Multi-Neuron Unleashes Expressivity of ReLU Networks Under Convex Relaxation

Neural work certification has established itself as a crucial tool for ensuring the robustness of neural networks. Certification methods typically rely on convex relaxations of the feasible output set to provide sound bounds. However, complete certification requires exact bounds, which strongly limits the expressivity of ReLU networks: even for the simple ``$\max$'' function in $\mathbb{R}^2$, there does not exist a ReLU network that expresses this function and can be exactly bounded by single-neuron relaxation methods. This raises the question whether there exists a convex relaxation that can provide exact bounds for general continuous piecewise linear functions in $\mathbb{R}^n$. In this work, we answer this question affirmatively by showing that (layer-wise) multi-neuron relaxation provides complete certification for general ReLU networks. Based on this novel result, we show that the expressivity of ReLU networks is no longer limited under multi-neuron relaxation. To the best of our knowledge, this is the first positive result on the completeness of convex relaxations, shedding light on the practice of certified robustness.

Average Certified Radius is a Poor Metric for Randomized Smoothing

Randomized smoothing is a popular approach for providing certified robustness guarantees against adversarial attacks, and has become a very active area of research. Over the past years, the average certified radius (ACR) has emerged as the single most important metric for comparing methods and tracking progress in the field. However, in this work, we show that ACR is an exceptionally poor metric for evaluating robustness guarantees provided by randomized smoothing. We theoretically show not only that a trivial classifier can have arbitrarily large ACR, but also that ACR is much more sensitive to improvements on easy samples than on hard ones. Empirically, we confirm that existing training strategies that improve ACR reduce the model's robustness on hard samples. Further, we show that by focusing on easy samples, we can effectively replicate the increase in ACR. We develop strategies, including explicitly discarding hard samples, reweighing the dataset with certified radius, and extreme optimization for easy samples, to achieve state-of-the-art ACR, although these strategies ignore robustness for the general data distribution. Overall, our results suggest that ACR has introduced a strong undesired bias to the field, and better metrics are required to holistically evaluate randomized smoothing.

CTBENCH: A Library and Benchmark for Certified Training

Training certifiably robust neural networks is an important but challenging task. While many algorithms for (deterministic) certified training have been proposed, they are often evaluated on different training schedules, certification methods, and systematically under-tuned hyperparameters, making it difficult to compare their performance. To address this challenge, we introduce CTBENCH, a unified library and a high-quality benchmark for certified training that evaluates all algorithms under fair settings and systematically tuned hyperparameters. We show that (1) almost all algorithms in CTBENCH surpass the corresponding reported performance in literature in the magnitude of algorithmic improvements, thus establishing new state-of-the-art, and (2) the claimed advantage of recent algorithms drops significantly when we enhance the outdated baselines with a fair training schedule, a fair certification method and well-tuned hyperparameters. Based on CTBENCH, we provide new insights into the current state of certified training and suggest future research directions. We are confident that CTBENCH will serve as a benchmark and testbed for future research in certified training.

Repeat-Aware Neighbor Sampling for Dynamic Graph Learning

Dynamic graph learning equips the edges with time attributes and allows multiple links between two nodes, which is a crucial technology for understanding evolving data scenarios like traffic prediction and recommendation systems. Existing works obtain the evolving patterns mainly depending on the most recent neighbor sequences. However, we argue that whether two nodes will have interaction with each other in the future is highly correlated with the same interaction that happened in the past. Only considering the recent neighbors overlooks the phenomenon of repeat behavior and fails to accurately capture the temporal evolution of interactions. To fill this gap, this paper presents RepeatMixer, which considers evolving patterns of first and high-order repeat behavior in the neighbor sampling strategy and temporal information learning. Firstly, we define the first-order repeat-aware nodes of the source node as the destination nodes that have interacted historically and extend this concept to high orders as nodes in the destination node's high-order neighbors. Then, we extract neighbors of the source node that interacted before the appearance of repeat-aware nodes with a slide window strategy as its neighbor sequence. Next, we leverage both the first and high-order neighbor sequences of source and destination nodes to learn temporal patterns of interactions via an MLP-based encoder. Furthermore, considering the varying temporal patterns on different orders, we introduce a time-aware aggregation mechanism that adaptively aggregates the temporal representations from different orders based on the significance of their interaction time sequences. Experimental results demonstrate the superiority of RepeatMixer over state-of-the-art models in link prediction tasks, underscoring the effectiveness of the proposed repeat-aware neighbor sampling strategy.

Overcoming the Paradox of Certified Training with Gaussian Smoothing

Training neural networks with high certified accuracy against adversarial examples remains an open problem despite significant efforts. While certification methods can effectively leverage tight convex relaxations for bound computation, in training, these methods perform worse than looser relaxations. Prior work hypothesized that this is caused by the discontinuity and perturbation sensitivity of the loss surface induced by these tighter relaxations. In this work, we show theoretically that Gaussian Loss Smoothing can alleviate both of these issues. We confirm this empirically by proposing a certified training method combining PGPE, an algorithm computing gradients of a smoothed loss, with different convex relaxations. When using this training method, we observe that tighter bounds indeed lead to strictly better networks that can outperform state-of-the-art methods on the same network. While scaling PGPE-based training remains challenging due to high computational cost, our results clearly demonstrate the promise of Gaussian Loss Smoothing for training certifiably robust neural networks.

