Yuanzhi Li

Yuanzhi Li, Sébastien Bubeck, Ronen Eldan et al. · microsoft-research

We continue the investigation into the power of smaller Transformer-based language models as initiated by \textbf{TinyStories} -- a 10 million parameter model that can produce coherent English -- and the follow-up work on \textbf{phi-1}, a 1.3 billion parameter model with Python coding performance close to the state-of-the-art. The latter work proposed to use existing Large Language Models (LLMs) to generate ``textbook quality" data as a way to enhance the learning process compared to traditional web data. We follow the ``Textbooks Are All You Need" approach, focusing this time on common sense reasoning in natural language, and create a new 1.3 billion parameter model named \textbf{phi-1.5}, with performance on natural language tasks comparable to models 5x larger, and surpassing most non-frontier LLMs on more complex reasoning tasks such as grade-school mathematics and basic coding. More generally, \textbf{phi-1.5} exhibits many of the traits of much larger LLMs, both good -- such as the ability to ``think step by step" or perform some rudimentary in-context learning -- and bad, including hallucinations and the potential for toxic and biased generations -- encouragingly though, we are seeing improvement on that front thanks to the absence of web data. We open-source \textbf{phi-1.5} to promote further research on these urgent topics.

Eric Luxenberg, Dhruv Malik, Yuanzhi Li et al. · stanford

We consider robust empirical risk minimization (ERM), where model parameters are chosen to minimize the worst-case empirical loss when each data point varies over a given convex uncertainty set. In some simple cases, such problems can be expressed in an analytical form. In general the problem can be made tractable via dualization, which turns a min-max problem into a min-min problem. Dualization requires expertise and is tedious and error-prone. We demonstrate how CVXPY can be used to automate this dualization procedure in a user-friendly manner. Our framework allows practitioners to specify and solve robust ERM problems with a general class of convex losses, capturing many standard regression and classification problems. Users can easily specify any complex uncertainty set that is representable via disciplined convex programming (DCP) constraints.

Zeyuan Allen-Zhu, Yuanzhi Li

Large language models (LLMs) can store a vast amount of world knowledge, often extractable via question-answering (e.g., "What is Abraham Lincoln's birthday?"). However, do they answer such questions based on exposure to similar questions during training (i.e., cheating), or by genuinely learning to extract knowledge from sources like Wikipedia? In this paper, we investigate this issue using a controlled biography dataset. We find a strong correlation between the model's ability to extract knowledge and various diversity measures of the training data. $\textbf{Essentially}$, for knowledge to be reliably extracted, it must be sufficiently augmented (e.g., through paraphrasing, sentence shuffling, translations) $\textit{during pretraining}$. Without such augmentation, knowledge may be memorized but not extractable, leading to 0% accuracy, regardless of subsequent instruction fine-tuning. To understand why this occurs, we employ (nearly) linear probing to demonstrate a strong connection between the observed correlation and how the model internally encodes knowledge -- whether it is linearly encoded in the hidden embeddings of entity names or distributed across other token embeddings in the training text. This paper provides $\textbf{several key recommendations for LLM pretraining in the industry}$: (1) rewrite the pretraining data -- using small, auxiliary models -- to provide knowledge augmentation, and (2) incorporate more instruction-finetuning data into the pretraining stage before it becomes too late.

Yue Wu, Yewen Fan, Paul Pu Liang et al.

High sample complexity has long been a challenge for RL. On the other hand, humans learn to perform tasks not only from interaction or demonstrations, but also by reading unstructured text documents, e.g., instruction manuals. Instruction manuals and wiki pages are among the most abundant data that could inform agents of valuable features and policies or task-specific environmental dynamics and reward structures. Therefore, we hypothesize that the ability to utilize human-written instruction manuals to assist learning policies for specific tasks should lead to a more efficient and better-performing agent. We propose the Read and Reward framework. Read and Reward speeds up RL algorithms on Atari games by reading manuals released by the Atari game developers. Our framework consists of a QA Extraction module that extracts and summarizes relevant information from the manual and a Reasoning module that evaluates object-agent interactions based on information from the manual. An auxiliary reward is then provided to a standard A2C RL agent, when interaction is detected. Experimentally, various RL algorithms obtain significant improvement in performance and training speed when assisted by our design.

Zixiang Chen, Yihe Deng, Yue Wu et al.

The Mixture-of-Experts (MoE) layer, a sparsely-activated model controlled by a router, has achieved great success in deep learning. However, the understanding of such architecture remains elusive. In this paper, we formally study how the MoE layer improves the performance of neural network learning and why the mixture model will not collapse into a single model. Our empirical results suggest that the cluster structure of the underlying problem and the non-linearity of the expert are pivotal to the success of MoE. To further understand this, we consider a challenging classification problem with intrinsic cluster structures, which is hard to learn using a single expert. Yet with the MoE layer, by choosing the experts as two-layer nonlinear convolutional neural networks (CNNs), we show that the problem can be learned successfully. Furthermore, our theory shows that the router can learn the cluster-center features, which helps divide the input complex problem into simpler linear classification sub-problems that individual experts can conquer. To our knowledge, this is the first result towards formally understanding the mechanism of the MoE layer for deep learning.

Zeyuan Allen-Zhu, Yuanzhi Li

Language models can store vast factual knowledge, yet their ability to flexibly use this knowledge for downstream tasks (e.g., via instruction finetuning) remains questionable. This paper investigates four fundamental knowledge manipulation tasks: retrieval (e.g., "What is person A's attribute X?"), classification (e.g., "Is A's attribute X even or odd?"), comparison (e.g., "Is A greater than B in attribute X?"), and inverse search (e.g., "Which person's attribute X equals T?"). We show that language models excel in knowledge retrieval but struggle even in the simplest classification or comparison tasks unless Chain of Thoughts (CoTs) are employed during both training and inference. Moreover, their performance in inverse knowledge search is virtually 0%, regardless of the prompts. Our primary contribution is a controlled, synthetic experiment that confirms these weaknesses are inherent to language models: they cannot efficiently manipulate knowledge from pre-training data, even when such knowledge is perfectly stored in the models, despite adequate training and sufficient model size. Our findings also apply to modern pretrained language models such as GPT-4, thus giving rise to many Turing tests to distinguish Humans from contemporary AIs.

Sitan Chen, Sinho Chewi, Jerry Li et al.

We provide theoretical convergence guarantees for score-based generative models (SGMs) such as denoising diffusion probabilistic models (DDPMs), which constitute the backbone of large-scale real-world generative models such as DALL$\cdot$E 2. Our main result is that, assuming accurate score estimates, such SGMs can efficiently sample from essentially any realistic data distribution. In contrast to prior works, our results (1) hold for an $L^2$-accurate score estimate (rather than $L^\infty$-accurate); (2) do not require restrictive functional inequality conditions that preclude substantial non-log-concavity; (3) scale polynomially in all relevant problem parameters; and (4) match state-of-the-art complexity guarantees for discretization of the Langevin diffusion, provided that the score error is sufficiently small. We view this as strong theoretical justification for the empirical success of SGMs. We also examine SGMs based on the critically damped Langevin diffusion (CLD). Contrary to conventional wisdom, we provide evidence that the use of the CLD does not reduce the complexity of SGMs.

