Which parts of a dataset will a given model find difficult? Recent work has shown that SGD-trained models have a bias towards simplicity, leading them to prioritize learning a majority class, or to rely upon harmful spurious correlations. Here, we show that the preference for "easy" runs far deeper: A model may prioritize any class or group of the dataset that it finds simple-at the expense of what it finds complex-as measured by performance difference on the test set. When subsets with different levels of complexity align with demographic groups, we term this difficulty disparity, a phenomenon that occurs even with balanced datasets that lack group/label associations. We show how difficulty disparity is a model-dependent quantity, and is further amplified in commonly-used models as selected by typical average performance scores. We quantify an amplification factor across a range of settings in order to compare disparity of different models on a fixed dataset. Finally, we present two real-world examples of difficulty amplification in action, resulting in worse-than-expected performance disparities between groups even when using a balanced dataset. The existence of such disparities in balanced datasets demonstrates that merely balancing sample sizes of groups is not sufficient to ensure unbiased performance. We hope this work presents a step towards measurable understanding of the role of model bias as it interacts with the structure of data, and call for additional model-dependent mitigation methods to be deployed alongside dataset audits.
Arjun Subramonian, Levent Sagun, Yizhou Sun · meta-ai
Graph neural network (GNN) link prediction is increasingly deployed in citation, collaboration, and online social networks to recommend academic literature, collaborators, and friends. While prior research has investigated the dyadic fairness of GNN link prediction, the within-group (e.g., queer women) fairness and "rich get richer" dynamics of link prediction remain underexplored. However, these aspects have significant consequences for degree and power imbalances in networks. In this paper, we shed light on how degree bias in networks affects Graph Convolutional Network (GCN) link prediction. In particular, we theoretically uncover that GCNs with a symmetric normalized graph filter have a within-group preferential attachment bias. We validate our theoretical analysis on real-world citation, collaboration, and online social networks. We further bridge GCN's preferential attachment bias with unfairness in link prediction and propose a new within-group fairness metric. This metric quantifies disparities in link prediction scores within social groups, towards combating the amplification of degree and power disparities. Finally, we propose a simple training-time strategy to alleviate within-group unfairness, and we show that it is effective on citation, social, and credit networks.
Samuel J. Bell, Diane Bouchacourt, Levent Sagun · meta-ai
Neural networks can fail when the data contains spurious correlations. To understand this phenomenon, researchers have proposed numerous spurious correlations benchmarks upon which to evaluate mitigation methods. However, we observe that these benchmarks exhibit substantial disagreement, with the best methods on one benchmark performing poorly on another. We explore this disagreement, and examine benchmark validity by defining three desiderata that a benchmark should satisfy in order to meaningfully evaluate methods. Our results have implications for both benchmarks and mitigations: we find that certain benchmarks are not meaningful measures of method performance, and that several methods are not sufficiently robust for widespread use. We present a simple recipe for practitioners to choose methods using the most similar benchmark to their given problem.
David Lopez-Paz, Diane Bouchacourt, Levent Sagun et al.
As learning machines increase their influence on decisions concerning human lives, analyzing their fairness properties becomes a subject of central importance. Yet, our best tools for measuring the fairness of learning systems are rigid fairness metrics encapsulated as mathematical one-liners, offer limited power to the stakeholders involved in the prediction task, and are easy to manipulate when we exhort excessive pressure to optimize them. To advance these issues, we propose to shift focus from shaping fairness metrics to curating the distributions of examples under which these are computed. In particular, we posit that every claim about fairness should be immediately followed by the tagline "Fair under what examples, and collected by whom?". By highlighting connections to the literature in domain generalization, we propose to measure fairness as the ability of the system to generalize under multiple stress tests -- distributions of examples with social relevance. We encourage each stakeholder to curate one or multiple stress tests containing examples reflecting their (possibly conflicting) interests. The machine passes or fails each stress test by falling short of or exceeding a pre-defined metric value. The test results involve all stakeholders in a discussion about how to improve the learning system, and provide flexible assessments of fairness dependent on context and based on interpretable data. We provide full implementation guidelines for stress testing, illustrate both the benefits and shortcomings of this framework, and introduce a cryptographic scheme to enable a degree of prediction accountability from system providers.
