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While the scaling laws of large language models (LLMs) training have been extensively studied, optimal inference configurations of LLMs remain underexplored. We study inference scaling laws and compute-optimal inference, focusing on the trade-offs between model sizes and generating additional tokens with different inference strategies. As a first step towards understanding and designing compute-optimal inference methods, we studied cost-performance trade-offs for inference strategies such as greedy search, majority voting, best-of-$n$, weighted voting, and two different tree search algorithms, using different model sizes and compute budgets. Our findings indicate smaller models (e.g., Llemma-7B) can outperform larger models given the same computation budgets, and that smaller models paired with advanced inference algorithms yield Pareto-optimal cost-performance trade-offs. For instance, the Llemma-7B model, equipped with our novel tree search algorithm, consistently outperforms Llemma-34B with standard majority voting on the MATH benchmark across all FLOPs budgets. We hope these findings contribute to a broader understanding of inference scaling laws for LLMs.

Preventing the performance decay of Transformers on inputs longer than those used for training has been an important challenge in extending the context length of these models. Though the Transformer architecture has fundamentally no limits on the input sequence lengths it can process, the choice of position encoding used during training can limit the performance of these models on longer inputs. We propose a novel functional relative position encoding with progressive interpolation, FIRE, to improve Transformer generalization to longer contexts. We theoretically prove that this can represent some of the popular relative position encodings, such as T5's RPE, Alibi, and Kerple. We next empirically show that FIRE models have better generalization to longer contexts on both zero-shot language modeling and long text benchmarks., Comment: 26 pages; ICLR 2024 camera ready version

We propose a new class of linear Transformers called FourierLearner-Transformers (FLTs), which incorporate a wide range of relative positional encoding mechanisms (RPEs). These include regular RPE techniques applied for sequential data, as well as novel RPEs operating on geometric data embedded in higher-dimensional Euclidean spaces. FLTs construct the optimal RPE mechanism implicitly by learning its spectral representation. As opposed to other architectures combining efficient low-rank linear attention with RPEs, FLTs remain practical in terms of their memory usage and do not require additional assumptions about the structure of the RPE mask. Besides, FLTs allow for applying certain structural inductive bias techniques to specify masking strategies, e.g. they provide a way to learn the so-called local RPEs introduced in this paper and give accuracy gains as compared with several other linear Transformers for language modeling. We also thoroughly test FLTs on other data modalities and tasks, such as image classification, 3D molecular modeling, and learnable optimizers. To the best of our knowledge, for 3D molecular data, FLTs are the first Transformer architectures providing linear attention and incorporating RPE masking., Comment: AISTATS 2024

The Physics-Informed Neural Network (PINN) approach is a new and promising way to solve partial differential equations using deep learning. The $L^2$ Physics-Informed Loss is the de-facto standard in training Physics-Informed Neural Networks. In this paper, we challenge this common practice by investigating the relationship between the loss function and the approximation quality of the learned solution. In particular, we leverage the concept of stability in the literature of partial differential equation to study the asymptotic behavior of the learned solution as the loss approaches zero. With this concept, we study an important class of high-dimensional non-linear PDEs in optimal control, the Hamilton-Jacobi-Bellman(HJB) Equation, and prove that for general $L^p$ Physics-Informed Loss, a wide class of HJB equation is stable only if $p$ is sufficiently large. Therefore, the commonly used $L^2$ loss is not suitable for training PINN on those equations, while $L^{\infty}$ loss is a better choice. Based on the theoretical insight, we develop a novel PINN training algorithm to minimize the $L^{\infty}$ loss for HJB equations which is in a similar spirit to adversarial training. The effectiveness of the proposed algorithm is empirically demonstrated through experiments. Our code is released at https://github.com/LithiumDA/L_inf-PINN.

Relative Positional Encoding (RPE), which encodes the relative distance between any pair of tokens, is one of the most successful modifications to the original Transformer. As far as we know, theoretical understanding of the RPE-based Transformers is largely unexplored. In this work, we mathematically analyze the power of RPE-based Transformers regarding whether the model is capable of approximating any continuous sequence-to-sequence functions. One may naturally assume the answer is in the affirmative -- RPE-based Transformers are universal function approximators. However, we present a negative result by showing there exist continuous sequence-to-sequence functions that RPE-based Transformers cannot approximate no matter how deep and wide the neural network is. One key reason lies in that most RPEs are placed in the softmax attention that always generates a right stochastic matrix. This restricts the network from capturing positional information in the RPEs and limits its capacity. To overcome the problem and make the model more powerful, we first present sufficient conditions for RPE-based Transformers to achieve universal function approximation. With the theoretical guidance, we develop a novel attention module, called Universal RPE-based (URPE) Attention, which satisfies the conditions. Therefore, the corresponding URPE-based Transformers become universal function approximators. Extensive experiments covering typical architectures and tasks demonstrate that our model is parameter-efficient and can achieve superior performance to strong baselines in a wide range of applications. The code will be made publicly available at https://github.com/lsj2408/URPE., Comment: 22 pages; NeurIPS 2022, Camera Ready Version

Physics-Informed Neural Network (PINN) has become a commonly used machine learning approach to solve partial differential equations (PDE). But, facing high-dimensional secondorder PDE problems, PINN will suffer from severe scalability issues since its loss includes second-order derivatives, the computational cost of which will grow along with the dimension during stacked back-propagation. In this work, we develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks. In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein's Identity, the second-order derivatives can be efficiently calculated without back-propagation. We further discuss the model capacity and provide variance reduction methods to address key limitations in the derivative estimation. Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is significantly faster. Our code is released at https://github.com/LithiumDA/PINN-without-Stacked-BP., Comment: AISTATS 2023

Several recent studies have demonstrated that attention-based networks, such as Vision Transformer (ViT), can outperform Convolutional Neural Networks (CNNs) on several computer vision tasks without using convolutional layers. This naturally leads to the following questions: Can a self-attention layer of ViT express any convolution operation? In this work, we prove that a single ViT layer with image patches as the input can perform any convolution operation constructively, where the multi-head attention mechanism and the relative positional encoding play essential roles. We further provide a lower bound on the number of heads for Vision Transformers to express CNNs. Corresponding with our analysis, experimental results show that the construction in our proof can help inject convolutional bias into Transformers and significantly improve the performance of ViT in low data regimes.

The attention module, which is a crucial component in Transformer, cannot scale efficiently to long sequences due to its quadratic complexity. Many works focus on approximating the dot-then-exponentiate softmax function in the original attention, leading to sub-quadratic or even linear-complexity Transformer architectures. However, we show that these methods cannot be applied to more powerful attention modules that go beyond the dot-then-exponentiate style, e.g., Transformers with relative positional encoding (RPE). Since in many state-of-the-art models, relative positional encoding is used as default, designing efficient Transformers that can incorporate RPE is appealing. In this paper, we propose a novel way to accelerate attention calculation for Transformers with RPE on top of the kernelized attention. Based upon the observation that relative positional encoding forms a Toeplitz matrix, we mathematically show that kernelized attention with RPE can be calculated efficiently using Fast Fourier Transform (FFT). With FFT, our method achieves $\mathcal{O}(n\log n)$ time complexity. Interestingly, we further demonstrate that properly using relative positional encoding can mitigate the training instability problem of vanilla kernelized attention. On a wide range of tasks, we empirically show that our models can be trained from scratch without any optimization issues. The learned model performs better than many efficient Transformer variants and is faster than standard Transformer in the long-sequence regime., Comment: NeurIPS 2021, camera ready version