Difan Zou, Yuan Cao, Yuanzhi Li et al.

Mixup, a simple data augmentation method that randomly mixes two data points via linear interpolation, has been extensively applied in various deep learning applications to gain better generalization. However, the theoretical underpinnings of its efficacy are not yet fully understood. In this paper, we aim to seek a fundamental understanding of the benefits of Mixup. We first show that Mixup using different linear interpolation parameters for features and labels can still achieve similar performance to the standard Mixup. This indicates that the intuitive linearity explanation in Zhang et al., (2018) may not fully explain the success of Mixup. Then we perform a theoretical study of Mixup from the feature learning perspective. We consider a feature-noise data model and show that Mixup training can effectively learn the rare features (appearing in a small fraction of data) from its mixture with the common features (appearing in a large fraction of data). In contrast, standard training can only learn the common features but fails to learn the rare features, thus suffering from bad generalization performance. Moreover, our theoretical analysis also shows that the benefits of Mixup for feature learning are mostly gained in the early training phase, based on which we propose to apply early stopping in Mixup. Experimental results verify our theoretical findings and demonstrate the effectiveness of the early-stopped Mixup training.

Yue Wu, Xuan Tang, Tom M. Mitchell et al.

Recent large language models (LLMs) have demonstrated great potential toward intelligent agents and next-gen automation, but there currently lacks a systematic benchmark for evaluating LLMs' abilities as agents. We introduce SmartPlay: both a challenging benchmark and a methodology for evaluating LLMs as agents. SmartPlay consists of 6 different games, including Rock-Paper-Scissors, Tower of Hanoi, Minecraft. Each game features a unique setting, providing up to 20 evaluation settings and infinite environment variations. Each game in SmartPlay uniquely challenges a subset of 9 important capabilities of an intelligent LLM agent, including reasoning with object dependencies, planning ahead, spatial reasoning, learning from history, and understanding randomness. The distinction between the set of capabilities each game test allows us to analyze each capability separately. SmartPlay serves not only as a rigorous testing ground for evaluating the overall performance of LLM agents but also as a road-map for identifying gaps in current methodologies. We release our benchmark at github.com/Microsoft/SmartPlay

Samy Jelassi, Stéphane d'Ascoli, Carles Domingo-Enrich et al.

We examine how transformers cope with two challenges: learning basic integer arithmetic, and generalizing to longer sequences than seen during training. We find that relative position embeddings enable length generalization for simple tasks, such as addition: models trained on $5$-digit numbers can perform $15$-digit sums. However, this method fails for multiplication, and we propose train set priming: adding a few ($10$ to $50$) long sequences to the training set. We show that priming allows models trained on $5$-digit $\times$ $3$-digit multiplications to generalize to $35\times 3$ examples. We also show that models can be primed for different generalization lengths, and that the priming sample size scales as the logarithm of the training set size. Finally, we discuss potential applications of priming beyond arithmetic.

Samy Jelassi, Michael E. Sander, Yuanzhi Li

Vision Transformers (ViTs) have achieved comparable or superior performance than Convolutional Neural Networks (CNNs) in computer vision. This empirical breakthrough is even more remarkable since, in contrast to CNNs, ViTs do not embed any visual inductive bias of spatial locality. Yet, recent works have shown that while minimizing their training loss, ViTs specifically learn spatially localized patterns. This raises a central question: how do ViTs learn these patterns by solely minimizing their training loss using gradient-based methods from random initialization? In this paper, we provide some theoretical justification of this phenomenon. We propose a spatially structured dataset and a simplified ViT model. In this model, the attention matrix solely depends on the positional encodings. We call this mechanism the positional attention mechanism. On the theoretical side, we consider a binary classification task and show that while the learning problem admits multiple solutions that generalize, our model implicitly learns the spatial structure of the dataset while generalizing: we call this phenomenon patch association. We prove that patch association helps to sample-efficiently transfer to downstream datasets that share the same structure as the pre-training one but differ in the features. Lastly, we empirically verify that a ViT with positional attention performs similarly to the original one on CIFAR-10/100, SVHN and ImageNet.

Michael Santacroce, Zixin Wen, Yelong Shen et al.

Auto-regressive large language models such as GPT-3 require enormous computational resources to use. Traditionally, structured pruning methods are employed to reduce resource usage. However, their application to and efficacy for generative language models is heavily under-explored. In this paper we conduct an comprehensive evaluation of common structured pruning methods, including magnitude, random, and movement pruning on the feed-forward layers in GPT-type models. Unexpectedly, random pruning results in performance that is comparable to the best established methods, across multiple natural language generation tasks. To understand these results, we provide a framework for measuring neuron-level redundancy of models pruned by different methods, and discover that established structured pruning methods do not take into account the distinctiveness of neurons, leaving behind excess redundancies. In view of this, we introduce Globally Unique Movement (GUM) to improve the uniqueness of neurons in pruned models. We then discuss the effects of our techniques on different redundancy metrics to explain the improved performance.

Yuchen Li, Yuanzhi Li, Andrej Risteski

While the successes of transformers across many domains are indisputable, accurate understanding of the learning mechanics is still largely lacking. Their capabilities have been probed on benchmarks which include a variety of structured and reasoning tasks -- but mathematical understanding is lagging substantially behind. Recent lines of work have begun studying representational aspects of this question: that is, the size/depth/complexity of attention-based networks to perform certain tasks. However, there is no guarantee the learning dynamics will converge to the constructions proposed. In our paper, we provide fine-grained mechanistic understanding of how transformers learn "semantic structure", understood as capturing co-occurrence structure of words. Precisely, we show, through a combination of mathematical analysis and experiments on Wikipedia data and synthetic data modeled by Latent Dirichlet Allocation (LDA), that the embedding layer and the self-attention layer encode the topical structure. In the former case, this manifests as higher average inner product of embeddings between same-topic words. In the latter, it manifests as higher average pairwise attention between same-topic words. The mathematical results involve several assumptions to make the analysis tractable, which we verify on data, and might be of independent interest as well.

Yuan Cao, Difan Zou, Yuanzhi Li et al.

