new

Get trending papers in your email inbox!

Subscribe

byAK and the research community

Mar 14

Cross-Entropy Loss Functions: Theoretical Analysis and Applications

Cross-entropy is a widely used loss function in applications. It coincides with the logistic loss applied to the outputs of a neural network, when the softmax is used. But, what guarantees can we rely on when using cross-entropy as a surrogate loss? We present a theoretical analysis of a broad family of loss functions, comp-sum losses, that includes cross-entropy (or logistic loss), generalized cross-entropy, the mean absolute error and other cross-entropy-like loss functions. We give the first H-consistency bounds for these loss functions. These are non-asymptotic guarantees that upper bound the zero-one loss estimation error in terms of the estimation error of a surrogate loss, for the specific hypothesis set H used. We further show that our bounds are tight. These bounds depend on quantities called minimizability gaps. To make them more explicit, we give a specific analysis of these gaps for comp-sum losses. We also introduce a new family of loss functions, smooth adversarial comp-sum losses, that are derived from their comp-sum counterparts by adding in a related smooth term. We show that these loss functions are beneficial in the adversarial setting by proving that they admit H-consistency bounds. This leads to new adversarial robustness algorithms that consist of minimizing a regularized smooth adversarial comp-sum loss. While our main purpose is a theoretical analysis, we also present an extensive empirical analysis comparing comp-sum losses. We further report the results of a series of experiments demonstrating that our adversarial robustness algorithms outperform the current state-of-the-art, while also achieving a superior non-adversarial accuracy.

Few-Shot Segmentation Without Meta-Learning: A Good Transductive Inference Is All You Need?

We show that the way inference is performed in few-shot segmentation tasks has a substantial effect on performances -- an aspect often overlooked in the literature in favor of the meta-learning paradigm. We introduce a transductive inference for a given query image, leveraging the statistics of its unlabeled pixels, by optimizing a new loss containing three complementary terms: i) the cross-entropy on the labeled support pixels; ii) the Shannon entropy of the posteriors on the unlabeled query-image pixels; and iii) a global KL-divergence regularizer based on the proportion of the predicted foreground. As our inference uses a simple linear classifier of the extracted features, its computational load is comparable to inductive inference and can be used on top of any base training. Foregoing episodic training and using only standard cross-entropy training on the base classes, our inference yields competitive performances on standard benchmarks in the 1-shot scenarios. As the number of available shots increases, the gap in performances widens: on PASCAL-5i, our method brings about 5% and 6% improvements over the state-of-the-art, in the 5- and 10-shot scenarios, respectively. Furthermore, we introduce a new setting that includes domain shifts, where the base and novel classes are drawn from different datasets. Our method achieves the best performances in this more realistic setting. Our code is freely available online: https://github.com/mboudiaf/RePRI-for-Few-Shot-Segmentation.

Scaling Laws for Autoregressive Generative Modeling

We identify empirical scaling laws for the cross-entropy loss in four domains: generative image modeling, video modeling, multimodal imageleftrightarrowtext models, and mathematical problem solving. In all cases autoregressive Transformers smoothly improve in performance as model size and compute budgets increase, following a power-law plus constant scaling law. The optimal model size also depends on the compute budget through a power-law, with exponents that are nearly universal across all data domains. The cross-entropy loss has an information theoretic interpretation as S(True) + D_{KL}(True||Model), and the empirical scaling laws suggest a prediction for both the true data distribution's entropy and the KL divergence between the true and model distributions. With this interpretation, billion-parameter Transformers are nearly perfect models of the YFCC100M image distribution downsampled to an 8times 8 resolution, and we can forecast the model size needed to achieve any given reducible loss (ie D_{KL}) in nats/image for other resolutions. We find a number of additional scaling laws in specific domains: (a) we identify a scaling relation for the mutual information between captions and images in multimodal models, and show how to answer the question "Is a picture worth a thousand words?"; (b) in the case of mathematical problem solving, we identify scaling laws for model performance when extrapolating beyond the training distribution; (c) we finetune generative image models for ImageNet classification and find smooth scaling of the classification loss and error rate, even as the generative loss levels off. Taken together, these results strengthen the case that scaling laws have important implications for neural network performance, including on downstream tasks.

Optimizing Calibration by Gaining Aware of Prediction Correctness

Model calibration aims to align confidence with prediction correctness. The Cross-Entropy (CE) loss is widely used for calibrator training, which enforces the model to increase confidence on the ground truth class. However, we find the CE loss has intrinsic limitations. For example, for a narrow misclassification, a calibrator trained by the CE loss often produces high confidence on the wrongly predicted class (e.g., a test sample is wrongly classified and its softmax score on the ground truth class is around 0.4), which is undesirable. In this paper, we propose a new post-hoc calibration objective derived from the aim of calibration. Intuitively, the proposed objective function asks that the calibrator decrease model confidence on wrongly predicted samples and increase confidence on correctly predicted samples. Because a sample itself has insufficient ability to indicate correctness, we use its transformed versions (e.g., rotated, greyscaled and color-jittered) during calibrator training. Trained on an in-distribution validation set and tested with isolated, individual test samples, our method achieves competitive calibration performance on both in-distribution and out-of-distribution test sets compared with the state of the art. Further, our analysis points out the difference between our method and commonly used objectives such as CE loss and mean square error loss, where the latters sometimes deviates from the calibration aim.

Scaling Laws for Neural Machine Translation

We present an empirical study of scaling properties of encoder-decoder Transformer models used in neural machine translation (NMT). We show that cross-entropy loss as a function of model size follows a certain scaling law. Specifically (i) We propose a formula which describes the scaling behavior of cross-entropy loss as a bivariate function of encoder and decoder size, and show that it gives accurate predictions under a variety of scaling approaches and languages; we show that the total number of parameters alone is not sufficient for such purposes. (ii) We observe different power law exponents when scaling the decoder vs scaling the encoder, and provide recommendations for optimal allocation of encoder/decoder capacity based on this observation. (iii) We also report that the scaling behavior of the model is acutely influenced by composition bias of the train/test sets, which we define as any deviation from naturally generated text (either via machine generated or human translated text). We observe that natural text on the target side enjoys scaling, which manifests as successful reduction of the cross-entropy loss. (iv) Finally, we investigate the relationship between the cross-entropy loss and the quality of the generated translations. We find two different behaviors, depending on the nature of the test data. For test sets which were originally translated from target language to source language, both loss and BLEU score improve as model size increases. In contrast, for test sets originally translated from source language to target language, the loss improves, but the BLEU score stops improving after a certain threshold. We release generated text from all models used in this study.

Cut Your Losses in Large-Vocabulary Language Models

As language models grow ever larger, so do their vocabularies. This has shifted the memory footprint of LLMs during training disproportionately to one single layer: the cross-entropy in the loss computation. Cross-entropy builds up a logit matrix with entries for each pair of input tokens and vocabulary items and, for small models, consumes an order of magnitude more memory than the rest of the LLM combined. We propose Cut Cross-Entropy (CCE), a method that computes the cross-entropy loss without materializing the logits for all tokens into global memory. Rather, CCE only computes the logit for the correct token and evaluates the log-sum-exp over all logits on the fly. We implement a custom kernel that performs the matrix multiplications and the log-sum-exp reduction over the vocabulary in flash memory, making global memory consumption for the cross-entropy computation negligible. This has a dramatic effect. Taking the Gemma 2 (2B) model as an example, CCE reduces the memory footprint of the loss computation from 24 GB to 1 MB, and the total training-time memory consumption of the classifier head from 28 GB to 1 GB. To improve the throughput of CCE, we leverage the inherent sparsity of softmax and propose to skip elements of the gradient computation that have a negligible (i.e., below numerical precision) contribution to the gradient. Experiments demonstrate that the dramatic reduction in memory consumption is accomplished without sacrificing training speed or convergence.

Label Distributionally Robust Losses for Multi-class Classification: Consistency, Robustness and Adaptivity

We study a family of loss functions named label-distributionally robust (LDR) losses for multi-class classification that are formulated from distributionally robust optimization (DRO) perspective, where the uncertainty in the given label information are modeled and captured by taking the worse case of distributional weights. The benefits of this perspective are several fold: (i) it provides a unified framework to explain the classical cross-entropy (CE) loss and SVM loss and their variants, (ii) it includes a special family corresponding to the temperature-scaled CE loss, which is widely adopted but poorly understood; (iii) it allows us to achieve adaptivity to the uncertainty degree of label information at an instance level. Our contributions include: (1) we study both consistency and robustness by establishing top-k (forall kgeq 1) consistency of LDR losses for multi-class classification, and a negative result that a top-1 consistent and symmetric robust loss cannot achieve top-k consistency simultaneously for all kgeq 2; (2) we propose a new adaptive LDR loss that automatically adapts the individualized temperature parameter to the noise degree of class label of each instance; (3) we demonstrate stable and competitive performance for the proposed adaptive LDR loss on 7 benchmark datasets under 6 noisy label and 1 clean settings against 13 loss functions, and on one real-world noisy dataset. The code is open-sourced at https://github.com/Optimization-AI/ICML2023_LDR.

