Yann Dauphin

LG
h-index117
25papers
11,105citations
Novelty50%
AI Score42

25 Papers

LGApr 27, 2023Code
JaxPruner: A concise library for sparsity research

Joo Hyung Lee, Wonpyo Park, Nicole Mitchell et al. · mila

This paper introduces JaxPruner, an open-source JAX-based pruning and sparse training library for machine learning research. JaxPruner aims to accelerate research on sparse neural networks by providing concise implementations of popular pruning and sparse training algorithms with minimal memory and latency overhead. Algorithms implemented in JaxPruner use a common API and work seamlessly with the popular optimization library Optax, which, in turn, enables easy integration with existing JAX based libraries. We demonstrate this ease of integration by providing examples in four different codebases: Scenic, t5x, Dopamine and FedJAX and provide baseline experiments on popular benchmarks.

LGNov 23, 2023
A density estimation perspective on learning from pairwise human preferences

Vincent Dumoulin, Daniel D. Johnson, Pablo Samuel Castro et al.

Learning from human feedback (LHF) -- and in particular learning from pairwise preferences -- has recently become a crucial ingredient in training large language models (LLMs), and has been the subject of much research. Most recent works frame it as a reinforcement learning problem, where a reward function is learned from pairwise preference data and the LLM is treated as a policy which is adapted to maximize the rewards, often under additional regularization constraints. We propose an alternative interpretation which centers on the generative process for pairwise preferences and treats LHF as a density estimation problem. We provide theoretical and empirical results showing that for a family of generative processes defined via preference behavior distribution equations, training a reward function on pairwise preferences effectively models an annotator's implicit preference distribution. Finally, we discuss and present findings on "annotator misspecification" -- failure cases where wrong modeling assumptions are made about annotator behavior, resulting in poorly-adapted models -- suggesting that approaches that learn from pairwise human preferences could have trouble learning from a population of annotators with diverse viewpoints.

CVApr 6, 2023
Robustmix: Improving Robustness by Regularizing the Frequency Bias of Deep Nets

Jonas Ngnawe, Marianne Abemgnigni Njifon, Jonathan Heek et al.

Deep networks have achieved impressive results on a range of well-curated benchmark datasets. Surprisingly, their performance remains sensitive to perturbations that have little effect on human performance. In this work, we propose a novel extension of Mixup called Robustmix that regularizes networks to classify based on lower-frequency spatial features. We show that this type of regularization improves robustness on a range of benchmarks such as Imagenet-C and Stylized Imagenet. It adds little computational overhead and, furthermore, does not require a priori knowledge of a large set of image transformations. We find that this approach further complements recent advances in model architecture and data augmentation, attaining a state-of-the-art mCE of 44.8 with an EfficientNet-B8 model and RandAugment, which is a reduction of 16 mCE compared to the baseline.

CLJul 7, 2025
Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities

Gheorghe Comanici, Eric Bieber, Mike Schaekermann et al. · amazon-science, baidu

In this report, we introduce the Gemini 2.X model family: Gemini 2.5 Pro and Gemini 2.5 Flash, as well as our earlier Gemini 2.0 Flash and Flash-Lite models. Gemini 2.5 Pro is our most capable model yet, achieving SoTA performance on frontier coding and reasoning benchmarks. In addition to its incredible coding and reasoning skills, Gemini 2.5 Pro is a thinking model that excels at multimodal understanding and it is now able to process up to 3 hours of video content. Its unique combination of long context, multimodal and reasoning capabilities can be combined to unlock new agentic workflows. Gemini 2.5 Flash provides excellent reasoning abilities at a fraction of the compute and latency requirements and Gemini 2.0 Flash and Flash-Lite provide high performance at low latency and cost. Taken together, the Gemini 2.X model generation spans the full Pareto frontier of model capability vs cost, allowing users to explore the boundaries of what is possible with complex agentic problem solving.

