HARP: A challenging human-annotated math reasoning benchmark

Math reasoning is becoming an ever increasing area of focus as we scale large language models. However, even the previously-toughest evals like MATH are now close to saturated by frontier models (90.0% for o1-mini and 86.5% for Gemini 1.5 Pro). We introduce HARP, Human Annotated Reasoning Problems (for Math), consisting of 5,409 problems from the US national math competitions (A(J)HSME, AMC, AIME, USA(J)MO). Of these, 4,780 have answers that are automatically check-able (with libraries such as SymPy). These problems range six difficulty levels, with frontier models performing relatively poorly on the hardest bracket of 197 problems (average accuracy 41.1% for o1-mini, and 9.6% for Gemini 1.5 Pro). Our dataset also features multiple choices (for 4,110 problems) and an average of two human-written, ground-truth solutions per problem, offering new avenues of research that we explore briefly. We report evaluations for many frontier models and share some interesting analyses, such as demonstrating that frontier models across families intrinsically scale their inference-time compute for more difficult problems. Finally, we open source all code used for dataset construction (including scraping) and all code for evaluation (including answer checking) to enable future research at: https://github.com/aadityasingh/HARP.

Tokenization counts: the impact of tokenization on arithmetic in frontier LLMs

Tokenization, the division of input text into input tokens, is an often overlooked aspect of the large language model (LLM) pipeline and could be the source of useful or harmful inductive biases. Historically, LLMs have relied on byte pair encoding, without care to specific input domains. With the increased use of LLMs for reasoning, various number-specific tokenization schemes have been adopted, with popular models like LLaMa and PaLM opting for single-digit tokenization while GPT-3.5 and GPT-4 have separate tokens for each 1-, 2-, and 3-digit numbers. In this work, we study the effect this choice has on numerical reasoning through the use of arithmetic tasks. We consider left-to-right and right-to-left tokenization for GPT-3.5 and -4, finding that right-to-left tokenization (enforced by comma separating numbers at inference time) leads to largely improved performance. Furthermore, we find that model errors when using standard left-to-right tokenization follow stereotyped error patterns, suggesting that model computations are systematic rather than approximate. We show that the model is able to convert between tokenizations easily, thus allowing chain-of-thought-inspired approaches to recover performance on left-to-right tokenized inputs. We also find the gap between tokenization directions decreases when models are scaled, possibly indicating that larger models are better able to override this tokenization-dependent inductive bias. In summary, our work performs the first study of how number tokenization choices lead to differences in model performance on arithmetic tasks, accompanied by a thorough analysis of error patterns. We hope this work inspires practitioners to more carefully ablate number tokenization-related choices when working towards general models of numerical reasoning.

Confronting Reward Model Overoptimization with Constrained RLHF

Large language models are typically aligned with human preferences by optimizing $\textit{reward models}$ (RMs) fitted to human feedback. However, human preferences are multi-faceted, and it is increasingly common to derive reward from a composition of simpler reward models which each capture a different aspect of language quality. This itself presents a challenge, as it is difficult to appropriately weight these component RMs when combining them. Compounding this difficulty, because any RM is only a proxy for human evaluation, this process is vulnerable to $\textit{overoptimization}$, wherein past a certain point, accumulating higher reward is associated with worse human ratings. In this paper, we perform, to our knowledge, the first study on overoptimization in composite RMs, showing that correlation between component RMs has a significant effect on the locations of these points. We then introduce an approach to solve this issue using constrained reinforcement learning as a means of preventing the agent from exceeding each RM's threshold of usefulness. Our method addresses the problem of weighting component RMs by learning dynamic weights, naturally expressed by Lagrange multipliers. As a result, each RM stays within the range at which it is an effective proxy, improving evaluation performance. Finally, we introduce an adaptive method using gradient-free optimization to identify and optimize towards these points during a single run.