Rethinking Fine-Tuning when Scaling Test-Time Compute: Limiting Confidence Improves Mathematical Reasoning

Recent progress in large language models (LLMs) highlights the power of scaling test-time compute to achieve strong performance on complex tasks, such as mathematical reasoning and code generation. This raises a critical question: how should model training be modified to optimize performance under a subsequent test-time compute strategy and budget? To explore this, we focus on pass@N, a simple test-time strategy that searches for a correct answer in N independent samples. We show, surprisingly, that training with cross-entropy (CE) loss can be {it misaligned} with pass@N in that pass@N accuracy {it decreases} with longer training. We explain the origins of this misalignment in terms of model overconfidence induced by CE, and experimentally verify our prediction of overconfidence as an impediment to scaling test-time compute via pass@N. Furthermore we suggest a principled, modified training loss that is better aligned to pass@N by limiting model confidence and rescuing pass@N test performance. Our algorithm demonstrates improved mathematical reasoning on MATH and MiniF2F benchmarks under several scenarios: (1) providing answers to math questions; and (2) proving theorems by searching over proof trees of varying shapes. Overall our work underscores the importance of co-designing two traditionally separate phases of LLM development: training-time protocols and test-time search and reasoning strategies.

Scaling Law with Learning Rate Annealing

We find that the cross-entropy loss curves of neural language models empirically adhere to a scaling law with learning rate (LR) annealing over training steps (s): $L(s) = L_0 + Acdot S_1^{-alpha} - Ccdot S_2 Where S_1 is forward area and S_2$ is learning rate annealing area. This formulation takes into account two factors: (1) The forward scaling defined as typical scaling law, and (2) the additional loss drop brought by LR annealing. Therefore, this formulation can describe the full loss curve at each step, rather than the single loss point at the end of training. Applying the scaling law with LR annealing and fitting only one or two training curves, we can accurately predict the loss of language model training at any given step and across any learning rate scheduler (LRS). Furthermore, this equation accurately describes the dynamics during training process, and provides a theoretical verification and explanation for numerous experimental findings of previous studies, particularly those focusing on LR schedule and LR annealing. The resulting insights, also serve as a guide for researchers to select critical LRS in advance by prediction using our equation. Most significantly, since all the points in a full training curve follow the equation, we can achieve accurate loss prediction at any given step across any learning rate scheduler, while expending less than 1\% of the computational cost required by the chinchilla scaling law to fit language modeling loss. This approach extremely democratizes scaling law fitting and predicting in developing large language models.

G-SimCLR : Self-Supervised Contrastive Learning with Guided Projection via Pseudo Labelling

In the realms of computer vision, it is evident that deep neural networks perform better in a supervised setting with a large amount of labeled data. The representations learned with supervision are not only of high quality but also helps the model in enhancing its accuracy. However, the collection and annotation of a large dataset are costly and time-consuming. To avoid the same, there has been a lot of research going on in the field of unsupervised visual representation learning especially in a self-supervised setting. Amongst the recent advancements in self-supervised methods for visual recognition, in SimCLR Chen et al. shows that good quality representations can indeed be learned without explicit supervision. In SimCLR, the authors maximize the similarity of augmentations of the same image and minimize the similarity of augmentations of different images. A linear classifier trained with the representations learned using this approach yields 76.5% top-1 accuracy on the ImageNet ILSVRC-2012 dataset. In this work, we propose that, with the normalized temperature-scaled cross-entropy (NT-Xent) loss function (as used in SimCLR), it is beneficial to not have images of the same category in the same batch. In an unsupervised setting, the information of images pertaining to the same category is missing. We use the latent space representation of a denoising autoencoder trained on the unlabeled dataset and cluster them with k-means to obtain pseudo labels. With this apriori information we batch images, where no two images from the same category are to be found. We report comparable performance enhancements on the CIFAR10 dataset and a subset of the ImageNet dataset. We refer to our method as G-SimCLR.

Stop Regressing: Training Value Functions via Classification for Scalable Deep RL

Value functions are a central component of deep reinforcement learning (RL). These functions, parameterized by neural networks, are trained using a mean squared error regression objective to match bootstrapped target values. However, scaling value-based RL methods that use regression to large networks, such as high-capacity Transformers, has proven challenging. This difficulty is in stark contrast to supervised learning: by leveraging a cross-entropy classification loss, supervised methods have scaled reliably to massive networks. Observing this discrepancy, in this paper, we investigate whether the scalability of deep RL can also be improved simply by using classification in place of regression for training value functions. We demonstrate that value functions trained with categorical cross-entropy significantly improves performance and scalability in a variety of domains. These include: single-task RL on Atari 2600 games with SoftMoEs, multi-task RL on Atari with large-scale ResNets, robotic manipulation with Q-transformers, playing Chess without search, and a language-agent Wordle task with high-capacity Transformers, achieving state-of-the-art results on these domains. Through careful analysis, we show that the benefits of categorical cross-entropy primarily stem from its ability to mitigate issues inherent to value-based RL, such as noisy targets and non-stationarity. Overall, we argue that a simple shift to training value functions with categorical cross-entropy can yield substantial improvements in the scalability of deep RL at little-to-no cost.

Toward a Theory of Tokenization in LLMs

While there has been a large body of research attempting to circumvent tokenization for language modeling (Clark et al., 2022; Xue et al., 2022), the current consensus is that it is a necessary initial step for designing state-of-the-art performant language models. In this paper, we investigate tokenization from a theoretical point of view by studying the behavior of transformers on simple data generating processes. When trained on data drawn from certain simple k^{th}-order Markov processes for k > 1, transformers exhibit a surprising phenomenon - in the absence of tokenization, they empirically fail to learn the right distribution and predict characters according to a unigram model (Makkuva et al., 2024). With the addition of tokenization, however, we empirically observe that transformers break through this barrier and are able to model the probabilities of sequences drawn from the source near-optimally, achieving small cross-entropy loss. With this observation as starting point, we study the end-to-end cross-entropy loss achieved by transformers with and without tokenization. With the appropriate tokenization, we show that even the simplest unigram models (over tokens) learnt by transformers are able to model the probability of sequences drawn from k^{th}-order Markov sources near optimally. Our analysis provides a justification for the use of tokenization in practice through studying the behavior of transformers on Markovian data.

Neural Collapse in Deep Linear Networks: From Balanced to Imbalanced Data

Modern deep neural networks have achieved impressive performance on tasks from image classification to natural language processing. Surprisingly, these complex systems with massive amounts of parameters exhibit the same structural properties in their last-layer features and classifiers across canonical datasets when training until convergence. In particular, it has been observed that the last-layer features collapse to their class-means, and those class-means are the vertices of a simplex Equiangular Tight Frame (ETF). This phenomenon is known as Neural Collapse (NC). Recent papers have theoretically shown that NC emerges in the global minimizers of training problems with the simplified "unconstrained feature model". In this context, we take a step further and prove the NC occurrences in deep linear networks for the popular mean squared error (MSE) and cross entropy (CE) losses, showing that global solutions exhibit NC properties across the linear layers. Furthermore, we extend our study to imbalanced data for MSE loss and present the first geometric analysis of NC under bias-free setting. Our results demonstrate the convergence of the last-layer features and classifiers to a geometry consisting of orthogonal vectors, whose lengths depend on the amount of data in their corresponding classes. Finally, we empirically validate our theoretical analyses on synthetic and practical network architectures with both balanced and imbalanced scenarios.

Robust Mean Teacher for Continual and Gradual Test-Time Adaptation

Since experiencing domain shifts during test-time is inevitable in practice, test-time adaption (TTA) continues to adapt the model after deployment. Recently, the area of continual and gradual test-time adaptation (TTA) emerged. In contrast to standard TTA, continual TTA considers not only a single domain shift, but a sequence of shifts. Gradual TTA further exploits the property that some shifts evolve gradually over time. Since in both settings long test sequences are present, error accumulation needs to be addressed for methods relying on self-training. In this work, we propose and show that in the setting of TTA, the symmetric cross-entropy is better suited as a consistency loss for mean teachers compared to the commonly used cross-entropy. This is justified by our analysis with respect to the (symmetric) cross-entropy's gradient properties. To pull the test feature space closer to the source domain, where the pre-trained model is well posed, contrastive learning is leveraged. Since applications differ in their requirements, we address several settings, including having source data available and the more challenging source-free setting. We demonstrate the effectiveness of our proposed method 'robust mean teacher' (RMT) on the continual and gradual corruption benchmarks CIFAR10C, CIFAR100C, and Imagenet-C. We further consider ImageNet-R and propose a new continual DomainNet-126 benchmark. State-of-the-art results are achieved on all benchmarks.

EnsLoss: Stochastic Calibrated Loss Ensembles for Preventing Overfitting in Classification

Empirical risk minimization (ERM) with a computationally feasible surrogate loss is a widely accepted approach for classification. Notably, the convexity and calibration (CC) properties of a loss function ensure consistency of ERM in maximizing accuracy, thereby offering a wide range of options for surrogate losses. In this article, we propose a novel ensemble method, namely EnsLoss, which extends the ensemble learning concept to combine loss functions within the ERM framework. A key feature of our method is the consideration on preserving the "legitimacy" of the combined losses, i.e., ensuring the CC properties. Specifically, we first transform the CC conditions of losses into loss-derivatives, thereby bypassing the need for explicit loss functions and directly generating calibrated loss-derivatives. Therefore, inspired by Dropout, EnsLoss enables loss ensembles through one training process with doubly stochastic gradient descent (i.e., random batch samples and random calibrated loss-derivatives). We theoretically establish the statistical consistency of our approach and provide insights into its benefits. The numerical effectiveness of EnsLoss compared to fixed loss methods is demonstrated through experiments on a broad range of 14 OpenML tabular datasets and 46 image datasets with various deep learning architectures. Python repository and source code are available on GitHub at https://github.com/statmlben/ensloss.

