Accuracy-on-the-line is a widely observed phenomenon in machine learning, where a model’s accuracy on in-distribution (ID) and out-of-distribution (OOD) data is positively correlated across different hyperparameters and data configurations. But when does this useful relationship break down? In this work, we explore its robustness. The key observation is that noisy data and the presence of nuisance features can be sufficient to shatter the Accuracy-on-the-line phenomenon. In these cases, ID and OOD accuracy can become negatively correlated, leading to "Accuracy-on-the-wrong-line". This phenomenon can also occur in the presence of spurious (shortcut) features, which tend to overshadow the more complex signal (core, non-spurious) features, resulting in a large nuisance feature space. Moreover, scaling to larger datasets does not mitigate this undesirable behaviour and may even exacerbate it. We formally prove a lower bound on OOD error in a linear classification model, characterising the conditions on the noise and nuisance features for a large OOD error. We finally demonstrate this phenomenon across both synthetic and real datasets with noisy data and nuisance features.

series models where we treat images as sequences of patch tokens and employ uni-directional language models to learn visual representations. This modeling paradigm allows us to process images in a recurrent formulation with linear complexity relative to the sequence length, which can effectively address the memory and computation explosion issues posed by high-resolution and fine-grained images. In detail, we introduce two simple designs that seamlessly integrate image inputs into the causal inference framework: a global pooling token placed at the beginning of the sequence and a flipping operation between every two layers. Extensive empirical studies highlight that compared with the existing plain architectures such as DeiT [46] and Vim [57], Adventurer offers an optimal efficiency-accuracy trade-off. For example, our Adventurer-Base attains a competitive test accuracy of 84.3% on the standard ImageNet-1k benchmark with 216 images/s training throughput, which is 3.8× and 6.2× faster than Vim and DeiT to achieve the same result. As Adventurer offers great computation and memory efficiency and allows scaling with linear complexity, we hope this architecture can benefit future explorations in modeling long sequences for high-resolution or fine-grained images. Code is available at https://github.com/wangf3014/Adventurer.

The attention operator is arguably the key distinguishing factor of transformer architectures, which have demonstrated state-of-the-art performance on a variety of tasks. However, transformer attention operators often impose a significant computational burden, with the computational complexity scaling quadratically with the number of tokens. In this work, we propose a novel transformer attention operator whose computational complexity scales linearly with the number of tokens. We derive our network architecture by extending prior work which has shown that a transformer style architecture naturally arises by "white-box" architecture design, where each layer of the network is designed to implement an incremental optimization step of a maximal coding rate reduction objective (MCR$^2$). Specifically, we derive a novel variational form of the MCR$^2$ objective and show that the architecture that results from unrolled gradient descent of this variational objective leads to a new attention module called Token Statistics Self-Attention ($\texttt{TSSA}$). $\texttt{TSSA}$ has $\textit{linear computational and memory complexity}$ and radically departs from the typical attention architecture that computes pairwise similarities between tokens. Experiments on vision, language, and long sequence tasks show that simply swapping $\texttt{TSSA}$ for standard self-attention, which we refer to as the Token Statistics Transformer ($\texttt{ToST}$), achieves competitive performance with conventional transformers while being significantly more computationally efficient and interpretable. Our results also somewhat call into question the conventional wisdom that pairwise similarity style attention mechanisms are critical to the success of transformer architectures.

Since the introduction of Vision Transformer (ViT), patchification has long been regarded as a common image pre-processing approach for plain visual architectures. By compressing the spatial size of images, this approach can effectively shorten the token sequence and reduce the computational cost of ViT-like plain architectures. In this work, we aim to thoroughly examine the information loss caused by this patchification-based compressive encoding paradigm and how it affects visual understanding. We conduct extensive patch size scaling experiments and excitedly observe an intriguing scaling law in patchification: the models can consistently benefit from decreased patch sizes and attain improved predictive performance, until it reaches the minimum patch size of 1*1, i.e., pixel tokenization. This conclusion is broadly applicable across different vision tasks, various input scales, and diverse architectures such as ViT and the recent Mamba models. Moreover, as a by-product, we discover that with smaller patches, task-specific decoder heads become less critical for dense prediction. In the experiments, we successfully scale up the visual sequence to an exceptional length of 50,176 tokens, achieving a competitive test accuracy of 84.6% with a base-sized model on the ImageNet-1k benchmark. We hope this study can provide insights and theoretical foundations for future works of building non-compressive vision models.