CVMay 22, 2023Code
Tied-Augment: Controlling Representation Similarity Improves Data Augmentation

Emirhan Kurtulus, Zichao Li, Yann Dauphin et al.

Data augmentation methods have played an important role in the recent advance of deep learning models, and have become an indispensable component of state-of-the-art models in semi-supervised, self-supervised, and supervised training for vision. Despite incurring no additional latency at test time, data augmentation often requires more epochs of training to be effective. For example, even the simple flips-and-crops augmentation requires training for more than 5 epochs to improve performance, whereas RandAugment requires more than 90 epochs. We propose a general framework called Tied-Augment, which improves the efficacy of data augmentation in a wide range of applications by adding a simple term to the loss that can control the similarity of representations under distortions. Tied-Augment can improve state-of-the-art methods from data augmentation (e.g. RandAugment, mixup), optimization (e.g. SAM), and semi-supervised learning (e.g. FixMatch). For example, Tied-RandAugment can outperform RandAugment by 2.0% on ImageNet. Notably, using Tied-Augment, data augmentation can be made to improve generalization even when training for a few epochs and when fine-tuning. We open source our code at https://github.com/ekurtulus/tied-augment/tree/main.

AIMar 21, 2025
Capturing Individual Human Preferences with Reward Features

André Barreto, Vincent Dumoulin, Yiran Mao et al.

Reinforcement learning from human feedback usually models preferences using a reward model that does not distinguish between people. We argue that this is unlikely to be a good design choice in contexts with high potential for disagreement, like in the training of large language models. We propose a method to specialise a reward model to a person or group of people. Our approach builds on the observation that individual preferences can be captured as a linear combination of a set of general reward features. We show how to learn such features and subsequently use them to quickly adapt the reward model to a specific individual, even if their preferences are not reflected in the training data. We present experiments with large language models comparing the proposed architecture with a non-adaptive reward model and also adaptive counterparts, including models that do in-context personalisation. Depending on how much disagreement there is in the training data, our model either significantly outperforms the baselines or matches their performance with a simpler architecture and more stable training.

LGFeb 4, 2025
Avoiding spurious sharpness minimization broadens applicability of SAM

Sidak Pal Singh, Hossein Mobahi, Atish Agarwala et al. · eth-zurich

Curvature regularization techniques like Sharpness Aware Minimization (SAM) have shown great promise in improving generalization on vision tasks. However, we find that SAM performs poorly in domains like natural language processing (NLP), often degrading performance -- even with twice the compute budget. We investigate the discrepancy across domains and find that in the NLP setting, SAM is dominated by regularization of the logit statistics -- instead of improving the geometry of the function itself. We use this observation to develop an alternative algorithm we call Functional-SAM, which regularizes curvature only through modification of the statistics of the overall function implemented by the neural network, and avoids spurious minimization through logit manipulation. Furthermore, we argue that preconditioning the SAM perturbation also prevents spurious minimization, and when combined with Functional-SAM, it gives further improvements. Our proposed algorithms show improved performance over AdamW and SAM baselines when trained for an equal number of steps, in both fixed-length and Chinchilla-style training settings, at various model scales (including billion-parameter scale). On the whole, our work highlights the importance of more precise characterizations of sharpness in broadening the applicability of curvature regularization to large language models (LLMs).

LGOct 21, 2024
Towards Optimal Adapter Placement for Efficient Transfer Learning

Aleksandra I. Nowak, Otniel-Bogdan Mercea, Anurag Arnab et al.

Parameter-efficient transfer learning (PETL) aims to adapt pre-trained models to new downstream tasks while minimizing the number of fine-tuned parameters. Adapters, a popular approach in PETL, inject additional capacity into existing networks by incorporating low-rank projections, achieving performance comparable to full fine-tuning with significantly fewer parameters. This paper investigates the relationship between the placement of an adapter and its performance. We observe that adapter location within a network significantly impacts its effectiveness, and that the optimal placement is task-dependent. To exploit this observation, we introduce an extended search space of adapter connections, including long-range and recurrent adapters. We demonstrate that even randomly selected adapter placements from this expanded space yield improved results, and that high-performing placements often correlate with high gradient rank. Our findings reveal that a small number of strategically placed adapters can match or exceed the performance of the common baseline of adding adapters in every block, opening a new avenue for research into optimal adapter placement strategies.

