The rise of foundation models fine-tuned on human feedback from potentially untrusted users has increased the risk of adversarial data poisoning, necessitating the study of robustness of learning algorithms against such attacks. Existing research on provable certified robustness against data poisoning attacks primarily focuses on certifying robustness for static adversaries who modify a fraction of the dataset used to train the model before the training algorithm is applied. In practice, particularly when learning from human feedback in an online sense, adversaries can observe and react to the learning process and inject poisoned samples that optimize adversarial objectives better than when they are restricted to poisoning a static dataset once, before the learning algorithm is applied. Indeed, it has been shown in prior work that online dynamic adversaries can be significantly more powerful than static ones. We present a novel framework for computing certified bounds on the impact of dynamic poisoning, and use these certificates to design robust learning algorithms. We give an illustration of the framework for the mean estimation problem and binary classification problems and outline directions for extending this in further work.

Human feedback plays a critical role in learning and refining reward models for text-to-image generation, but the optimal form the feedback should take for learning an accurate reward function has not been conclusively established. This paper investigates the effectiveness of fine-grained feedback which captures nuanced distinctions in image quality and prompt-alignment, compared to traditional coarse-grained feedback (for example, thumbs up/down or ranking between a set of options). While fine-grained feedback holds promise, particularly for systems catering to diverse societal preferences, we show that demonstrating its superiority to coarse-grained feedback is not automatic. Through experiments on real and synthetic preference data, we surface the complexities of building effective models due to the interplay of model choice, feedback type, and the alignment between human judgment and computational interpretation. We identify key challenges in eliciting and utilizing fine-grained feedback, prompting a reassessment of its assumed benefits and practicality. Our findings -- e.g., that fine-grained feedback can lead to worse models for a fixed budget, in some settings; however, in controlled settings with known attributes, fine grained rewards can indeed be more helpful -- call for careful consideration of feedback attributes and potentially beckon novel modeling approaches to appropriately unlock the potential value of fine-grained feedback in-the-wild.

Recent Text-to-Image (T2I) generation models such as Stable Diffusion and Imagen have made significant progress in generating high-resolution images based on text descriptions. However, many generated images still suffer from issues such as artifacts/implausibility, misalignment with text descriptions, and low aesthetic quality. Inspired by the success of Reinforcement Learning with Human Feedback (RLHF) for large language models, prior works collected human-provided scores as feedback on generated images and trained a reward model to improve the T2I generation. In this paper, we enrich the feedback signal by (i) marking image regions that are implausible or misaligned with the text, and (ii) annotating which words in the text prompt are misrepresented or missing on the image. We collect such rich human feedback on 18K generated images (RichHF-18K) and train a multimodal transformer to predict the rich feedback automatically. We show that the predicted rich human feedback can be leveraged to improve image generation, for example, by selecting high-quality training data to finetune and improve the generative models, or by creating masks with predicted heatmaps to inpaint the problematic regions. Notably, the improvements generalize to models (Muse) beyond those used to generate the images on which human feedback data were collected (Stable Diffusion variants). The RichHF-18K data set will be released in our GitHub repository: https://github.com/google-research/google-research/tree/master/richhf_18k.

In the task of differentially private (DP) continual counting, we receive a stream of increments and our goal is to output an approximate running total of these increments, without revealing too much about any specific increment. Despite its simplicity, differentially private continual counting has attracted significant attention both in theory and in practice. Existing algorithms for differentially private continual counting are either inefficient in terms of their space usage or add an excessive amount of noise, inducing suboptimal utility. The most practical DP continual counting algorithms add carefully correlated Gaussian noise to the values. The task of choosing the covariance for this noise can be expressed in terms of factoring the lower-triangular matrix of ones (which computes prefix sums). We present two approaches from this class (for different parameter regimes) that achieve near-optimal utility for DP continual counting and only require logarithmic or polylogarithmic space (and time). Our first approach is based on a space-efficient streaming matrix multiplication algorithm for a class of Toeplitz matrices. We show that to instantiate this algorithm for DP continual counting, it is sufficient to find a low-degree rational function that approximates the square root on a circle in the complex plane. We then apply and extend tools from approximation theory to achieve this. We also derive efficient closed-forms for the objective function for arbitrarily many steps, and show direct numerical optimization yields a highly practical solution to the problem. Our second approach combines our first approach with a recursive construction similar to the binary tree mechanism.