Don't Take It Literally: An Edit-Invariant Sequence Loss for Text Generation

Neural text generation models are typically trained by maximizing log-likelihood with the sequence cross entropy (CE) loss, which encourages an exact token-by-token match between a target sequence with a generated sequence. Such training objective is sub-optimal when the target sequence is not perfect, e.g., when the target sequence is corrupted with noises, or when only weak sequence supervision is available. To address the challenge, we propose a novel Edit-Invariant Sequence Loss (EISL), which computes the matching loss of a target n-gram with all n-grams in the generated sequence. EISL is designed to be robust to various noises and edits in the target sequences. Moreover, the EISL computation is essentially an approximate convolution operation with target n-grams as kernels, which is easy to implement and efficient to compute with existing libraries. To demonstrate the effectiveness of EISL, we conduct experiments on a wide range of tasks, including machine translation with noisy target sequences, unsupervised text style transfer with only weak training signals, and non-autoregressive generation with non-predefined generation order. Experimental results show our method significantly outperforms the common CE loss and other strong baselines on all the tasks. EISL has a simple API that can be used as a drop-in replacement of the CE loss: https://github.com/guangyliu/EISL.

Cauchy-Schwarz Divergence Information Bottleneck for Regression

The information bottleneck (IB) approach is popular to improve the generalization, robustness and explainability of deep neural networks. Essentially, it aims to find a minimum sufficient representation t by striking a trade-off between a compression term I(x;t) and a prediction term I(y;t), where I(cdot;cdot) refers to the mutual information (MI). MI is for the IB for the most part expressed in terms of the Kullback-Leibler (KL) divergence, which in the regression case corresponds to prediction based on mean squared error (MSE) loss with Gaussian assumption and compression approximated by variational inference. In this paper, we study the IB principle for the regression problem and develop a new way to parameterize the IB with deep neural networks by exploiting favorable properties of the Cauchy-Schwarz (CS) divergence. By doing so, we move away from MSE-based regression and ease estimation by avoiding variational approximations or distributional assumptions. We investigate the improved generalization ability of our proposed CS-IB and demonstrate strong adversarial robustness guarantees. We demonstrate its superior performance on six real-world regression tasks over other popular deep IB approaches. We additionally observe that the solutions discovered by CS-IB always achieve the best trade-off between prediction accuracy and compression ratio in the information plane. The code is available at https://github.com/SJYuCNEL/Cauchy-Schwarz-Information-Bottleneck.

The Nature of Mathematical Modeling and Probabilistic Optimization Engineering in Generative AI

In this paper, we give an in-depth analysis on the mathematical problem formulations and the probabilistic optimization explorations for some of the key components in Transformer model [33] in the field of generative AI. We explore and discuss some potential further enhancement for current state of the art methods for some key underlying technologies of generative AI models from algorithmic and probabilistic optimization perspective. In particular, we present an optimal solution for sub-word encoding (SWE) based on similar initial settings as that of byte-pair encoding (BPE) algorithm in [9] with similar objectives as that of WordPiece approach in [28, 31] to maximize the likelihood of the training data. We also present cross entropy optimization method to optimize hyperparameters for word2vec model [17]. In addition, we propose a factored combination of rotary positional encoding (RoPE) [32] and attention with linear biases (ALiBi) [23] with a harmonic series. We also present a probabilistic FlashAttention [6, 7] (PrFlashAttention) method with a probability distribution over block distances in the matrix to decide which block is likely to participate in a given round of attention computation while maintaining the lower triangle shape of the tensor for autoregressive language models by re-shaping the tensors. Finally, we present staircase adaptive quantization (SAQ) of key-value (KV) cache for multi-query attention (MQA) based on the framework presented in [16] to have gradual quantization degradation while achieving reasonable model quality and cost savings.

SuSana Distancia is all you need: Enforcing class separability in metric learning via two novel distance-based loss functions for few-shot image classification

Few-shot learning is a challenging area of research that aims to learn new concepts with only a few labeled samples of data. Recent works based on metric-learning approaches leverage the meta-learning approach, which is encompassed by episodic tasks that make use a support (training) and query set (test) with the objective of learning a similarity comparison metric between those sets. Due to the lack of data, the learning process of the embedding network becomes an important part of the few-shot task. Previous works have addressed this problem using metric learning approaches, but the properties of the underlying latent space and the separability of the difference classes on it was not entirely enforced. In this work, we propose two different loss functions which consider the importance of the embedding vectors by looking at the intra-class and inter-class distance between the few data. The first loss function is the Proto-Triplet Loss, which is based on the original triplet loss with the modifications needed to better work on few-shot scenarios. The second loss function, which we dub ICNN loss is based on an inter and intra class nearest neighbors score, which help us to assess the quality of embeddings obtained from the trained network. Our results, obtained from a extensive experimental setup show a significant improvement in accuracy in the miniImagenNet benchmark compared to other metric-based few-shot learning methods by a margin of 2%, demonstrating the capability of these loss functions to allow the network to generalize better to previously unseen classes. In our experiments, we demonstrate competitive generalization capabilities to other domains, such as the Caltech CUB, Dogs and Cars datasets compared with the state of the art.

Pretraining Data Detection for Large Language Models: A Divergence-based Calibration Method

As the scale of training corpora for large language models (LLMs) grows, model developers become increasingly reluctant to disclose details on their data. This lack of transparency poses challenges to scientific evaluation and ethical deployment. Recently, pretraining data detection approaches, which infer whether a given text was part of an LLM's training data through black-box access, have been explored. The Min-K\% Prob method, which has achieved state-of-the-art results, assumes that a non-training example tends to contain a few outlier words with low token probabilities. However, the effectiveness may be limited as it tends to misclassify non-training texts that contain many common words with high probabilities predicted by LLMs. To address this issue, we introduce a divergence-based calibration method, inspired by the divergence-from-randomness concept, to calibrate token probabilities for pretraining data detection. We compute the cross-entropy (i.e., the divergence) between the token probability distribution and the token frequency distribution to derive a detection score. We have developed a Chinese-language benchmark, PatentMIA, to assess the performance of detection approaches for LLMs on Chinese text. Experimental results on English-language benchmarks and PatentMIA demonstrate that our proposed method significantly outperforms existing methods. Our code and PatentMIA benchmark are available at https://github.com/zhang-wei-chao/DC-PDD.

Dual-Encoders for Extreme Multi-Label Classification

Dual-encoder (DE) models are widely used in retrieval tasks, most commonly studied on open QA benchmarks that are often characterized by multi-class and limited training data. In contrast, their performance in multi-label and data-rich retrieval settings like extreme multi-label classification (XMC), remains under-explored. Current empirical evidence indicates that DE models fall significantly short on XMC benchmarks, where SOTA methods linearly scale the number of learnable parameters with the total number of classes (documents in the corpus) by employing per-class classification head. To this end, we first study and highlight that existing multi-label contrastive training losses are not appropriate for training DE models on XMC tasks. We propose decoupled softmax loss - a simple modification to the InfoNCE loss - that overcomes the limitations of existing contrastive losses. We further extend our loss design to a soft top-k operator-based loss which is tailored to optimize top-k prediction performance. When trained with our proposed loss functions, standard DE models alone can match or outperform SOTA methods by up to 2% at Precision@1 even on the largest XMC datasets while being 20x smaller in terms of the number of trainable parameters. This leads to more parameter-efficient and universally applicable solutions for retrieval tasks. Our code and models are publicly available at https://github.com/nilesh2797/dexml.

Learning Physical Models that Can Respect Conservation Laws

Recent work in scientific machine learning (SciML) has focused on incorporating partial differential equation (PDE) information into the learning process. Much of this work has focused on relatively ``easy'' PDE operators (e.g., elliptic and parabolic), with less emphasis on relatively ``hard'' PDE operators (e.g., hyperbolic). Within numerical PDEs, the latter problem class requires control of a type of volume element or conservation constraint, which is known to be challenging. Delivering on the promise of SciML requires seamlessly incorporating both types of problems into the learning process. To address this issue, we propose ProbConserv, a framework for incorporating conservation constraints into a generic SciML architecture. To do so, ProbConserv combines the integral form of a conservation law with a Bayesian update. We provide a detailed analysis of ProbConserv on learning with the Generalized Porous Medium Equation (GPME), a widely-applicable parameterized family of PDEs that illustrates the qualitative properties of both easier and harder PDEs. ProbConserv is effective for easy GPME variants, performing well with state-of-the-art competitors; and for harder GPME variants it outperforms other approaches that do not guarantee volume conservation. ProbConserv seamlessly enforces physical conservation constraints, maintains probabilistic uncertainty quantification (UQ), and deals well with shocks and heteroscedasticities. In each case, it achieves superior predictive performance on downstream tasks.

Learning from Aggregate responses: Instance Level versus Bag Level Loss Functions

Due to the rise of privacy concerns, in many practical applications the training data is aggregated before being shared with the learner, in order to protect privacy of users' sensitive responses. In an aggregate learning framework, the dataset is grouped into bags of samples, where each bag is available only with an aggregate response, providing a summary of individuals' responses in that bag. In this paper, we study two natural loss functions for learning from aggregate responses: bag-level loss and the instance-level loss. In the former, the model is learnt by minimizing a loss between aggregate responses and aggregate model predictions, while in the latter the model aims to fit individual predictions to the aggregate responses. In this work, we show that the instance-level loss can be perceived as a regularized form of the bag-level loss. This observation lets us compare the two approaches with respect to bias and variance of the resulting estimators, and introduce a novel interpolating estimator which combines the two approaches. For linear regression tasks, we provide a precise characterization of the risk of the interpolating estimator in an asymptotic regime where the size of the training set grows in proportion to the features dimension. Our analysis allows us to theoretically understand the effect of different factors, such as bag size on the model prediction risk. In addition, we propose a mechanism for differentially private learning from aggregate responses and derive the optimal bag size in terms of prediction risk-privacy trade-off. We also carry out thorough experiments to corroborate our theory and show the efficacy of the interpolating estimator.