Arjun Subramonian, Adina Williams, Maximilian Nickel et al. · meta-ai
The expressive power of graph neural networks is usually measured by comparing how many pairs of graphs or nodes an architecture can possibly distinguish as non-isomorphic to those distinguishable by the $k$-dimensional Weisfeiler-Leman ($k$-WL) test. In this paper, we uncover misalignments between graph machine learning practitioners' conceptualizations of expressive power and $k$-WL through a systematic analysis of the reliability and validity of $k$-WL. We conduct a survey ($n = 18$) of practitioners to surface their conceptualizations of expressive power and their assumptions about $k$-WL. In contrast to practitioners' beliefs, our analysis (which draws from graph theory and benchmark auditing) reveals that $k$-WL does not guarantee isometry, can be irrelevant to real-world graph tasks, and may not promote generalization or trustworthiness. We argue for extensional definitions and measurement of expressive power based on benchmarks. We further contribute guiding questions for constructing such benchmarks, which is critical for graph machine learning practitioners to develop and transparently communicate our understandings of expressive power.
Existing works show that although modern neural networks achieve remarkable generalization performance on the in-distribution (ID) dataset, the accuracy drops significantly on the out-of-distribution (OOD) datasets \cite{recht2018cifar, recht2019imagenet}. To understand why a variety of models consistently make more mistakes in the OOD datasets, we propose a new metric to quantify the difficulty of the test images (either ID or OOD) that depends on the interaction of the training dataset and the model. In particular, we introduce \textit{confusion score} as a label-free measure of image difficulty which quantifies the amount of disagreement on a given test image based on the class conditional probabilities estimated by an ensemble of trained models. Using the confusion score, we investigate CIFAR-10 and its OOD derivatives. Next, by partitioning test and OOD datasets via their confusion scores, we predict the relationship between ID and OOD accuracies for various architectures. This allows us to obtain an estimator of the OOD accuracy of a given model only using ID test labels. Our observations indicate that the biggest contribution to the accuracy drop comes from images with high confusion scores. Upon further inspection, we report on the nature of the misclassified images grouped by their confusion scores: \textit{(i)} images with high confusion scores contain \textit{weak spurious correlations} that appear in multiple classes in the training data and lack clear \textit{class-specific features}, and \textit{(ii)} images with low confusion scores exhibit spurious correlations that belong to another class, namely \textit{class-specific spurious correlations}.
Anaelia Ovalle, Candace Ross, Sebastian Ruder et al.
Large language models demonstrate strong reasoning capabilities through chain-of-thought prompting, but whether this reasoning quality transfers across languages remains underexplored. We introduce a human-validated framework to evaluate whether model-generated reasoning traces logically support their conclusions across languages. Analyzing 65k reasoning traces from GlobalMMLU questions across 6 languages and 6 frontier models, we uncover a critical blind spot: while models achieve high task accuracy, their reasoning can fail to support their conclusions. Reasoning traces in non-Latin scripts show at least twice as much misalignment between their reasoning and conclusions than those in Latin scripts. We develop an error taxonomy through human annotation to characterize these failures, finding they stem primarily from evidential errors (unsupported claims, ambiguous facts) followed by illogical reasoning steps. Our findings demonstrate that current multilingual evaluation practices provide an incomplete picture of model reasoning capabilities and highlight the need for reasoning-aware evaluation frameworks.
Stéphane d'Ascoli, Hugo Touvron, Matthew Leavitt et al.