Differentially private learning algorithms inject noise into the learning process. While the most common private learning algorithm, DP-SGD, adds independent Gaussian noise in each iteration, recent work on matrix factorization mechanisms has shown empirically that introducing correlations in the noise can greatly improve their utility. We characterize the asymptotic learning utility for any choice of the correlation function, giving precise analytical bounds for linear regression and as the solution to a convex program for general convex functions. We show, using these bounds, how correlated noise provably improves upon vanilla DP-SGD as a function of problem parameters such as the effective dimension and condition number. Moreover, our analytical expression for the near-optimal correlation function circumvents the cubic complexity of the semi-definite program used to optimize the noise correlation matrix in previous work. We validate these theoretical results with experiments on private deep learning. Our work matches or outperforms prior work while being efficient both in terms of computation and memory.

Fine-tuning text-to-image models with reward functions trained on human feedback data has proven effective for aligning model behavior with human intent. However, excessive optimization with such reward models, which serve as mere proxy objectives, can compromise the performance of fine-tuned models, a phenomenon known as reward overoptimization. To investigate this issue in depth, we introduce the Text-Image Alignment Assessment (TIA2) benchmark, which comprises a diverse collection of text prompts, images, and human annotations. Our evaluation of several state-of-the-art reward models on this benchmark reveals their frequent misalignment with human assessment. We empirically demonstrate that overoptimization occurs notably when a poorly aligned reward model is used as the fine-tuning objective. To address this, we propose TextNorm, a simple method that enhances alignment based on a measure of reward model confidence estimated across a set of semantically contrastive text prompts. We demonstrate that incorporating the confidence-calibrated rewards in fine-tuning effectively reduces overoptimization, resulting in twice as many wins in human evaluation for text-image alignment compared against the baseline reward models.

In order to train networks for verified adversarial robustness, it is common to over-approximate the worst-case loss over perturbation regions, resulting in networks that attain verifiability at the expense of standard performance. As shown in recent work, better trade-offs between accuracy and robustness can be obtained by carefully coupling adversarial training with over-approximations. We hypothesize that the expressivity of a loss function, which we formalize as the ability to span a range of trade-offs between lower and upper bounds to the worst-case loss through a single parameter (the over-approximation coefficient), is key to attaining state-of-the-art performance. To support our hypothesis, we show that trivial expressive losses, obtained via convex combinations between adversarial attacks and IBP bounds, yield state-of-the-art results across a variety of settings in spite of their conceptual simplicity. We provide a detailed analysis of the relationship between the over-approximation coefficient and performance profiles across different expressive losses, showing that, while expressivity is essential, better approximations of the worst-case loss are not necessarily linked to superior robustness-accuracy trade-offs.

We provide an improved analysis of standard differentially private gradient descent for linear regression under the squared error loss. Under modest assumptions on the input, we characterize the distribution of the iterate at each time step. Our analysis leads to new results on the algorithm’s accuracy: for a proper fixed choice of hyperparameters, the sample complexity depends only linearly on the dimension of the data. This matches the dimension-dependence of the (non-private) ordinary least squares estimator as well as that of recent private algorithms that rely on sophisticated adaptive gradient-clipping schemes (Varshney et al., 2022; Liu et al., 2023). Our analysis of the iterates’ distribution also allows us to construct confidence intervals for the empirical optimizer which adapt automatically to the variance of the algorithm on a particular data set. We validate our theorems through experiments on synthetic data.

We introduce the first model-stealing attack that extracts precise, nontrivial information from black-box production language models like OpenAI's ChatGPT or Google's PaLM-2. Specifically, our attack recovers the embedding projection layer (up to symmetries) of a transformer model, given typical API access. For under $20 USD, our attack extracts the entire projection matrix of OpenAI's Ada and Babbage language models. We thereby confirm, for the first time, that these black-box models have a hidden dimension of 1024 and 2048, respectively. We also recover the exact hidden dimension size of the GPT-3.5-turbo model, and estimate it would cost under \\$2,000 in queries to recover the entire projection matrix. We conclude with potential defenses and mitigations, and discuss the implications of possible future work that could extend our attack.

