Exploring the Generalization Capabilities of AID-based Bi-level Optimization

Bi-level optimization has achieved considerable success in contemporary machine learning applications, especially for given proper hyperparameters. However, due to the two-level optimization structure, commonly, researchers focus on two types of bi-level optimization methods: approximate implicit differentiation (AID)-based and iterative differentiation (ITD)-based approaches. ITD-based methods can be readily transformed into single-level optimization problems, facilitating the study of their generalization capabilities. In contrast, AID-based methods cannot be easily transformed similarly but must stay in the two-level structure, leaving their generalization properties enigmatic. In this paper, although the outer-level function is nonconvex, we ascertain the uniform stability of AID-based methods, which achieves similar results to a single-level nonconvex problem. We conduct a convergence analysis for a carefully chosen step size to maintain stability. Combining the convergence and stability results, we give the generalization ability of AID-based bi-level optimization methods. Furthermore, we carry out an ablation study of the parameters and assess the performance of these methods on real-world tasks. Our experimental results corroborate the theoretical findings, demonstrating the effectiveness and potential applications of these methods.

Entropic Distribution Matching in Supervised Fine-tuning of LLMs: Less Overfitting and Better Diversity

Large language models rely on Supervised Fine-Tuning (SFT) to specialize in downstream tasks. Cross Entropy (CE) loss is the de facto choice in SFT, but it often leads to overfitting and limited output diversity due to its aggressive updates to the data distribution. This paper aim to address these issues by introducing the maximum entropy principle, which favors models with flatter distributions that still effectively capture the data. Specifically, we develop a new distribution matching method called GEM, which solves reverse Kullback-Leibler divergence minimization with an entropy regularizer. For the SFT of Llama-3-8B models, GEM outperforms CE in several aspects. First, when applied to the UltraFeedback dataset to develop general instruction-following abilities, GEM exhibits reduced overfitting, evidenced by lower perplexity and better performance on the IFEval benchmark. Furthermore, GEM enhances output diversity, leading to performance gains of up to 7 points on math reasoning and code generation tasks using best-of-n sampling, even without domain-specific data. Second, when fine-tuning with domain-specific datasets for math reasoning and code generation, GEM also shows less overfitting and improvements of up to 10 points compared with CE.

Adam-mini: Use Fewer Learning Rates To Gain More

We propose Adam-mini, an optimizer that achieves on-par or better performance than AdamW with 45% to 50% less memory footprint. Adam-mini reduces memory by cutting down the learning rate resources in Adam (i.e., $1/\sqrt{v}$). We find that $\geq$ 90% of these learning rates in $v$ could be harmlessly removed if we (1) carefully partition the parameters into blocks following our proposed principle on Hessian structure; (2) assign a single but good learning rate to each parameter block. We further find that, for each of these parameter blocks, there exists a single high-quality learning rate that can outperform Adam, provided that sufficient resources are available to search it out. We then provide one cost-effective way to find good learning rates and propose Adam-mini. Empirically, we verify that Adam-mini performs on par or better than AdamW on various language models sized from 125M to 7B for pre-training, supervised fine-tuning, and RLHF. The reduced memory footprint of Adam-mini also alleviates communication overheads among GPUs and CPUs, thereby increasing throughput. For instance, Adam-mini achieves 49.6% higher throughput than AdamW when pre-training Llama2-7B on $2\times$ A800-80GB GPUs, which saves 33% wall-clock time for pre-training.

Why Transformers Need Adam: A Hessian Perspective

SGD performs worse than Adam by a significant margin on Transformers, but the reason remains unclear. In this work, we provide an explanation of SGD's failure on Transformers through the lens of Hessian: (i) Transformers are ``heterogeneous'': the Hessian spectrum across parameter blocks vary dramatically, a phenomenon we call ``block heterogeneity"; (ii) Heterogeneity hampers SGD: SGD performs badly on problems with block heterogeneity. To validate that heterogeneity hampers SGD, we check various Transformers, CNNs, MLPs, and quadratic problems, and find that SGD works well on problems without block heterogeneity but performs badly when the heterogeneity exists. Our initial theoretical analysis indicates that SGD fails because it applies one single learning rate for all blocks, which cannot handle the heterogeneity among blocks. The failure could be rescued if we could assign different learning rates across blocks, as designed in Adam.

Rethinking SIGN Training: Provable Nonconvex Acceleration without First- and Second-Order Gradient Lipschitz

Sign-based stochastic methods have gained attention due to their ability to achieve robust performance despite using only the sign information for parameter updates. However, the current convergence analysis of sign-based methods relies on the strong assumptions of first-order gradient Lipschitz and second-order gradient Lipschitz, which may not hold in practical tasks like deep neural network training that involve high non-smoothness. In this paper, we revisit sign-based methods and analyze their convergence under more realistic assumptions of first- and second-order smoothness. We first establish the convergence of the sign-based method under weak first-order Lipschitz. Motivated by the weak first-order Lipschitz, we propose a relaxed second-order condition that still allows for nonconvex acceleration in sign-based methods. Based on our theoretical results, we gain insights into the computational advantages of the recently developed LION algorithm. In distributed settings, we prove that this nonconvex acceleration persists with linear speedup in the number of nodes, when utilizing fast communication compression gossip protocols. The novelty of our theoretical results lies in that they are derived under much weaker assumptions, thereby expanding the provable applicability of sign-based algorithms to a wider range of problems.