Convolutional architectures have proven extremely successful for vision tasks. Their hard inductive biases enable sample-efficient learning, but come at the cost of a potentially lower performance ceiling. Vision Transformers (ViTs) rely on more flexible self-attention layers, and have recently outperformed CNNs for image classification. However, they require costly pre-training on large external datasets or distillation from pre-trained convolutional networks. In this paper, we ask the following question: is it possible to combine the strengths of these two architectures while avoiding their respective limitations? To this end, we introduce gated positional self-attention (GPSA), a form of positional self-attention which can be equipped with a ``soft" convolutional inductive bias. We initialise the GPSA layers to mimic the locality of convolutional layers, then give each attention head the freedom to escape locality by adjusting a gating parameter regulating the attention paid to position versus content information. The resulting convolutional-like ViT architecture, ConViT, outperforms the DeiT on ImageNet, while offering a much improved sample efficiency. We further investigate the role of locality in learning by first quantifying how it is encouraged in vanilla self-attention layers, then analysing how it is escaped in GPSA layers. We conclude by presenting various ablations to better understand the success of the ConViT. Our code and models are released publicly at https://github.com/facebookresearch/convit.
Anaelia Ovalle, Krunoslav Lehman Pavasovic, Louis Martin et al.
Natural-language assistants are designed to provide users with helpful responses while avoiding harmful outputs, largely achieved through alignment to human preferences. Yet there is limited understanding of whether alignment techniques may inadvertently perpetuate or even amplify harmful biases inherited from their pre-aligned base models. This issue is compounded by the choice of bias evaluation benchmarks in popular preference-finetuned models, which predominantly focus on dominant social categories, such as binary gender, thereby limiting insights into biases affecting underrepresented groups. Towards addressing this gap, we center transgender, nonbinary, and other gender-diverse identities to investigate how alignment procedures interact with pre-existing gender-diverse bias in LLMs. Our key contributions include: 1) a comprehensive survey of bias evaluation modalities across leading preference-finetuned LLMs, highlighting critical gaps in gender-diverse representation, 2) systematic evaluation of gender-diverse biases across 16 models spanning Direct Preference Optimization (DPO) stages, uncovering harms popular bias benchmarks fail to detect, and 3) a flexible framework for measuring harmful biases in implicit reward signals applicable to other social contexts. Our findings reveal that DPO-aligned models are particularly sensitive to supervised finetuning (SFT), and can amplify two forms of real-world gender-diverse harms from their base models: stigmatization and gender non-affirmative language. We conclude with recommendations tailored to DPO and broader alignment practices, advocating for the adoption of community-informed bias evaluation frameworks to more effectively identify and address underrepresented harms in LLMs.
Krunoslav Lehman Pavasovic, David Lopez-Paz, Giulio Biroli et al.
In this paper, we leverage existing statistical methods to better understand feature learning from data. We tackle this by modifying the model-free variable selection method, Feature Ordering by Conditional Independence (FOCI), which is introduced in \cite{azadkia2021simple}. While FOCI is based on a non-parametric coefficient of conditional dependence, we introduce its parametric, differentiable approximation. With this approximate coefficient of correlation, we present a new algorithm called difFOCI, which is applicable to a wider range of machine learning problems thanks to its differentiable nature and learnable parameters. We present difFOCI in three contexts: (1) as a variable selection method with baseline comparisons to FOCI, (2) as a trainable model parametrized with a neural network, and (3) as a generic, widely applicable neural network regularizer, one that improves feature learning with better management of spurious correlations. We evaluate difFOCI on increasingly complex problems ranging from basic variable selection in toy examples to saliency map comparisons in convolutional networks. We then show how difFOCI can be incorporated in the context of fairness to facilitate classifications without relying on sensitive data.
Large language models (LLMs) are often fine-tuned for use on downstream tasks, though this can degrade capabilities learned during previous training. This phenomenon, often referred to as catastrophic forgetting, has important potential implications for the safety of deployed models. In this work, we first show that models trained on downstream tasks forget their safety tuning to a greater extent than models trained in the opposite order. Second, we show that forgetting disproportionately impacts safety information about certain groups. To quantify this phenomenon, we define a new metric we term biased forgetting. We conduct a systematic evaluation of the effects of task ordering on forgetting and apply mitigations that can help the model recover from the forgetting observed. We hope our findings can better inform methods for chaining the finetuning of LLMs in continual learning settings to enable training of safer and less toxic models.