We study the implicit bias of batch normalization trained by gradient descent. We show that when learning a linear model with batch normalization for binary classification, gradient descent converges to a uniform margin classifier on the training data with an $\exp(-Ω(\log^2 t))$ convergence rate. This distinguishes linear models with batch normalization from those without batch normalization in terms of both the type of implicit bias and the convergence rate. We further extend our result to a class of two-layer, single-filter linear convolutional neural networks, and show that batch normalization has an implicit bias towards a patch-wise uniform margin. Based on two examples, we demonstrate that patch-wise uniform margin classifiers can outperform the maximum margin classifiers in certain learning problems. Our results contribute to a better theoretical understanding of batch normalization.

Marah Abdin, Jyoti Aneja, Hany Awadalla et al. · microsoft-research, stanford

We introduce phi-3-mini, a 3.8 billion parameter language model trained on 3.3 trillion tokens, whose overall performance, as measured by both academic benchmarks and internal testing, rivals that of models such as Mixtral 8x7B and GPT-3.5 (e.g., phi-3-mini achieves 69% on MMLU and 8.38 on MT-bench), despite being small enough to be deployed on a phone. Our training dataset is a scaled-up version of the one used for phi-2, composed of heavily filtered publicly available web data and synthetic data. The model is also further aligned for robustness, safety, and chat format. We also provide parameter-scaling results with a 7B, 14B models trained for 4.8T tokens, called phi-3-small, phi-3-medium, both significantly more capable than phi-3-mini (e.g., respectively 75%, 78% on MMLU, and 8.7, 8.9 on MT-bench). To enhance multilingual, multimodal, and long-context capabilities, we introduce three models in the phi-3.5 series: phi-3.5-mini, phi-3.5-MoE, and phi-3.5-Vision. The phi-3.5-MoE, a 16 x 3.8B MoE model with 6.6 billion active parameters, achieves superior performance in language reasoning, math, and code tasks compared to other open-source models of similar scale, such as Llama 3.1 and the Mixtral series, and on par with Gemini-1.5-Flash and GPT-4o-mini. Meanwhile, phi-3.5-Vision, a 4.2 billion parameter model derived from phi-3.5-mini, excels in reasoning tasks and is adept at handling both single-image and text prompts, as well as multi-image and text prompts.

Samy Jelassi, Yuanzhi Li

Stochastic gradient descent (SGD) with momentum is widely used for training modern deep learning architectures. While it is well-understood that using momentum can lead to faster convergence rate in various settings, it has also been observed that momentum yields higher generalization. Prior work argue that momentum stabilizes the SGD noise during training and this leads to higher generalization. In this paper, we adopt another perspective and first empirically show that gradient descent with momentum (GD+M) significantly improves generalization compared to gradient descent (GD) in some deep learning problems. From this observation, we formally study how momentum improves generalization. We devise a binary classification setting where a one-hidden layer (over-parameterized) convolutional neural network trained with GD+M provably generalizes better than the same network trained with GD, when both algorithms are similarly initialized. The key insight in our analysis is that momentum is beneficial in datasets where the examples share some feature but differ in their margin. Contrary to GD that memorizes the small margin data, GD+M still learns the feature in these data thanks to its historical gradients. Lastly, we empirically validate our theoretical findings.

Yunwei Ren, Yuanzhi Li

Recently, contrastive learning approaches (e.g., CLIP (Radford et al., 2021)) have received huge success in multimodal learning, where the model tries to minimize the distance between the representations of different views (e.g., image and its caption) of the same data point while keeping the representations of different data points away from each other. However, from a theoretical perspective, it is unclear how contrastive learning can learn the representations from different views efficiently, especially when the data is not isotropic. In this work, we analyze the training dynamics of a simple multimodal contrastive learning model and show that contrastive pairs are important for the model to efficiently balance the learned representations. In particular, we show that the positive pairs will drive the model to align the representations at the cost of increasing the condition number, while the negative pairs will reduce the condition number, keeping the learned representations balanced.

Sitan Chen, Jerry Li, Yuanzhi Li

Motivated by the recent empirical successes of deep generative models, we study the computational complexity of the following unsupervised learning problem. For an unknown neural network $F:\mathbb{R}^d\to\mathbb{R}^{d'}$, let $D$ be the distribution over $\mathbb{R}^{d'}$ given by pushing the standard Gaussian $\mathcal{N}(0,\textrm{Id}_d)$ through $F$. Given i.i.d. samples from $D$, the goal is to output any distribution close to $D$ in statistical distance. We show under the statistical query (SQ) model that no polynomial-time algorithm can solve this problem even when the output coordinates of $F$ are one-hidden-layer ReLU networks with $\log(d)$ neurons. Previously, the best lower bounds for this problem simply followed from lower bounds for supervised learning and required at least two hidden layers and $\mathrm{poly}(d)$ neurons [Daniely-Vardi '21, Chen-Gollakota-Klivans-Meka '22]. The key ingredient in our proof is an ODE-based construction of a compactly supported, piecewise-linear function $f$ with polynomially-bounded slopes such that the pushforward of $\mathcal{N}(0,1)$ under $f$ matches all low-degree moments of $\mathcal{N}(0,1)$.

Samy Jelassi, David Dobre, Arthur Mensch et al.

Adaptive methods are a crucial component widely used for training generative adversarial networks (GANs). While there has been some work to pinpoint the "marginal value of adaptive methods" in standard tasks, it remains unclear why they are still critical for GAN training. In this paper, we formally study how adaptive methods help train GANs; inspired by the grafting method proposed in arXiv:2002.11803 [cs.LG], we separate the magnitude and direction components of the Adam updates, and graft them to the direction and magnitude of SGDA updates respectively. By considering an update rule with the magnitude of the Adam update and the normalized direction of SGD, we empirically show that the adaptive magnitude of Adam is key for GAN training. This motivates us to have a closer look at the class of normalized stochastic gradient descent ascent (nSGDA) methods in the context of GAN training. We propose a synthetic theoretical framework to compare the performance of nSGDA and SGDA for GAN training with neural networks. We prove that in that setting, GANs trained with nSGDA recover all the modes of the true distribution, whereas the same networks trained with SGDA (and any learning rate configuration) suffer from mode collapse. The critical insight in our analysis is that normalizing the gradients forces the discriminator and generator to be updated at the same pace. We also experimentally show that for several datasets, Adam's performance can be recovered with nSGDA methods.

Sitan Chen, Jerry Li, Yuanzhi Li et al.

We consider the problem of learning high dimensional polynomial transformations of Gaussians. Given samples of the form $p(x)$, where $x\sim N(0, \mathrm{Id}_r)$ is hidden and $p: \mathbb{R}^r \to \mathbb{R}^d$ is a function where every output coordinate is a low-degree polynomial, the goal is to learn the distribution over $p(x)$. This problem is natural in its own right, but is also an important special case of learning deep generative models, namely pushforwards of Gaussians under two-layer neural networks with polynomial activations. Understanding the learnability of such generative models is crucial to understanding why they perform so well in practice. Our first main result is a polynomial-time algorithm for learning quadratic transformations of Gaussians in a smoothed setting. Our second main result is a polynomial-time algorithm for learning constant-degree polynomial transformations of Gaussian in a smoothed setting, when the rank of the associated tensors is small. In fact our results extend to any rotation-invariant input distribution, not just Gaussian. These are the first end-to-end guarantees for learning a pushforward under a neural network with more than one layer. Along the way, we also give the first polynomial-time algorithms with provable guarantees for tensor ring decomposition, a popular generalization of tensor decomposition that is used in practice to implicitly store large tensors.

