Expressive Losses for Verified Robustness via Convex Combinations

In order to train networks for verified adversarial robustness, previous work typically over-approximates the worst-case loss over (subsets of) perturbation regions or induces verifiability on top of adversarial training. The key to state-of-the-art performance lies in the expressivity of the employed loss function, which should be able to match the tightness of the verifiers to be employed post-training. We formalize a definition of expressivity, and show that it can be satisfied via simple convex combinations between adversarial attacks and IBP bounds. We then show that the resulting algorithms, named CC-IBP and MTL-IBP, yield state-of-the-art results across a variety of settings in spite of their conceptual simplicity. In particular, for $\ell_\infty$ perturbations of radius $\frac{1}{255}$ on TinyImageNet and downscaled ImageNet, MTL-IBP improves on the best standard and verified accuracies from the literature by from $1.98\%$ to $3.92\%$ points while only relying on single-step adversarial attacks.

IBP Regularization for Verified Adversarial Robustness via Branch-and-Bound

Recent works have tried to increase the verifiability of adversarially trained networks by running the attacks over domains larger than the original perturbations and adding various regularization terms to the objective. However, these algorithms either underperform or require complex and expensive stage-wise training procedures, hindering their practical applicability. We present IBP-R, a novel verified training algorithm that is both simple and effective. IBP-R induces network verifiability by coupling adversarial attacks on enlarged domains with a regularization term, based on inexpensive interval bound propagation, that minimizes the gap between the non-convex verification problem and its approximations. By leveraging recent branch-and-bound frameworks, we show that IBP-R obtains state-of-the-art verified robustness-accuracy trade-offs for small perturbations on CIFAR-10 while training significantly faster than relevant previous work. Additionally, we present UPB, a novel branching strategy that, relying on a simple heuristic based on $\beta$-CROWN, reduces the cost of state-of-the-art branching algorithms while yielding splits of comparable quality.

In Defense of the Unitary Scalarization for Deep Multi-Task Learning

Recent multi-task learning research argues against unitary scalarization, where training simply minimizes the sum of the task losses. Several ad-hoc multi-task optimization algorithms have instead been proposed, inspired by various hypotheses about what makes multi-task settings difficult. The majority of these optimizers require per-task gradients, and introduce significant memory, runtime, and implementation overhead. We present a theoretical analysis suggesting that many specialized multi-task optimizers can be interpreted as forms of regularization. Moreover, we show that, when coupled with standard regularization and stabilization techniques from single-task learning, unitary scalarization matches or improves upon the performance of complex multi-task optimizers in both supervised and reinforcement learning settings. We believe our results call for a critical reevaluation of recent research in the area.

Improved Branch and Bound for Neural Network Verification via Lagrangian Decomposition

We improve the scalability of Branch and Bound (BaB) algorithms for formally proving input-output properties of neural networks. First, we propose novel bounding algorithms based on Lagrangian Decomposition. Previous works have used off-the-shelf solvers to solve relaxations at each node of the BaB tree, or constructed weaker relaxations that can be solved efficiently, but lead to unnecessarily weak bounds. Our formulation restricts the optimization to a subspace of the dual domain that is guaranteed to contain the optimum, resulting in accelerated convergence. Furthermore, it allows for a massively parallel implementation, which is amenable to GPU acceleration via modern deep learning frameworks. Second, we present a novel activation-based branching strategy. By coupling an inexpensive heuristic with fast dual bounding, our branching scheme greatly reduces the size of the BaB tree compared to previous heuristic methods. Moreover, it performs competitively with a recent strategy based on learning algorithms, without its large offline training cost. Finally, we design a BaB framework, named Branch and Dual Network Bound (BaDNB), based on our novel bounding and branching algorithms. We show that BaDNB outperforms previous complete verification systems by a large margin, cutting average verification times by factors up to 50 on adversarial robustness properties.