Melting Pot 2.0

Multi-agent artificial intelligence research promises a path to develop intelligent technologies that are more human-like and more human-compatible than those produced by "solipsistic" approaches, which do not consider interactions between agents. Melting Pot is a research tool developed to facilitate work on multi-agent artificial intelligence, and provides an evaluation protocol that measures generalization to novel social partners in a set of canonical test scenarios. Each scenario pairs a physical environment (a "substrate") with a reference set of co-players (a "background population"), to create a social situation with substantial interdependence between the individuals involved. For instance, some scenarios were inspired by institutional-economics-based accounts of natural resource management and public-good-provision dilemmas. Others were inspired by considerations from evolutionary biology, game theory, and artificial life. Melting Pot aims to cover a maximally diverse set of interdependencies and incentives. It includes the commonly-studied extreme cases of perfectly-competitive (zero-sum) motivations and perfectly-cooperative (shared-reward) motivations, but does not stop with them. As in real-life, a clear majority of scenarios in Melting Pot have mixed incentives. They are neither purely competitive nor purely cooperative and thus demand successful agents be able to navigate the resulting ambiguity. Here we describe Melting Pot 2.0, which revises and expands on Melting Pot. We also introduce support for scenarios with asymmetric roles, and explain how to integrate them into the evaluation protocol. This report also contains: (1) details of all substrates and scenarios; (2) a complete description of all baseline algorithms and results. Our intention is for it to serve as a reference for researchers using Melting Pot 2.0.

In-context Reinforcement Learning with Algorithm Distillation

We propose Algorithm Distillation (AD), a method for distilling reinforcement learning (RL) algorithms into neural networks by modeling their training histories with a causal sequence model. Algorithm Distillation treats learning to reinforcement learn as an across-episode sequential prediction problem. A dataset of learning histories is generated by a source RL algorithm, and then a causal transformer is trained by autoregressively predicting actions given their preceding learning histories as context. Unlike sequential policy prediction architectures that distill post-learning or expert sequences, AD is able to improve its policy entirely in-context without updating its network parameters. We demonstrate that AD can reinforcement learn in-context in a variety of environments with sparse rewards, combinatorial task structure, and pixel-based observations, and find that AD learns a more data-efficient RL algorithm than the one that generated the source data.

Semantic Exploration from Language Abstractions and Pretrained Representations

Continuous first-person 3D environments pose unique exploration challenges to reinforcement learning (RL) agents because of their high-dimensional state and action spaces. These challenges can be ameliorated by using semantically meaningful state abstractions to define novelty for exploration. We propose that learned representations shaped by natural language provide exactly this form of abstraction. In particular, we show that vision-language representations, when pretrained on image captioning datasets sampled from the internet, can drive meaningful, task-relevant exploration and improve performance on 3D simulated environments. We also characterize why and how language provides useful abstractions for exploration by comparing the impacts of using representations from a pretrained model, a language oracle, and several ablations. We demonstrate the benefits of our approach in two very different task domains -- one that stresses the identification and manipulation of everyday objects, and one that requires navigational exploration in an expansive world -- as well as two popular deep RL algorithms: Impala and R2D2. Our results suggest that using language-shaped representations could improve exploration for various algorithms and agents in challenging environments.

Collaborating with Humans without Human Data

Collaborating with humans requires rapidly adapting to their individual strengths, weaknesses, and preferences. Unfortunately, most standard multi-agent reinforcement learning techniques, such as self-play (SP) or population play (PP), produce agents that overfit to their training partners and do not generalize well to humans. Alternatively, researchers can collect human data, train a human model using behavioral cloning, and then use that model to train "human-aware" agents ("behavioral cloning play", or BCP). While such an approach can improve the generalization of agents to new human co-players, it involves the onerous and expensive step of collecting large amounts of human data first. Here, we study the problem of how to train agents that collaborate well with human partners without using human data. We argue that the crux of the problem is to produce a diverse set of training partners. Drawing inspiration from successful multi-agent approaches in competitive domains, we find that a surprisingly simple approach is highly effective. We train our agent partner as the best response to a population of self-play agents and their past checkpoints taken throughout training, a method we call Fictitious Co-Play (FCP). Our experiments focus on a two-player collaborative cooking simulator that has recently been proposed as a challenge problem for coordination with humans. We find that FCP agents score significantly higher than SP, PP, and BCP when paired with novel agent and human partners. Furthermore, humans also report a strong subjective preference to partnering with FCP agents over all baselines.