Patrick Haller, Mark Ibrahim, Polina Kirichenko et al.
For Large Language Models (LLMs) to be reliable, they must learn robust knowledge that can be generally applied in diverse settings -- often unlike those seen during training. Yet, extensive research has shown that LLM performance can be brittle, with models exhibiting excessive sensitivity to trivial input variations. In this work, we explore whether this brittleness is a direct result of unstable internal knowledge representations. To explore this question, we build on previous work showing that LLM representations encode statement truthfulness -- i.e., true, factual statements can be easily separated from false, inaccurate ones. Specifically, we test the robustness of learned knowledge by evaluating representation separability on samples that have undergone superficial transformations to drive them out-of-distribution (OOD), such as typos or reformulations. By applying semantically-preserving perturbations, we study how separability degrades as statements become more OOD, across four LLM families, five evaluation datasets, and three knowledge probing methods. Our results reveal that internal representations of statement truthfulness collapse as the samples' presentations become less similar to those seen during pre-training. While LLMs can often distinguish between true and false statements when they closely resemble the pre-training data, this ability is highly dependent on the statement's exact surface form. These findings offer a possible explanation for brittle benchmark performance: LLMs may learn shallow, non-robust knowledge representations that allow for only limited generalizability. Our work presents a fundamental challenge for the utility of truthfulness probes, and more broadly, calls for further research on improving the robustness of learned knowledge representations.
Samuel J. Bell, Mariano Coria Meglioli, Megan Richards et al.
Text toxicity detection systems exhibit significant biases, producing disproportionate rates of false positives on samples mentioning demographic groups. But what about toxicity detection in speech? To investigate the extent to which text-based biases are mitigated by speech-based systems, we produce a set of high-quality group annotations for the multilingual MuTox dataset, and then leverage these annotations to systematically compare speech- and text-based toxicity classifiers. Our findings indicate that access to speech data during inference supports reduced bias against group mentions, particularly for ambiguous and disagreement-inducing samples. Our results also suggest that improving classifiers, rather than transcription pipelines, is more helpful for reducing group bias. We publicly release our annotations and provide recommendations for future toxicity dataset construction.
Discriminative self-supervised learning allows training models on any random group of internet images, and possibly recover salient information that helps differentiate between the images. Applied to ImageNet, this leads to object centric features that perform on par with supervised features on most object-centric downstream tasks. In this work, we question if using this ability, we can learn any salient and more representative information present in diverse unbounded set of images from across the globe. To do so, we train models on billions of random images without any data pre-processing or prior assumptions about what we want the model to learn. We scale our model size to dense 10 billion parameters to avoid underfitting on a large data size. We extensively study and validate our model performance on over 50 benchmarks including fairness, robustness to distribution shift, geographical diversity, fine grained recognition, image copy detection and many image classification datasets. The resulting model, not only captures well semantic information, it also captures information about artistic style and learns salient information such as geolocations and multilingual word embeddings based on visual content only. More importantly, we discover that such model is more robust, more fair, less harmful and less biased than supervised models or models trained on object centric datasets such as ImageNet.
Priya Goyal, Adriana Romero Soriano, Caner Hazirbas et al.
Does everyone equally benefit from computer vision systems? Answers to this question become more and more important as computer vision systems are deployed at large scale, and can spark major concerns when they exhibit vast performance discrepancies between people from various demographic and social backgrounds. Systematic diagnosis of fairness, harms, and biases of computer vision systems is an important step towards building socially responsible systems. To initiate an effort towards standardized fairness audits, we propose three fairness indicators, which aim at quantifying harms and biases of visual systems. Our indicators use existing publicly available datasets collected for fairness evaluations, and focus on three main types of harms and bias identified in the literature, namely harmful label associations, disparity in learned representations of social and demographic traits, and biased performance on geographically diverse images from across the world.We define precise experimental protocols applicable to a wide range of computer vision models. These indicators are part of an ever-evolving suite of fairness probes and are not intended to be a substitute for a thorough analysis of the broader impact of the new computer vision technologies. Yet, we believe it is a necessary first step towards (1) facilitating the widespread adoption and mandate of the fairness assessments in computer vision research, and (2) tracking progress towards building socially responsible models. To study the practical effectiveness and broad applicability of our proposed indicators to any visual system, we apply them to off-the-shelf models built using widely adopted model training paradigms which vary in their ability to whether they can predict labels on a given image or only produce the embeddings. We also systematically study the effect of data domain and model size.