LGOct 20, 2021
No One Representation to Rule Them All: Overlapping Features of Training Methods

Raphael Gontijo-Lopes, Yann Dauphin, Ekin D. Cubuk

Despite being able to capture a range of features of the data, high accuracy models trained with supervision tend to make similar predictions. This seemingly implies that high-performing models share similar biases regardless of training methodology, which would limit ensembling benefits and render low-accuracy models as having little practical use. Against this backdrop, recent work has developed quite different training techniques, such as large-scale contrastive learning, yielding competitively high accuracy on generalization and robustness benchmarks. This motivates us to revisit the assumption that models necessarily learn similar functions. We conduct a large-scale empirical study of models across hyper-parameters, architectures, frameworks, and datasets. We find that model pairs that diverge more in training methodology display categorically different generalization behavior, producing increasingly uncorrelated errors. We show these models specialize in subdomains of the data, leading to higher ensemble performance: with just 2 models (each with ImageNet accuracy ~76.5%), we can create ensembles with 83.4% (+7% boost). Surprisingly, we find that even significantly low-accuracy models can be used to improve high-accuracy models. Finally, we show diverging training methodology yield representations that capture overlapping (but not supersetting) feature sets which, when combined, lead to increased downstream performance.

LGAug 25, 2021
Auxiliary Task Update Decomposition: The Good, The Bad and The Neutral

Lucio M. Dery, Yann Dauphin, David Grangier

While deep learning has been very beneficial in data-rich settings, tasks with smaller training set often resort to pre-training or multitask learning to leverage data from other tasks. In this case, careful consideration is needed to select tasks and model parameterizations such that updates from the auxiliary tasks actually help the primary task. We seek to alleviate this burden by formulating a model-agnostic framework that performs fine-grained manipulation of the auxiliary task gradients. We propose to decompose auxiliary updates into directions which help, damage or leave the primary task loss unchanged. This allows weighting the update directions differently depending on their impact on the problem of interest. We present a novel and efficient algorithm for that purpose and show its advantage in practice. Our method leverages efficient automatic differentiation procedures and randomized singular value decomposition for scalability. We show that our framework is generic and encompasses some prior work as particular cases. Our approach consistently outperforms strong and widely used baselines when leveraging out-of-distribution data for Text and Image classification tasks.

CVJul 26, 2021
Continental-Scale Building Detection from High Resolution Satellite Imagery

Wojciech Sirko, Sergii Kashubin, Marvin Ritter et al.

Identifying the locations and footprints of buildings is vital for many practical and scientific purposes. Such information can be particularly useful in developing regions where alternative data sources may be scarce. In this work, we describe a model training pipeline for detecting buildings across the entire continent of Africa, using 50 cm satellite imagery. Starting with the U-Net model, widely used in satellite image analysis, we study variations in architecture, loss functions, regularization, pre-training, self-training and post-processing that increase instance segmentation performance. Experiments were carried out using a dataset of 100k satellite images across Africa containing 1.75M manually labelled building instances, and further datasets for pre-training and self-training. We report novel methods for improving performance of building detection with this type of model, including the use of mixup (mAP +0.12) and self-training with soft KL loss (mAP +0.06). The resulting pipeline obtains good results even on a wide variety of challenging rural and urban contexts, and was used to create the Open Buildings dataset of 516M Africa-wide detected footprints.

LGOct 14, 2020
Temperature check: theory and practice for training models with softmax-cross-entropy losses

Atish Agarwala, Jeffrey Pennington, Yann Dauphin et al.