It Takes Two to Tango: Mixup for Deep Metric Learning

Metric learning involves learning a discriminative representation such that embeddings of similar classes are encouraged to be close, while embeddings of dissimilar classes are pushed far apart. State-of-the-art methods focus mostly on sophisticated loss functions or mining strategies. On the one hand, metric learning losses consider two or more examples at a time. On the other hand, modern data augmentation methods for classification consider two or more examples at a time. The combination of the two ideas is under-studied. In this work, we aim to bridge this gap and improve representations using mixup, which is a powerful data augmentation approach interpolating two or more examples and corresponding target labels at a time. This task is challenging because unlike classification, the loss functions used in metric learning are not additive over examples, so the idea of interpolating target labels is not straightforward. To the best of our knowledge, we are the first to investigate mixing both examples and target labels for deep metric learning. We develop a generalized formulation that encompasses existing metric learning loss functions and modify it to accommodate for mixup, introducing Metric Mix, or Metrix. We also introduce a new metric - utilization, to demonstrate that by mixing examples during training, we are exploring areas of the embedding space beyond the training classes, thereby improving representations. To validate the effect of improved representations, we show that mixing inputs, intermediate representations or embeddings along with target labels significantly outperforms state-of-the-art metric learning methods on four benchmark deep metric learning datasets.

Minimum Entropy Coupling with Bottleneck

This paper investigates a novel lossy compression framework operating under logarithmic loss, designed to handle situations where the reconstruction distribution diverges from the source distribution. This framework is especially relevant for applications that require joint compression and retrieval, and in scenarios involving distributional shifts due to processing. We show that the proposed formulation extends the classical minimum entropy coupling framework by integrating a bottleneck, allowing for a controlled degree of stochasticity in the coupling. We explore the decomposition of the Minimum Entropy Coupling with Bottleneck (MEC-B) into two distinct optimization problems: Entropy-Bounded Information Maximization (EBIM) for the encoder, and Minimum Entropy Coupling (MEC) for the decoder. Through extensive analysis, we provide a greedy algorithm for EBIM with guaranteed performance, and characterize the optimal solution near functional mappings, yielding significant theoretical insights into the structural complexity of this problem. Furthermore, we illustrate the practical application of MEC-B through experiments in Markov Coding Games (MCGs) under rate limits. These games simulate a communication scenario within a Markov Decision Process, where an agent must transmit a compressed message from a sender to a receiver through its actions. Our experiments highlight the trade-offs between MDP rewards and receiver accuracy across various compression rates, showcasing the efficacy of our method compared to conventional compression baseline.

EQ-Net: Elastic Quantization Neural Networks

Current model quantization methods have shown their promising capability in reducing storage space and computation complexity. However, due to the diversity of quantization forms supported by different hardware, one limitation of existing solutions is that usually require repeated optimization for different scenarios. How to construct a model with flexible quantization forms has been less studied. In this paper, we explore a one-shot network quantization regime, named Elastic Quantization Neural Networks (EQ-Net), which aims to train a robust weight-sharing quantization supernet. First of all, we propose an elastic quantization space (including elastic bit-width, granularity, and symmetry) to adapt to various mainstream quantitative forms. Secondly, we propose the Weight Distribution Regularization Loss (WDR-Loss) and Group Progressive Guidance Loss (GPG-Loss) to bridge the inconsistency of the distribution for weights and output logits in the elastic quantization space gap. Lastly, we incorporate genetic algorithms and the proposed Conditional Quantization-Aware Accuracy Predictor (CQAP) as an estimator to quickly search mixed-precision quantized neural networks in supernet. Extensive experiments demonstrate that our EQ-Net is close to or even better than its static counterparts as well as state-of-the-art robust bit-width methods. Code can be available at https://github.com/xuke225/EQ-Net.git{https://github.com/xuke225/EQ-Net}.

Source-Free Domain Adaptation for Image Segmentation

Domain adaptation (DA) has drawn high interest for its capacity to adapt a model trained on labeled source data to perform well on unlabeled or weakly labeled target data from a different domain. Most common DA techniques require concurrent access to the input images of both the source and target domains. However, in practice, privacy concerns often impede the availability of source images in the adaptation phase. This is a very frequent DA scenario in medical imaging, where, for instance, the source and target images could come from different clinical sites. We introduce a source-free domain adaptation for image segmentation. Our formulation is based on minimizing a label-free entropy loss defined over target-domain data, which we further guide with a domain-invariant prior on the segmentation regions. Many priors can be derived from anatomical information. Here, a class ratio prior is estimated from anatomical knowledge and integrated in the form of a Kullback Leibler (KL) divergence in our overall loss function. Furthermore, we motivate our overall loss with an interesting link to maximizing the mutual information between the target images and their label predictions. We show the effectiveness of our prior aware entropy minimization in a variety of domain-adaptation scenarios, with different modalities and applications, including spine, prostate, and cardiac segmentation. Our method yields comparable results to several state of the art adaptation techniques, despite having access to much less information, as the source images are entirely absent in our adaptation phase. Our straightforward adaptation strategy uses only one network, contrary to popular adversarial techniques, which are not applicable to a source-free DA setting. Our framework can be readily used in a breadth of segmentation problems, and our code is publicly available: https://github.com/mathilde-b/SFDA

The Z-loss: a shift and scale invariant classification loss belonging to the Spherical Family

Despite being the standard loss function to train multi-class neural networks, the log-softmax has two potential limitations. First, it involves computations that scale linearly with the number of output classes, which can restrict the size of problems we are able to tackle with current hardware. Second, it remains unclear how close it matches the task loss such as the top-k error rate or other non-differentiable evaluation metrics which we aim to optimize ultimately. In this paper, we introduce an alternative classification loss function, the Z-loss, which is designed to address these two issues. Unlike the log-softmax, it has the desirable property of belonging to the spherical loss family (Vincent et al., 2015), a class of loss functions for which training can be performed very efficiently with a complexity independent of the number of output classes. We show experimentally that it significantly outperforms the other spherical loss functions previously investigated. Furthermore, we show on a word language modeling task that it also outperforms the log-softmax with respect to certain ranking scores, such as top-k scores, suggesting that the Z-loss has the flexibility to better match the task loss. These qualities thus makes the Z-loss an appealing candidate to train very efficiently large output networks such as word-language models or other extreme classification problems. On the One Billion Word (Chelba et al., 2014) dataset, we are able to train a model with the Z-loss 40 times faster than the log-softmax and more than 4 times faster than the hierarchical softmax.

Hyp-OC: Hyperbolic One Class Classification for Face Anti-Spoofing

Face recognition technology has become an integral part of modern security systems and user authentication processes. However, these systems are vulnerable to spoofing attacks and can easily be circumvented. Most prior research in face anti-spoofing (FAS) approaches it as a two-class classification task where models are trained on real samples and known spoof attacks and tested for detection performance on unknown spoof attacks. However, in practice, FAS should be treated as a one-class classification task where, while training, one cannot assume any knowledge regarding the spoof samples a priori. In this paper, we reformulate the face anti-spoofing task from a one-class perspective and propose a novel hyperbolic one-class classification framework. To train our network, we use a pseudo-negative class sampled from the Gaussian distribution with a weighted running mean and propose two novel loss functions: (1) Hyp-PC: Hyperbolic Pairwise Confusion loss, and (2) Hyp-CE: Hyperbolic Cross Entropy loss, which operate in the hyperbolic space. Additionally, we employ Euclidean feature clipping and gradient clipping to stabilize the training in the hyperbolic space. To the best of our knowledge, this is the first work extending hyperbolic embeddings for face anti-spoofing in a one-class manner. With extensive experiments on five benchmark datasets: Rose-Youtu, MSU-MFSD, CASIA-MFSD, Idiap Replay-Attack, and OULU-NPU, we demonstrate that our method significantly outperforms the state-of-the-art, achieving better spoof detection performance.

Unraveling the Mystery of Scaling Laws: Part I

Scaling law principles indicate a power-law correlation between loss and variables such as model size, dataset size, and computational resources utilized during training. These principles play a vital role in optimizing various aspects of model pre-training, ultimately contributing to the success of large language models such as GPT-4, Llama and Gemini. However, the original scaling law paper by OpenAI did not disclose the complete details necessary to derive the precise scaling law formulas, and their conclusions are only based on models containing up to 1.5 billion parameters. Though some subsequent works attempt to unveil these details and scale to larger models, they often neglect the training dependency of important factors such as the learning rate, context length and batch size, leading to their failure to establish a reliable formula for predicting the test loss trajectory. In this technical report, we confirm that the scaling law formulations proposed in the original OpenAI paper remain valid when scaling the model size up to 33 billion, but the constant coefficients in these formulas vary significantly with the experiment setup. We meticulously identify influential factors and provide transparent, step-by-step instructions to estimate all constant terms in scaling-law formulas by training on models with only 1M~60M parameters. Using these estimated formulas, we showcase the capability to accurately predict various attributes for models with up to 33B parameters before their training, including (1) the minimum possible test loss; (2) the minimum required training steps and processed tokens to achieve a specific loss; (3) the critical batch size with an optimal time/computation trade-off at any loss value; and (4) the complete test loss trajectory with arbitrary batch size.