Despite the popularity of transformers in practice, their architectures are empirically designed and neither mathematically justified nor interpretable. Moreover, as indicated by many empirical studies, some components of transformer architectures may be redundant. To derive a fully interpretable transformer architecture with only necessary components, we contend that the goal of representation learning is to compress a set of noisy initial token representations towards a mixture of low-dimensional subspaces. To compress these noisy token representations, an associated denoising operation naturally takes the form of a multi-head (subspace) self-attention. By unrolling such iterative denoising operations into a deep network, we arrive at a highly compact architecture that consists of only self-attention operators with skip connections at each layer. Moreover, we show that each layer performs highly efficient denoising: it improves the signal-to-noise ratio of token representations at a linear rate with respect to the number of layers. Despite its simplicity, extensive experiments on vision and language tasks demonstrate that such a transformer achieves performance close to that of standard transformer architectures such as GPT-2 and CRATE.

Pharmacogenomics studies are attracting an increasing amount of interest from researchers in precision medicine. The advances in high-throughput experiments and multiplexed approaches allow the large-scale quantification of drug sensitivities in molecularly characterized cancer cell lines (CCLs), resulting in a number of open drug sensitivity datasets for drug biomarker discovery. However, a significant inconsistency in drug sensitivity values among these datasets has been noted. Such inconsistency indicates the presence of substantial noise, subsequently hindering downstream analyses. To address the noise in drug sensitivity data, we introduce a robust and scalable deep learning framework, Residual Thresholded Deep Matrix Factorization (RT-DMF). This method takes a single drug sensitivity data matrix as its sole input and outputs a corrected and imputed matrix. Deep matrix factorization (DMF) excels at uncovering subtle patterns, due to its minimal reliance on data structure assumptions. This attribute significantly boosts DMF's ability to identify complex hidden patterns among nuisance effects in the data, thereby facilitating the detection of signals that are therapeutically relevant. Furthermore, RT-DMF incorporates an iterative residual thresholding procedure, which plays a crucial role in retaining signals more likely to hold therapeutic importance. Validation using simulated datasets and real pharmacogenomics datasets demonstrates the effectiveness of our approach in correcting noise and imputing missing data in drug sensitivity datasets (open-source package available at https://github.com/tomwhoooo/rtdmf).

Modern learning frameworks often train deep neural networks with massive amounts of unlabeled data to learn representations by solving simple pretext tasks, then use the representations as foundations for downstream tasks. These networks are empirically designed; as such, they are usually not interpretable, their representations are not structured, and their designs are potentially redundant. White-box deep networks, in which each layer explicitly identifies and transforms structures in the data, present a promising alternative. However, existing white-box architectures have only been shown to work at scale in supervised settings with labeled data, such as classification. In this work, we provide the first instantiation of the white-box design paradigm that can be applied to large-scale unsupervised representation learning. We do this by exploiting a fundamental connection between diffusion, compression, and (masked) completion, deriving a deep transformer-like masked autoencoder architecture, called CRATE-MAE, in which the role of each layer is mathematically fully interpretable: they transform the data distribution to and from a structured representation. Extensive empirical evaluations confirm our analytical insights. CRATE-MAE demonstrates highly promising performance on large-scale imagery datasets while using only ~30% of the parameters compared to the standard masked autoencoder with the same model configuration. The representations learned by CRATE-MAE have explicit structure and also contain semantic meaning.

The maximal coding rate reduction (MCR$^2$) objective for learning structured and compact deep representations is drawing increasing attention, especially after its recent usage in the derivation of fully explainable and highly effective deep network architectures. However, it lacks a complete theoretical justification: only the properties of its global optima are known, and its global landscape has not been studied. In this work, we give a complete characterization of the properties of all its local and global optima as well as other types of critical points. Specifically, we show that each (local or global) maximizer of the MCR$^2$ problem corresponds to a low-dimensional, discriminative, and diverse representation, and furthermore, each critical point of the objective is either a local maximizer or a strict saddle point. Such a favorable landscape makes MCR$^2$ a natural choice of objective for learning diverse and discriminative representations via first-order optimization. To further verify our theoretical findings, we illustrate these properties with extensive experiments on both synthetic and real data sets.

