On the Expressive Power of a Variant of the Looped Transformer

Besides natural language processing, transformers exhibit extraordinary performance in solving broader applications, including scientific computing and computer vision. Previous works try to explain this from the expressive power and capability perspectives that standard transformers are capable of performing some algorithms. To empower transformers with algorithmic capabilities and motivated by the recently proposed looped transformer (Yang et al., 2024; Giannou et al., 2023), we design a novel transformer block, dubbed Algorithm Transformer (abbreviated as AlgoFormer). Compared with the standard transformer and vanilla looped transformer, the proposed AlgoFormer can achieve significantly higher expressiveness in algorithm representation when using the same number of parameters. In particular, inspired by the structure of human-designed learning algorithms, our transformer block consists of a pre-transformer that is responsible for task pre-processing, a looped transformer for iterative optimization algorithms, and a post-transformer for producing the desired results after post-processing. We provide theoretical evidence of the expressive power of the AlgoFormer in solving some challenging problems, mirroring human-designed algorithms. Furthermore, some theoretical and empirical results are presented to show that the designed transformer has the potential to be smarter than human-designed algorithms. Experimental results demonstrate the empirical superiority of the proposed transformer in that it outperforms the standard transformer and vanilla looped transformer in some challenging tasks.

Diffusion of Thoughts: Chain-of-Thought Reasoning in Diffusion Language Models

Diffusion models have gained attention in text processing, offering many potential advantages over traditional autoregressive models. This work explores the integration of diffusion models and Chain-of-Thought (CoT), a well-established technique to improve the reasoning ability in autoregressive language models. We propose Diffusion-of-Thought (DoT), allowing reasoning steps to diffuse over time through the diffusion process. In contrast to traditional autoregressive language models that make decisions in a left-to-right, token-by-token manner, DoT offers more flexibility in the trade-off between computation and reasoning performance. Our experimental results demonstrate the effectiveness of DoT in multi-digit multiplication and grade school math problems. Additionally, DoT showcases promising self-correction abilities and benefits from existing reasoning-enhancing techniques like self-consistency decoding. Our findings contribute to the understanding and development of reasoning capabilities in diffusion language models.

LEGO-Prover: Neural Theorem Proving with Growing Libraries

Despite the success of large language models (LLMs), the task of theorem proving still remains one of the hardest reasoning tasks that is far from being fully solved. Prior methods using language models have demonstrated promising results, but they still struggle to prove even middle school level theorems. One common limitation of these methods is that they assume a fixed theorem library during the whole theorem proving process. However, as we all know, creating new useful theorems or even new theories is not only helpful but crucial and necessary for advancing mathematics and proving harder and deeper results. In this work, we present LEGO-Prover, which employs a growing skill library containing verified lemmas as skills to augment the capability of LLMs used in theorem proving. By constructing the proof modularly, LEGO-Prover enables LLMs to utilize existing skills retrieved from the library and to create new skills during the proving process. These skills are further evolved (by prompting an LLM) to enrich the library on another scale. Modular and reusable skills are constantly added to the library to enable tackling increasingly intricate mathematical problems. Moreover, the learned library further bridges the gap between human proofs and formal proofs by making it easier to impute missing steps. LEGO-Prover advances the state-of-the-art pass rate on miniF2F-valid (48.0% to 57.0%) and miniF2F-test (45.5% to 47.1%). During the proving process, LEGO-Prover also manages to generate over 20,000 skills (theorems/lemmas) and adds them to the growing library. Our ablation study indicates that these newly added skills are indeed helpful for proving theorems, resulting in an improvement from a success rate of 47.1% to 50.4%. We also release our code and all the generated skills.

Effective and Parameter-Efficient Reusing Fine-Tuned Models

Many pre-trained large-scale models provided online have become highly effective in transferring to downstream tasks. At the same time, various task-specific models fine-tuned on these pre-trained models are available online for public use. In practice, as collecting task-specific data is labor-intensive and fine-tuning the large pre-trained models is computationally expensive, one can reuse task-specific finetuned models to deal with downstream tasks. However, using a model per task causes a heavy burden on storage and serving. Recently, many training-free and parameter-efficient methods have been proposed for reusing multiple fine-tuned task-specific models into a single multi-task model. However, these methods exhibit a large accuracy gap compared with using a fine-tuned model per task. In this paper, we propose Parameter-Efficient methods for ReUsing (PERU) fine-tuned models. For reusing Fully Fine-Tuned (FFT) models, we propose PERU-FFT by injecting a sparse task vector into a merged model by magnitude pruning. For reusing LoRA fine-tuned models, we propose PERU-LoRA use a lower-rank matrix to approximate the LoRA matrix by singular value decomposition. Both PERUFFT and PERU-LoRA are training-free. Extensive experiments conducted on computer vision and natural language process tasks demonstrate the effectiveness and parameter-efficiency of the proposed methods. The proposed PERU-FFT and PERU-LoRA outperform existing reusing model methods by a large margin and achieve comparable performance to using a fine-tuned model per task.

MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models

Large language models (LLMs) have pushed the limits of natural language understanding and exhibited excellent problem-solving ability. Despite the great success, most existing open-source LLMs (e.g., LLaMA-2) are still far away from satisfactory for solving mathematical problem due to the complex reasoning procedures. To bridge this gap, we propose MetaMath, a fine-tuned language model that specializes in mathematical reasoning. Specifically, we start by bootstrapping mathematical questions by rewriting the question from multiple perspectives without extra knowledge, which results in a new dataset called MetaMathQA. Then we fine-tune the LLaMA-2 models on MetaMathQA. Experimental results on two popular benchmarks (i.e., GSM8K and MATH) for mathematical reasoning demonstrate that MetaMath outperforms a suite of open-source LLMs by a significant margin. Our MetaMath-7B model achieves 66.4% on GSM8K and 19.4% on MATH, exceeding the state-of-the-art models of the same size by 11.5% and 8.7%. Particularly, MetaMath-70B achieves an accuracy of 82.3% on GSM8K, slightly better than GPT-3.5-Turbo. We release the MetaMathQA dataset, the MetaMath models with different model sizes and the training code for public use.

Forward-Backward Reasoning in Large Language Models for Verification

Chain-of-Though (CoT) prompting has shown promising performance in various reasoning tasks. Recently, Self-Consistency \citep{wang2023selfconsistency} proposes to sample a diverse set of reasoning chains which may lead to different answers while the answer that receives the most votes is selected. In this paper, we propose a novel method to use backward reasoning in verifying candidate answers. We mask a token in the question by ${\bf x}$ and ask the LLM to predict the masked token when a candidate answer is provided by \textit{a simple template}, i.e., "\textit{\textbf{If we know the answer of the above question is \{a candidate answer\}, what is the value of unknown variable ${\bf x}$?}}" Intuitively, the LLM is expected to predict the masked token successfully if the provided candidate answer is correct. We further propose FOBAR to combine forward and backward reasoning for estimating the probability of candidate answers. We conduct extensive experiments on six data sets and three LLMs. Experimental results demonstrate that FOBAR achieves state-of-the-art performance on various reasoning benchmarks.

GrowCLIP: Data-aware Automatic Model Growing for Large-scale Contrastive Language-Image Pre-training

Cross-modal pre-training has shown impressive performance on a wide range of downstream tasks, benefiting from massive image-text pairs collected from the Internet. In practice, online data are growing constantly, highlighting the importance of the ability of pre-trained model to learn from data that is continuously growing. Existing works on cross-modal pre-training mainly focus on training a network with fixed architecture. However, it is impractical to limit the model capacity when considering the continuously growing nature of pre-training data in real-world applications. On the other hand, it is important to utilize the knowledge in the current model to obtain efficient training and better performance. To address the above issues, in this paper, we propose GrowCLIP, a data-driven automatic model growing algorithm for contrastive language-image pre-training with continuous image-text pairs as input. Specially, we adopt a dynamic growth space and seek out the optimal architecture at each growth step to adapt to online learning scenarios. And the shared encoder is proposed in our growth space to enhance the degree of cross-modal fusion. Besides, we explore the effect of growth in different dimensions, which could provide future references for the design of cross-modal model architecture. Finally, we employ parameter inheriting with momentum (PIM) to maintain the previous knowledge and address the issue of the local minimum dilemma. Compared with the existing methods, GrowCLIP improves 2.3% average top-1 accuracy on zero-shot image classification of 9 downstream tasks. As for zero-shot image retrieval, GrowCLIP can improve 1.2% for top-1 image-to-text recall on Flickr30K dataset.

Backward Reasoning in Large Language Models for Verification

Chain-of-Though (CoT) prompting has shown promising performance in various reasoning tasks. Recently, Self-Consistency \citep{wang2023selfconsistency} proposes to sample a diverse set of reasoning chains which may lead to different answers while the answer that receives the most votes is selected. In this paper, we propose a novel method to use backward reasoning in verifying candidate answers. We mask a token in the question by ${\bf x}$ and ask the LLM to predict the masked token when a candidate answer is provided by \textit{a simple template}, i.e., ``\textit{\textbf{If we know the answer of the above question is \{a candidate answer\}, what is the value of unknown variable ${\bf x}$?}}'' Intuitively, the LLM is expected to predict the masked token successfully if the provided candidate answer is correct. We further propose FOBAR to combine forward and backward reasoning for estimating the probability of candidate answers. We conduct extensive experiments on six data sets and three LLMs. Experimental results demonstrate that FOBAR achieves state-of-the-art performance on various reasoning benchmarks.

DiffFlow: A Unified SDE Framework for Score-Based Diffusion Models and Generative Adversarial Networks

Generative models can be categorized into two types: explicit generative models that define explicit density forms and allow exact likelihood inference, such as score-based diffusion models (SDMs) and normalizing flows; implicit generative models that directly learn a transformation from the prior to the data distribution, such as generative adversarial nets (GANs). While these two types of models have shown great success, they suffer from respective limitations that hinder them from achieving fast sampling and high sample quality simultaneously. In this paper, we propose a unified theoretic framework for SDMs and GANs. We shown that: i) the learning dynamics of both SDMs and GANs can be described as a novel SDE named Discriminator Denoising Diffusion Flow (DiffFlow) where the drift can be determined by some weighted combinations of scores of the real data and the generated data; ii) By adjusting the relative weights between different score terms, we can obtain a smooth transition between SDMs and GANs while the marginal distribution of the SDE remains invariant to the change of the weights; iii) we prove the asymptotic optimality and maximal likelihood training scheme of the DiffFlow dynamics; iv) under our unified theoretic framework, we introduce several instantiations of the DiffFLow that provide new algorithms beyond GANs and SDMs with exact likelihood inference and have potential to achieve flexible trade-off between high sample quality and fast sampling speed.