Leon Bottou

Randall Balestriero, Leon Bottou, Yann LeCun

Regularization is a fundamental technique to prevent over-fitting and to improve generalization performances by constraining a model's complexity. Current Deep Networks heavily rely on regularizers such as Data-Augmentation (DA) or weight-decay, and employ structural risk minimization, i.e. cross-validation, to select the optimal regularization hyper-parameters. In this study, we demonstrate that techniques such as DA or weight decay produce a model with a reduced complexity that is unfair across classes. The optimal amount of DA or weight decay found from cross-validation leads to disastrous model performances on some classes e.g. on Imagenet with a resnet50, the "barn spider" classification test accuracy falls from $68\%$ to $46\%$ only by introducing random crop DA during training. Even more surprising, such performance drop also appears when introducing uninformative regularization techniques such as weight decay. Those results demonstrate that our search for ever increasing generalization performance -- averaged over all classes and samples -- has left us with models and regularizers that silently sacrifice performances on some classes. This scenario can become dangerous when deploying a model on downstream tasks e.g. an Imagenet pre-trained resnet50 deployed on INaturalist sees its performances fall from $70\%$ to $30\%$ on class \#8889 when introducing random crop DA during the Imagenet pre-training phase. Those results demonstrate that designing novel regularizers without class-dependent bias remains an open research question.

Alberto Bietti, Vivien Cabannes, Diane Bouchacourt et al.

Large language models based on transformers have achieved great empirical successes. However, as they are deployed more widely, there is a growing need to better understand their internal mechanisms in order to make them more reliable. These models appear to store vast amounts of knowledge from their training data, and to adapt quickly to new information provided in their context or prompt. We study how transformers balance these two types of knowledge by considering a synthetic setup where tokens are generated from either global or context-specific bigram distributions. By a careful empirical analysis of the training process on a simplified two-layer transformer, we illustrate the fast learning of global bigrams and the slower development of an "induction head" mechanism for the in-context bigrams. We highlight the role of weight matrices as associative memories, provide theoretical insights on how gradients enable their learning during training, and study the role of data-distributional properties.

Vivien Cabannes, Leon Bottou, Yann Lecun et al.

Self-Supervised Learning (SSL) has emerged as the solution of choice to learn transferable representations from unlabeled data. However, SSL requires to build samples that are known to be semantically akin, i.e. positive views. Requiring such knowledge is the main limitation of SSL and is often tackled by ad-hoc strategies e.g. applying known data-augmentations to the same input. In this work, we formalize and generalize this principle through Positive Active Learning (PAL) where an oracle queries semantic relationships between samples. PAL achieves three main objectives. First, it unveils a theoretically grounded learning framework beyond SSL, based on similarity graphs, that can be extended to tackle supervised and semi-supervised learning depending on the employed oracle. Second, it provides a consistent algorithm to embed a priori knowledge, e.g. some observed labels, into any SSL losses without any change in the training pipeline. Third, it provides a proper active learning framework yielding low-cost solutions to annotate datasets, arguably bringing the gap between theory and practice of active learning that is based on simple-to-answer-by-non-experts queries of semantic relationships between inputs.

Zhuoming Chen, Ranajoy Sadhukhan, Zihao Ye et al. · uw

Large language models (LLMs) with long context windows have gained significant attention. However, the KV cache, stored to avoid re-computation, becomes a bottleneck. Various dynamic sparse or TopK-based attention approximation methods have been proposed to leverage the common insight that attention is sparse. In this paper, we first show that TopK attention itself suffers from quality degradation in certain downstream tasks because attention is not always as sparse as expected. Rather than selecting the keys and values with the highest attention scores, sampling with theoretical guarantees can provide a better estimation for attention output. To make the sampling-based approximation practical in LLM generation, we propose MagicPIG, a heterogeneous system based on Locality Sensitive Hashing (LSH). MagicPIG significantly reduces the workload of attention computation while preserving high accuracy for diverse tasks. MagicPIG stores the LSH hash tables and runs the attention computation on the CPU, which allows it to serve longer contexts and larger batch sizes with high approximation accuracy. MagicPIG can improve decoding throughput by up to $5\times$ across various GPU hardware and achieve 54ms decoding latency on a single RTX 4090 for Llama-3.1-8B-Instruct model with a context of 96k tokens. The code is available at https://github.com/Infini-AI-Lab/MagicPIG.

Benjamin Aubin, Agnieszka Słowik, Martin Arjovsky et al.