Differentially private (DP) machine learning is considered the gold-standard solution for training a model from sensitive data while still preserving privacy. However, a major barrier to achieving this ideal is its sub-optimal privacy-accuracy trade-off, which is particularly visible in DP representation learning. Specifically, it has been shown that under modest privacy budgets, most models learn representations that are not significantly better than hand-crafted features. In this work, we show that effective DP representation learning can be done via image captioning and scaling up to internet-scale multimodal datasets. Through a series of engineering tricks, we successfully train a DP image captioner (DP-Cap) on a 233M subset of LAION-2B from scratch using a reasonable amount of computation, and obtaining unprecedented high-quality image features that can be used in a variety of downstream vision and vision-language tasks. For example, under a privacy budget of $\varepsilon=8$ for the LAION dataset, a linear classifier trained on top of learned DP-Cap features attains $65.8\%$ accuracy on ImageNet-1K, considerably improving the previous SOTA of $56.5\%$. Our work challenges the prevailing sentiment that high-utility DP representation learning cannot be achieved by training from scratch.

Artificial intelligence (AI) has seen a tremendous surge in capabilities thanks to the use of foundation models trained on internet-scale data. On the flip side, the uncurated nature of internet-scale data also poses significant privacy and legal risks, as they often contain personal information or copyrighted material that should not be trained on without permission. In this work, we propose as a mitigation measure a recipe to train foundation vision models via self-supervised learning with differential privacy (DP) guarantee. We identify masked autoencoders as a suitable learning algorithm that aligns well with DP-SGD, and train *ViP*---a **Vi**sion transformer with differential **P**rivacy---under a strict privacy budget of $\epsilon=8$ on the LAION400M dataset. We evaluate the quality of representation learned by ViP using standard downstream vision tasks; in particular, ViP achieves a (non-private) linear probing accuracy of 55.7% on ImageNet, comparable to that of end-to-end trained AlexNet (trained and evaluated on ImageNet). Our result suggests that scaling to internet-scale data can be practical for private learning. Code and DP pre-trained models are available at https://github.com/facebookresearch/ViP-MAE.

Hanlin Zhang, YiFan Zhang, Yaodong Yu, Dhruv Madeka, Dean Foster, Eric Xing, Himabindu Lakkaraju, Sham Kakade. Proceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers). 2024.

CRATE, a white-box transformer architecture designed to learn compressed and sparse representations, offers an intriguing alternative to standard vision transformers (ViTs) due to its inherent mathematical interpretability. Despite extensive investigations into the scaling behaviors of language and vision transformers, the scalability of CRATE remains an open question which this paper aims to address. Specifically, we propose CRATE-$\alpha$, featuring strategic yet minimal modifications to the sparse coding block in the CRATE architecture design, and a light training recipe designed to improve the scalability of CRATE. Through extensive experiments, we demonstrate that CRATE-$\alpha$ can effectively scale with larger model sizes and datasets. For example, our CRATE-$\alpha$-B substantially outperforms the prior best CRATE-B model accuracy on ImageNet classification by 3.7%, achieving an accuracy of 83.2%. Meanwhile, when scaling further, our CRATE-$\alpha$-L obtains an ImageNet classification accuracy of 85.1%. More notably, these model performance improvements are achieved while preserving, and potentially even enhancing the interpretability of learned CRATE models, as we demonstrate through showing that the learned token representations of increasingly larger trained CRATE-$\alpha$ models yield increasingly higher-quality unsupervised object segmentation of images.