Recent work provides promising evidence that Physics-Informed Neural Networks (PINN) can efficiently solve partial differential equations (PDE). However, previous works have failed to provide guarantees on the worst-case residual error of a PINN across the spatio-temporal domain - a measure akin to the tolerance of numerical solvers - focusing instead on point-wise comparisons between their solution and the ones obtained by a solver on a set of inputs. In real-world applications, one cannot consider tests on a finite set of points to be sufficient grounds for deployment, as the performance could be substantially worse on a different set. To alleviate this issue, we establish guaranteed error-based conditions for PINNs over their continuous applicability domain. To verify the extent to which they hold, we introduce $\partial$-CROWN: a general, efficient and scalable post-training framework to bound PINN residual errors. We demonstrate its effectiveness in obtaining tight certificates by applying it to two classically studied PINNs – Burgers’ and Schrödinger’s equations –, and two more challenging ones with real-world applications – the Allan-Cahn and Diffusion-Sorption equations.

This paper presents a new *bi-Lipschitz* invertible neural network, the BiLipNet, which has the ability to smoothly control both its *Lipschitzness* (output sensitivity to input perturbations) and *inverse Lipschitzness* (input distinguishability from different outputs). The second main contribution is a new scalar-output network, the PLNet, which is a composition of a BiLipNet and a quadratic potential. We show that PLNet satisfies the Polyak-Łojasiewicz condition and can be applied to learn non-convex surrogate losses with a unique and efficiently-computable global minimum. The central technical element in these networks is a novel invertible residual layer with certified strong monotonicity and Lipschitzness, which we compose with orthogonal layers to build the BiLipNet. The certification of these properties is based on incremental quadratic constraints, resulting in much tighter bounds than can be achieved with spectral normalization. Moreover, we formulate the calculation of the inverse of a BiLipNet -- and hence the minimum of a PLNet -- as a series of three-operator splitting problems, for which fast algorithms can be applied.

Concept bottleneck models (CBMs) are interpretable neural networks that first predict labels for human-interpretable concepts relevant to the prediction task, and then predict the final label based on the concept label predictions. We extend CBMs to interactive prediction settings where the model can query a human collaborator for the label to some concepts. We develop an interaction policy that, at prediction time, chooses which concepts to request a label for so as to maximally improve the final prediction. We demonstrate that a simple policy combining concept prediction uncertainty and influence of the concept on the final prediction achieves strong performance and outperforms static approaches as well as active feature acquisition methods proposed in the literature. We show that the interactive CBM can achieve accuracy gains of 5-10% with only 5 interactions over competitive baselines on the Caltech-UCSD Birds, CheXpert and OAI datasets.

Placing a human in the loop may help abate the risks of deploying AI systems in safety-critical settings (e.g., a clinician working with a medical AI system). However, mitigating risks arising from human error and uncertainty within such human-AI interactions is an important and understudied issue. In this work, we study human uncertainty in the context of concept-based models, a family of AI systems that enable human feedback via concept interventions where an expert intervenes on human-interpretable concepts relevant to the task. Prior work in this space often assumes that humans are oracles who are always certain and correct. Yet, real-world decision-making by humans is prone to occasional mistakes and uncertainty. We study how existing concept-based models deal with uncertain interventions from humans using two novel datasets: UMNIST, a visual dataset with controlled simulated uncertainty based on the MNIST dataset, and CUB-S, a relabeling of the popular CUB concept dataset with rich, densely-annotated soft labels from humans. We show that training with uncertain concept labels may help mitigate weaknesses of concept-based systems when handling uncertain interventions. These results allow us to identify several open challenges, which we argue can be tackled through future multidisciplinary research on building interactive uncertainty-aware systems. To facilitate further research, we release a new elicitation platform, UElic, to collect uncertain feedback from humans in collaborative prediction tasks.