Yuanzhi Li, Raghu Meka, Rina Panigrahy et al.

Supervised and unsupervised learning using deep neural networks typically aims to exploit the underlying structure in the training data; this structure is often explained using a latent generative process that produces the data, and the generative process is often hierarchical, involving latent concepts. Despite the significant work on understanding the learning of the latent structure and underlying concepts using theory and experiments, a framework that mathematically captures the definition of a concept and provides ways to operate on concepts is missing. In this work, we propose to characterize a simple primitive concept by the zero set of a collection of polynomials and use moment statistics of the data to uniquely represent the concepts; we show how this view can be used to obtain a signature of the concept. These signatures can be used to discover a common structure across the set of concepts and could recursively produce the signature of higher-level concepts from the signatures of lower-level concepts. To utilize such desired properties, we propose a method by keeping a dictionary of concepts and show that the proposed method can learn different types of hierarchical structures of the data.

Junpeng Liu, Yifan Song, Bill Yuchen Lin et al. · pku

Multimodal Large Language models (MLLMs) have shown promise in web-related tasks, but evaluating their performance in the web domain remains a challenge due to the lack of comprehensive benchmarks. Existing benchmarks are either designed for general multimodal tasks, failing to capture the unique characteristics of web pages, or focus on end-to-end web agent tasks, unable to measure fine-grained abilities such as OCR, understanding, and grounding. In this paper, we introduce \bench{}, a multimodal benchmark designed to assess the capabilities of MLLMs across a variety of web tasks. \bench{} consists of seven tasks, and comprises 1.5K human-curated instances from 139 real websites, covering 87 sub-domains. We evaluate 14 open-source MLLMs, Gemini Pro, Claude-3 series, and GPT-4V(ision) on \bench{}, revealing significant challenges and performance gaps. Further analysis highlights the limitations of current MLLMs, including inadequate grounding in text-rich environments and subpar performance with low-resolution image inputs. We believe \bench{} will serve as a valuable resource for the research community and contribute to the creation of more powerful and versatile MLLMs for web-related applications.

Zixiang Chen, Yihe Deng, Yuanzhi Li et al.

Multi-modal learning has become increasingly popular due to its ability to leverage information from different data sources (e.g., text and images) to improve the model performance. Recently, CLIP has emerged as an effective approach that employs vision-language contrastive pretraining to learn joint image and text representations and exhibits remarkable performance in zero-shot learning and text-guided natural image generation. Despite the huge practical success of CLIP, its theoretical understanding remains elusive. In this paper, we formally study transferrable representation learning underlying CLIP and demonstrate how features from different modalities get aligned. We also analyze its zero-shot transfer performance on the downstream tasks. Inspired by our analysis, we propose a new CLIP-type approach, which achieves better performance than CLIP and other state-of-the-art methods on benchmark datasets.

Kaiqi Jiang, Dhruv Malik, Yuanzhi Li

Adaptive optimization methods are well known to achieve superior convergence relative to vanilla gradient methods. The traditional viewpoint in optimization, particularly in convex optimization, explains this improved performance by arguing that, unlike vanilla gradient schemes, adaptive algorithms mimic the behavior of a second-order method by adapting to the global geometry of the loss function. We argue that in the context of neural network optimization, this traditional viewpoint is insufficient. Instead, we advocate for a local trajectory analysis. For iterate trajectories produced by running a generic optimization algorithm OPT, we introduce $R^{\text{OPT}}_{\text{med}}$, a statistic that is analogous to the condition number of the loss Hessian evaluated at the iterates. Through extensive experiments, we show that adaptive methods such as Adam bias the trajectories towards regions where $R^{\text{Adam}}_{\text{med}}$ is small, where one might expect faster convergence. By contrast, vanilla gradient methods like SGD bias the trajectories towards regions where $R^{\text{SGD}}_{\text{med}}$ is comparatively large. We complement these empirical observations with a theoretical result that provably demonstrates this phenomenon in the simplified setting of a two-layer linear network. We view our findings as evidence for the need of a new explanation of the success of adaptive methods, one that is different than the conventional wisdom.

Zixin Wen, Yuanzhi Li

Recently the surprising discovery of the Bootstrap Your Own Latent (BYOL) method by Grill et al. shows the negative term in contrastive loss can be removed if we add the so-called prediction head to the network. This initiated the research of non-contrastive self-supervised learning. It is mysterious why even when there exist trivial collapsed global optimal solutions, neural networks trained by (stochastic) gradient descent can still learn competitive representations. This phenomenon is a typical example of implicit bias in deep learning and remains little understood. In this work, we present our empirical and theoretical discoveries on non-contrastive self-supervised learning. Empirically, we find that when the prediction head is initialized as an identity matrix with only its off-diagonal entries being trainable, the network can learn competitive representations even though the trivial optima still exist in the training objective. Theoretically, we present a framework to understand the behavior of the trainable, but identity-initialized prediction head. Under a simple setting, we characterized the substitution effect and acceleration effect of the prediction head. The substitution effect happens when learning the stronger features in some neurons can substitute for learning these features in other neurons through updating the prediction head. And the acceleration effect happens when the substituted features can accelerate the learning of other weaker features to prevent them from being ignored. These two effects enable the neural networks to learn all the features rather than focus only on learning the stronger features, which is likely the cause of the dimensional collapse phenomenon. To the best of our knowledge, this is also the first end-to-end optimization guarantee for non-contrastive methods using nonlinear neural networks with a trainable prediction head and normalization.

Ahmet Cagri Duzgun, Samy Jelassi, Yuanzhi Li

Overparameterization, the condition where models have more parameters than necessary to fit their training loss, is a crucial factor for the success of deep learning. However, the characteristics of the features learned by overparameterized networks are not well understood. In this work, we explore this question by comparing models with the same architecture but different widths. We first examine the expressivity of the features of these models, and show that the feature space of overparameterized networks cannot be spanned by concatenating many underparameterized features, and vice versa. This reveals that both overparameterized and underparameterized networks acquire some distinctive features. We then evaluate the performance of these models, and find that overparameterized networks outperform underparameterized networks, even when many of the latter are concatenated. We corroborate these findings using a VGG-16 and ResNet18 on CIFAR-10 and a Transformer on the MNLI classification dataset. Finally, we propose a toy setting to explain how overparameterized networks can learn some important features that the underparamaterized networks cannot learn.