In this paper, we contend that a natural objective of representation learning is to compress and transform the distribution of the data, say sets of tokens, towards a low-dimensional Gaussian mixture supported on incoherent subspaces. The goodness of such a representation can be evaluated by a principled measure, called sparse rate reduction, that simultaneously maximizes the intrinsic information gain and extrinsic sparsity of the learned representation. From this perspective, popular deep network architectures, including transformers, can be viewed as realizing iterative schemes to optimize this measure. Particularly, we derive a transformer block from alternating optimization on parts of this objective: the multi-head self-attention operator compresses the representation by implementing an approximate gradient descent step on the coding rate of the features, and the subsequent multi-layer perceptron sparsifies the features. This leads to a family of white-box transformer-like deep network architectures, named CRATE, which are mathematically fully interpretable. We show, by way of a novel connection between denoising and compression, that the inverse to the aforementioned compressive encoding can be realized by the same class of CRATE architectures. Thus, the so-derived white-box architectures are universal to both encoders and decoders. Experiments show that these networks, despite their simplicity, indeed learn to compress and sparsify representations of large-scale real-world image and text datasets, and achieve strong performance across different settings: ViT, MAE, DINO, BERT, and GPT2. We believe the proposed computational framework demonstrates great potential in bridging the gap between theory and practice of deep learning, from a unified perspective of data compression.

Conformal prediction is emerging as a popular paradigm for providing rigorous uncertainty quantification in machine learning since it can be easily applied as a post-processing step to already trained models. In this paper, we extend conformal prediction to the federated learning setting. The main challenge we face is data heterogeneity across the clients --- this violates the fundamental tenet of *exchangeability* required for conformal prediction. We propose a weaker notion of *partial exchangeability*, better suited to the FL setting, and use it to develop the Federated Conformal Prediction (FCP) framework. We show FCP enjoys rigorous theoretical guarantees and excellent empirical performance on several computer vision and medical imaging datasets. Our results demonstrate a practical approach to incorporating meaningful uncertainty quantification in distributed and heterogeneous environments. We provide code used in our experiments https://github.com/clu5/federated-conformal.

In this paper, we contend that the objective of representation learning is to compress and transform the distribution of the data, say sets of tokens, towards a mixture of low-dimensional Gaussian distributions supported on incoherent subspaces. The quality of the final representation can be measured by a unified objective function called sparse rate reduction. From this perspective, popular deep networks such as transformers can be naturally viewed as realizing iterative schemes to optimize this objective incrementally. Particularly, we show that the standard transformer block can be derived from alternating optimization on complementary parts of this objective: the multi-head self-attention operator can be viewed as a gradient descent step to compress the token sets by minimizing their lossy coding rate, and the subsequent multi-layer perceptron can be viewed as attempting to sparsify the representation of the tokens. This leads to a family of white-box transformer-like deep network architectures which are mathematically fully interpretable. Despite their simplicity, experiments show that these networks indeed learn to optimize the designed objective: they compress and sparsify representations of large-scale real-world vision datasets such as ImageNet, and achieve performance very close to thoroughly engineered transformers such as ViT. Code is at https://github.com/Ma-Lab-Berkeley/CRATE.

We study the stochastic bilinear minimax optimization problem, presenting an analysis of the same-sample Stochastic ExtraGradient (SEG) method with constant step size, and presenting variations of the method that yield favorable convergence. In sharp contrasts with the basic SEG method whose last iterate only contracts to a fixed neighborhood of the Nash equilibrium, SEG augmented with iteration averaging provably converges to the Nash equilibrium under the same standard settings, and such a rate is further improved by incorporating a scheduled restarting procedure. In the interpolation setting where noise vanishes at the Nash equilibrium, we achieve an optimal convergence rate up to tight constants. We present numerical experiments that validate our theoretical findings and demonstrate the effectiveness of the SEG method when equipped with iteration averaging and restarting.

Distributionally robust supervised learning (DRSL) is emerging as a key paradigm for building reliable machine learning systems for real-world applications–reflecting the need for classifiers and predictive models that are robust to the distribution shifts that arise from phenomena such as selection bias or nonstationarity. Existing algorithms for solving Wasserstein DRSL— one of the most popular DRSL frameworks based around robustness to perturbations in the Wasserstein distance—have serious limitations that limit their use in large-scale problems—in particular they involve solving complex subproblems and they fail to make use of stochastic gradients. We revisit Wasserstein DRSL through the lens of min-max optimization and derive scalable and efficiently implementable stochastic extra-gradient algorithms which provably achieve faster convergence rates than existing approaches. We demonstrate their effectiveness on synthetic and real data when compared to existing DRSL approaches. Key to our results is the use of variance reduction and random reshuffling to accelerate stochastic min-max optimization, the analysis of which may be of independent interest.