Most work on the formal verification of neural networks has focused on bounding the set of outputs that correspond to a given set of inputs (for example, bounded perturbations of a nominal input). However, many use cases of neural network verification require solving the inverse problem, or over-approximating the set of inputs that lead to certain outputs. We present the INVPROP algorithm for verifying properties over the preimage of a linearly constrained output set, which can be combined with branch-and-bound to increase precision. Contrary to other approaches, our efficient algorithm is GPU-accelerated and does not require a linear programming solver. We demonstrate our algorithm for identifying safe control regions for a dynamical system via backward reachability analysis, verifying adversarial robustness, and detecting out-of-distribution inputs to a neural network. Our results show that in certain settings, we find over-approximations over $2500\times$ tighter than prior work while being $2.5\times$ faster. By strengthening robustness verification with output constraints, we consistently verify more properties than the previous state-of-the-art on multiple benchmarks, including a large model with 167k neurons in VNN-COMP 2023. Our algorithm has been incorporated into the $\alpha,\beta$-CROWN verifier, available at https://abcrown.org.

Training reliable deep learning models which avoid making overconfident but incorrect predictions is a longstanding challenge. This challenge is further exacerbated when learning has to be differentially private: protection provided to sensitive data comes at the price of injecting additional randomness into the learning process. In this work, we conduct a thorough empirical investigation of selective classifiers---that can abstain under uncertainty---under a differential privacy constraint. We find that some popular selective prediction approaches are ineffective in a differentially private setting because they increase the risk of privacy leakage. At the same time, we identify that a recent approach that only uses checkpoints produced by an off-the-shelf private learning algorithm stands out as particularly suitable under DP. Further, we show that differential privacy does not just harm utility but also degrades selective classification performance. To analyze this effect across privacy levels, we propose a novel evaluation mechanism which isolates selective prediction performance across model utility levels at full coverage. Our experimental results show that recovering the performance level attainable by non-private models is possible but comes at a considerable coverage cost as the privacy budget decreases.

Concept Bottleneck Models (CBMs) tackle the opacity of neural architectures by constructing and explaining their predictions using a set of high-level concepts. A special property of these models is that they permit concept interventions, wherein users can correct mispredicted concepts and thus improve the model's performance. Recent work, however, has shown that intervention efficacy can be highly dependent on the order in which concepts are intervened on and on the model's architecture and training hyperparameters. We argue that this is rooted in a CBM's lack of train-time incentives for the model to be appropriately receptive to concept interventions. To address this, we propose Intervention-aware Concept Embedding models (IntCEMs), a novel CBM-based architecture and training paradigm that improves a model's receptiveness to test-time interventions. Our model learns a concept intervention policy in an end-to-end fashion from where it can sample meaningful intervention trajectories at train-time. This conditions IntCEMs to effectively select and receive concept interventions when deployed at test-time. Our experiments show that IntCEMs significantly outperform state-of-the-art concept-interpretable models when provided with test-time concept interventions, demonstrating the effectiveness of our approach.

Recent work has shown the potential benefit of selective prediction systems that can learn to defer to a human when the predictions of the AI are unreliable, particularly to improve the reliability of AI systems in high-stakes applications like healthcare or conservation. However, most prior work assumes that human behavior remains unchanged when they solve a prediction task as part of a human-AI team as opposed to by themselves. We show that this is not the case by performing experiments to quantify human-AI interaction in the context of selective prediction. In particular, we study the impact of communicating different types of information to humans about the AI system's decision to defer. Using real-world conservation data and a selective prediction system that improves expected accuracy over that of the human or AI system working individually, we show that this messaging has a significant impact on the accuracy of human judgements. Our results study two components of the messaging strategy: 1) Whether humans are informed about the prediction of the AI system and 2) Whether they are informed about the decision of the selective prediction system to defer. By manipulating these messaging components, we show that it is possible to significantly boost human performance by informing the human of the decision to defer, but not revealing the prediction of the AI. We therefore show that it is vital to consider how the decision to defer is communicated to a human when designing selective prediction systems, and that the composite accuracy of a human-AI team must be carefully evaluated using a human-in-the-loop framework.