Efficient Nearest Neighbor Search for Cross-Encoder Models using Matrix Factorization

Efficient k-nearest neighbor search is a fundamental task, foundational for many problems in NLP. When the similarity is measured by dot-product between dual-encoder vectors or ell_2-distance, there already exist many scalable and efficient search methods. But not so when similarity is measured by more accurate and expensive black-box neural similarity models, such as cross-encoders, which jointly encode the query and candidate neighbor. The cross-encoders' high computational cost typically limits their use to reranking candidates retrieved by a cheaper model, such as dual encoder or TF-IDF. However, the accuracy of such a two-stage approach is upper-bounded by the recall of the initial candidate set, and potentially requires additional training to align the auxiliary retrieval model with the cross-encoder model. In this paper, we present an approach that avoids the use of a dual-encoder for retrieval, relying solely on the cross-encoder. Retrieval is made efficient with CUR decomposition, a matrix decomposition approach that approximates all pairwise cross-encoder distances from a small subset of rows and columns of the distance matrix. Indexing items using our approach is computationally cheaper than training an auxiliary dual-encoder model through distillation. Empirically, for k > 10, our approach provides test-time recall-vs-computational cost trade-offs superior to the current widely-used methods that re-rank items retrieved using a dual-encoder or TF-IDF.

Sparse Concept Bottleneck Models: Gumbel Tricks in Contrastive Learning

We propose a novel architecture and method of explainable classification with Concept Bottleneck Models (CBMs). While SOTA approaches to Image Classification task work as a black box, there is a growing demand for models that would provide interpreted results. Such a models often learn to predict the distribution over class labels using additional description of this target instances, called concepts. However, existing Bottleneck methods have a number of limitations: their accuracy is lower than that of a standard model and CBMs require an additional set of concepts to leverage. We provide a framework for creating Concept Bottleneck Model from pre-trained multi-modal encoder and new CLIP-like architectures. By introducing a new type of layers known as Concept Bottleneck Layers, we outline three methods for training them: with ell_1-loss, contrastive loss and loss function based on Gumbel-Softmax distribution (Sparse-CBM), while final FC layer is still trained with Cross-Entropy. We show a significant increase in accuracy using sparse hidden layers in CLIP-based bottleneck models. Which means that sparse representation of concepts activation vector is meaningful in Concept Bottleneck Models. Moreover, with our Concept Matrix Search algorithm we can improve CLIP predictions on complex datasets without any additional training or fine-tuning. The code is available at: https://github.com/Andron00e/SparseCBM.

Constructing interval variables via faceted Rasch measurement and multitask deep learning: a hate speech application

We propose a general method for measuring complex variables on a continuous, interval spectrum by combining supervised deep learning with the Constructing Measures approach to faceted Rasch item response theory (IRT). We decompose the target construct, hate speech in our case, into multiple constituent components that are labeled as ordinal survey items. Those survey responses are transformed via IRT into a debiased, continuous outcome measure. Our method estimates the survey interpretation bias of the human labelers and eliminates that influence on the generated continuous measure. We further estimate the response quality of each labeler using faceted IRT, allowing responses from low-quality labelers to be removed. Our faceted Rasch scaling procedure integrates naturally with a multitask deep learning architecture for automated prediction on new data. The ratings on the theorized components of the target outcome are used as supervised, ordinal variables for the neural networks' internal concept learning. We test the use of an activation function (ordinal softmax) and loss function (ordinal cross-entropy) designed to exploit the structure of ordinal outcome variables. Our multitask architecture leads to a new form of model interpretation because each continuous prediction can be directly explained by the constituent components in the penultimate layer. We demonstrate this new method on a dataset of 50,000 social media comments sourced from YouTube, Twitter, and Reddit and labeled by 11,000 U.S.-based Amazon Mechanical Turk workers to measure a continuous spectrum from hate speech to counterspeech. We evaluate Universal Sentence Encoders, BERT, and RoBERTa as language representation models for the comment text, and compare our predictive accuracy to Google Jigsaw's Perspective API models, showing significant improvement over this standard benchmark.

Scaling Laws for Downstream Task Performance of Large Language Models

Scaling laws provide important insights that can guide the design of large language models (LLMs). Existing work has primarily focused on studying scaling laws for pretraining (upstream) loss. However, in transfer learning settings, in which LLMs are pretrained on an unsupervised dataset and then finetuned on a downstream task, we often also care about the downstream performance. In this work, we study the scaling behavior in a transfer learning setting, where LLMs are finetuned for machine translation tasks. Specifically, we investigate how the choice of the pretraining data and its size affect downstream performance (translation quality) as judged by two metrics: downstream cross-entropy and BLEU score. Our experiments indicate that the size of the finetuning dataset and the distribution alignment between the pretraining and downstream data significantly influence the scaling behavior. With sufficient alignment, both downstream cross-entropy and BLEU score improve monotonically with more pretraining data. In such cases, we show that it is possible to predict the downstream BLEU score with good accuracy using a log-law. However, there are also cases where moderate misalignment causes the BLEU score to fluctuate or get worse with more pretraining, whereas downstream cross-entropy monotonically improves. By analyzing these observations, we provide new practical insights for choosing appropriate pretraining data.

Dual-Head Knowledge Distillation: Enhancing Logits Utilization with an Auxiliary Head

Traditional knowledge distillation focuses on aligning the student's predicted probabilities with both ground-truth labels and the teacher's predicted probabilities. However, the transition to predicted probabilities from logits would obscure certain indispensable information. To address this issue, it is intuitive to additionally introduce a logit-level loss function as a supplement to the widely used probability-level loss function, for exploiting the latent information of logits. Unfortunately, we empirically find that the amalgamation of the newly introduced logit-level loss and the previous probability-level loss will lead to performance degeneration, even trailing behind the performance of employing either loss in isolation. We attribute this phenomenon to the collapse of the classification head, which is verified by our theoretical analysis based on the neural collapse theory. Specifically, the gradients of the two loss functions exhibit contradictions in the linear classifier yet display no such conflict within the backbone. Drawing from the theoretical analysis, we propose a novel method called dual-head knowledge distillation, which partitions the linear classifier into two classification heads responsible for different losses, thereby preserving the beneficial effects of both losses on the backbone while eliminating adverse influences on the classification head. Extensive experiments validate that our method can effectively exploit the information inside the logits and achieve superior performance against state-of-the-art counterparts.

Fire Together Wire Together: A Dynamic Pruning Approach with Self-Supervised Mask Prediction

Dynamic model pruning is a recent direction that allows for the inference of a different sub-network for each input sample during deployment. However, current dynamic methods rely on learning a continuous channel gating through regularization by inducing sparsity loss. This formulation introduces complexity in balancing different losses (e.g task loss, regularization loss). In addition, regularization based methods lack transparent tradeoff hyperparameter selection to realize a computational budget. Our contribution is two-fold: 1) decoupled task and pruning losses. 2) Simple hyperparameter selection that enables FLOPs reduction estimation before training. Inspired by the Hebbian theory in Neuroscience: "neurons that fire together wire together", we propose to predict a mask to process k filters in a layer based on the activation of its previous layer. We pose the problem as a self-supervised binary classification problem. Each mask predictor module is trained to predict if the log-likelihood for each filter in the current layer belongs to the top-k activated filters. The value k is dynamically estimated for each input based on a novel criterion using the mass of heatmaps. We show experiments on several neural architectures, such as VGG, ResNet and MobileNet on CIFAR and ImageNet datasets. On CIFAR, we reach similar accuracy to SOTA methods with 15% and 24% higher FLOPs reduction. Similarly in ImageNet, we achieve lower drop in accuracy with up to 13% improvement in FLOPs reduction.

AEM: Attention Entropy Maximization for Multiple Instance Learning based Whole Slide Image Classification

Multiple Instance Learning (MIL) has demonstrated effectiveness in analyzing whole slide images (WSIs), yet it often encounters overfitting challenges in real-world applications, particularly in the form of attention over-concentration. While existing methods to alleviate this issue introduce complex modules or processing steps, such as multiple-stage training and teacher-student distillation, this paper proposes a simple yet effective regularization: Attention Entropy Maximization (AEM). Motivated by our investigation revealing a positive correlation between attention entropy and model performance, AEM incorporates a negative entropy loss for attention values into the standard MIL framework, penalizing overly concentrated attention and encouraging the model to consider a broader range of informative regions in WSIs, potentially improving its generalization capabilities. Compared to existing overfitting mitigation methods, our AEM approach offers advantages of simplicity, efficiency, and versatility. It requires no additional modules or processing steps, involves only one hyperparameter, and demonstrates compatibility with MIL frameworks and techniques. These advantages make AEM particularly attractive for practical applications. We evaluate AEM on three benchmark datasets, demonstrating consistent performance improvements over existing methods. Furthermore, AEM shows high versatility, integrating effectively with four feature extractors, two advanced MIL frameworks, three attention mechanisms, and Subsampling augmentation technique. The source code is available at https://github.com/dazhangyu123/AEM.

Analysis on Riemann Hypothesis with Cross Entropy Optimization and Reasoning

In this paper, we present a novel framework for the analysis of Riemann Hypothesis [27], which is composed of three key components: a) probabilistic modeling with cross entropy optimization and reasoning; b) the application of the law of large numbers; c) the application of mathematical inductions. The analysis is mainly conducted by virtue of probabilistic modeling of cross entropy optimization and reasoning with rare event simulation techniques. The application of the law of large numbers [2, 3, 6] and the application of mathematical inductions make the analysis of Riemann Hypothesis self-contained and complete to make sure that the whole complex plane is covered as conjectured in Riemann Hypothesis. We also discuss the method of enhanced top-p sampling with large language models (LLMs) for reasoning, where next token prediction is not just based on the estimated probabilities of each possible token in the current round but also based on accumulated path probabilities among multiple top-k chain of thoughts (CoTs) paths. The probabilistic modeling of cross entropy optimization and reasoning may suit well with the analysis of Riemann Hypothesis as Riemann Zeta functions are inherently dealing with the sums of infinite components of a complex number series. We hope that our analysis in this paper could shed some light on some of the insights of Riemann Hypothesis. The framework and techniques presented in this paper, coupled with recent developments with chain of thought (CoT) or diagram of thought (DoT) reasoning in large language models (LLMs) with reinforcement learning (RL) [1, 7, 18, 21, 24, 34, 39-41], could pave the way for eventual proof of Riemann Hypothesis [27].