Motivated by applications to online learning in sparse estimation and Bayesian optimization, we consider the problem of online unconstrained nonsubmodular minimization with delayed costs in both full information and bandit feedback settings. In contrast to previous works on online unconstrained submodular minimization, we focus on a class of nonsubmodular functions with special structure, and prove regret guarantees for several variants of the online and approximate online bandit gradient descent algorithms in static and delayed scenarios. We derive bounds for the agent’s regret in the full information and bandit feedback setting, even if the delay between choosing a decision and receiving the incurred cost is unbounded. Key to our approach is the notion of $(\alpha, \beta)$-regret and the extension of the generic convex relaxation model from \citet{El-2020-Optimal}, the analysis of which is of independent interest. We conduct and showcase several simulation studies to demonstrate the efficacy of our algorithms.

We propose a metric—Projection Norm—to predict a model’s performance on out-of-distribution (OOD) data without access to ground truth labels. Projection Norm first uses model predictions to pseudo-label test samples and then trains a new model on the pseudo-labels. The more the new model’s parameters differ from an in-distribution model, the greater the predicted OOD error. Empirically, our approach outperforms existing methods on both image and text classification tasks and across different network architectures. Theoretically, we connect our approach to a bound on the test error for overparameterized linear models. Furthermore, we find that Projection Norm is the only approach that achieves non-trivial detection performance on adversarial examples. Our code is available at \url{https://github.com/yaodongyu/ProjNorm}.

Having similar behavior at training time and test time—what we call a “What You See Is What You Get” (WYSIWYG) property—is desirable in machine learning. Models trained with standard stochastic gradient descent (SGD), however, do not necessarily have this property, as their complex behaviors such as robustness or subgroup performance can differ drastically between training and test time. In contrast, we show that Differentially-Private (DP) training provably ensures the high-level WYSIWYG property, which we quantify using a notion of distributional generalization. Applying this connection, we introduce new conceptual tools for designing deep-learning methods by reducing generalization concerns to optimization ones: to mitigate unwanted behavior at test time, it is provably sufficient to mitigate this behavior on the training data. By applying this novel design principle, which bypasses “pathologies” of SGD, we construct simple algorithms that are competitive with SOTA in several distributional-robustness applications, significantly improve the privacy vs. disparate impact trade-off of DP-SGD, and mitigate robust overfitting in adversarial training. Finally, we also improve on theoretical bounds relating DP, stability, and distributional generalization.

State-of-the-art federated learning methods can perform far worse than their centralized counterparts when clients have dissimilar data distributions. For neural networks, even when centralized SGD easily finds a solution that is simultaneously performant for all clients, current federated optimization methods fail to converge to a comparable solution. We show that this performance disparity can largely be attributed to optimization challenges presented by nonconvexity. Specifically, we find that the early layers of the network do learn useful features, but the final layers fail to make use of them. That is, federated optimization applied to this non-convex problem distorts the learning of the final layers. Leveraging this observation, we propose a Train-Convexify-Train (TCT) procedure to sidestep this issue: first, learn features using off-the-shelf methods (e.g., FedAvg); then, optimize a convexified problem obtained from the network's empirical neural tangent kernel approximation. Our technique yields accuracy improvements of up to $+36\%$ on FMNIST and $+37\%$ on CIFAR10 when clients have dissimilar data.

Uncertainty quantification is essential for the reliable deployment of machine learning models to high-stakes application domains. Uncertainty quantification is all the more challenging when training distribution and test distribution are different, even if the distribution shifts are mild. Despite the ubiquity of distribution shifts in real-world applications, existing uncertainty quantification approaches mainly study the in-distribution setting where the train and test distributions are the same. In this paper, we develop a systematic calibration model to handle distribution shifts by leveraging data from multiple domains. Our proposed method---multi-domain temperature scaling---uses the heterogeneity in the domains to improve calibration robustness under distribution shift. Through experiments on three benchmark data sets, we find our proposed method outperforms existing methods as measured on both in-distribution and out-of-distribution test sets.