The future distribution systems expose to an unprecedented level of uncertainties due to renewable resources, nontraditional loads, aging infrastructure, etc., posing potential risks to secure operation of the system.This article proposes a new technique to estimate the voltage stability margin of the distribution systems with high penetration of renewables.Its convergence and robustness under complex and stressed working conditions are guaranteed in theory. This technique is handy for the operation as it features self-adaptive step size and is applicable to general system topology. It leverages a newly derived analytical solvability certificate based on the Kantorovich fixed-point theorem. A fast version of the proposed technique is duly proposed to speed up the computation up to 8 times while maintaining high accuracy, which lends itself to online and time-sensitive emergency tasks. Numerical simulations with various IEEE test feeders verify the performance of the techniques.

Recent progress in neural network verification has challenged the notion of a convex barrier, that is, an inherent weakness in the convex relaxation of the output of a neural network. Specifically, there now exists a tight relaxation for verifying the robustness of a neural network to $\ell_\infty$ input perturbations, as well as efficient primal and dual solvers for the relaxation. Buoyed by this success, we consider the problem of developing similar techniques for verifying robustness to input perturbations within the probability simplex. We prove a somewhat surprising result that, in this case, not only can one design a tight relaxation that overcomes the convex barrier, but the size of the relaxation remains linear in the number of neurons, thereby leading to simpler and more efficient algorithms. We establish the scalability of our overall approach via the specification of $\ell_1$ robustness for CIFAR-10 and MNIST classification, where our approach improves the state of the art verified accuracy by up to $14.4\%$. Furthermore, we establish its accuracy on a novel and highly challenging task of verifying the robustness of a multi-modal (text and image) classifier to arbitrary changes in its textual input.

Most real world applications require dealing with stochasticity like sensor noise or predictive uncertainty, where formal specifications of desired behavior are inherently probabilistic. Despite the promise of formal verification in ensuring the reliability of neural networks, progress in the direction of probabilistic specifications has been limited. In this direction, we first introduce a general formulation of probabilistic specifications for neural networks, which captures both probabilistic networks (e.g., Bayesian neural networks, MC-Dropout networks) and uncertain inputs (distributions over inputs arising from sensor noise or other perturbations). We then propose a general technique to verify such specifications by generalizing the notion of Lagrangian duality, replacing standard Lagrangian multipliers with "functional multipliers" that can be arbitrary functions of the activations at a given layer. We show that an optimal choice of functional multipliers leads to exact verification (i.e., sound and complete verification), and for specific forms of multipliers, we develop tractable practical verification algorithms. We empirically validate our algorithms by applying them to Bayesian Neural Networks (BNNs) and MC Dropout Networks, and certifying properties such as adversarial robustness and robust detection of out-of-distribution (OOD) data. On these tasks we are able to provide significantly stronger guarantees when compared to prior work -- for instance, for a VGG-64 MC-Dropout CNN trained on CIFAR-10 in a verification-agnostic manner, we improve the certified AUC (a verified lower bound on the true AUC) for robust OOD detection (on CIFAR-100) from $0 \% \rightarrow 29\%$. Similarly, for a BNN trained on MNIST, we improve on the $\ell_\infty$ robust accuracy from $60.2 \% \rightarrow 74.6\%$. Further, on a novel specification -- distributionally robust OOD detection -- we improve on the certified AUC from $5\% \rightarrow 23\%$.

We consider a robust optimization problem in an electric power system under uncertain demand and availability of renewable energy resources. Solving the deterministic alternating current (AC) optimal power flow (ACOPF) problem has been considered challenging since the 1960s due to its nonconvexity. Linear approximation of the AC power flow system sees pervasive use, but does not guarantee a physically feasible system configuration. In recent years, various convex relaxation schemes for the ACOPF problem have been investigated, and under some assumptions, a physically feasible solution can be recovered. Based on these convex relaxations, we construct a robust convex optimization problem with recourse to solve for optimal controllable injections (fossil fuel, nuclear, etc.) in electric power systems under uncertainty (renewable energy generation, demand fluctuation, etc.). We propose a cutting-plane method to solve this robust optimization problem, and we establish convergence and other desirable properties. Experimental results indicate that our robust convex relaxation of the ACOPF problem can provide a tight lower bound.