Why do Learning Rates Transfer? Reconciling Optimization and Scaling Limits for Deep Learning

Recently, there has been growing evidence that if the width and depth of a neural network are scaled toward the so-called rich feature learning limit (muP and its depth extension), then some hyperparameters - such as the learning rate - exhibit transfer from small to very large models, thus reducing the cost of hyperparameter tuning. From an optimization perspective, this phenomenon is puzzling, as it implies that the loss landscape is remarkably consistent across very different model sizes. In this work, we find empirical evidence that learning rate transfer can be attributed to the fact that under muP and its depth extension, the largest eigenvalue of the training loss Hessian (i.e. the sharpness) is largely independent of the width and depth of the network for a sustained period of training time. On the other hand, we show that under the neural tangent kernel (NTK) regime, the sharpness exhibits very different dynamics at different scales, thus preventing learning rate transfer. But what causes these differences in the sharpness dynamics? Through a connection between the spectra of the Hessian and the NTK matrix, we argue that the cause lies in the presence (for muP) or progressive absence (for the NTK regime) of feature learning, which results in a different evolution of the NTK, and thus of the sharpness. We corroborate our claims with a substantial suite of experiments, covering a wide range of datasets and architectures: from ResNets and Vision Transformers trained on benchmark vision datasets to Transformers-based language models trained on WikiText

Small-scale proxies for large-scale Transformer training instabilities

Teams that have trained large Transformer-based models have reported training instabilities at large scale that did not appear when training with the same hyperparameters at smaller scales. Although the causes of such instabilities are of scientific interest, the amount of resources required to reproduce them has made investigation difficult. In this work, we seek ways to reproduce and study training stability and instability at smaller scales. First, we focus on two sources of training instability described in previous work: the growth of logits in attention layers (Dehghani et al., 2023) and divergence of the output logits from the log probabilities (Chowdhery et al., 2022). By measuring the relationship between learning rate and loss across scales, we show that these instabilities also appear in small models when training at high learning rates, and that mitigations previously employed at large scales are equally effective in this regime. This prompts us to investigate the extent to which other known optimizer and model interventions influence the sensitivity of the final loss to changes in the learning rate. To this end, we study methods such as warm-up, weight decay, and the muParam (Yang et al., 2022), and combine techniques to train small models that achieve similar losses across orders of magnitude of learning rate variation. Finally, to conclude our exploration we study two cases where instabilities can be predicted before they emerge by examining the scaling behavior of model activation and gradient norms.

A Neural PDE Solver with Temporal Stencil Modeling

Numerical simulation of non-linear partial differential equations plays a crucial role in modeling physical science and engineering phenomena, such as weather, climate, and aerodynamics. Recent Machine Learning (ML) models trained on low-resolution spatio-temporal signals have shown new promises in capturing important dynamics in high-resolution signals, under the condition that the models can effectively recover the missing details. However, this study shows that significant information is often lost in the low-resolution down-sampled features. To address such issues, we propose a new approach, namely Temporal Stencil Modeling (TSM), which combines the strengths of advanced time-series sequence modeling (with the HiPPO features) and state-of-the-art neural PDE solvers (with learnable stencil modeling). TSM aims to recover the lost information from the PDE trajectories and can be regarded as a temporal generalization of classic finite volume methods such as WENO. Our experimental results show that TSM achieves the new state-of-the-art simulation accuracy for 2-D incompressible Navier-Stokes turbulent flows: it significantly outperforms the previously reported best results by 19.9% in terms of the highly-correlated duration time and reduces the inference latency into 80%. We also show a strong generalization ability of the proposed method to various out-of-distribution turbulent flow settings. Our code is available at "https://github.com/Edward-Sun/TSM-PDE".

Information Bottleneck Analysis of Deep Neural Networks via Lossy Compression

The Information Bottleneck (IB) principle offers an information-theoretic framework for analyzing the training process of deep neural networks (DNNs). Its essence lies in tracking the dynamics of two mutual information (MI) values: one between the hidden layer and the class label, and the other between the hidden layer and the DNN input. According to the hypothesis put forth by Shwartz-Ziv and Tishby (2017), the training process consists of two distinct phases: fitting and compression. The latter phase is believed to account for the good generalization performance exhibited by DNNs. Due to the challenging nature of estimating MI between high-dimensional random vectors, this hypothesis has only been verified for toy NNs or specific types of NNs, such as quantized NNs and dropout NNs. In this paper, we introduce a comprehensive framework for conducting IB analysis of general NNs. Our approach leverages the stochastic NN method proposed by Goldfeld et al. (2019) and incorporates a compression step to overcome the obstacles associated with high dimensionality. In other words, we estimate the MI between the compressed representations of high-dimensional random vectors. The proposed method is supported by both theoretical and practical justifications. Notably, we demonstrate the accuracy of our estimator through synthetic experiments featuring predefined MI values. Finally, we perform IB analysis on a close-to-real-scale convolutional DNN, which reveals new features of the MI dynamics.

Language models scale reliably with over-training and on downstream tasks

Scaling laws are useful guides for developing language models, but there are still gaps between current scaling studies and how language models are ultimately trained and evaluated. For instance, scaling is usually studied in the compute-optimal training regime (i.e., "Chinchilla optimal" regime); however, in practice, models are often over-trained to reduce inference costs. Moreover, scaling laws mostly predict loss on next-token prediction, but ultimately models are compared based on downstream task performance. In this paper, we address both shortcomings. To do so, we create a testbed of 104 models with 0.011B to 6.9B parameters trained with various numbers of tokens on three data distributions. First, we investigate scaling in the over-trained regime. We fit scaling laws that extrapolate in both the number of model parameters and the ratio of training tokens to parameters. This enables us to predict the validation loss of a 1.4B parameter, 900B token run (i.e., 32times over-trained) and a 6.9B parameter, 138B token runx2014each from experiments that take 300times less compute. Second, we relate the perplexity of a language model to its downstream task performance via a power law. We use this law to predict top-1 error averaged over downstream tasks for the two aforementioned models using experiments that take 20times less compute. Our experiments are available at https://github.com/mlfoundations/scaling.

Refined Regret for Adversarial MDPs with Linear Function Approximation

We consider learning in an adversarial Markov Decision Process (MDP) where the loss functions can change arbitrarily over K episodes and the state space can be arbitrarily large. We assume that the Q-function of any policy is linear in some known features, that is, a linear function approximation exists. The best existing regret upper bound for this setting (Luo et al., 2021) is of order mathcal O(K^{2/3}) (omitting all other dependencies), given access to a simulator. This paper provides two algorithms that improve the regret to mathcal O(sqrt K) in the same setting. Our first algorithm makes use of a refined analysis of the Follow-the-Regularized-Leader (FTRL) algorithm with the log-barrier regularizer. This analysis allows the loss estimators to be arbitrarily negative and might be of independent interest. Our second algorithm develops a magnitude-reduced loss estimator, further removing the polynomial dependency on the number of actions in the first algorithm and leading to the optimal regret bound (up to logarithmic terms and dependency on the horizon). Moreover, we also extend the first algorithm to simulator-free linear MDPs, which achieves mathcal O(K^{8/9}) regret and greatly improves over the best existing bound mathcal O(K^{14/15}). This algorithm relies on a better alternative to the Matrix Geometric Resampling procedure by Neu & Olkhovskaya (2020), which could again be of independent interest.

A Three-regime Model of Network Pruning

Recent work has highlighted the complex influence training hyperparameters, e.g., the number of training epochs, can have on the prunability of machine learning models. Perhaps surprisingly, a systematic approach to predict precisely how adjusting a specific hyperparameter will affect prunability remains elusive. To address this gap, we introduce a phenomenological model grounded in the statistical mechanics of learning. Our approach uses temperature-like and load-like parameters to model the impact of neural network (NN) training hyperparameters on pruning performance. A key empirical result we identify is a sharp transition phenomenon: depending on the value of a load-like parameter in the pruned model, increasing the value of a temperature-like parameter in the pre-pruned model may either enhance or impair subsequent pruning performance. Based on this transition, we build a three-regime model by taxonomizing the global structure of the pruned NN loss landscape. Our model reveals that the dichotomous effect of high temperature is associated with transitions between distinct types of global structures in the post-pruned model. Based on our results, we present three case-studies: 1) determining whether to increase or decrease a hyperparameter for improved pruning; 2) selecting the best model to prune from a family of models; and 3) tuning the hyperparameter of the Sharpness Aware Minimization method for better pruning performance.

