Daily Papers

Transformer-based language models spread FLOPs uniformly across input sequences. In this work we demonstrate that transformers can instead learn to dynamically allocate FLOPs (or compute) to specific positions in a sequence, optimising the allocation along the sequence for different layers across the model depth. Our method enforces a total compute budget by capping the number of tokens (k) that can participate in the self-attention and MLP computations at a given layer. The tokens to be processed are determined by the network using a top-k routing mechanism. Since k is defined a priori, this simple procedure uses a static computation graph with known tensor sizes, unlike other conditional computation techniques. Nevertheless, since the identities of the k tokens are fluid, this method can expend FLOPs non-uniformly across the time and model depth dimensions. Thus, compute expenditure is entirely predictable in sum total, but dynamic and context-sensitive at the token-level. Not only do models trained in this way learn to dynamically allocate compute, they do so efficiently. These models match baseline performance for equivalent FLOPS and wall-clock times to train, but require a fraction of the FLOPs per forward pass, and can be upwards of 50\% faster to step during post-training sampling.

Autoregressive large language models (LLMs) have made remarkable progress in various natural language generation tasks. However, they incur high computation cost and latency resulting from the autoregressive token-by-token generation. To address this issue, several approaches have been proposed to reduce computational cost using early-exit strategies. These strategies enable faster text generation using reduced computation without applying the full computation graph to each token. While existing token-level early exit methods show promising results for online inference, they cannot be readily applied for batch inferencing and Key-Value caching. This is because they have to wait until the last token in a batch exits before they can stop computing. This severely limits the practical application of such techniques. In this paper, we propose a simple and effective token-level early exit method, SkipDecode, designed to work seamlessly with batch inferencing and KV caching. It overcomes prior constraints by setting up a singular exit point for every token in a batch at each sequence position. It also guarantees a monotonic decrease in exit points, thereby eliminating the need to recompute KV Caches for preceding tokens. Rather than terminating computation prematurely as in prior works, our approach bypasses lower to middle layers, devoting most of the computational resources to upper layers, allowing later tokens to benefit from the compute expenditure by earlier tokens. Our experimental results show that SkipDecode can obtain 2x to 5x inference speedups with negligible regression across a variety of tasks. This is achieved using OPT models of 1.3 billion and 6.7 billion parameters, all the while being directly compatible with batching and KV caching optimization techniques.

The costs of training frontier AI models have grown dramatically in recent years, but there is limited public data on the magnitude and growth of these expenses. This paper develops a detailed cost model to address this gap, estimating training costs using three approaches that account for hardware, energy, cloud rental, and staff expenses. The analysis reveals that the amortized cost to train the most compute-intensive models has grown precipitously at a rate of 2.4x per year since 2016 (95% CI: 2.0x to 3.1x). For key frontier models, such as GPT-4 and Gemini, the most significant expenses are AI accelerator chips and staff costs, each costing tens of millions of dollars. Other notable costs include server components (15-22%), cluster-level interconnect (9-13%), and energy consumption (2-6%). If the trend of growing development costs continues, the largest training runs will cost more than a billion dollars by 2027, meaning that only the most well-funded organizations will be able to finance frontier AI models.

Scaling data and compute is critical to the success of machine learning. However, scaling demands predictability: we want methods to not only perform well with more compute or data, but also have their performance be predictable from small-scale runs, without running the large-scale experiment. In this paper, we show that value-based off-policy RL methods are predictable despite community lore regarding their pathological behavior. First, we show that data and compute requirements to attain a given performance level lie on a Pareto frontier, controlled by the updates-to-data (UTD) ratio. By estimating this frontier, we can predict this data requirement when given more compute, and this compute requirement when given more data. Second, we determine the optimal allocation of a total resource budget across data and compute for a given performance and use it to determine hyperparameters that maximize performance for a given budget. Third, this scaling behavior is enabled by first estimating predictable relationships between hyperparameters, which is used to manage effects of overfitting and plasticity loss unique to RL. We validate our approach using three algorithms: SAC, BRO, and PQL on DeepMind Control, OpenAI gym, and IsaacGym, when extrapolating to higher levels of data, compute, budget, or performance.