Recent research has made the surprising finding that state-of-the-art deep learning models sometimes fail to generalize to small variations of the input. Adversarial training has been shown to be an effective approach to overcome this problem. However, its application has been limited to enforcing invariance to analytically defined transformations like lp-norm bounded perturbations. Such perturbations do not necessarily cover plausible real-world variations that preserve the semantics of the input (such as a change in lighting conditions). In this paper, we propose a novel approach to express and formalize robustness to these kinds of real-world transformations of the input. The two key ideas underlying our formulation are (1) leveraging disentangled representations of the input to define different factors of variations, and (2) generating new input images by adversarially composing the representations of different images. We use a StyleGAN model to demonstrate the efficacy of this framework. Specifically, we leverage the disentangled latent representations computed by a StyleGAN model to generate perturbations of an image that are similar to real-world variations (like adding make-up, or changing the skin-tone of a person) and train models to be invariant to these perturbations. Extensive experiments show that our method improves generalization and reduces the effect of spurious correlations (reducing the error rate of a "smile" detector by 21% for example).

Modern recommender systems need to deal with multiple objectives like balancing user engagement with recommending diverse and fresh content. An appealing way to optimally trade these off is by imposing constraints on the ranking according to which items are presented to a user. This results in a constrained ranking optimization problem that can be solved as a linear program (LP). However, off-the-shelf LP solvers are unable to meet the severe latency constraints in systems that serve live traffic. To address this challenge, we exploit the structure of the dual optimization problem to develop a fast solver. We analyze theoretical properties of our solver and show experimentally that it is able to solve constrained ranking problems on synthetic and real-world recommendation datasets an order of magnitude faster than off-the-shelf solvers, thereby enabling their deployment under severe latency constraints.

Convex optimization problems with staged structure appear in several contexts, including optimal control, verification of deep neural networks, and isotonic regression. Off-the-shelf solvers can solve these problems but may scale poorly. We develop a nonconvex reformulation designed to exploit this staged structure. Our reformulation has only simple bound constraints, enabling solution via projected gradient methods and their accelerated variants. The method automatically generates a sequence of primal and dual feasible solutions to the original convex problem, making optimality certification easy. We establish theoretical properties of the nonconvex formulation, showing that it is (almost) free of spurious local minima and has the same global optimum as the convex problem. We modify projected gradient descent to avoid spurious local minimizers so it always converges to the global minimizer. For neural network verification, our approach obtains small duality gaps in only a few gradient steps. Consequently, it can provide tight duality gaps for many large-scale verification problems where both off-the-shelf and specialized solvers struggle.

Does a Variational AutoEncoder (VAE) consistently encode typical samples generated from its decoder? This paper shows that the perhaps surprising answer to this question is `No'; a (nominally trained) VAE does not necessarily amortize inference for typical samples that it is capable of generating. We study the implications of this behaviour on the learned representations and also the consequences of fixing it by introducing a notion of self consistency. Our approach hinges on an alternative construction of the variational approximation distribution to the true posterior of an extended VAE model with a Markov chain alternating between the encoder and the decoder. The method can be used to train a VAE model from scratch or given an already trained VAE, it can be run as a post processing step in an entirely self supervised way without access to the original training data. Our experimental analysis reveals that encoders trained with our self-consistency approach lead to representations that are robust (insensitive) to perturbations in the input introduced by adversarial attacks. We provide experimental results on the ColorMnist and CelebA benchmark datasets that quantify the properties of the learned representations and compare the approach with a baseline that is specifically trained for the desired property.

