrasts with other NLP tasks, where performance improves with model size. However, this result is expected if false answers are learned from the training distribution. We suggest that scaling up models alone is less promising for improving truthfulness than fine-tuning using training objectives other than imitation of text from the web., Comment: ACL 2022 (main conference); the TruthfulQA benchmark and evaluation code is available at https://github.com/sylinrl/TruthfulQA

The NeurIPS 2020 Procgen Competition was designed as a centralized benchmark with clearly defined tasks for measuring Sample Efficiency and Generalization in Reinforcement Learning. Generalization remains one of the most fundamental challenges in deep reinforcement learning, and yet we do not have enough benchmarks to measure the progress of the community on Generalization in Reinforcement Learning. We present the design of a centralized benchmark for Reinforcement Learning which can help measure Sample Efficiency and Generalization in Reinforcement Learning by doing end to end evaluation of the training and rollout phases of thousands of user submitted code bases in a scalable way. We designed the benchmark on top of the already existing Procgen Benchmark by defining clear tasks and standardizing the end to end evaluation setups. The design aims to maximize the flexibility available for researchers who wish to design future iterations of such benchmarks, and yet imposes necessary practical constraints to allow for a system like this to scale. This paper presents the competition setup and the details and analysis of the top solutions identified through this setup in context of 2020 iteration of the competition at NeurIPS.

We introduce Phasic Policy Gradient (PPG), a reinforcement learning framework which modifies traditional on-policy actor-critic methods by separating policy and value function training into distinct phases. In prior methods, one must choose between using a shared network or separate networks to represent the policy and value function. Using separate networks avoids interference between objectives, while using a shared network allows useful features to be shared. PPG is able to achieve the best of both worlds by splitting optimization into two phases, one that advances training and one that distills features. PPG also enables the value function to be more aggressively optimized with a higher level of sample reuse. Compared to PPO, we find that PPG significantly improves sample efficiency on the challenging Procgen Benchmark.

We introduce Procgen Benchmark, a suite of 16 procedurally generated game-like environments designed to benchmark both sample efficiency and generalization in reinforcement learning. We believe that the community will benefit from increased access to high quality training environments, and we provide detailed experimental protocols for using this benchmark. We empirically demonstrate that diverse environment distributions are essential to adequately train and evaluate RL agents, thereby motivating the extensive use of procedural content generation. We then use this benchmark to investigate the effects of scaling model size, finding that larger models significantly improve both sample efficiency and generalization.

We study the topological version of the partition calculus in the setting of countable ordinals. Let $\alpha$ and $\beta$ be ordinals and let $k$ be a positive integer. We write $\beta\to_{top}(\alpha,k)^2$ to mean that, for every red-blue coloring of the collection of 2-sized subsets of $\beta$, there is either a red-homogeneous set homeomorphic to $\alpha$ or a blue-homogeneous set of size $k$. The least such $\beta$ is the topological Ramsey number $R^{top}(\alpha,k)$. We prove a topological version of the Erd\H{o}s-Milner theorem, namely that $R^{top}(\alpha,k)$ is countable whenever $\alpha$ is countable. More precisely, we prove that $R^{top}(\omega^{\omega^\beta},k+1)\leq\omega^{\omega^{\beta\cdot k}}$ for all countable ordinals $\beta$ and finite $k$. Our proof is modeled on a new easy proof of a weak version of the Erd\H{o}s-Milner theorem that may be of independent interest. We also provide more careful upper bounds for certain small values of $\alpha$, proving among other results that $R^{top}(\omega+1,k+1)=\omega^k+1$, $R^{top}(\alpha,k)< \omega^\omega$ whenever $\alpha<\omega^2$, $R^{top}(\omega^2,k)\leq\omega^\omega$ and $R^{top}(\omega^2+1,k+2)\leq\omega^{\omega\cdot k}+1$ for all finite $k$. Our computations use a variety of techniques, including a topological pigeonhole principle for ordinals, considerations of a tree ordering based on the Cantor normal form of ordinals, and some ultrafilter arguments., Comment: Final version

Given a cardinal $\kappa$ and a sequence $\left(\alpha_i\right)_{i\in\kappa}$ of ordinals, we determine the least ordinal $\beta$ (when one exists) such that the topological partition relation \[\beta\rightarrow\left(top\,\alpha_i\right)^1_{i\in\kappa}\] holds, including an independence result for one class of cases. Here the prefix "$top$" means that the homogeneous set must have the correct topology rather than the correct order type. The answer is linked to the non-topological pigeonhole principle of Milner and Rado., Comment: 24 pages