Machine Learning

A closed-form formula for the Kullback-Leibler divergence between Cauchy distributions

We report a closed-form expression for the Kullback-Leibler divergence between Cauchy distributions which involves the calculation of a novel definite integral. The formula shows that the Kullback-Leibler divergence between Cauchy densities is always...

ADADELTA: An Adaptive Learning Rate Method

We present a novel per-dimension learning rate method for gradient descent called ADADELTA. The method dynamically adapts over time using only first order information and has minimal computational overhead beyond vanilla stochastic gradient descent. ...

Adafactor: Adaptive Learning Rates with Sublinear Memory Cost

In several recently proposed stochastic optimization methods (e.g. RMSProp, Adam, Adadelta), parameter updates are scaled by the inverse square roots of exponential moving averages of squared past gradients. Maintaining these per-parameter second-mom...

Adam: A Method for Stochastic Optimization

We introduce Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments. The method is straightforward to implement, is computationally efficient, has little me...

Averaging Weights Leads to Wider Optima and Better Generalization

Deep neural networks are typically trained by optimizing a loss function with an SGD variant, in conjunction with a decaying learning rate, until convergence. We show that simple averaging of multiple points along the trajectory of SGD, with a cyclic...

Chakra: Advancing Performance Benchmarking and Co-design using Standardized Execution Traces

Benchmarking and co-design are essential for driving optimizations and innovation around ML models, ML software, and next-generation hardware. Full workload benchmarks, e.g. MLPerf, play an essential role in enabling fair comparison across different ...

Cyclical Learning Rates for Training Neural Networks

It is known that the learning rate is the most important hyper-parameter to tune for training deep neural networks. This paper describes a new method for setting the learning rate, named cyclical learning rates, which practically eliminates the need ...

Decoupled Weight Decay Regularization

L$_2$ regularization and weight decay regularization are equivalent for standard stochastic gradient descent (when rescaled by the learning rate), but as we demonstrate this is \emph{not} the case for adaptive gradient algorithms, such as Adam. While...

Efficient Simulation of the von Mises Distribution

FlashAttention-2: Faster Attention with Better Parallelism and Work Partitioning

Scaling Transformers to longer sequence lengths has been a major problem in the last several years, promising to improve performance in language modeling and high-resolution image understanding, as well as to unlock new applications in code, audio, a...

Generating Sequences With Recurrent Neural Networks

This paper shows how Long Short-term Memory recurrent neural networks can be used to generate complex sequences with long-range structure, simply by predicting one data point at a time. The approach is demonstrated for text (where the data are discre...

Mish: A Self Regularized Non-Monotonic Activation Function

We propose $\textit{Mish}$, a novel self-regularized non-monotonic activation function which can be mathematically defined as: $f(x)=x\tanh(softplus(x))$. As activation functions play a crucial role in the performance and training dynamics in neural ...

Muon is Scalable for LLM Training

Recently, the Muon optimizer based on matrix orthogonalization has demonstrated strong results in training small-scale language models, but the scalability to larger models has not been proven. We identify two crucial techniques for scaling up Muon: ...

On the Variance of the Adaptive Learning Rate and Beyond

The learning rate warmup heuristic achieves remarkable success in stabilizing training, accelerating convergence and improving generalization for adaptive stochastic optimization algorithms like RMSprop and Adam. Here, we study its mechanism in detai...

Rethinking the Inception Architecture for Computer Vision

Convolutional networks are at the core of most state-of-the-art computer vision solutions for a wide variety of tasks. Since 2014 very deep convolutional networks started to become mainstream, yielding substantial gains in various benchmarks. Althoug...

Spatial Transformer Networks

Convolutional Neural Networks define an exceptionally powerful class of models, but are still limited by the lack of ability to be spatially invariant to the input data in a computationally and parameter efficient manner. In this work we introduce a ...

Stochastic Weight Averaging in Parallel: Large-Batch Training that Generalizes Well

We propose Stochastic Weight Averaging in Parallel (SWAP), an algorithm to accelerate DNN training. Our algorithm uses large mini-batches to compute an approximate solution quickly and then refines it by averaging the weights of multiple models compu...

Super-Convergence: Very Fast Training of Neural Networks Using Large Learning Rates

In this paper, we describe a phenomenon, which we named "super-convergence", where neural networks can be trained an order of magnitude faster than with standard training methods. The existence of super-convergence is relevant to understanding why de...

SWALP : Stochastic Weight Averaging in Low-Precision Training

Low precision operations can provide scalability, memory savings, portability, and energy efficiency. This paper proposes SWALP, an approach to low precision training that averages low-precision SGD iterates with a modified learning rate schedule. SW...

There Are Many Consistent Explanations of Unlabeled Data: Why You Should Average

Presently the most successful approaches to semi-supervised learning are based on consistency regularization, whereby a model is trained to be robust to small perturbations of its inputs and parameters. To understand consistency regularization, we co...