Zhengzhong Liu, Bowen Tan, Hongyi Wang et al.

We detail the training of the LLM360 K2-65B model, scaling up our 360-degree OPEN SOURCE approach to the largest and most powerful models under project LLM360. While open-source LLMs continue to advance, the answer to "How are the largest LLMs trained?" remains unclear within the community. The implementation details for such high-capacity models are often protected due to business considerations associated with their high cost. This lack of transparency prevents LLM researchers from leveraging valuable insights from prior experience, e.g., "What are the best practices for addressing loss spikes?" The LLM360 K2 project addresses this gap by providing full transparency and access to resources accumulated during the training of LLMs at the largest scale. This report highlights key elements of the K2 project, including our first model, K2 DIAMOND, a 65 billion-parameter LLM that surpasses LLaMA-65B and rivals LLaMA2-70B, while requiring fewer FLOPs and tokens. We detail the implementation steps and present a longitudinal analysis of K2 DIAMOND's capabilities throughout its training process. We also outline ongoing projects such as TXT360, setting the stage for future models in the series. By offering previously unavailable resources, the K2 project also resonates with the 360-degree OPEN SOURCE principles of transparency, reproducibility, and accessibility, which we believe are vital in the era of resource-intensive AI research.

Ruiqian Nai, Zixin Wen, Ji Li et al.

In representation learning, a disentangled representation is highly desirable as it encodes generative factors of data in a separable and compact pattern. Researchers have advocated leveraging disentangled representations to complete downstream tasks with encouraging empirical evidence. This paper further investigates the necessity of disentangled representation in downstream applications. Specifically, we show that dimension-wise disentangled representations are unnecessary on a fundamental downstream task, abstract visual reasoning. We provide extensive empirical evidence against the necessity of disentanglement, covering multiple datasets, representation learning methods, and downstream network architectures. Furthermore, our findings suggest that the informativeness of representations is a better indicator of downstream performance than disentanglement. Finally, the positive correlation between informativeness and disentanglement explains the claimed usefulness of disentangled representations in previous works. The source code is available at https://github.com/Richard-coder-Nai/disentanglement-lib-necessity.git.

Binghui Li, Yuanzhi Li

Similar to surprising performance in the standard deep learning, deep nets trained by adversarial training also generalize well for unseen clean data (natural data). However, despite adversarial training can achieve low robust training error, there exists a significant robust generalization gap. We call this phenomenon the Clean Generalization and Robust Overfitting (CGRO). In this work, we study the CGRO phenomenon in adversarial training from two views: representation complexity and training dynamics. Specifically, we consider a binary classification setting with $N$ separated training data points. First, we prove that, based on the assumption that we assume there is $\operatorname{poly}(D)$-size clean classifier (where $D$ is the data dimension), ReLU net with only $O(N D)$ extra parameters is able to leverages robust memorization to achieve the CGRO, while robust classifier still requires exponential representation complexity in worst case. Next, we focus on a structured-data case to analyze training dynamics, where we train a two-layer convolutional network with $O(N D)$ width against adversarial perturbation. We then show that a three-stage phase transition occurs during learning process and the network provably converges to robust memorization regime, which thereby results in the CGRO. Besides, we also empirically verify our theoretical analysis by experiments in real-image recognition datasets.

Zeyuan Allen-Zhu, Yuanzhi Li

Scaling laws describe the relationship between the size of language models and their capabilities. Unlike prior studies that evaluate a model's capability via loss or benchmarks, we estimate the number of knowledge bits a model stores. We focus on factual knowledge represented as tuples, such as (USA, capital, Washington D.C.) from a Wikipedia page. Through multiple controlled datasets, we establish that language models can and only can store 2 bits of knowledge per parameter, even when quantized to int8, and such knowledge can be flexibly extracted for downstream applications. Consequently, a 7B model can store 14B bits of knowledge, surpassing the English Wikipedia and textbooks combined based on our estimation. More broadly, we present 12 results on how (1) training duration, (2) model architecture, (3) quantization, (4) sparsity constraints such as MoE, and (5) data signal-to-noise ratio affect a model's knowledge storage capacity. Notable insights include: * The GPT-2 architecture, with rotary embedding, matches or even surpasses LLaMA/Mistral architectures in knowledge storage, particularly over shorter training durations. This arises because LLaMA/Mistral uses GatedMLP, which is less stable and harder to train. * Prepending training data with domain names (e.g., wikipedia.org) significantly increases a model's knowledge capacity. Language models can autonomously identify and prioritize domains rich in knowledge, optimizing their storage capacity.

Edward J. Hu, Yelong Shen, Phillip Wallis et al.

An important paradigm of natural language processing consists of large-scale pre-training on general domain data and adaptation to particular tasks or domains. As we pre-train larger models, full fine-tuning, which retrains all model parameters, becomes less feasible. Using GPT-3 175B as an example -- deploying independent instances of fine-tuned models, each with 175B parameters, is prohibitively expensive. We propose Low-Rank Adaptation, or LoRA, which freezes the pre-trained model weights and injects trainable rank decomposition matrices into each layer of the Transformer architecture, greatly reducing the number of trainable parameters for downstream tasks. Compared to GPT-3 175B fine-tuned with Adam, LoRA can reduce the number of trainable parameters by 10,000 times and the GPU memory requirement by 3 times. LoRA performs on-par or better than fine-tuning in model quality on RoBERTa, DeBERTa, GPT-2, and GPT-3, despite having fewer trainable parameters, a higher training throughput, and, unlike adapters, no additional inference latency. We also provide an empirical investigation into rank-deficiency in language model adaptation, which sheds light on the efficacy of LoRA. We release a package that facilitates the integration of LoRA with PyTorch models and provide our implementations and model checkpoints for RoBERTa, DeBERTa, and GPT-2 at https://github.com/microsoft/LoRA.

Jeremy M. Cohen, Simran Kaur, Yuanzhi Li et al.

We empirically demonstrate that full-batch gradient descent on neural network training objectives typically operates in a regime we call the Edge of Stability. In this regime, the maximum eigenvalue of the training loss Hessian hovers just above the numerical value $2 / \text{(step size)}$, and the training loss behaves non-monotonically over short timescales, yet consistently decreases over long timescales. Since this behavior is inconsistent with several widespread presumptions in the field of optimization, our findings raise questions as to whether these presumptions are relevant to neural network training. We hope that our findings will inspire future efforts aimed at rigorously understanding optimization at the Edge of Stability. Code is available at https://github.com/locuslab/edge-of-stability.