Training-Free Unsupervised Prompt for Vision-Language Models

Prompt learning has become the most effective paradigm for adapting large pre-trained vision-language models (VLMs) to downstream tasks. Recently, unsupervised prompt tuning methods, such as UPL and POUF, directly leverage pseudo-labels as supervisory information to fine-tune additional adaptation modules on unlabeled data. However, inaccurate pseudo labels easily misguide the tuning process and result in poor representation capabilities. In light of this, we propose Training-Free Unsupervised Prompts (TFUP), which maximally preserves the inherent representation capabilities and enhances them with a residual connection to similarity-based prediction probabilities in a training-free and labeling-free manner. Specifically, we integrate both instance confidence and prototype scores to select representative samples, which are used to customize a reliable Feature Cache Model (FCM) for training-free inference. Then, we design a Multi-level Similarity Measure (MSM) that considers both feature-level and semantic-level similarities to calculate the distance between each test image and the cached sample as the weight of the corresponding cached label to generate similarity-based prediction probabilities. In this way, TFUP achieves surprising performance, even surpassing the training-base method on multiple classification datasets. Based on our TFUP, we propose a training-based approach (TFUP-T) to further boost the adaptation performance. In addition to the standard cross-entropy loss, TFUP-T adopts an additional marginal distribution entropy loss to constrain the model from a global perspective. Our TFUP-T achieves new state-of-the-art classification performance compared to unsupervised and few-shot adaptation approaches on multiple benchmarks. In particular, TFUP-T improves the classification accuracy of POUF by 3.3% on the most challenging Domain-Net dataset.

Cross-Domain Keyword Extraction with Keyness Patterns

Domain dependence and annotation subjectivity pose challenges for supervised keyword extraction. Based on the premises that second-order keyness patterns are existent at the community level and learnable from annotated keyword extraction datasets, this paper proposes a supervised ranking approach to keyword extraction that ranks keywords with keyness patterns consisting of independent features (such as sublanguage domain and term length) and three categories of dependent features -- heuristic features, specificity features, and representavity features. The approach uses two convolutional-neural-network based models to learn keyness patterns from keyword datasets and overcomes annotation subjectivity by training the two models with bootstrap sampling strategy. Experiments demonstrate that the approach not only achieves state-of-the-art performance on ten keyword datasets in general supervised keyword extraction with an average top-10-F-measure of 0.316 , but also robust cross-domain performance with an average top-10-F-measure of 0.346 on four datasets that are excluded in the training process. Such cross-domain robustness is attributed to the fact that community-level keyness patterns are limited in number and temperately independent of language domains, the distinction between independent features and dependent features, and the sampling training strategy that balances excess risk and lack of negative training data.

Grokking as the Transition from Lazy to Rich Training Dynamics

We propose that the grokking phenomenon, where the train loss of a neural network decreases much earlier than its test loss, can arise due to a neural network transitioning from lazy training dynamics to a rich, feature learning regime. To illustrate this mechanism, we study the simple setting of vanilla gradient descent on a polynomial regression problem with a two layer neural network which exhibits grokking without regularization in a way that cannot be explained by existing theories. We identify sufficient statistics for the test loss of such a network, and tracking these over training reveals that grokking arises in this setting when the network first attempts to fit a kernel regression solution with its initial features, followed by late-time feature learning where a generalizing solution is identified after train loss is already low. We provide an asymptotic theoretical description of the grokking dynamics in this model using dynamical mean field theory (DMFT) for high dimensional data. We find that the key determinants of grokking are the rate of feature learning -- which can be controlled precisely by parameters that scale the network output -- and the alignment of the initial features with the target function y(x). We argue this delayed generalization arises when (1) the top eigenvectors of the initial neural tangent kernel and the task labels y(x) are misaligned, but (2) the dataset size is large enough so that it is possible for the network to generalize eventually, but not so large that train loss perfectly tracks test loss at all epochs, and (3) the network begins training in the lazy regime so does not learn features immediately. We conclude with evidence that this transition from lazy (linear model) to rich training (feature learning) can control grokking in more general settings, like on MNIST, one-layer Transformers, and student-teacher networks.

Tackling Data Heterogeneity in Federated Learning via Loss Decomposition

Federated Learning (FL) is a rising approach towards collaborative and privacy-preserving machine learning where large-scale medical datasets remain localized to each client. However, the issue of data heterogeneity among clients often compels local models to diverge, leading to suboptimal global models. To mitigate the impact of data heterogeneity on FL performance, we start with analyzing how FL training influence FL performance by decomposing the global loss into three terms: local loss, distribution shift loss and aggregation loss. Remarkably, our loss decomposition reveals that existing local training-based FL methods attempt to reduce the distribution shift loss, while the global aggregation-based FL methods propose better aggregation strategies to reduce the aggregation loss. Nevertheless, a comprehensive joint effort to minimize all three terms is currently limited in the literature, leading to subpar performance when dealing with data heterogeneity challenges. To fill this gap, we propose a novel FL method based on global loss decomposition, called FedLD, to jointly reduce these three loss terms. Our FedLD involves a margin control regularization in local training to reduce the distribution shift loss, and a principal gradient-based server aggregation strategy to reduce the aggregation loss. Notably, under different levels of data heterogeneity, our strategies achieve better and more robust performance on retinal and chest X-ray classification compared to other FL algorithms. Our code is available at https://github.com/Zeng-Shuang/FedLD.

xMEN: A Modular Toolkit for Cross-Lingual Medical Entity Normalization

Objective: To improve performance of medical entity normalization across many languages, especially when fewer language resources are available compared to English. Materials and Methods: We introduce xMEN, a modular system for cross-lingual medical entity normalization, which performs well in both low- and high-resource scenarios. When synonyms in the target language are scarce for a given terminology, we leverage English aliases via cross-lingual candidate generation. For candidate ranking, we incorporate a trainable cross-encoder model if annotations for the target task are available. We also evaluate cross-encoders trained in a weakly supervised manner based on machine-translated datasets from a high resource domain. Our system is publicly available as an extensible Python toolkit. Results: xMEN improves the state-of-the-art performance across a wide range of multilingual benchmark datasets. Weakly supervised cross-encoders are effective when no training data is available for the target task. Through the compatibility of xMEN with the BigBIO framework, it can be easily used with existing and prospective datasets. Discussion: Our experiments show the importance of balancing the output of general-purpose candidate generators with subsequent trainable re-rankers, which we achieve through a rank regularization term in the loss function of the cross-encoder. However, error analysis reveals that multi-word expressions and other complex entities are still challenging. Conclusion: xMEN exhibits strong performance for medical entity normalization in multiple languages, even when no labeled data and few terminology aliases for the target language are available. Its configuration system and evaluation modules enable reproducible benchmarks. Models and code are available online at the following URL: https://github.com/hpi-dhc/xmen

The Fine Line: Navigating Large Language Model Pretraining with Down-streaming Capability Analysis

Uncovering early-stage metrics that reflect final model performance is one core principle for large-scale pretraining. The existing scaling law demonstrates the power-law correlation between pretraining loss and training flops, which serves as an important indicator of the current training state for large language models. However, this principle only focuses on the model's compression properties on the training data, resulting in an inconsistency with the ability improvements on the downstream tasks. Some follow-up works attempted to extend the scaling-law to more complex metrics (such as hyperparameters), but still lacked a comprehensive analysis of the dynamic differences among various capabilities during pretraining. To address the aforementioned limitations, this paper undertakes a comprehensive comparison of model capabilities at various pretraining intermediate checkpoints. Through this analysis, we confirm that specific downstream metrics exhibit similar training dynamics across models of different sizes, up to 67 billion parameters. In addition to our core findings, we've reproduced Amber and OpenLLaMA, releasing their intermediate checkpoints. This initiative offers valuable resources to the research community and facilitates the verification and exploration of LLM pretraining by open-source researchers. Besides, we provide empirical summaries, including performance comparisons of different models and capabilities, and tuition of key metrics for different training phases. Based on these findings, we provide a more user-friendly strategy for evaluating the optimization state, offering guidance for establishing a stable pretraining process.

Respecting causality is all you need for training physics-informed neural networks

While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date PINNs have not been successful in simulating dynamical systems whose solution exhibits multi-scale, chaotic or turbulent behavior. In this work we attribute this shortcoming to the inability of existing PINNs formulations to respect the spatio-temporal causal structure that is inherent to the evolution of physical systems. We argue that this is a fundamental limitation and a key source of error that can ultimately steer PINN models to converge towards erroneous solutions. We address this pathology by proposing a simple re-formulation of PINNs loss functions that can explicitly account for physical causality during model training. We demonstrate that this simple modification alone is enough to introduce significant accuracy improvements, as well as a practical quantitative mechanism for assessing the convergence of a PINNs model. We provide state-of-the-art numerical results across a series of benchmarks for which existing PINNs formulations fail, including the chaotic Lorenz system, the Kuramoto-Sivashinsky equation in the chaotic regime, and the Navier-Stokes equations in the turbulent regime. To the best of our knowledge, this is the first time that PINNs have been successful in simulating such systems, introducing new opportunities for their applicability to problems of industrial complexity.

EMO: Earth Mover Distance Optimization for Auto-Regressive Language Modeling

Neural language models are probabilistic models of human text. They are predominantly trained using maximum likelihood estimation (MLE), which is equivalent to minimizing the forward cross-entropy between the empirical data distribution and the model distribution. However, various degeneration phenomena are still widely observed when decoding from the distributions learned by such models. We establish that the forward cross-entropy is suboptimal as a distance metric for aligning human and model distribution due to its (1) recall-prioritization (2) negative diversity ignorance and (3) train-test mismatch. In this paper, we propose Earth Mover Distance Optimization (EMO) for auto-regressive language modeling. EMO capitalizes on the inherent properties of earth mover distance to address the aforementioned challenges. Due to the high complexity of direct computation, we further introduce a feasible upper bound for EMO to ease end-to-end training. Upon extensive evaluation of language models trained using EMO and MLE. We find that EMO demonstrates a consistently better language modeling performance than MLE across domains. Moreover, EMO demonstrates noteworthy enhancements in downstream performance with minimal fine-tuning on merely 25,000 sentences. This highlights the tremendous potential of EMO as a lightweight calibration method for enhancing large-scale pre-trained language models.