Convex relaxations have emerged as a promising approach for verifying properties of neural networks, but widely used using Linear Programming (LP) relaxations only provide meaningful certificates when networks are specifically trained to facilitate verification. This precludes many important applications which involve \emph{verification-agnostic} networks that are not trained specifically to promote verifiability. On the other hand, semidefinite programming (SDP) relaxations have shown success on verification-agnostic networks, such as adversarially trained image classifiers without additional regularization, but do not currently scale beyond small networks due to poor time and space asymptotics. In this work, we propose a first-order dual SDP algorithm that provides (1) any-time bounds (2) requires memory only linear in the total number of network activations and (3) has per-iteration complexity that scales linearly with the complexity of a forward/backward pass through the network. By exploiting iterative eigenvector methods, we express all solver operations in terms of forward and backward passes through the network, enabling efficient use of hardware optimized for deep learning. This allows us to dramatically improve the magnitude of $\ell_\infty$ perturbations for which we can verify robustness verification-agnostic networks ($1\% \to 88\%$ on MNIST, $6\%\to 40\%$ on CIFAR-10). We also demonstrate tight verification for a quadratic stability specification for the decoder of a variational autoencoder.

A fundamental component of neural network verification is the computation of bounds on the values their outputs can take. Previous methods have either used off-the-shelf solvers, discarding the problem structure, or relaxed the problem even further, making the bounds unnecessarily loose. We propose a novel approach based on Lagrangian Decomposition. Our formulation admits an efficient supergradient ascent algorithm, as well as an improved proximal algorithm. Both the algorithms offer three advantages: (i) they yield bounds that are provably at least as tight as previous dual algorithms relying on Lagrangian relaxations; (ii) they are based on operations analogous to forward/backward pass of neural networks layers and are therefore easily parallelizable, amenable to GPU implementation and able to take advantage of the convolutional structure of problems; and (iii) they allow for anytime stopping while still providing valid bounds. Empirically, we show that we obtain bounds comparable with off-the-shelf solvers in a fraction of their running time, and obtain tighter bounds in the same time as previous dual algorithms. This results in an overall speed-up when employing the bounds for formal verification. Code for our algorithms is available at https://github.com/oval-group/decomposition-plnn-bounds.

Neural architectures able to generate variable-length outputs are extremely effective for applications like Machine Translation and Image Captioning. In this paper, we study the vulnerability of these models to attacks aimed at changing the output-size that can have undesirable consequences including increased computation and inducing faults in downstream modules that expect outputs of a certain length. We show the existence and construction of such attacks with two key contributions. First, to overcome the difficulties of discrete search space and the non-differentiable adversarial objective function, we develop an easy-to-compute differentiable proxy objective that can be used with gradient-based algorithms to find output-lengthening inputs. Second, we develop a verification approach to formally prove that the network cannot produce outputs greater than a certain length. Experimental results on Machine Translation and Image Captioning models show that our adversarial output-lengthening approach can produce outputs that are 50 times longer than the input, while our verification approach can, given a model and input domain, prove that the output length is below a certain size.

Po-Sen Huang, Robert Stanforth, Johannes Welbl, Chris Dyer, Dani Yogatama, Sven Gowal, Krishnamurthy Dvijotham, Pushmeet Kohli. Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). 2019.

Recent work has shown that it is possible to train deep neural networks that are provably robust to norm-bounded adversarial perturbations. Most of these methods are based on minimizing an upper bound on the worst-case loss over all possible adversarial perturbations. While these techniques show promise, they often result in difficult optimization procedures that remain hard to scale to larger networks. Through a comprehensive analysis, we show how a simple bounding technique, interval bound propagation (IBP), can be exploited to train large provably robust neural networks that beat the state-of-the-art in verified accuracy. While the upper bound computed by IBP can be quite weak for general networks, we demonstrate that an appropriate loss and clever hyper-parameter schedule allow the network to adapt such that the IBP bound is tight. This results in a fast and stable learning algorithm that outperforms more sophisticated methods and achieves state-of-the-art results on MNIST, CIFAR-10 and SVHN. It also allows us to train the largest model to be verified beyond vacuous bounds on a downscaled version of IMAGENET.

This paper addressed the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (e.g., robustness to bounded norm adversarial perturbations). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications. We formulate verification as an optimization problem (seeking to find the largest violation of the specification) and solve a Lagrangian relaxation of the optimization problem to obtain an upper bound on the worst case violation of the specification being verified. Our approach is anytime, i.e., it can be stopped at any time and a valid bound on the maximum violation can be obtained. Finally, we highlight how this approach can be used to train models that are amenable to verification.