Sitan Chen, Yuanzhi Li

The multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. Given a sequence length $k$, attention matrices $\mathbfΘ_1,\ldots,\mathbfΘ_m\in\mathbb{R}^{d\times d}$, and projection matrices $\mathbf{W}_1,\ldots,\mathbf{W}_m\in\mathbb{R}^{d\times d}$, the corresponding multi-head attention layer $F: \mathbb{R}^{k\times d}\to \mathbb{R}^{k\times d}$ transforms length-$k$ sequences of $d$-dimensional tokens $\mathbf{X}\in\mathbb{R}^{k\times d}$ via $F(\mathbf{X}) \triangleq \sum^m_{i=1} \mathrm{softmax}(\mathbf{X}\mathbfΘ_i\mathbf{X}^\top)\mathbf{X}\mathbf{W}_i$. In this work, we initiate the study of provably learning a multi-head attention layer from random examples and give the first nontrivial upper and lower bounds for this problem: - Provided $\{\mathbf{W}_i, \mathbfΘ_i\}$ satisfy certain non-degeneracy conditions, we give a $(dk)^{O(m^3)}$-time algorithm that learns $F$ to small error given random labeled examples drawn uniformly from $\{\pm 1\}^{k\times d}$. - We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable. We focus on Boolean $\mathbf{X}$ to mimic the discrete nature of tokens in large language models, though our techniques naturally extend to standard continuous settings, e.g. Gaussian. Our algorithm, which is centered around using examples to sculpt a convex body containing the unknown parameters, is a significant departure from existing provable algorithms for learning feedforward networks, which predominantly exploit algebraic and rotation invariance properties of the Gaussian distribution. In contrast, our analysis is more flexible as it primarily relies on various upper and lower tail bounds for the input distribution and "slices" thereof.

Samy Jelassi, Clara Mohri, David Brandfonbrener et al. · harvard, microsoft-research

The Mixture-of-Experts (MoE) architecture enables a significant increase in the total number of model parameters with minimal computational overhead. However, it is not clear what performance tradeoffs, if any, exist between MoEs and standard dense transformers. In this paper, we show that as we increase the number of experts (while fixing the number of active parameters), the memorization performance consistently increases while the reasoning capabilities saturate. We begin by analyzing the theoretical limitations of MoEs at reasoning. We prove that there exist graph problems that cannot be solved by any number of experts of a certain width; however, the same task can be easily solved by a dense model with a slightly larger width. On the other hand, we find that on memory-intensive tasks, MoEs can effectively leverage a small number of active parameters with a large number of experts to memorize the data. We empirically validate these findings on synthetic graph problems and memory-intensive closed book retrieval tasks. Lastly, we pre-train a series of MoEs and dense transformers and evaluate them on commonly used benchmarks in math and natural language. We find that increasing the number of experts helps solve knowledge-intensive tasks, but fails to yield the same benefits for reasoning tasks.

Binghui Li, Yuanzhi Li

Adversarial training is a widely-applied approach to training deep neural networks to be robust against adversarial perturbation. However, although adversarial training has achieved empirical success in practice, it still remains unclear why adversarial examples exist and how adversarial training methods improve model robustness. In this paper, we provide a theoretical understanding of adversarial examples and adversarial training algorithms from the perspective of feature learning theory. Specifically, we focus on a multiple classification setting, where the structured data can be composed of two types of features: the robust features, which are resistant to perturbation but sparse, and the non-robust features, which are susceptible to perturbation but dense. We train a two-layer smoothed ReLU convolutional neural network to learn our structured data. First, we prove that by using standard training (gradient descent over the empirical risk), the network learner primarily learns the non-robust feature rather than the robust feature, which thereby leads to the adversarial examples that are generated by perturbations aligned with negative non-robust feature directions. Then, we consider the gradient-based adversarial training algorithm, which runs gradient ascent to find adversarial examples and runs gradient descent over the empirical risk at adversarial examples to update models. We show that the adversarial training method can provably strengthen the robust feature learning and suppress the non-robust feature learning to improve the network robustness. Finally, we also empirically validate our theoretical findings with experiments on real-image datasets, including MNIST, CIFAR10 and SVHN.

Yuanzhi Li, Lebin Zhou, Nam Ling et al.

The gaming and entertainment industry is rapidly evolving, driven by immersive experiences and the integration of generative AI (GAI) technologies. Training such models effectively requires large-scale datasets that capture the diversity and context of gaming environments. However, existing datasets are often limited to specific domains or rely on artificial degradations, which do not accurately capture the unique characteristics of gaming content. Moreover, benchmarks for controllable video generation remain absent. To address these limitations, we introduce $\mathtt{M^3VIR}$, a large-scale, multi-modal, multi-view dataset specifically designed to overcome the shortcomings of current resources. Unlike existing datasets, $\mathtt{M^3VIR}$ provides diverse, high-fidelity gaming content rendered with Unreal Engine 5, offering authentic ground-truth LR-HR paired and multi-view frames across 80 scenes in 8 categories. It includes $\mathtt{M^3VIR\_MR}$ for super-resolution (SR), novel view synthesis (NVS), and combined NVS+SR tasks, and $\mathtt{M^3VIR\_{MS}}$, the first multi-style, object-level ground-truth set enabling research on controlled video generation. Additionally, we benchmark several state-of-the-art SR and NVS methods to establish performance baselines. While no existing approaches directly handle controlled video generation, $\mathtt{M^3VIR}$ provides a benchmark for advancing this area. By releasing the dataset, we aim to facilitate research in AI-powered restoration, compression, and controllable content generation for next-generation cloud gaming and entertainment.

Kaiqi Jiang, Jeremy Cohen, Yuanzhi Li

The study of Neural Tangent Kernels (NTKs) in deep learning has drawn increasing attention in recent years. NTKs typically actively change during training and are related to feature learning. In parallel, recent work on Gradient Descent (GD) has found a phenomenon called Edge of Stability (EoS), in which the largest eigenvalue of the NTK oscillates around a value inversely proportional to the step size. However, although follow-up works have explored the underlying mechanism of such eigenvalue behavior in depth, the understanding of the behavior of the NTK eigenvectors during EoS is still missing. This paper examines the dynamics of NTK eigenvectors during EoS in detail. Across different architectures, we observe that larger learning rates cause the leading eigenvectors of the final NTK, as well as the full NTK matrix, to have greater alignment with the training target. We then study the underlying mechanism of this phenomenon and provide a theoretical analysis for a two-layer linear network. Our study enhances the understanding of GD training dynamics in deep learning.

Aakash Lahoti, Stefani Karp, Ezra Winston et al.