This work attempts to provide a plausible theoretical framework that aims to interpret modern deep (convolutional) networks from the principles of data compression and discriminative representation. We argue that for high-dimensional multi-class data, the optimal linear discriminative representation maximizes the coding rate difference between the whole dataset and the average of all the subsets. We show that the basic iterative gradient ascent scheme for optimizing the rate reduction objective naturally leads to a multi-layer deep network, named ReduNet, which shares common characteristics of modern deep networks. The deep layered architectures, linear and nonlinear operators, and even parameters of the network are all explicitly constructed layer-by-layer via forward propagation, although they are amenable to fine-tuning via back propagation. All components of so-obtained “white-box” network have precise optimization, statistical, and geometric interpretation. Moreover, all linear operators of the so-derived network naturally become multi-channel convolutions when we enforce classification to be rigorously shift-invariant. The derivation in the invariant setting suggests a trade-off between sparsity and invariance, and also indicates that such a deep convolution network is significantly more efficient to construct and learn in the spectral domain. Our preliminary simulations and experiments clearly verify the effectiveness of both the rate reduction objective and the associated ReduNet. All code and data are available at https://github.com/Ma-Lab-Berkeley.

The classical bias-variance trade-off predicts that bias decreases and variance increase with model complexity, leading to a U-shaped risk curve. Recent work calls this into question for neural networks and other over-parameterized models, for which it is often observed that larger models generalize better. We provide a simple explanation of this by measuring the bias and variance of neural networks: while the bias is \emph{monotonically decreasing} as in the classical theory, the variance is \emph{unimodal} or bell-shaped: it increases then decreases with the width of the network. We vary the network architecture, loss function, and choice of dataset and confirm that variance unimodality occurs robustly for all models we considered. The risk curve is the sum of the bias and variance curves and displays different qualitative shapes depending on the relative scale of bias and variance, with the double descent in the recent literature as a special case. We corroborate these empirical results with a theoretical analysis of two-layer linear networks with random first layer. Finally, evaluation on out-of-distribution data shows that most of the drop in accuracy comes from increased bias while variance increases by a relatively small amount. Moreover, we find that deeper models decrease bias and increase variance for both in-distribution and out-of-distribution data.

Robustness of machine learning models to various adversarial and non-adversarial corruptions continues to be of interest. In this paper, we introduce the notion of the boundary thickness of a classifier, and we describe its connection with and usefulness for model robustness. Thick decision boundaries lead to improved performance, while thin decision boundaries lead to overfitting (e.g., measured by the robust generalization gap between training and testing) and lower robustness. We show that a thicker boundary helps improve robustness against adversarial examples (e.g., improving the robust test accuracy of adversarial training), as well as so-called out-of-distribution (OOD) transforms, and we show that many commonly-used regularization and data augmentation procedures can increase boundary thickness. On the theoretical side, we establish that maximizing boundary thickness is akin to minimizing the so-called mixup loss. Using these observations, we can show that noise-augmentation on mixup training further increases boundary thickness, thereby combating vulnerability to various forms of adversarial attacks and OOD transforms. We can also show that the performance improvement in several recent lines of work happens in conjunction with a thicker boundary.

To learn intrinsic low-dimensional structures from high-dimensional data that most discriminate between classes, we propose the principle of {\em Maximal Coding Rate Reduction} ($\text{MCR}^2$), an information-theoretic measure that maximizes the coding rate difference between the whole dataset and the sum of each individual class. We clarify its relationships with most existing frameworks such as cross-entropy, information bottleneck, information gain, contractive and contrastive learning, and provide theoretical guarantees for learning diverse and discriminative features. The coding rate can be accurately computed from finite samples of degenerate subspace-like distributions and can learn intrinsic representations in supervised, self-supervised, and unsupervised settings in a unified manner. Empirically, the representations learned using this principle alone are significantly more robust to label corruptions in classification than those using cross-entropy, and can lead to state-of-the-art results in clustering mixed data from self-learned invariant features.