The softmax function combined with a cross-entropy loss is a principled approach to modeling probability distributions that has become ubiquitous in deep learning. The softmax function is defined by a lone hyperparameter, the temperature, that is commonly set to one or regarded as a way to tune model confidence after training; however, less is known about how the temperature impacts training dynamics or generalization performance. In this work we develop a theory of early learning for models trained with softmax-cross-entropy loss and show that the learning dynamics depend crucially on the inverse-temperature $β$ as well as the magnitude of the logits at initialization, $||β{\bf z}||_{2}$. We follow up these analytic results with a large-scale empirical study of a variety of model architectures trained on CIFAR10, ImageNet, and IMDB sentiment analysis. We find that generalization performance depends strongly on the temperature, but only weakly on the initial logit magnitude. We provide evidence that the dependence of generalization on $β$ is not due to changes in model confidence, but is a dynamical phenomenon. It follows that the addition of $β$ as a tunable hyperparameter is key to maximizing model performance. Although we find the optimal $β$ to be sensitive to the architecture, our results suggest that tuning $β$ over the range $10^{-2}$ to $10^1$ improves performance over all architectures studied. We find that smaller $β$ may lead to better peak performance at the cost of learning stability.

LGOct 7, 2020
Gradient Flow in Sparse Neural Networks and How Lottery Tickets Win

Utku Evci, Yani A. Ioannou, Cem Keskin et al.

Sparse Neural Networks (NNs) can match the generalization of dense NNs using a fraction of the compute/storage for inference, and also have the potential to enable efficient training. However, naively training unstructured sparse NNs from random initialization results in significantly worse generalization, with the notable exceptions of Lottery Tickets (LTs) and Dynamic Sparse Training (DST). Through our analysis of gradient flow during training we attempt to answer: (1) why training unstructured sparse networks from random initialization performs poorly and; (2) what makes LTs and DST the exceptions? We show that sparse NNs have poor gradient flow at initialization and demonstrate the importance of using sparsity-aware initialization. Furthermore, we find that DST methods significantly improve gradient flow during training over traditional sparse training methods. Finally, we show that LTs do not improve gradient flow, rather their success lies in re-learning the pruning solution they are derived from - however, this comes at the cost of learning novel solutions.

LGMar 24, 2020
Robust and On-the-fly Dataset Denoising for Image Classification

Jiaming Song, Lunjia Hu, Michael Auli et al.

Memorization in over-parameterized neural networks could severely hurt generalization in the presence of mislabeled examples. However, mislabeled examples are hard to avoid in extremely large datasets collected with weak supervision. We address this problem by reasoning counterfactually about the loss distribution of examples with uniform random labels had they were trained with the real examples, and use this information to remove noisy examples from the training set. First, we observe that examples with uniform random labels have higher losses when trained with stochastic gradient descent under large learning rates. Then, we propose to model the loss distribution of the counterfactual examples using only the network parameters, which is able to model such examples with remarkable success. Finally, we propose to remove examples whose loss exceeds a certain quantile of the modeled loss distribution. This leads to On-the-fly Data Denoising (ODD), a simple yet effective algorithm that is robust to mislabeled examples, while introducing almost zero computational overhead compared to standard training. ODD is able to achieve state-of-the-art results on a wide range of datasets including real-world ones such as WebVision and Clothing1M.

LGNov 13, 2019
What Do Compressed Deep Neural Networks Forget?

Sara Hooker, Aaron Courville, Gregory Clark et al.

Deep neural network pruning and quantization techniques have demonstrated it is possible to achieve high levels of compression with surprisingly little degradation to test set accuracy. However, this measure of performance conceals significant differences in how different classes and images are impacted by model compression techniques. We find that models with radically different numbers of weights have comparable top-line performance metrics but diverge considerably in behavior on a narrow subset of the dataset. This small subset of data points, which we term Pruning Identified Exemplars (PIEs) are systematically more impacted by the introduction of sparsity. Compression disproportionately impacts model performance on the underrepresented long-tail of the data distribution. PIEs over-index on atypical or noisy images that are far more challenging for both humans and algorithms to classify. Our work provides intuition into the role of capacity in deep neural networks and the trade-offs incurred by compression. An understanding of this disparate impact is critical given the widespread deployment of compressed models in the wild.