Parallel Learning by Multitasking Neural Networks

A modern challenge of Artificial Intelligence is learning multiple patterns at once (i.e.parallel learning). While this can not be accomplished by standard Hebbian associative neural networks, in this paper we show how the Multitasking Hebbian Network (a variation on theme of the Hopfield model working on sparse data-sets) is naturally able to perform this complex task. We focus on systems processing in parallel a finite (up to logarithmic growth in the size of the network) amount of patterns, mirroring the low-storage level of standard associative neural networks at work with pattern recognition. For mild dilution in the patterns, the network handles them hierarchically, distributing the amplitudes of their signals as power-laws w.r.t. their information content (hierarchical regime), while, for strong dilution, all the signals pertaining to all the patterns are raised with the same strength (parallel regime). Further, confined to the low-storage setting (i.e., far from the spin glass limit), the presence of a teacher neither alters the multitasking performances nor changes the thresholds for learning: the latter are the same whatever the training protocol is supervised or unsupervised. Results obtained through statistical mechanics, signal-to-noise technique and Monte Carlo simulations are overall in perfect agreement and carry interesting insights on multiple learning at once: for instance, whenever the cost-function of the model is minimized in parallel on several patterns (in its description via Statistical Mechanics), the same happens to the standard sum-squared error Loss function (typically used in Machine Learning).

Improving Knowledge Distillation via Regularizing Feature Norm and Direction

Knowledge distillation (KD) exploits a large well-trained model (i.e., teacher) to train a small student model on the same dataset for the same task. Treating teacher features as knowledge, prevailing methods of knowledge distillation train student by aligning its features with the teacher's, e.g., by minimizing the KL-divergence between their logits or L2 distance between their intermediate features. While it is natural to believe that better alignment of student features to the teacher better distills teacher knowledge, simply forcing this alignment does not directly contribute to the student's performance, e.g., classification accuracy. In this work, we propose to align student features with class-mean of teacher features, where class-mean naturally serves as a strong classifier. To this end, we explore baseline techniques such as adopting the cosine distance based loss to encourage the similarity between student features and their corresponding class-means of the teacher. Moreover, we train the student to produce large-norm features, inspired by other lines of work (e.g., model pruning and domain adaptation), which find the large-norm features to be more significant. Finally, we propose a rather simple loss term (dubbed ND loss) to simultaneously (1) encourage student to produce large-norm features, and (2) align the direction of student features and teacher class-means. Experiments on standard benchmarks demonstrate that our explored techniques help existing KD methods achieve better performance, i.e., higher classification accuracy on ImageNet and CIFAR100 datasets, and higher detection precision on COCO dataset. Importantly, our proposed ND loss helps the most, leading to the state-of-the-art performance on these benchmarks. The source code is available at https://github.com/WangYZ1608/Knowledge-Distillation-via-ND.

Transformer as Linear Expansion of Learngene

We propose expanding the shared Transformer module to produce and initialize Transformers of varying depths, enabling adaptation to diverse resource constraints. Drawing an analogy to genetic expansibility, we term such module as learngene. To identify the expansion mechanism, we delve into the relationship between the layer's position and its corresponding weight value, and find that linear function appropriately approximates this relationship. Building on this insight, we present Transformer as Linear Expansion of learnGene (TLEG), a novel approach for flexibly producing and initializing Transformers of diverse depths. Specifically, to learn learngene, we firstly construct an auxiliary Transformer linearly expanded from learngene, after which we train it through employing soft distillation. Subsequently, we can produce and initialize Transformers of varying depths via linearly expanding the well-trained learngene, thereby supporting diverse downstream scenarios. Extensive experiments on ImageNet-1K demonstrate that TLEG achieves comparable or better performance in contrast to many individual models trained from scratch, while reducing around 2x training cost. When transferring to several downstream classification datasets, TLEG surpasses existing initialization methods by a large margin (e.g., +6.87% on iNat 2019 and +7.66% on CIFAR-100). Under the situation where we need to produce models of varying depths adapting for different resource constraints, TLEG achieves comparable results while reducing around 19x parameters stored to initialize these models and around 5x pre-training costs, in contrast to the pre-training and fine-tuning approach. When transferring a fixed set of parameters to initialize different models, TLEG presents better flexibility and competitive performance while reducing around 2.9x parameters stored to initialize, compared to the pre-training approach.

Signal-to-Noise Ratio: A Robust Distance Metric for Deep Metric Learning

Deep metric learning, which learns discriminative features to process image clustering and retrieval tasks, has attracted extensive attention in recent years. A number of deep metric learning methods, which ensure that similar examples are mapped close to each other and dissimilar examples are mapped farther apart, have been proposed to construct effective structures for loss functions and have shown promising results. In this paper, different from the approaches on learning the loss structures, we propose a robust SNR distance metric based on Signal-to-Noise Ratio (SNR) for measuring the similarity of image pairs for deep metric learning. By exploring the properties of our SNR distance metric from the view of geometry space and statistical theory, we analyze the properties of our metric and show that it can preserve the semantic similarity between image pairs, which well justify its suitability for deep metric learning. Compared with Euclidean distance metric, our SNR distance metric can further jointly reduce the intra-class distances and enlarge the inter-class distances for learned features. Leveraging our SNR distance metric, we propose Deep SNR-based Metric Learning (DSML) to generate discriminative feature embeddings. By extensive experiments on three widely adopted benchmarks, including CARS196, CUB200-2011 and CIFAR10, our DSML has shown its superiority over other state-of-the-art methods. Additionally, we extend our SNR distance metric to deep hashing learning, and conduct experiments on two benchmarks, including CIFAR10 and NUS-WIDE, to demonstrate the effectiveness and generality of our SNR distance metric.

Q-VLM: Post-training Quantization for Large Vision-Language Models

In this paper, we propose a post-training quantization framework of large vision-language models (LVLMs) for efficient multi-modal inference. Conventional quantization methods sequentially search the layer-wise rounding functions by minimizing activation discretization errors, which fails to acquire optimal quantization strategy without considering cross-layer dependency. On the contrary, we mine the cross-layer dependency that significantly influences discretization errors of the entire vision-language model, and embed this dependency into optimal quantization strategy searching with low search cost. Specifically, we observe the strong correlation between the activation entropy and the cross-layer dependency concerning output discretization errors. Therefore, we employ the entropy as the proxy to partition blocks optimally, which aims to achieve satisfying trade-offs between discretization errors and the search cost. Moreover, we optimize the visual encoder to disentangle the cross-layer dependency for fine-grained decomposition of search space, so that the search cost is further reduced without harming the quantization accuracy. Experimental results demonstrate that our method compresses the memory by 2.78x and increase generate speed by 1.44x about 13B LLaVA model without performance degradation on diverse multi-modal reasoning tasks. Code is available at https://github.com/ChangyuanWang17/QVLM.

Revisiting Discriminative vs. Generative Classifiers: Theory and Implications

A large-scale deep model pre-trained on massive labeled or unlabeled data transfers well to downstream tasks. Linear evaluation freezes parameters in the pre-trained model and trains a linear classifier separately, which is efficient and attractive for transfer. However, little work has investigated the classifier in linear evaluation except for the default logistic regression. Inspired by the statistical efficiency of naive Bayes, the paper revisits the classical topic on discriminative vs. generative classifiers. Theoretically, the paper considers the surrogate loss instead of the zero-one loss in analyses and generalizes the classical results from binary cases to multiclass ones. We show that, under mild assumptions, multiclass naive Bayes requires O(log n) samples to approach its asymptotic error while the corresponding multiclass logistic regression requires O(n) samples, where n is the feature dimension. To establish it, we present a multiclass H-consistency bound framework and an explicit bound for logistic loss, which are of independent interests. Simulation results on a mixture of Gaussian validate our theoretical findings. Experiments on various pre-trained deep vision models show that naive Bayes consistently converges faster as the number of data increases. Besides, naive Bayes shows promise in few-shot cases and we observe the "two regimes" phenomenon in pre-trained supervised models. Our code is available at https://github.com/ML-GSAI/Revisiting-Dis-vs-Gen-Classifiers.

Adversarial Retriever-Ranker for dense text retrieval

Current dense text retrieval models face two typical challenges. First, they adopt a siamese dual-encoder architecture to encode queries and documents independently for fast indexing and searching, while neglecting the finer-grained term-wise interactions. This results in a sub-optimal recall performance. Second, their model training highly relies on a negative sampling technique to build up the negative documents in their contrastive losses. To address these challenges, we present Adversarial Retriever-Ranker (AR2), which consists of a dual-encoder retriever plus a cross-encoder ranker. The two models are jointly optimized according to a minimax adversarial objective: the retriever learns to retrieve negative documents to cheat the ranker, while the ranker learns to rank a collection of candidates including both the ground-truth and the retrieved ones, as well as providing progressive direct feedback to the dual-encoder retriever. Through this adversarial game, the retriever gradually produces harder negative documents to train a better ranker, whereas the cross-encoder ranker provides progressive feedback to improve retriever. We evaluate AR2 on three benchmarks. Experimental results show that AR2 consistently and significantly outperforms existing dense retriever methods and achieves new state-of-the-art results on all of them. This includes the improvements on Natural Questions R@5 to 77.9%(+2.1%), TriviaQA R@5 to 78.2%(+1.4), and MS-MARCO MRR@10 to 39.5%(+1.3%). Code and models are available at https://github.com/microsoft/AR2.