Expressivity of ReLU-Networks under Convex Relaxations

Convex relaxations are a key component of training and certifying provably safe neural networks. However, despite substantial progress, a wide and poorly understood accuracy gap to standard networks remains, raising the question of whether this is due to fundamental limitations of convex relaxations. Initial work investigating this question focused on the simple and widely used IBP relaxation. It revealed that some univariate, convex, continuous piecewise linear (CPWL) functions cannot be encoded by any ReLU network such that its IBP-analysis is precise. To explore whether this limitation is shared by more advanced convex relaxations, we conduct the first in-depth study on the expressive power of ReLU networks across all commonly used convex relaxations. We show that: (i) more advanced relaxations allow a larger class of univariate functions to be expressed as precisely analyzable ReLU networks, (ii) more precise relaxations can allow exponentially larger solution spaces of ReLU networks encoding the same functions, and (iii) even using the most precise single-neuron relaxations, it is impossible to construct precisely analyzable ReLU networks that express multivariate, convex, monotone CPWL functions.

Understanding Certified Training with Interval Bound Propagation

As robustness verification methods are becoming more precise, training certifiably robust neural networks is becoming ever more relevant. To this end, certified training methods compute and then optimize an upper bound on the worst-case loss over a robustness specification. Curiously, training methods based on the imprecise interval bound propagation (IBP) consistently outperform those leveraging more precise bounding methods. Still, we lack an understanding of the mechanisms making IBP so successful. In this work, we thoroughly investigate these mechanisms by leveraging a novel metric measuring the tightness of IBP bounds. We first show theoretically that, for deep linear models, tightness decreases with width and depth at initialization, but improves with IBP training, given sufficient network width. We, then, derive sufficient and necessary conditions on weight matrices for IBP bounds to become exact and demonstrate that these impose strong regularization, explaining the empirically observed trade-off between robustness and accuracy in certified training. Our extensive experimental evaluation validates our theoretical predictions for ReLU networks, including that wider networks improve performance, yielding state-of-the-art results. Interestingly, we observe that while all IBP-based training methods lead to high tightness, this is neither sufficient nor necessary to achieve high certifiable robustness. This hints at the existence of new training methods that do not induce the strong regularization required for tight IBP bounds, leading to improved robustness and standard accuracy.

SMAP: A Novel Heterogeneous Information Framework for Scenario-based Optimal Model Assignment

The increasing maturity of big data applications has led to a proliferation of models targeting the same objectives within the same scenarios and datasets. However, selecting the most suitable model that considers model's features while taking specific requirements and constraints into account still poses a significant challenge. Existing methods have focused on worker-task assignments based on crowdsourcing, they neglect the scenario-dataset-model assignment problem. To address this challenge, a new problem named the Scenario-based Optimal Model Assignment (SOMA) problem is introduced and a novel framework entitled Scenario and Model Associative percepts (SMAP) is developed. SMAP is a heterogeneous information framework that can integrate various types of information to intelligently select a suitable dataset and allocate the optimal model for a specific scenario. To comprehensively evaluate models, a new score function that utilizes multi-head attention mechanisms is proposed. Moreover, a novel memory mechanism named the mnemonic center is developed to store the matched heterogeneous information and prevent duplicate matching. Six popular traffic scenarios are selected as study cases and extensive experiments are conducted on a dataset to verify the effectiveness and efficiency of SMAP and the score function.

TAPS: Connecting Certified and Adversarial Training

Training certifiably robust neural networks remains a notoriously hard problem. On one side, adversarial training optimizes under-approximations of the worst-case loss, which leads to insufficient regularization for certification, while on the other, sound certified training methods optimize loose over-approximations, leading to over-regularization and poor (standard) accuracy. In this work we propose TAPS, an (unsound) certified training method that combines IBP and PGD training to yield precise, although not necessarily sound, worst-case loss approximations, reducing over-regularization and increasing certified and standard accuracies. Empirically, TAPS achieves a new state-of-the-art in many settings, e.g., reaching a certified accuracy of $22\%$ on TinyImageNet for $\ell_\infty$-perturbations with radius $\epsilon=1/255$.

"Is your explanation stable?": A Robustness Evaluation Framework for Feature Attribution

Understanding the decision process of neural networks is hard. One vital method for explanation is to attribute its decision to pivotal features. Although many algorithms are proposed, most of them solely improve the faithfulness to the model. However, the real environment contains many random noises, which may leads to great fluctuations in the explanations. More seriously, recent works show that explanation algorithms are vulnerable to adversarial attacks. All of these make the explanation hard to trust in real scenarios. To bridge this gap, we propose a model-agnostic method \emph{Median Test for Feature Attribution} (MeTFA) to quantify the uncertainty and increase the stability of explanation algorithms with theoretical guarantees. MeTFA has the following two functions: (1) examine whether one feature is significantly important or unimportant and generate a MeTFA-significant map to visualize the results; (2) compute the confidence interval of a feature attribution score and generate a MeTFA-smoothed map to increase the stability of the explanation. Experiments show that MeTFA improves the visual quality of explanations and significantly reduces the instability while maintaining the faithfulness. To quantitatively evaluate the faithfulness of an explanation under different noise settings, we further propose several robust faithfulness metrics. Experiment results show that the MeTFA-smoothed explanation can significantly increase the robust faithfulness. In addition, we use two scenarios to show MeTFA's potential in the applications. First, when applied to the SOTA explanation method to locate context bias for semantic segmentation models, MeTFA-significant explanations use far smaller regions to maintain 99\%+ faithfulness. Second, when tested with different explanation-oriented attacks, MeTFA can help defend vanilla, as well as adaptive, adversarial attacks against explanations.