Large neural networks spend most computation on floating point tensor multiplications. In this work, we find that a floating point multiplier can be approximated by one integer adder with high precision. We propose the linear-complexity multiplication L-Mul algorithm that approximates floating point number multiplication with integer addition operations. The new algorithm costs significantly less computation resource than 8-bit floating point multiplication but achieves higher precision. Compared to 8-bit floating point multiplications, the proposed method achieves higher precision but consumes significantly less bit-level computation. Since multiplying floating point numbers requires substantially higher energy compared to integer addition operations, applying the L-Mul operation in tensor processing hardware can potentially reduce 95% energy cost by element-wise floating point tensor multiplications and 80% energy cost of dot products. We calculated the theoretical error expectation of L-Mul, and evaluated the algorithm on a wide range of textual, visual, and symbolic tasks, including natural language understanding, structural reasoning, mathematics, and commonsense question answering. Our numerical analysis experiments agree with the theoretical error estimation, which indicates that L-Mul with 4-bit mantissa achieves comparable precision as float8_e4m3 multiplications, and L-Mul with 3-bit mantissa outperforms float8_e5m2. Evaluation results on popular benchmarks show that directly applying L-Mul to the attention mechanism is almost lossless. We further show that replacing all floating point multiplications with 3-bit mantissa L-Mul in a transformer model achieves equivalent precision as using float8_e4m3 as accumulation precision in both fine-tuning and inference.

Major Internet advertising platforms offer budget pacing tools as a standard service for advertisers to manage their ad campaigns. Given the inherent non-stationarity in an advertiser's value and also competing advertisers' values over time, a commonly used approach is to learn a target expenditure plan that specifies a target spend as a function of time, and then run a controller that tracks this plan. This raises the question: how many historical samples are required to learn a good expenditure plan? We study this question by considering an advertiser repeatedly participating in T second-price auctions, where the tuple of her value and the highest competing bid is drawn from an unknown time-varying distribution. The advertiser seeks to maximize her total utility subject to her budget constraint. Prior work has shown the sufficiency of Tlog T samples per distribution to achieve the optimal O(T)-regret. We dramatically improve this state-of-the-art and show that just one sample per distribution is enough to achieve the near-optimal tilde O(T)-regret, while still being robust to noise in the sampling distributions.

Solving arithmetic tasks is a simple and fundamental skill, yet modern Large Language Models (LLMs) have great difficulty with them. We introduce the Integrated Gated Calculator (IGC), a module that enables LLMs to perform arithmetic by emulating a calculator on the GPU. We finetune a Llama model with our module and test it on the BigBench Arithmetic benchmark, where it beats the State of the Art, outperforming all models on the benchmark, including models almost two orders of magnitude larger. Our approach takes only a single iteration to run and requires no external tools. It performs arithmetic operations entirely inside the LLM without the need to produce intermediate tokens. It is computationally efficient, interpretable, and avoids side-effects on tasks that do not require arithmetic operations. It reliably achieves 98\% to 99\% accuracy across multiple training runs and for all subtasks, including the substantially harder subtask of multiplication, which was previously unsolved.

In recent years, researchers have focused on reducing the model size and number of computations (measured as "multiply-accumulate" or MAC operations) of DNNs. The energy consumption of a DNN depends on both the number of MAC operations and the energy efficiency of each MAC operation. The former can be estimated at design time; however, the latter depends on the intricate data reuse patterns and underlying hardware architecture. Hence, estimating it at design time is challenging. This work shows that the conventional approach to estimate the data reuse, viz. arithmetic intensity, does not always correctly estimate the degree of data reuse in DNNs since it gives equal importance to all the data types. We propose a novel model, termed "data type aware weighted arithmetic intensity" (DI), which accounts for the unequal importance of different data types in DNNs. We evaluate our model on 25 state-of-the-art DNNs on two GPUs. We show that our model accurately models data-reuse for all possible data reuse patterns for different types of convolution and different types of layers. We show that our model is a better indicator of the energy efficiency of DNNs. We also show its generality using the central limit theorem.

Large models represent a groundbreaking advancement in multiple application fields, enabling remarkable achievements across various tasks. However, their unprecedented scale comes with significant computational costs. These models, often consisting of billions of parameters, require vast amounts of computational resources for execution. Especially, the expansive scale and computational demands pose considerable challenges when customizing them for particular downstream tasks, particularly over the hardware platforms constrained by computational capabilities. Parameter Efficient Fine-Tuning (PEFT) provides a practical solution by efficiently adapt the large models over the various downstream tasks. In particular, PEFT refers to the process of adjusting the parameters of a pre-trained large models to adapt it to a specific task while minimizing the number of additional parameters introduced or computational resources required. This approach is particularly important when dealing with large language models with high parameter counts, as fine-tuning these models from scratch can be computationally expensive and resource-intensive, posing considerable challenges in the supporting system platform design. In this survey, we present comprehensive studies of various PEFT algorithms, examining their performance and computational overhead. Moreover, we provide an overview of applications developed using different PEFT algorithms and discuss common techniques employed to mitigate computation costs for PEFT. In addition to the algorithmic perspective, we overview various real-world system designs to investigate the implementation costs associated with different PEFT algorithms. This survey serves as an indispensable resource for researchers aiming to understand both the PEFT algorithm and its system implementation, offering detailed insights into recent advancements and practical applications.