LGMar 12, 2019
On the Pitfalls of Measuring Emergent Communication

Ryan Lowe, Jakob Foerster, Y-Lan Boureau et al.

How do we know if communication is emerging in a multi-agent system? The vast majority of recent papers on emergent communication show that adding a communication channel leads to an increase in reward or task success. This is a useful indicator, but provides only a coarse measure of the agent's learned communication abilities. As we move towards more complex environments, it becomes imperative to have a set of finer tools that allow qualitative and quantitative insights into the emergence of communication. This may be especially useful to allow humans to monitor agents' behaviour, whether for fault detection, assessing performance, or even building trust. In this paper, we examine a few intuitive existing metrics for measuring communication, and show that they can be misleading. Specifically, by training deep reinforcement learning agents to play simple matrix games augmented with a communication channel, we find a scenario where agents appear to communicate (their messages provide information about their subsequent action), and yet the messages do not impact the environment or other agent in any way. We explain this phenomenon using ablation studies and by visualizing the representations of the learned policies. We also survey some commonly used metrics for measuring emergent communication, and provide recommendations as to when these metrics should be used.

CLFeb 4, 2019
Strategies for Structuring Story Generation

Angela Fan, Mike Lewis, Yann Dauphin

Writers generally rely on plans or sketches to write long stories, but most current language models generate word by word from left to right. We explore coarse-to-fine models for creating narrative texts of several hundred words, and introduce new models which decompose stories by abstracting over actions and entities. The model first generates the predicate-argument structure of the text, where different mentions of the same entity are marked with placeholder tokens. It then generates a surface realization of the predicate-argument structure, and finally replaces the entity placeholders with context-sensitive names and references. Human judges prefer the stories from our models to a wide range of previous approaches to hierarchical text generation. Extensive analysis shows that our methods can help improve the diversity and coherence of events and entities in generated stories.

CLMay 13, 2018
Hierarchical Neural Story Generation

Angela Fan, Mike Lewis, Yann Dauphin

We explore story generation: creative systems that can build coherent and fluent passages of text about a topic. We collect a large dataset of 300K human-written stories paired with writing prompts from an online forum. Our dataset enables hierarchical story generation, where the model first generates a premise, and then transforms it into a passage of text. We gain further improvements with a novel form of model fusion that improves the relevance of the story to the prompt, and adding a new gated multi-scale self-attention mechanism to model long-range context. Experiments show large improvements over strong baselines on both automated and human evaluations. Human judges prefer stories generated by our approach to those from a strong non-hierarchical model by a factor of two to one.

LGJun 14, 2017
Empirical Analysis of the Hessian of Over-Parametrized Neural Networks

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.

LGJun 9, 2017
Tackling Over-pruning in Variational Autoencoders

Serena Yeung, Anitha Kannan, Yann Dauphin et al.

Variational autoencoders (VAE) are directed generative models that learn factorial latent variables. As noted by Burda et al. (2015), these models exhibit the problem of factor over-pruning where a significant number of stochastic factors fail to learn anything and become inactive. This can limit their modeling power and their ability to learn diverse and meaningful latent representations. In this paper, we evaluate several methods to address this problem and propose a more effective model-based approach called the epitomic variational autoencoder (eVAE). The so-called epitomes of this model are groups of mutually exclusive latent factors that compete to explain the data. This approach helps prevent inactive units since each group is pressured to explain the data. We compare the approaches with qualitative and quantitative results on MNIST and TFD datasets. Our results show that eVAE makes efficient use of model capacity and generalizes better than VAE.

MLApr 28, 2017
Parseval Networks: Improving Robustness to Adversarial Examples

Moustapha Cisse, Piotr Bojanowski, Edouard Grave et al.