Scaling the Convex Barrier with Sparse Dual Algorithms

Tight and efficient neural network bounding is crucial to the scaling of neural network verification systems. Many efficient bounding algorithms have been presented recently, but they are often too loose to verify more challenging properties. This is due to the weakness of the employed relaxation, which is usually a linear program of size linear in the number of neurons. While a tighter linear relaxation for piecewise-linear activations exists, it comes at the cost of exponentially many constraints and currently lacks an efficient customized solver. We alleviate this deficiency by presenting two novel dual algorithms: one operates a subgradient method on a small active set of dual variables, the other exploits the sparsity of Frank-Wolfe type optimizers to incur only a linear memory cost. Both methods recover the strengths of the new relaxation: tightness and a linear separation oracle. At the same time, they share the benefits of previous dual approaches for weaker relaxations: massive parallelism, GPU implementation, low cost per iteration and valid bounds at any time. As a consequence, we can obtain better bounds than off-the-shelf solvers in only a fraction of their running time, attaining significant formal verification speed-ups.

Scaling the Convex Barrier with Active Sets

Tight and efficient neural network bounding is of critical importance for the scaling of neural network verification systems. A number of efficient specialised dual solvers for neural network bounds have been presented recently, but they are often too loose to verify more challenging properties. This lack of tightness is linked to the weakness of the employed relaxation, which is usually a linear program of size linear in the number of neurons. While a tighter linear relaxation for piecewise linear activations exists, it comes at the cost of exponentially many constraints and thus currently lacks an efficient customised solver. We alleviate this deficiency via a novel dual algorithm that realises the full potential of the new relaxation by operating on a small active set of dual variables. Our method recovers the strengths of the new relaxation in the dual space: tightness and a linear separation oracle. At the same time, it shares the benefits of previous dual approaches for weaker relaxations: massive parallelism, GPU implementation, low cost per iteration and valid bounds at any time. As a consequence, we obtain better bounds than off-the-shelf solvers in only a fraction of their running time and recover the speed-accuracy trade-offs of looser dual solvers if the computational budget is small. We demonstrate that this results in significant formal verification speed-ups.

Lagrangian Decomposition for Neural Network Verification

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.

Sampling Acquisition Functions for Batch Bayesian Optimization

This paper presents Acquisition Thompson Sampling (ATS), a novel algorithm for batch Bayesian Optimization (BO) based on the idea of sampling multiple acquisition functions from a stochastic process. We define this process through the dependency of the acquisition functions on a set of model parameters. ATS is conceptually simple, straightforward to implement and, unlike other batch BO methods, it can be employed to parallelize any sequential acquisition function. In order to improve performance for multi-modal tasks, we show that ATS can be combined with existing techniques in order to realize different explore-exploit trade-offs and take into account pending function evaluations. We present experiments on a variety of benchmark functions and on the hyper-parameter optimization of a popular gradient boosting tree algorithm. These demonstrate the competitiveness of our algorithm with two state-of-the-art batch BO methods, and its advantages to classical parallel Thompson Sampling for BO.

Benchmarking and Optimization of Gradient Boosting Decision Tree Algorithms

Gradient boosting decision trees (GBDTs) have seen widespread adoption in academia, industry and competitive data science due to their state-of-the-art performance in many machine learning tasks. One relative downside to these models is the large number of hyper-parameters that they expose to the end-user. To maximize the predictive power of GBDT models, one must either manually tune the hyper-parameters, or utilize automated techniques such as those based on Bayesian optimization. Both of these approaches are time-consuming since they involve repeatably training the model for different sets of hyper-parameters. A number of software GBDT packages have started to offer GPU acceleration which can help to alleviate this problem. In this paper, we consider three such packages: XGBoost, LightGBM and Catboost. Firstly, we evaluate the performance of the GPU acceleration provided by these packages using large-scale datasets with varying shapes, sparsities and learning tasks. Then, we compare the packages in the context of hyper-parameter optimization, both in terms of how quickly each package converges to a good validation score, and in terms of generalization performance.