The learnable, linear neural network layers between tensor power spaces of R^{n} that are equivariant to the orthogonal group, O(n), the special orthogonal group, SO(n), and the symplectic group, Sp(n), were characterised in arXiv:2212.08630. We present an algorithm for multiplying a vector by any weight matrix for each of these groups, using category theoretic constructions to implement the procedure. We achieve a significant reduction in computational cost compared with a naive implementation by making use of Kronecker product matrices to perform the multiplication. We show that our approach extends to the symmetric group, S_n, recovering the algorithm of arXiv:2303.06208 in the process.

Using fault-tolerant constructions, computations performed with unreliable components can simulate their noiseless counterparts though the introduction of a modest amount of redundancy. Given the modest overhead required to achieve fault-tolerance, and the fact that increasing the reliability of basic components often comes at a cost, are there situations where fault-tolerance may be more economical? We present a general framework to account for this overhead cost in order to effectively compare fault-tolerant to non-fault-tolerant approaches for computation, in the limit of small logical error rates. Using this detailed accounting, we determine explicit boundaries at which fault-tolerant designs become more efficient than designs that achieve comparable reliability through direct consumption of resources. We find that the fault-tolerant construction is always preferred in the limit of high reliability in cases where the resources required to construct a basic unit grows faster than log(1 / epsilon) asymptotically for small epsilon.

We introduce Goat, a fine-tuned LLaMA model that significantly outperforms GPT-4 on a range of arithmetic tasks. Fine-tuned on a synthetically generated dataset, Goat achieves state-of-the-art performance on BIG-bench arithmetic sub-task. In particular, the zero-shot Goat-7B matches or even surpasses the accuracy achieved by the few-shot PaLM-540B. Surprisingly, Goat can achieve near-perfect accuracy on large-number addition and subtraction through supervised fine-tuning only, which is almost impossible with previous pretrained language models, such as Bloom, OPT, GPT-NeoX, etc. We attribute Goat's exceptional performance to LLaMA's consistent tokenization of numbers. To tackle more challenging tasks like large-number multiplication and division, we propose an approach that classifies tasks based on their learnability, and subsequently decomposes unlearnable tasks, such as multi-digit multiplication and division, into a series of learnable tasks by leveraging basic arithmetic principles. We thoroughly examine the performance of our model, offering a comprehensive evaluation of the effectiveness of our proposed decomposition steps. Additionally, Goat-7B can be easily trained using LoRA on a 24GB VRAM GPU, facilitating reproducibility for other researchers. We release our model, dataset, and the Python script for dataset generation.

Tensor computations, with matrix multiplication being the primary operation, serve as the fundamental basis for data analysis, physics, machine learning, and deep learning. As the scale and complexity of data continue to grow rapidly, the demand for tensor computations has also increased significantly. To meet this demand, several research institutions have started developing dedicated hardware for tensor computations. To further improve the computational performance of tensor process units, we have reexamined the issue of computation reuse that was previously overlooked in existing architectures. As a result, we propose a novel EN-T architecture that can reduce chip area and power consumption. Furthermore, our method is compatible with existing tensor processing units. We evaluated our method on prevalent microarchitectures, the results demonstrate an average improvement in area efficiency of 8.7\%, 12.2\%, and 11.0\% for tensor computing units at computational scales of 256 GOPS, 1 TOPS, and 4 TOPS, respectively. Similarly, there were energy efficiency enhancements of 13.0\%, 17.5\%, and 15.5\%.