Stéphane d'Ascoli, Levent Sagun, Giulio Biroli et al.
Vision Transformers (ViT) have recently emerged as a powerful alternative to convolutional networks (CNNs). Although hybrid models attempt to bridge the gap between these two architectures, the self-attention layers they rely on induce a strong computational bottleneck, especially at large spatial resolutions. In this work, we explore the idea of reducing the time spent training these layers by initializing them as convolutional layers. This enables us to transition smoothly from any pre-trained CNN to its functionally identical hybrid model, called Transformed CNN (T-CNN). With only 50 epochs of fine-tuning, the resulting T-CNNs demonstrate significant performance gains over the CNN (+2.2% top-1 on ImageNet-1k for a ResNet50-RS) as well as substantially improved robustness (+11% top-1 on ImageNet-C). We analyze the representations learnt by the T-CNN, providing deeper insights into the fruitful interplay between convolutions and self-attention. Finally, we experiment initializing the T-CNN from a partially trained CNN, and find that it reaches better performance than the corresponding hybrid model trained from scratch, while reducing training time.
Stéphane d'Ascoli, Marylou Gabrié, Levent Sagun et al.
One of the central puzzles in modern machine learning is the ability of heavily overparametrized models to generalize well. Although the low-dimensional structure of typical datasets is key to this behavior, most theoretical studies of overparametrization focus on isotropic inputs. In this work, we instead consider an analytically tractable model of structured data, where the input covariance is built from independent blocks allowing us to tune the saliency of low-dimensional structures and their alignment with respect to the target function. Using methods from statistical physics, we derive a precise asymptotic expression for the train and test error achieved by random feature models trained to classify such data, which is valid for any convex loss function. We study in detail how the data structure affects the double descent curve, and show that in the over-parametrized regime, its impact is greater for logistic loss than for mean-squared loss: the easier the task, the wider the gap in performance at the advantage of the logistic loss. Our insights are confirmed by numerical experiments on MNIST and CIFAR10.
Levent Sagun, Caglar Gulcehre, Adriana Romero et al.
Science meets Engineering in Deep Learning took place in Vancouver as part of the Workshop section of NeurIPS 2019. As organizers of the workshop, we created the following report in an attempt to isolate emerging topics and recurring themes that have been presented throughout the event. Deep learning can still be a complex mix of art and engineering despite its tremendous success in recent years. The workshop aimed at gathering people across the board to address seemingly contrasting challenges in the problems they are working on. As part of the call for the workshop, particular attention has been given to the interdependence of architecture, data, and optimization that gives rise to an enormous landscape of design and performance intricacies that are not well-understood. This year, our goal was to emphasize the following directions in our community: (i) identify obstacles in the way to better models and algorithms; (ii) identify the general trends from which we would like to build scientific and potentially theoretical understanding; and (iii) the rigorous design of scientific experiments and experimental protocols whose purpose is to resolve and pinpoint the origin of mysteries while ensuring reproducibility and robustness of conclusions. In the event, these topics emerged and were broadly discussed, matching our expectations and paving the way for new studies in these directions. While we acknowledge that the text is naturally biased as it comes through our lens, here we present an attempt to do a fair job of highlighting the outcome of the workshop.