We study the problem of learning one-hidden-layer neural networks with Rectified Linear Unit (ReLU) activation function, where the inputs are sampled from standard Gaussian distribution and the outputs are generated from a noisy teacher network. We analyze the performance of gradient descent for training such kind of neural networks based on empirical risk minimization, and provide algorithm-dependent guarantees. In particular, we prove that tensor initialization followed by gradient descent can converge to the ground-truth parameters at a linear rate up to some statistical error. To the best of our knowledge, this is the first work characterizing the recovery guarantee for practical learning of one-hidden-layer ReLU networks with multiple neurons. Numerical experiments verify our theoretical findings.

We identify a trade-off between robustness and accuracy that serves as a guiding principle in the design of defenses against adversarial examples. Although this problem has been widely studied empirically, much remains unknown concerning the theory underlying this trade-off. In this work, we decompose the prediction error for adversarial examples (robust error) as the sum of the natural (classification) error and boundary error, and provide a differentiable upper bound using the theory of classification-calibrated loss, which is shown to be the tightest possible upper bound uniform over all probability distributions and measurable predictors. Inspired by our theoretical analysis, we also design a new defense method, TRADES, to trade adversarial robustness off against accuracy. Our proposed algorithm performs well experimentally in real-world datasets. The methodology is the foundation of our entry to the NeurIPS 2018 Adversarial Vision Challenge in which we won the 1st place out of 2,000 submissions, surpassing the runner-up approach by 11.41% in terms of mean L_2 perturbation distance.

Multi-task learning (MTL) is a machine learning paradigm that improves the performance of each task by exploiting useful information contained in multiple related tasks. However, the relatedness of tasks can be exploited by attackers to launch data poisoning attacks, which has been demonstrated a big threat to single-task learning. In this paper, we provide the first study on the vulnerability of MTL. Specifically, we focus on multi-task relationship learning (MTRL) models, a popular subclass of MTL models where task relationships are quantized and are learned directly from training data. We formulate the problem of computing optimal poisoning attacks on MTRL as a bilevel program that is adaptive to arbitrary choice of target tasks and attacking tasks. We propose an efficient algorithm called PATOM for computing optimal attack strategies. PATOM leverages the optimality conditions of the subproblem of MTRL to compute the implicit gradients of the upper level objective function. Experimental results on real-world datasets show that MTRL models are very sensitive to poisoning attacks and the attacker can significantly degrade the performance of target tasks, by either directly poisoning the target tasks or indirectly poisoning the related tasks exploiting the task relatedness. We also found that the tasks being attacked are always strongly correlated, which provides a clue for defending against such attacks.

We propose a primal-dual based framework for analyzing the global optimality of nonconvex low-rank matrix recovery. Our analysis are based on the restricted strongly convex and smooth conditions, which can be verified for a broad family of loss functions. In addition, our analytic framework can directly handle the widely-used incoherence constraints through the lens of duality. We illustrate the applicability of the proposed framework to matrix completion and one-bit matrix completion, and prove that all these problems have no spurious local minima. Our results not only improve the sample complexity required for characterizing the global optimality of matrix completion, but also resolve an open problem in Ge et al. (2017) regarding one-bit matrix completion. Numerical experiments show that primal-dual based algorithm can successfully recover the global optimum for various low-rank problems.

We propose stochastic optimization algorithms that can find local minima faster than existing algorithms for nonconvex optimization problems, by exploiting the third-order smoothness to escape non-degenerate saddle points more efficiently. More specifically, the proposed algorithm only needs $\tilde{O}(\epsilon^{-10/3})$ stochastic gradient evaluations to converge to an approximate local minimum $\mathbf{x}$, which satisfies $\|\nabla f(\mathbf{x})\|_2\leq\epsilon$ and $\lambda_{\min}(\nabla^2 f(\mathbf{x}))\geq -\sqrt{\epsilon}$ in unconstrained stochastic optimization, where $\tilde{O}(\cdot)$ hides logarithm polynomial terms and constants. This improves upon the $\tilde{O}(\epsilon^{-7/2})$ gradient complexity achieved by the state-of-the-art stochastic local minima finding algorithms by a factor of $\tilde{O}(\epsilon^{-1/6})$. Experiments on two nonconvex optimization problems demonstrate the effectiveness of our algorithm and corroborate our theory.