Weight decay can significantly impact the optimization dynamics of deep neural networks. In certain situations the effects of weight decay and gradient updates on the magnitude of a parameter vector cancel out on average, forming a state known as equilibrium. This causes the expected rotation of the vector in each update to remain constant along with its magnitude. Importantly, equilibrium can arise independently for the weight vectors of different layers and neurons. These equilibria are highly homogeneous for some optimizer and normalization configurations, effectively balancing the average rotation--a proxy for the effective learning rate--across network components. In this work we explore the equilibrium states of multiple optimizers including AdamW and SGD with momentum, providing insights into interactions between the learning rate, weight decay, initialization, normalization and learning rate schedule. We show how rotational equilibrium can be enforced throughout training, eliminating the chaotic transient phase corresponding to the transition towards equilibrium, thus simplifying the training dynamics. Finally, we show that rotational behavior may play a key role in the effectiveness of AdamW compared to Adam with L2-regularization, the performance of different normalization layers, and the need for learning rate warmup.

We study stochastic Cubic Newton methods for solving general possibly non-convex minimization problems. We propose a new framework, which we call the helper framework, that provides a unified view of the stochastic and variance-reduced second-order algorithms equipped with global complexity guarantees. It can also be applied to learning with auxiliary information. Our helper framework offers the algorithm designer high flexibility for constructing and analyzing the stochastic Cubic Newton methods, allowing arbitrary size batches, and the use of noisy and possibly biased estimates of the gradients and Hessians, incorporating both the variance reduction and the lazy Hessian updates. We recover the best-known complexities for the stochastic and variance-reduced Cubic Newton, under weak assumptions on the noise. A direct consequence of our theory is the new lazy stochastic second-order method, which significantly improves the arithmetic complexity for large dimension problems. We also establish complexity bounds for the classes of gradient-dominated objectives, that include convex and strongly convex problems. For Auxiliary Learning, we show that using a helper (auxiliary function) can outperform training alone if a given similarity measure is small.

Stochastic Gradient Descent (SGD) algorithms are widely used in optimizing neural networks, with Random Reshuffling (RR) and Single Shuffle (SS) being popular choices for cycling through random or single permutations of the training data. However, the convergence properties of these algorithms in the non-convex case are not fully understood. Existing results suggest that, in realistic training scenarios where the number of epochs is smaller than the training set size, RR may perform worse than SGD. In this paper, we analyze a general SGD algorithm that allows for arbitrary data orderings and show improved convergence rates for non-convex functions. Specifically, our analysis reveals that SGD with random and single shuffling is always faster or at least as good as classical SGD with replacement, regardless of the number of iterations. Overall, our study highlights the benefits of using SGD with random/single shuffling and provides new insights into its convergence properties for non-convex optimization.

This paper studies first-order algorithms for solving fully composite optimization problems over convex and compact sets. We leverage the structure of the objective by handling its differentiable and non-differentiable components separately, linearizing only the smooth parts. This provides us with new generalizations of the classical Frank-Wolfe method and the Conditional Gradient Sliding algorithm, that cater to a subclass of non-differentiable problems. Our algorithms rely on a stronger version of the linear minimization oracle, which can be efficiently implemented in several practical applications. We provide the basic version of our method with an affine-invariant analysis and prove global convergence rates for both convex and non-convex objectives. Furthermore, in the convex case, we propose an accelerated method with correspondingly improved complexity. Finally, we provide illustrative experiments to support our theoretical results.

The unprecedented growth in DNN model complexity, size, and amount of training data has led to a commensurate increase in demand for computing and a search for minimal encoding. Recent research advocates Hybrid Block Floating Point (HBFP) to minimize silicon provisioning in accelerators by converting the majority of arithmetic operations in training to 8-bit fixed point. In this paper, we perform a full-scale exploration of the HBFP design space using mathematical tools to study the interplay among various parameters and identify opportunities for even smaller encodings across layers and epochs. Based on our findings, we propose Accuracy Boosters, an epoch-driven mixed-mantissa HBFP technique that uses 6-bit mantissas only in the last epoch and first/last layers, and 4-bit mantissas for $99.7\%$ of all other arithmetic operations in training. Using analytic models, we show Accuracy Boosters enable increasing arithmetic density for an HBFP training accelerator by up to $21.3\times$ compared to FP32 and up to $4.4\times$ compared to another SOTA format Bfloat16, while preserving or outperforming FP32 accuracy.