A recent line of research has highlighted the existence of a "double descent" phenomenon in deep learning, whereby increasing the number of training examples $N$ causes the generalization error of neural networks to peak when $N$ is of the same order as the number of parameters $P$. In earlier works, a similar phenomenon was shown to exist in simpler models such as linear regression, where the peak instead occurs when $N$ is equal to the input dimension $D$. Since both peaks coincide with the interpolation threshold, they are often conflated in the litterature. In this paper, we show that despite their apparent similarity, these two scenarios are inherently different. In fact, both peaks can co-exist when neural networks are applied to noisy regression tasks. The relative size of the peaks is then governed by the degree of nonlinearity of the activation function. Building on recent developments in the analysis of random feature models, we provide a theoretical ground for this sample-wise triple descent. As shown previously, the nonlinear peak at $N\!=\!P$ is a true divergence caused by the extreme sensitivity of the output function to both the noise corrupting the labels and the initialization of the random features (or the weights in neural networks). This peak survives in the absence of noise, but can be suppressed by regularization. In contrast, the linear peak at $N\!=\!D$ is solely due to overfitting the noise in the labels, and forms earlier during training. We show that this peak is implicitly regularized by the nonlinearity, which is why it only becomes salient at high noise and is weakly affected by explicit regularization. Throughout the paper, we compare analytical results obtained in the random feature model with the outcomes of numerical experiments involving deep neural networks.
Umut Şimşekli, Mert Gürbüzbalaban, Thanh Huy Nguyen et al.
The gradient noise (GN) in the stochastic gradient descent (SGD) algorithm is often considered to be Gaussian in the large data regime by assuming that the \emph{classical} central limit theorem (CLT) kicks in. This assumption is often made for mathematical convenience, since it enables SGD to be analyzed as a stochastic differential equation (SDE) driven by a Brownian motion. We argue that the Gaussianity assumption might fail to hold in deep learning settings and hence render the Brownian motion-based analyses inappropriate. Inspired by non-Gaussian natural phenomena, we consider the GN in a more general context and invoke the \emph{generalized} CLT, which suggests that the GN converges to a \emph{heavy-tailed} $α$-stable random vector, where \emph{tail-index} $α$ determines the heavy-tailedness of the distribution. Accordingly, we propose to analyze SGD as a discretization of an SDE driven by a Lévy motion. Such SDEs can incur `jumps', which force the SDE and its discretization \emph{transition} from narrow minima to wider minima, as proven by existing metastability theory and the extensions that we proved recently. In this study, under the $α$-stable GN assumption, we further establish an explicit connection between the convergence rate of SGD to a local minimum and the tail-index $α$. To validate the $α$-stable assumption, we conduct experiments on common deep learning scenarios and show that in all settings, the GN is highly non-Gaussian and admits heavy-tails. We investigate the tail behavior in varying network architectures and sizes, loss functions, and datasets. Our results open up a different perspective and shed more light on the belief that SGD prefers wide minima.
Stéphane d'Ascoli, Levent Sagun, Joan Bruna et al.
Despite the phenomenal success of deep neural networks in a broad range of learning tasks, there is a lack of theory to understand the way they work. In particular, Convolutional Neural Networks (CNNs) are known to perform much better than Fully-Connected Networks (FCNs) on spatially structured data: the architectural structure of CNNs benefits from prior knowledge on the features of the data, for instance their translation invariance. The aim of this work is to understand this fact through the lens of dynamics in the loss landscape. We introduce a method that maps a CNN to its equivalent FCN (denoted as eFCN). Such an embedding enables the comparison of CNN and FCN training dynamics directly in the FCN space. We use this method to test a new training protocol, which consists in training a CNN, embedding it to FCN space at a certain ``relax time'', then resuming the training in FCN space. We observe that for all relax times, the deviation from the CNN subspace is small, and the final performance reached by the eFCN is higher than that reachable by a standard FCN of same architecture. More surprisingly, for some intermediate relax times, the eFCN outperforms the CNN it stemmed, by combining the prior information of the CNN and the expressivity of the FCN in a complementary way. The practical interest of our protocol is limited by the very large size of the highly sparse eFCN. However, it offers interesting insights into the persistence of architectural bias under stochastic gradient dynamics. It shows the existence of some rare basins in the FCN loss landscape associated with very good generalization. These can only be accessed thanks to the CNN prior, which helps navigate the landscape during the early stages of optimization.