We introduce Parseval networks, a form of deep neural networks in which the Lipschitz constant of linear, convolutional and aggregation layers is constrained to be smaller than 1. Parseval networks are empirically and theoretically motivated by an analysis of the robustness of the predictions made by deep neural networks when their input is subject to an adversarial perturbation. The most important feature of Parseval networks is to maintain weight matrices of linear and convolutional layers to be (approximately) Parseval tight frames, which are extensions of orthogonal matrices to non-square matrices. We describe how these constraints can be maintained efficiently during SGD. We show that Parseval networks match the state-of-the-art in terms of accuracy on CIFAR-10/100 and Street View House Numbers (SVHN) while being more robust than their vanilla counterpart against adversarial examples. Incidentally, Parseval networks also tend to train faster and make a better usage of the full capacity of the networks.

LGMar 5, 2015
EmoNets: Multimodal deep learning approaches for emotion recognition in video

Samira Ebrahimi Kahou, Xavier Bouthillier, Pascal Lamblin et al.

The task of the emotion recognition in the wild (EmotiW) Challenge is to assign one of seven emotions to short video clips extracted from Hollywood style movies. The videos depict acted-out emotions under realistic conditions with a large degree of variation in attributes such as pose and illumination, making it worthwhile to explore approaches which consider combinations of features from multiple modalities for label assignment. In this paper we present our approach to learning several specialist models using deep learning techniques, each focusing on one modality. Among these are a convolutional neural network, focusing on capturing visual information in detected faces, a deep belief net focusing on the representation of the audio stream, a K-Means based "bag-of-mouths" model, which extracts visual features around the mouth region and a relational autoencoder, which addresses spatio-temporal aspects of videos. We explore multiple methods for the combination of cues from these modalities into one common classifier. This achieves a considerably greater accuracy than predictions from our strongest single-modality classifier. Our method was the winning submission in the 2013 EmotiW challenge and achieved a test set accuracy of 47.67% on the 2014 dataset.

LGJun 10, 2014
Identifying and attacking the saddle point problem in high-dimensional non-convex optimization

Yann Dauphin, Razvan Pascanu, Caglar Gulcehre et al.

A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, neural network theory, and empirical evidence, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new approach to second-order optimization, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. We apply this algorithm to deep or recurrent neural network training, and provide numerical evidence for its superior optimization performance.

LGJul 18, 2012
Better Mixing via Deep Representations

Yoshua Bengio, Grégoire Mesnil, Yann Dauphin et al.

It has previously been hypothesized, and supported with some experimental evidence, that deeper representations, when well trained, tend to do a better job at disentangling the underlying factors of variation. We study the following related conjecture: better representations, in the sense of better disentangling, can be exploited to produce faster-mixing Markov chains. Consequently, mixing would be more efficient at higher levels of representation. To better understand why and how this is happening, we propose a secondary conjecture: the higher-level samples fill more uniformly the space they occupy and the high-density manifolds tend to unfold when represented at higher levels. The paper discusses these hypotheses and tests them experimentally through visualization and measurements of mixing and interpolating between samples.

LGJun 27, 2012
A Generative Process for Sampling Contractive Auto-Encoders

Salah Rifai, Yoshua Bengio, Yann Dauphin et al.

The contractive auto-encoder learns a representation of the input data that captures the local manifold structure around each data point, through the leading singular vectors of the Jacobian of the transformation from input to representation. The corresponding singular values specify how much local variation is plausible in directions associated with the corresponding singular vectors, while remaining in a high-density region of the input space. This paper proposes a procedure for generating samples that are consistent with the local structure captured by a contractive auto-encoder. The associated stochastic process defines a distribution from which one can sample, and which experimentally appears to converge quickly and mix well between modes, compared to Restricted Boltzmann Machines and Deep Belief Networks. The intuitions behind this procedure can also be used to train the second layer of contraction that pools lower-level features and learns to be invariant to the local directions of variation discovered in the first layer. We show that this can help learn and represent invariances present in the data and improve classification error.