Vision tasks are characterized by the properties of locality and translation invariance. The superior performance of convolutional neural networks (CNNs) on these tasks is widely attributed to the inductive bias of locality and weight sharing baked into their architecture. Existing attempts to quantify the statistical benefits of these biases in CNNs over locally connected convolutional neural networks (LCNs) and fully connected neural networks (FCNs) fall into one of the following categories: either they disregard the optimizer and only provide uniform convergence upper bounds with no separating lower bounds, or they consider simplistic tasks that do not truly mirror the locality and translation invariance as found in real-world vision tasks. To address these deficiencies, we introduce the Dynamic Signal Distribution (DSD) classification task that models an image as consisting of $k$ patches, each of dimension $d$, and the label is determined by a $d$-sparse signal vector that can freely appear in any one of the $k$ patches. On this task, for any orthogonally equivariant algorithm like gradient descent, we prove that CNNs require $\tilde{O}(k+d)$ samples, whereas LCNs require $Ω(kd)$ samples, establishing the statistical advantages of weight sharing in translation invariant tasks. Furthermore, LCNs need $\tilde{O}(k(k+d))$ samples, compared to $Ω(k^2d)$ samples for FCNs, showcasing the benefits of locality in local tasks. Additionally, we develop information theoretic tools for analyzing randomized algorithms, which may be of interest for statistical research.

Michael Santacroce, Yadong Lu, Han Yu et al.

Reinforcement Learning with Human Feedback (RLHF) has revolutionized language modeling by aligning models with human preferences. However, the RL stage, Proximal Policy Optimization (PPO), requires over 3x the memory of Supervised Fine-Tuning (SFT), making it infeasible to use for most practitioners. To address this issue, we present a comprehensive analysis the memory usage, performance, and training time of memory-savings techniques for PPO. We introduce Hydra-RLHF by first integrating the SFT and Reward models and then dynamically turning LoRA "off" during training. Our experiments show: 1. Using LoRA during PPO reduces its memory usage to be smaller than SFT while improving alignment across four public benchmarks, and 2. Hydra-PPO reduces the latency per sample of LoRA-PPO by up to 65% while maintaining its performance. Our results demonstrate that Hydra-PPO is a simple and promising solution for enabling more widespread usage of RLHF.

Yan Pan, Yuanzhi Li

While stochastic gradient descent (SGD) is still the most popular optimization algorithm in deep learning, adaptive algorithms such as Adam have established empirical advantages over SGD in some deep learning applications such as training transformers. However, it remains a question that why Adam converges significantly faster than SGD in these scenarios. In this paper, we propose one explanation of why Adam converges faster than SGD using a new concept directional sharpness. We argue that the performance of optimization algorithms is closely related to the directional sharpness of the update steps, and show SGD has much worse directional sharpness compared to adaptive algorithms. We further observe that only a small fraction of the coordinates causes the bad sharpness and slow convergence of SGD, and propose to use coordinate-wise clipping as a solution to SGD and other optimization algorithms. We demonstrate the effect of coordinate-wise clipping on sharpness reduction and speeding up the convergence of optimization algorithms under various settings. We show that coordinate-wise clipping improves the local loss reduction when only a small fraction of the coordinates has bad sharpness. We conclude that the sharpness reduction effect of adaptive coordinate-wise scaling is the reason for Adam's success in practice and suggest the use of coordinate-wise clipping as a universal technique to speed up deep learning optimization.

Yue Wu, Shrimai Prabhumoye, So Yeon Min et al.

Open-world survival games pose significant challenges for AI algorithms due to their multi-tasking, deep exploration, and goal prioritization requirements. Despite reinforcement learning (RL) being popular for solving games, its high sample complexity limits its effectiveness in complex open-world games like Crafter or Minecraft. We propose a novel approach, SPRING, to read the game's original academic paper and use the knowledge learned to reason and play the game through a large language model (LLM). Prompted with the LaTeX source as game context and a description of the agent's current observation, our SPRING framework employs a directed acyclic graph (DAG) with game-related questions as nodes and dependencies as edges. We identify the optimal action to take in the environment by traversing the DAG and calculating LLM responses for each node in topological order, with the LLM's answer to final node directly translating to environment actions. In our experiments, we study the quality of in-context "reasoning" induced by different forms of prompts under the setting of the Crafter open-world environment. Our experiments suggest that LLMs, when prompted with consistent chain-of-thought, have great potential in completing sophisticated high-level trajectories. Quantitatively, SPRING with GPT-4 outperforms all state-of-the-art RL baselines, trained for 1M steps, without any training. Finally, we show the potential of games as a test bed for LLMs.

Zeyuan Allen-Zhu, Yuanzhi Li

Transformer-based language models are effective but complex, and understanding their inner workings and reasoning mechanisms is a significant challenge. Previous research has primarily explored how these models handle simple tasks like name copying or selection, and we extend this by investigating how these models perform recursive language structure reasoning defined by context-free grammars (CFGs). We introduce a family of synthetic CFGs that produce hierarchical rules, capable of generating lengthy sentences (e.g., hundreds of tokens) that are locally ambiguous and require dynamic programming to parse. Despite this complexity, we demonstrate that generative models like GPT can accurately learn and reason over CFG-defined hierarchies and generate sentences based on it. We explore the model's internals, revealing that its hidden states precisely capture the structure of CFGs, and its attention patterns resemble the information passing in a dynamic programming algorithm. This paper also presents several corollaries, including showing why absolute positional embeddings is inferior to relative and rotary embeddings; uniform attention alone is surprisingly effective (motivating our follow-up work on Canon layers); encoder-only models (e.g., BERT, DeBERTa) struggle with deep structure reasoning on CFGs compared to autoregressive models (e.g., GPT); and injecting structural or syntactic noise into pretraining data markedly improves robustness to corrupted language prompts.

Sitan Chen, Sinho Chewi, Holden Lee et al.

We provide the first polynomial-time convergence guarantees for the probability flow ODE implementation (together with a corrector step) of score-based generative modeling. Our analysis is carried out in the wake of recent results obtaining such guarantees for the SDE-based implementation (i.e., denoising diffusion probabilistic modeling or DDPM), but requires the development of novel techniques for studying deterministic dynamics without contractivity. Through the use of a specially chosen corrector step based on the underdamped Langevin diffusion, we obtain better dimension dependence than prior works on DDPM ($O(\sqrt{d})$ vs. $O(d)$, assuming smoothness of the data distribution), highlighting potential advantages of the ODE framework.

Ronen Eldan, Yuanzhi Li

Language models (LMs) are powerful tools for natural language processing, but they often struggle to produce coherent and fluent text when they are small. Models with around 125M parameters such as GPT-Neo (small) or GPT-2 (small) can rarely generate coherent and consistent English text beyond a few words even after extensive training. This raises the question of whether the emergence of the ability to produce coherent English text only occurs at larger scales (with hundreds of millions of parameters or more) and complex architectures (with many layers of global attention). In this work, we introduce TinyStories, a synthetic dataset of short stories that only contain words that a typical 3 to 4-year-olds usually understand, generated by GPT-3.5 and GPT-4. We show that TinyStories can be used to train and evaluate LMs that are much smaller than the state-of-the-art models (below 10 million total parameters), or have much simpler architectures (with only one transformer block), yet still produce fluent and consistent stories with several paragraphs that are diverse and have almost perfect grammar, and demonstrate reasoning capabilities. We also introduce a new paradigm for the evaluation of language models: We suggest a framework which uses GPT-4 to grade the content generated by these models as if those were stories written by students and graded by a (human) teacher. This new paradigm overcomes the flaws of standard benchmarks which often requires the model's output to be very structures, and moreover provides a multidimensional score for the model, providing scores for different capabilities such as grammar, creativity and consistency. We hope that TinyStories can facilitate the development, analysis and research of LMs, especially for low-resource or specialized domains, and shed light on the emergence of language capabilities in LMs.