Adam Can Converge Without Any Modification on Update Rules

Ever since Reddi et al. 2018 pointed out the divergence issue of Adam, many new variants have been designed to obtain convergence. However, vanilla Adam remains exceptionally popular and it works well in practice. Why is there a gap between theory and practice? We point out there is a mismatch between the settings of theory and practice: Reddi et al. 2018 pick the problem after picking the hyperparameters of Adam, i.e., $(\beta_1, \beta_2)$; while practical applications often fix the problem first and then tune $(\beta_1, \beta_2)$. Due to this observation, we conjecture that the empirical convergence can be theoretically justified, only if we change the order of picking the problem and hyperparameter. In this work, we confirm this conjecture. We prove that, when $\beta_2$ is large and $\beta_1 < \sqrt{\beta_2}<1$, Adam converges to the neighborhood of critical points. The size of the neighborhood is propositional to the variance of stochastic gradients. Under an extra condition (strong growth condition), Adam converges to critical points. As $\beta_2$ increases, our convergence result can cover any $\beta_1 \in [0,1)$ including $\beta_1=0.9$, which is the default setting in deep learning libraries. To our knowledge, this is the first result showing that Adam can converge under a wide range of hyperparameters {\it without any modification} on its update rules. Further, our analysis does not require assumptions of bounded gradients or bounded 2nd-order momentum. When $\beta_2$ is small, we further point out a large region of $(\beta_1,\beta_2)$ where Adam can diverge to infinity. Our divergence result considers the same setting as our convergence result, indicating a phase transition from divergence to convergence when increasing $\beta_2$. These positive and negative results can provide suggestions on how to tune Adam hyperparameters.

Efficient-Adam: Communication-Efficient Distributed Adam with Complexity Analysis

Distributed adaptive stochastic gradient methods have been widely used for large-scale nonconvex optimization, such as training deep learning models. However, their communication complexity on finding $\varepsilon$-stationary points has rarely been analyzed in the nonconvex setting. In this work, we present a novel communication-efficient distributed Adam in the parameter-server model for stochastic nonconvex optimization, dubbed {\em Efficient-Adam}. Specifically, we incorporate a two-way quantization scheme into Efficient-Adam to reduce the communication cost between the workers and server. Simultaneously, we adopt a two-way error feedback strategy to reduce the biases caused by the two-way quantization on both the server and workers, respectively. In addition, we establish the iteration complexity for the proposed Efficient-Adam with a class of quantization operators, and further characterize its communication complexity between the server and workers when an $\varepsilon$-stationary point is achieved. Finally, we apply Efficient-Adam to solve a toy stochastic convex optimization problem and train deep learning models on real-world vision and language tasks. Extensive experiments together with a theoretical guarantee justify the merits of Efficient Adam.

Towards Practical Adam: Non-Convexity, Convergence Theory, and Mini-Batch Acceleration

Adam is one of the most influential adaptive stochastic algorithms for training deep neural networks, which has been pointed out to be divergent even in the simple convex setting via a few simple counterexamples. Many attempts, such as decreasing an adaptive learning rate, adopting a big batch size, incorporating a temporal decorrelation technique, seeking an analogous surrogate, \textit{etc.}, have been tried to promote Adam-type algorithms to converge. In contrast with existing approaches, we introduce an alternative easy-to-check sufficient condition, which merely depends on the parameters of the base learning rate and combinations of historical second-order moments, to guarantee the global convergence of generic Adam for solving large-scale non-convex stochastic optimization. This observation coupled with this sufficient condition gives much deeper interpretations on the divergence of Adam. On the other hand, in practice, mini-Adam and distributed-Adam are widely used without theoretical guarantee, we further give an analysis on how will the batch size or the number of nodes in the distributed system will affect the convergence of Adam, which theoretically shows that mini-batch and distributed Adam can be linearly accelerated by using a larger mini-batch size or more number of nodes. At last, we apply the generic Adam and mini-batch Adam with a sufficient condition for solving the counterexample and training several different neural networks on various real-world datasets. Experimental results are exactly in accord with our theoretical analysis.

Quantized Adam with Error Feedback

In this paper, we present a distributed variant of adaptive stochastic gradient method for training deep neural networks in the parameter-server model. To reduce the communication cost among the workers and server, we incorporate two types of quantization schemes, i.e., gradient quantization and weight quantization, into the proposed distributed Adam. Besides, to reduce the bias introduced by quantization operations, we propose an error-feedback technique to compensate for the quantized gradient. Theoretically, in the stochastic nonconvex setting, we show that the distributed adaptive gradient method with gradient quantization and error-feedback converges to the first-order stationary point, and that the distributed adaptive gradient method with weight quantization and error-feedback converges to the point related to the quantized level under both the single-worker and multi-worker modes. At last, we apply the proposed distributed adaptive gradient methods to train deep neural networks. Experimental results demonstrate the efficacy of our methods.

Arbitrary Style Transfer with Deep Feature Reshuffle

This paper introduces a novel method by reshuffling deep features (i.e., permuting the spacial locations of a feature map) of the style image for arbitrary style transfer. We theoretically prove that our new style loss based on reshuffle connects both global and local style losses respectively used by most parametric and non-parametric neural style transfer methods. This simple idea can effectively address the challenging issues in existing style transfer methods. On one hand, it can avoid distortions in local style patterns, and allow semantic-level transfer, compared with neural parametric methods. On the other hand, it can preserve globally similar appearance to the style image, and avoid wash-out artifacts, compared with neural non-parametric methods. Based on the proposed loss, we also present a progressive feature-domain optimization approach. The experiments show that our method is widely applicable to various styles, and produces better quality than existing methods.