There is an increasing interest in algorithms to learn invariant correlations across training environments. A big share of the current proposals find theoretical support in the causality literature but, how useful are they in practice? The purpose of this note is to propose six linear low-dimensional problems -- unit tests -- to evaluate different types of out-of-distribution generalization in a precise manner. Following initial experiments, none of the three recently proposed alternatives passes all tests. By providing the code to automatically replicate all the results in this manuscript (https://www.github.com/facebookresearch/InvarianceUnitTests), we hope that our unit tests become a standard steppingstone for researchers in out-of-distribution generalization.

Alexander Peysakhovich, Anna Klimovskaia Susmel, Leon Bottou

Dot product embeddings take a graph and construct vectors for nodes such that dot products between two vectors give the strength of the edge. Dot products make a strong transitivity assumption, however, many important forces generating graphs in the real world lead to non-transitive relationships. We remove the transitivity assumption by embedding nodes into a pseudo-Euclidean space - giving each node an attract and a repel vector. The inner product between two nodes is defined by taking the dot product in attract vectors and subtracting the dot product in repel vectors. Pseudo-Euclidean embeddings can compress networks efficiently, allow for multiple notions of nearest neighbors each with their own interpretation, and can be `slotted' into existing models such as exponential family embeddings or graph neural networks for better link prediction.

Rachel Ward, Xiaoxia Wu, Leon Bottou

Adaptive gradient methods such as AdaGrad and its variants update the stepsize in stochastic gradient descent on the fly according to the gradients received along the way; such methods have gained widespread use in large-scale optimization for their ability to converge robustly, without the need to fine-tune the stepsize schedule. Yet, the theoretical guarantees to date for AdaGrad are for online and convex optimization. We bridge this gap by providing theoretical guarantees for the convergence of AdaGrad for smooth, nonconvex functions. We show that the norm version of AdaGrad (AdaGrad-Norm) converges to a stationary point at the $\mathcal{O}(\log(N)/\sqrt{N})$ rate in the stochastic setting, and at the optimal $\mathcal{O}(1/N)$ rate in the batch (non-stochastic) setting -- in this sense, our convergence guarantees are 'sharp'. In particular, the convergence of AdaGrad-Norm is robust to the choice of all hyper-parameters of the algorithm, in contrast to stochastic gradient descent whose convergence depends crucially on tuning the step-size to the (generally unknown) Lipschitz smoothness constant and level of stochastic noise on the gradient. Extensive numerical experiments are provided to corroborate our theory; moreover, the experiments suggest that the robustness of AdaGrad-Norm extends to state-of-the-art models in deep learning, without sacrificing generalization.

Leon Bottou, Martin Arjovsky, David Lopez-Paz et al.

Learning algorithms for implicit generative models can optimize a variety of criteria that measure how the data distribution differs from the implicit model distribution, including the Wasserstein distance, the Energy distance, and the Maximum Mean Discrepancy criterion. A careful look at the geometries induced by these distances on the space of probability measures reveals interesting differences. In particular, we can establish surprising approximate global convergence guarantees for the $1$-Wasserstein distance,even when the parametric generator has a nonconvex parametrization.

Levent Sagun, Utku Evci, V. Ugur Guney et al.

We study the properties of common loss surfaces through their Hessian matrix. In particular, in the context of deep learning, we empirically show that the spectrum of the Hessian is composed of two parts: (1) the bulk centered near zero, (2) and outliers away from the bulk. We present numerical evidence and mathematical justifications to the following conjectures laid out by Sagun et al. (2016): Fixing data, increasing the number of parameters merely scales the bulk of the spectrum; fixing the dimension and changing the data (for instance adding more clusters or making the data less separable) only affects the outliers. We believe that our observations have striking implications for non-convex optimization in high dimensions. First, the flatness of such landscapes (which can be measured by the singularity of the Hessian) implies that classical notions of basins of attraction may be quite misleading. And that the discussion of wide/narrow basins may be in need of a new perspective around over-parametrization and redundancy that are able to create large connected components at the bottom of the landscape. Second, the dependence of small number of large eigenvalues to the data distribution can be linked to the spectrum of the covariance matrix of gradients of model outputs. With this in mind, we may reevaluate the connections within the data-architecture-algorithm framework of a model, hoping that it would shed light into the geometry of high-dimensional and non-convex spaces in modern applications. In particular, we present a case that links the two observations: small and large batch gradient descent appear to converge to different basins of attraction but we show that they are in fact connected through their flat region and so belong to the same basin.