The gradient noise (GN) in the stochastic gradient descent (SGD) algorithm is often considered to be Gaussian in the large data regime by assuming that the classical central limit theorem (CLT) kicks in. This assumption is often made for mathematical convenience, since it enables SGD to be analyzed as a stochastic differential equation (SDE) driven by a Brownian motion. We argue that the Gaussianity assumption might fail to hold in deep learning settings and hence render the Brownian motion-based analyses inappropriate. Inspired by non-Gaussian natural phenomena, we consider the GN in a more general context and invoke the generalized CLT (GCLT), which suggests that the GN converges to a heavy-tailed $α$-stable random variable. Accordingly, we propose to analyze SGD as an SDE driven by a Lévy motion. Such SDEs can incur `jumps', which force the SDE transition from narrow minima to wider minima, as proven by existing metastability theory. To validate the $α$-stable assumption, we conduct extensive experiments on common deep learning architectures and show that in all settings, the GN is highly non-Gaussian and admits heavy-tails. We further investigate the tail behavior in varying network architectures and sizes, loss functions, and datasets. Our results open up a different perspective and shed more light on the belief that SGD prefers wide minima.
Mario Geiger, Arthur Jacot, Stefano Spigler et al.
Supervised deep learning involves the training of neural networks with a large number $N$ of parameters. For large enough $N$, in the so-called over-parametrized regime, one can essentially fit the training data points. Sparsity-based arguments would suggest that the generalization error increases as $N$ grows past a certain threshold $N^{*}$. Instead, empirical studies have shown that in the over-parametrized regime, generalization error keeps decreasing with $N$. We resolve this paradox through a new framework. We rely on the so-called Neural Tangent Kernel, which connects large neural nets to kernel methods, to show that the initialization causes finite-size random fluctuations $\|f_{N}-\bar{f}_{N}\|\sim N^{-1/4}$ of the neural net output function $f_{N}$ around its expectation $\bar{f}_{N}$. These affect the generalization error $ε_{N}$ for classification: under natural assumptions, it decays to a plateau value $ε_{\infty}$ in a power-law fashion $\sim N^{-1/2}$. This description breaks down at a so-called jamming transition $N=N^{*}$. At this threshold, we argue that $\|f_{N}\|$ diverges. This result leads to a plausible explanation for the cusp in test error known to occur at $N^{*}$. Our results are confirmed by extensive empirical observations on the MNIST and CIFAR image datasets. Our analysis finally suggests that, given a computational envelope, the smallest generalization error is obtained using several networks of intermediate sizes, just beyond $N^{*}$, and averaging their outputs.
Mario Geiger, Stefano Spigler, Stéphane d'Ascoli et al.
Deep learning has been immensely successful at a variety of tasks, ranging from classification to AI. Learning corresponds to fitting training data, which is implemented by descending a very high-dimensional loss function. Understanding under which conditions neural networks do not get stuck in poor minima of the loss, and how the landscape of that loss evolves as depth is increased remains a challenge. Here we predict, and test empirically, an analogy between this landscape and the energy landscape of repulsive ellipses. We argue that in FC networks a phase transition delimits the over- and under-parametrized regimes where fitting can or cannot be achieved. In the vicinity of this transition, properties of the curvature of the minima of the loss are critical. This transition shares direct similarities with the jamming transition by which particles form a disordered solid as the density is increased, which also occurs in certain classes of computational optimization and learning problems such as the perceptron. Our analysis gives a simple explanation as to why poor minima of the loss cannot be encountered in the overparametrized regime, and puts forward the surprising result that the ability of fully connected networks to fit random data is independent of their depth. Our observations suggests that this independence also holds for real data. We also study a quantity $Δ$ which characterizes how well ($Δ<0$) or badly ($Δ>0$) a datum is learned. At the critical point it is power-law distributed, $P_+(Δ)\simΔ^θ$ for $Δ>0$ and $P_-(Δ)\sim(-Δ)^{-γ}$ for $Δ<0$, with $θ\approx0.3$ and $γ\approx0.2$. This observation suggests that near the transition the loss landscape has a hierarchical structure and that the learning dynamics is prone to avalanche-like dynamics, with abrupt changes in the set of patterns that are learned.