During early optimization passes, compilers must make predictions for machine-dependent characteristics such as execution unit utilization, number of register spills, latency, throughput etc. to generate better code. Often a hand-written static/analytical hardware cost model is built into the compiler. However, the need for more sophisticated and varied predictions has become more pronounced with the development of deep learning compilers which need to optimize dataflow graphs. Such compilers usually employ a much higher level MLIR form as an IR representation before lowering to traditional LLVM-IR. A static/analytical cost model in such a scenario is cumbersome and error prone as the opcodes represent very high level algebraic/arithmetic operations. Hence, we develop a machine learning-based cost model for high-level MLIR which can predict different target variables of interest such as CPU/GPU/xPU utilization, instructions executed, register usage etc. By considering the incoming MLIR as a text input a la NLP models we can apply well-known techniques from modern NLP research to help predict hardware characteristics more accurately. We expect such precise ML-driven hardware cost models to guide our deep learning compiler in graph level optimizations around operator fusion, local memory allocation, kernel scheduling etc. as well as in many kernel-level optimizations such as loop interchange, LICM and unroll. We report early work-in -progress results of developing such models on high-level MLIR representing dataflow graphs emitted by Pytorch/Tensorflow-like frameworks as well as lower-level dialects like affine. We show that these models can provide reasonably good estimates with low error bounds for various hardware characteristics of interest and can be a go-to mechanism for hardware cost modelling in the future.

The term GreenAI refers to a novel approach to Deep Learning, that is more aware of the ecological impact and the computational efficiency of its methods. The promoters of GreenAI suggested the use of Floating Point Operations (FLOPs) as a measure of the computational cost of Neural Networks; however, that measure does not correlate well with the energy consumption of hardware equipped with massively parallel processing units like GPUs or TPUs. In this article, we propose a simple refinement of the formula used to compute floating point operations for convolutional layers, called {\alpha}-FLOPs, explaining and correcting the traditional discrepancy with respect to different layers, and closer to reality. The notion of {\alpha}-FLOPs relies on the crucial insight that, in case of inputs with multiple dimensions, there is no reason to believe that the speedup offered by parallelism will be uniform along all different axes.

Test-time scaling is a promising new approach to language modeling that uses extra test-time compute to improve performance. Recently, OpenAI's o1 model showed this capability but did not publicly share its methodology, leading to many replication efforts. We seek the simplest approach to achieve test-time scaling and strong reasoning performance. First, we curate a small dataset s1K of 1,000 questions paired with reasoning traces relying on three criteria we validate through ablations: difficulty, diversity, and quality. Second, we develop budget forcing to control test-time compute by forcefully terminating the model's thinking process or lengthening it by appending "Wait" multiple times to the model's generation when it tries to end. This can lead the model to double-check its answer, often fixing incorrect reasoning steps. After supervised finetuning the Qwen2.5-32B-Instruct language model on s1K and equipping it with budget forcing, our model s1 exceeds o1-preview on competition math questions by up to 27% (MATH and AIME24). Further, scaling s1 with budget forcing allows extrapolating beyond its performance without test-time intervention: from 50% to 57% on AIME24. Our model, data, and code are open-source at https://github.com/simplescaling/s1.

Large Language Models (LLMs) have demonstrated remarkable capabilities across a wide range of natural language processing and reasoning tasks. However, their performance in the foundational domain of arithmetic remains unsatisfactory. When dealing with arithmetic tasks, LLMs often memorize specific examples rather than learning the underlying computational logic, limiting their ability to generalize to new problems. In this paper, we propose a Composable Arithmetic Execution Framework (CAEF) that enables LLMs to learn to execute step-by-step computations by emulating Turing Machines, thereby gaining a genuine understanding of computational logic. Moreover, the proposed framework is highly scalable, allowing composing learned operators to significantly reduce the difficulty of learning complex operators. In our evaluation, CAEF achieves nearly 100% accuracy across seven common mathematical operations on the LLaMA 3.1-8B model, effectively supporting computations involving operands with up to 100 digits, a level where GPT-4o falls short noticeably in some settings.

As opposed to evaluating computation and logic-based reasoning, current benchmarks for evaluating large language models (LLMs) in medicine are primarily focused on question-answering involving domain knowledge and descriptive reasoning. While such qualitative capabilities are vital to medical diagnosis, in real-world scenarios, doctors frequently use clinical calculators that follow quantitative equations and rule-based reasoning paradigms for evidence-based decision support. To this end, we propose MedCalc-Bench, a first-of-its-kind dataset focused on evaluating the medical calculation capability of LLMs. MedCalc-Bench contains an evaluation set of over 1000 manually reviewed instances from 55 different medical calculation tasks. Each instance in MedCalc-Bench consists of a patient note, a question requesting to compute a specific medical value, a ground truth answer, and a step-by-step explanation showing how the answer is obtained. While our evaluation results show the potential of LLMs in this area, none of them are effective enough for clinical settings. Common issues include extracting the incorrect entities, not using the correct equation or rules for a calculation task, or incorrectly performing the arithmetic for the computation. We hope our study highlights the quantitative knowledge and reasoning gaps in LLMs within medical settings, encouraging future improvements of LLMs for various clinical calculation tasks.