Adversarial training is an effective methodology for training deep neural networks that are robust against adversarial, norm-bounded perturbations. However, the computational cost of adversarial training grows prohibitively as the size of the model and number of input dimensions increase. Further, training against less expensive and therefore weaker adversaries produces models that are robust against weak attacks but break down under attacks that are stronger. This is often attributed to the phenomenon of gradient obfuscation; such models have a highly non-linear loss surface in the vicinity of training examples, making it hard for gradient-based attacks to succeed even though adversarial examples still exist. In this work, we introduce a novel regularizer that encourages the loss to behave linearly in the vicinity of the training data, thereby penalizing gradient obfuscation while encouraging robustness. We show via extensive experiments on CIFAR-10 and ImageNet, that models trained with our regularizer avoid gradient obfuscation and can be trained significantly faster than adversarial training. Using this regularizer, we exceed current state of the art and achieve 47% adversarial accuracy for ImageNet with L-infinity norm adversarial perturbations of radius 4/255 under an untargeted, strong, white-box attack. Additionally, we match state of the art results for CIFAR-10 at 8/255.

In both general equilibrium theory and game theory, the dominant mathematical models rest on a fully rational solution concept in which every player's action is a best-response to the actions of the other players. In both theories there is less agreement on suitable out- of-equilibrium modeling, but one attractive approach is the level k model in which a level 0 player adopts a very simple response to current conditions, a level 1 player best-responds to a model in which others take level 0 actions, and so forth. (This is analogous to k-ply exploration of game trees in AI, and to receding-horizon control in control theory.) If players have deterministic mental models with this kind of finite-level response, there is obviously no way their mental models can all be consistent. Nevertheless, there is experimental evidence that people act this way in many situations, motivating the question of what the dynamics of such interactions lead to. We address the problem of out-of-equilibrium price dynamics in the setting of Fisher markets. We develop a general framework in which sellers have (a) a set of atomic price update rules which are simple responses to a price vector; (b) a belief-formation procedure that simulates actions of other sellers (themselves using the atomic price updates) to some finite horizon in the future. In this framework, sellers use an atomic price update rule to respond to a price vector they generate with the belief formation procedure. The framework is general and allows sellers to have inconsistent and time- varying beliefs about each other. Under certain assumptions on the atomic update rules, we show that despite the inconsistent and time-varying nature of beliefs, the market converges to a unique equilibrium. (If the price updates are driven by weak-gross substitutes demands, this is the same equilibrium point predicted by those demands.) This result holds for both synchronous and asynchronous discrete-time updates. Moreover, the result is computationally feasible in the sense that the convergence rate is linear, i.e., the distance to equilibrium decays exponentially fast. To the best of our knowledge, this is the first result that demonstrates, in Fisher markets, convergence at any rate for dynamics driven by a plausible model of seller incentives. We then specialize our results to Fisher markets with elastic demands (a further special case corresponds to demand generated by buyers with constant elasticity of substitution (CES) utilities, in the weak gross substitutes (WGS) regime) and show that the atomic update rule in which a seller uses the best-response (=profit- maximizing) update given the prices of all other sellers, satisfies the assumptions required on atomic price update rules in our framework. We can even characterize the convergence rate (as a function of elasticity parameters of the demand function). Our results apply also to settings where, to the best of our knowledge, there exists no previous demonstration of efficient convergence of any discrete dynamic of price updates. Even for the simple case of (level 0) best- response dynamics, our result is the first to demonstrate a linear rate of convergence.

The problem of minimizing the rank of a matrix subject to linear equality constraints arises in applications in machine learning, dimensionality reduction, and control theory, and is known to be NP-hard. A popular heuristic minimizes the nuclear norm (sum of the singular values) of the matrix instead of the rank, and was recently shown to give an exact solution in several scenarios. In this paper, we present a new analysis for this heuristic based on a property of the nullspace of the operator defining the constraints, called the spherical section property. We give conditions for the exact recovery of all matrices up to a certain rank, and show that these conditions hold with high probability for operators generated from random Gaussian ensembles. Our analysis provides simpler proofs than existing isometry-based methods, as well as robust recovery results when the matrix is not exactly low-rank.