We analyze numerically the training dynamics of deep neural networks (DNN) by using methods developed in statistical physics of glassy systems. The two main issues we address are (1) the complexity of the loss landscape and of the dynamics within it, and (2) to what extent DNNs share similarities with glassy systems. Our findings, obtained for different architectures and datasets, suggest that during the training process the dynamics slows down because of an increasingly large number of flat directions. At large times, when the loss is approaching zero, the system diffuses at the bottom of the landscape. Despite some similarities with the dynamics of mean-field glassy systems, in particular, the absence of barrier crossing, we find distinctive dynamical behaviors in the two cases, showing that the statistical properties of the corresponding loss and energy landscapes are different. In contrast, when the network is under-parametrized we observe a typical glassy behavior, thus suggesting the existence of different phases depending on whether the network is under-parametrized or over-parametrized.
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.
We publicly release a new large-scale dataset, called SearchQA, for machine comprehension, or question-answering. Unlike recently released datasets, such as DeepMind CNN/DailyMail and SQuAD, the proposed SearchQA was constructed to reflect a full pipeline of general question-answering. That is, we start not from an existing article and generate a question-answer pair, but start from an existing question-answer pair, crawled from J! Archive, and augment it with text snippets retrieved by Google. Following this approach, we built SearchQA, which consists of more than 140k question-answer pairs with each pair having 49.6 snippets on average. Each question-answer-context tuple of the SearchQA comes with additional meta-data such as the snippet's URL, which we believe will be valuable resources for future research. We conduct human evaluation as well as test two baseline methods, one simple word selection and the other deep learning based, on the SearchQA. We show that there is a meaningful gap between the human and machine performances. This suggests that the proposed dataset could well serve as a benchmark for question-answering.
Andrew J. Ballard, Ritankar Das, Stefano Martiniani et al.
Machine learning techniques are being increasingly used as flexible non-linear fitting and prediction tools in the physical sciences. Fitting functions that exhibit multiple solutions as local minima can be analysed in terms of the corresponding machine learning landscape. Methods to explore and visualise molecular potential energy landscapes can be applied to these machine learning landscapes to gain new insight into the solution space involved in training and the nature of the corresponding predictions. In particular, we can define quantities analogous to molecular structure, thermodynamics, and kinetics, and relate these emergent properties to the structure of the underlying landscape. This Perspective aims to describe these analogies with examples from recent applications, and suggest avenues for new interdisciplinary research.
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.
Pratik Chaudhari, Anna Choromanska, Stefano Soatto et al.
This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative eigenvalues. We leverage upon this observation to construct a local-entropy-based objective function that favors well-generalizable solutions lying in large flat regions of the energy landscape, while avoiding poorly-generalizable solutions located in the sharp valleys. Conceptually, our algorithm resembles two nested loops of SGD where we use Langevin dynamics in the inner loop to compute the gradient of the local entropy before each update of the weights. We show that the new objective has a smoother energy landscape and show improved generalization over SGD using uniform stability, under certain assumptions. Our experiments on convolutional and recurrent networks demonstrate that Entropy-SGD compares favorably to state-of-the-art techniques in terms of generalization error and training time.
Levent Sagun, V. Ugur Guney, Gerard Ben Arous et al.
Finding minima of a real valued non-convex function over a high dimensional space is a major challenge in science. We provide evidence that some such functions that are defined on high dimensional domains have a narrow band of values whose pre-image contains the bulk of its critical points. This is in contrast with the low dimensional picture in which this band is wide. Our simulations agree with the previous theoretical work on spin glasses that proves the existence of such a band when the dimension of the domain tends to infinity. Furthermore our experiments on teacher-student networks with the MNIST dataset establish a similar phenomenon in deep networks. We finally observe that both the gradient descent and the stochastic gradient descent methods can reach this level within the same number of steps.