Levent Sagun, Leon Bottou, Yann LeCun

We look at the eigenvalues of the Hessian of a loss function before and after training. The eigenvalue distribution is seen to be composed of two parts, the bulk which is concentrated around zero, and the edges which are scattered away from zero. We present empirical evidence for the bulk indicating how over-parametrized the system is, and for the edges that depend on the input data.

Alekh Agarwal, Leon Bottou

This paper presents a lower bound for optimizing a finite sum of $n$ functions, where each function is $L$-smooth and the sum is $μ$-strongly convex. We show that no algorithm can reach an error $ε$ in minimizing all functions from this class in fewer than $Ω(n + \sqrt{n(κ-1)}\log(1/ε))$ iterations, where $κ=L/μ$ is a surrogate condition number. We then compare this lower bound to upper bounds for recently developed methods specializing to this setting. When the functions involved in this sum are not arbitrary, but based on i.i.d. random data, then we further contrast these complexity results with those for optimal first-order methods to directly optimize the sum. The conclusion we draw is that a lot of caution is necessary for an accurate comparison, and identify machine learning scenarios where the new methods help computationally.

Patrice Simard, David Chickering, Aparna Lakshmiratan et al.

Quick interaction between a human teacher and a learning machine presents numerous benefits and challenges when working with web-scale data. The human teacher guides the machine towards accomplishing the task of interest. The learning machine leverages big data to find examples that maximize the training value of its interaction with the teacher. When the teacher is restricted to labeling examples selected by the machine, this problem is an instance of active learning. When the teacher can provide additional information to the machine (e.g., suggestions on what examples or predictive features should be used) as the learning task progresses, then the problem becomes one of interactive learning. To accommodate the two-way communication channel needed for efficient interactive learning, the teacher and the machine need an environment that supports an interaction language. The machine can access, process, and summarize more examples than the teacher can see in a lifetime. Based on the machine's output, the teacher can revise the definition of the task or make it more precise. Both the teacher and the machine continuously learn and benefit from the interaction. We have built a platform to (1) produce valuable and deployable models and (2) support research on both the machine learning and user interface challenges of the interactive learning problem. The platform relies on a dedicated, low-latency, distributed, in-memory architecture that allows us to construct web-scale learning machines with quick interaction speed. The purpose of this paper is to describe this architecture and demonstrate how it supports our research efforts. Preliminary results are presented as illustrations of the architecture but are not the primary focus of the paper.

Dhruv Mahajan, S. Sathiya Keerthi, S. Sundararajan et al.

This paper proposes a novel parallel stochastic gradient descent (SGD) method that is obtained by applying parallel sets of SGD iterations (each set operating on one node using the data residing in it) for finding the direction in each iteration of a batch descent method. The method has strong convergence properties. Experiments on datasets with high dimensional feature spaces show the value of this method.

Dhruv Mahajan, Nikunj Agrawal, S. Sathiya Keerthi et al.

Scalable machine learning over big data is an important problem that is receiving a lot of attention in recent years. On popular distributed environments such as Hadoop running on a cluster of commodity machines, communication costs are substantial and algorithms need to be designed suitably considering those costs. In this paper we give a novel approach to the distributed training of linear classifiers (involving smooth losses and L2 regularization) that is designed to reduce the total communication costs. At each iteration, the nodes minimize locally formed approximate objective functions; then the resulting minimizers are combined to form a descent direction to move. Our approach gives a lot of freedom in the formation of the approximate objective function as well as in the choice of methods to solve them. The method is shown to have $O(log(1/ε))$ time convergence. The method can be viewed as an iterative parameter mixing method. A special instantiation yields a parallel stochastic gradient descent method with strong convergence. When communication times between nodes are large, our method is much faster than the Terascale method (Agarwal et al., 2011), which is a state of the art distributed solver based on the statistical query model (Chuet al., 2006) that computes function and gradient values in a distributed fashion. We also evaluate against other recent distributed methods and demonstrate superior performance of our method.

Alekh Agarwal, Leon Bottou, Miroslav Dudik et al.

Training examples are not all equally informative. Active learning strategies leverage this observation in order to massively reduce the number of examples that need to be labeled. We leverage the same observation to build a generic strategy for parallelizing learning algorithms. This strategy is effective because the search for informative examples is highly parallelizable and because we show that its performance does not deteriorate when the sifting process relies on a slightly outdated model. Parallel active learning is particularly attractive to train nonlinear models with non-linear representations because there are few practical parallel learning algorithms for such models. We report preliminary experiments using both kernel SVMs and SGD-trained neural networks.