Yue Wu, So Yeon Min, Yonatan Bisk et al.

Pre-trained large language models (LLMs) capture procedural knowledge about the world. Recent work has leveraged LLM's ability to generate abstract plans to simplify challenging control tasks, either by action scoring, or action modeling (fine-tuning). However, the transformer architecture inherits several constraints that make it difficult for the LLM to directly serve as the agent: e.g. limited input lengths, fine-tuning inefficiency, bias from pre-training, and incompatibility with non-text environments. To maintain compatibility with a low-level trainable actor, we propose to instead use the knowledge in LLMs to simplify the control problem, rather than solving it. We propose the Plan, Eliminate, and Track (PET) framework. The Plan module translates a task description into a list of high-level sub-tasks. The Eliminate module masks out irrelevant objects and receptacles from the observation for the current sub-task. Finally, the Track module determines whether the agent has accomplished each sub-task. On the AlfWorld instruction following benchmark, the PET framework leads to a significant 15% improvement over SOTA for generalization to human goal specifications.

Sitan Chen, Jerry Li, Yuanzhi Li et al.

Arguably the most fundamental question in the theory of generative adversarial networks (GANs) is to understand to what extent GANs can actually learn the underlying distribution. Theoretical and empirical evidence suggests local optimality of the empirical training objective is insufficient. Yet, it does not rule out the possibility that achieving a true population minimax optimal solution might imply distribution learning. In this paper, we show that standard cryptographic assumptions imply that this stronger condition is still insufficient. Namely, we show that if local pseudorandom generators (PRGs) exist, then for a large family of natural continuous target distributions, there are ReLU network generators of constant depth and polynomial size which take Gaussian random seeds so that (i) the output is far in Wasserstein distance from the target distribution, but (ii) no polynomially large Lipschitz discriminator ReLU network can detect this. This implies that even achieving a population minimax optimal solution to the Wasserstein GAN objective is likely insufficient for distribution learning in the usual statistical sense. Our techniques reveal a deep connection between GANs and PRGs, which we believe will lead to further insights into the computational landscape of GANs.

Yuanzhi Li, Ruosong Wang, Lin F. Yang

Recently there is a surge of interest in understanding the horizon-dependence of the sample complexity in reinforcement learning (RL). Notably, for an RL environment with horizon length $H$, previous work have shown that there is a probably approximately correct (PAC) algorithm that learns an $O(1)$-optimal policy using $\mathrm{polylog}(H)$ episodes of environment interactions when the number of states and actions is fixed. It is yet unknown whether the $\mathrm{polylog}(H)$ dependence is necessary or not. In this work, we resolve this question by developing an algorithm that achieves the same PAC guarantee while using only $O(1)$ episodes of environment interactions, completely settling the horizon-dependence of the sample complexity in RL. We achieve this bound by (i) establishing a connection between value functions in discounted and finite-horizon Markov decision processes (MDPs) and (ii) a novel perturbation analysis in MDPs. We believe our new techniques are of independent interest and could be applied in related questions in RL.

Zehao Dou, Yuanzhi Li

Adam-type methods, the extension of adaptive gradient methods, have shown great performance in the training of both supervised and unsupervised machine learning models. In particular, Adam-type optimizers have been widely used empirically as the default tool for training generative adversarial networks (GANs). On the theory side, however, despite the existence of theoretical results showing the efficiency of Adam-type methods in minimization problems, the reason of their wonderful performance still remains absent in GAN's training. In existing works, the fast convergence has long been considered as one of the most important reasons and multiple works have been proposed to give a theoretical guarantee of the convergence to a critical point of min-max optimization algorithms under certain assumptions. In this paper, we firstly argue empirically that in GAN's training, Adam does not converge to a critical point even upon successful training: Only the generator is converging while the discriminator's gradient norm remains high throughout the training. We name this one-sided convergence. Then we bridge the gap between experiments and theory by showing that Adam-type algorithms provably converge to a one-sided first order stationary points in min-max optimization problems under the one-sided MVI condition. We also empirically verify that such one-sided MVI condition is satisfied for standard GANs after trained over standard data sets. To the best of our knowledge, this is the very first result which provides an empirical observation and a strict theoretical guarantee on the one-sided convergence of Adam-type algorithms in min-max optimization.

Difan Zou, Yuan Cao, Yuanzhi Li et al.

Adaptive gradient methods such as Adam have gained increasing popularity in deep learning optimization. However, it has been observed that compared with (stochastic) gradient descent, Adam can converge to a different solution with a significantly worse test error in many deep learning applications such as image classification, even with a fine-tuned regularization. In this paper, we provide a theoretical explanation for this phenomenon: we show that in the nonconvex setting of learning over-parameterized two-layer convolutional neural networks starting from the same random initialization, for a class of data distributions (inspired from image data), Adam and gradient descent (GD) can converge to different global solutions of the training objective with provably different generalization errors, even with weight decay regularization. In contrast, we show that if the training objective is convex, and the weight decay regularization is employed, any optimization algorithms including Adam and GD will converge to the same solution if the training is successful. This suggests that the inferior generalization performance of Adam is fundamentally tied to the nonconvex landscape of deep learning optimization.

Dhruv Malik, Aldo Pacchiano, Vishwak Srinivasan et al.

Reinforcement learning (RL) is empirically successful in complex nonlinear Markov decision processes (MDPs) with continuous state spaces. By contrast, the majority of theoretical RL literature requires the MDP to satisfy some form of linear structure, in order to guarantee sample efficient RL. Such efforts typically assume the transition dynamics or value function of the MDP are described by linear functions of the state features. To resolve this discrepancy between theory and practice, we introduce the Effective Planning Window (EPW) condition, a structural condition on MDPs that makes no linearity assumptions. We demonstrate that the EPW condition permits sample efficient RL, by providing an algorithm which provably solves MDPs satisfying this condition. Our algorithm requires minimal assumptions on the policy class, which can include multi-layer neural networks with nonlinear activation functions. Notably, the EPW condition is directly motivated by popular gaming benchmarks, and we show that many classic Atari games satisfy this condition. We additionally show the necessity of conditions like EPW, by demonstrating that simple MDPs with slight nonlinearities cannot be solved sample efficiently.