We analyze Newton's method with lazy Hessian updates for solving general possibly non-convex optimization problems. We propose to reuse a previously seen Hessian for several iterations while computing new gradients at each step of the method. This significantly reduces the overall arithmetical complexity of second-order optimization schemes. By using the cubic regularization technique, we establish fast global convergence of our method to a second-order stationary point, while the Hessian does not need to be updated each iteration. For convex problems, we justify global and local superlinear rates for lazy Newton steps with quadratic regularization, which is easier to compute. The optimal frequency for updating the Hessian is once every $d$ iterations, where $d$ is the dimension of the problem. This provably improves the total arithmetical complexity of second-order algorithms by a factor $\sqrt{d}$.

Transformer-based language models have found many diverse applications requiring them to process sequences of increasing length. For these applications, the causal self-attention -- which is the only component scaling quadratically w.r.t. the sequence length -- becomes a central concern. While many works have proposed schemes to sparsify the attention patterns and reduce the computational overhead of self-attention, those are often limited by implementations concerns and end up imposing a simple and static structure over the attention matrix. Conversely, implementing more dynamic sparse attentions often results in runtimes significantly slower than computing the full attention using the Flash implementation from Dao et al. (2022). We extend FlashAttention to accommodate a large class of attention sparsity patterns that, in particular, encompass key/query dropping and hashing-based attention. This leads to implementations with no computational complexity overhead and a multi-fold runtime speedup on top of FlashAttention. Even with relatively low degrees of sparsity, our method improves visibly upon FlashAttention as the sequence length increases. Without sacrificing perplexity, we increase the training speed of a transformer language model by $2.0\times$ and $3.3\times$ for sequences of respectively $8k$ and $16k$ tokens.

Federated Learning (FL) is a novel approach enabling several clients holding sensitive data to collaboratively train machine learning models, without centralizing data. The cross-silo FL setting corresponds to the case of few ($2$--$50$) reliable clients, each holding medium to large datasets, and is typically found in applications such as healthcare, finance, or industry. While previous works have proposed representative datasets for cross-device FL, few realistic healthcare cross-silo FL datasets exist, thereby slowing algorithmic research in this critical application. In this work, we propose a novel cross-silo dataset suite focused on healthcare, FLamby (Federated Learning AMple Benchmark of Your cross-silo strategies), to bridge the gap between theory and practice of cross-silo FL. FLamby encompasses 7 healthcare datasets with natural splits, covering multiple tasks, modalities, and data volumes, each accompanied with baseline training code. As an illustration, we additionally benchmark standard FL algorithms on all datasets. Our flexible and modular suite allows researchers to easily download datasets, reproduce results and re-use the different components for their research. FLamby is available at~\url{www.github.com/owkin/flamby}.

When training neural networks, it has been widely observed that a large step size is essential in stochastic gradient descent (SGD) for obtaining superior models. However, the effect of large step sizes on the success of SGD is not well understood theoretically. Several previous works have attributed this success to the stochastic noise present in SGD. However, we show through a novel set of experiments that the stochastic noise is not sufficient to explain good non-convex training, and that instead the effect of a large learning rate itself is essential for obtaining best performance.We demonstrate the same effects also in the noise-less case, i.e. for full-batch GD. We formally prove that GD with large step size -- on certain non-convex function classes -- follows a different trajectory than GD with a small step size, which can lead to convergence to a global minimum instead of a local one. Our settings provide a framework for future analysis which allows comparing algorithms based on behaviors that can not be observed in the traditional settings.

In data-parallel optimization of machine learning models, workers collaborate to improve their estimates of the model: more accurate gradients allow them to use larger learning rates and optimize faster. In the decentralized setting, in which workers communicate over a sparse graph, current theory fails to capture important aspects of real-world behavior. First, the `spectral gap' of the communication graph is not predictive of its empirical performance in (deep) learning. Second, current theory does not explain that collaboration enables larger learning rates than training alone. In fact, it prescribes smaller learning rates, which further decrease as graphs become larger, failing to explain convergence dynamics in infinite graphs. This paper aims to paint an accurate picture of sparsely-connected distributed optimization. We quantify how the graph topology influences convergence in a quadratic toy problem and provide theoretical results for general smooth and (strongly) convex objectives. Our theory matches empirical observations in deep learning, and accurately describes the relative merits of different graph topologies. This paper is an extension of the conference paper by Vogels et. al. (2022). Code: https://github.com/epfml/topology-in-decentralized-learning.