Learning more skills through optimistic exploration

Unsupervised skill learning objectives (Gregor et al., 2016, Eysenbach et al., 2018) allow agents to learn rich repertoires of behavior in the absence of extrinsic rewards. They work by simultaneously training a policy to produce distinguishable latent-conditioned trajectories, and a discriminator to evaluate distinguishability by trying to infer latents from trajectories. The hope is for the agent to explore and master the environment by encouraging each skill (latent) to reliably reach different states. However, an inherent exploration problem lingers: when a novel state is actually encountered, the discriminator will necessarily not have seen enough training data to produce accurate and confident skill classifications, leading to low intrinsic reward for the agent and effective penalization of the sort of exploration needed to actually maximize the objective. To combat this inherent pessimism towards exploration, we derive an information gain auxiliary objective that involves training an ensemble of discriminators and rewarding the policy for their disagreement. Our objective directly estimates the epistemic uncertainty that comes from the discriminator not having seen enough training examples, thus providing an intrinsic reward more tailored to the true objective compared to pseudocount-based methods (Burda et al., 2019). We call this exploration bonus discriminator disagreement intrinsic reward, or DISDAIN. We demonstrate empirically that DISDAIN improves skill learning both in a tabular grid world (Four Rooms) and the 57 games of the Atari Suite (from pixels). Thus, we encourage researchers to treat pessimism with DISDAIN.

A Neural Architecture for Designing Truthful and Efficient Auctions

Auctions are protocols to allocate goods to buyers who have preferences over them, and collect payments in return. Economists have invested significant effort in designing auction rules that result in allocations of the goods that are desirable for the group as a whole. However, for settings where participants' valuations of the items on sale are their private information, the rules of the auction must deter buyers from misreporting their preferences, so as to maximize their own utility, since misreported preferences hinder the ability for the auctioneer to allocate goods to those who want them most. Manual auction design has yielded excellent mechanisms for specific settings, but requires significant effort when tackling new domains. We propose a deep learning based approach to automatically design auctions in a wide variety of domains, shifting the design work from human to machine. We assume that participants' valuations for the items for sale are independently sampled from an unknown but fixed distribution. Our system receives a data-set consisting of such valuation samples, and outputs an auction rule encoding the desired incentive structure. We focus on producing truthful and efficient auctions that minimize the economic burden on participants. We evaluate the auctions designed by our framework on well-studied domains, such as multi-unit and combinatorial auctions, showing that they outperform known auction designs in terms of the economic burden placed on participants.

Intrinsic Social Motivation via Causal Influence in Multi-Agent RL

We derive a new intrinsic social motivation for multi-agent reinforcement learning (MARL), in which agents are rewarded for having causal influence over another agent's actions. Causal influence is assessed using counterfactual reasoning. The reward does not depend on observing another agent's reward function, and is thus a more realistic approach to MARL than taken in previous work. We show that the causal influence reward is related to maximizing the mutual information between agents' actions. We test the approach in challenging social dilemma environments, where it consistently leads to enhanced cooperation between agents and higher collective reward. Moreover, we find that rewarding influence can lead agents to develop emergent communication protocols. We therefore employ influence to train agents to use an explicit communication channel, and find that it leads to more effective communication and higher collective reward. Finally, we show that influence can be computed by equipping each agent with an internal model that predicts the actions of other agents. This allows the social influence reward to be computed without the use of a centralised controller, and as such represents a significantly more general and scalable inductive bias for MARL with independent agents.