We propose a new algorithm for training deep neural networks (DNNs) with binary weights. In particular, we first cast the problem of training binary neural networks (BiNNs) as a bilevel optimization instance and subsequently construct flexible relaxations of this bilevel program. The resulting training method shares its algorithmic simplicity with several existing approaches to train BiNNs, in particular with the straight-through gradient estimator successfully employed in BinaryConnect and subsequent methods. In fact, our proposed method can be interpreted as an adaptive variant of the original straight-through estimator that conditionally (but not always) acts like a linear mapping in the backward pass of error propagation. Experimental results demonstrate that our new algorithm offers favorable performance compared to existing approaches.
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Kernel-based models such as kernel ridge regression and Gaussian processes are ubiquitous in machine learning applications for regression and optimization. It is well known that a serious downside for kernel-based models is the high computational cost; given a dataset of $n$ samples, the cost grows as $\mathcal{O}(n^3)$. Existing sparse approximation methods can yield a significant reduction in the computational cost, effectively reducing the real world cost down to as low as $\mathcal{O}(n)$ in certain cases. Despite this remarkable empirical success, significant gaps remain in the existing results for the analytical confidence bounds on the error due to approximation. In this work, we provide novel confidence intervals for the Nystr\"om method and the sparse variational Gaussian processes approximation method. Our confidence intervals lead to improved error bounds in both regression and optimization. We establish these confidence intervals using novel interpretations of the approximate (surrogate) posterior variance of the models.
We study ROUND-UFP and ROUND-SAP, two generalizations of the classical BIN PACKING problem that correspond to the unsplittable flow problem on a path (UFP) and the storage allocation problem (SAP), respectively. We are given a path with capacities on its edges and a set of tasks where for each task we are given a demand and a subpath. In ROUND-UFP, the goal is to find a packing of all tasks into a minimum number of copies (rounds) of the given path such that for each copy, the total demand of tasks on any edge does not exceed the capacity of the respective edge. In ROUND-SAP, the tasks are considered to be rectangles and the goal is to find a non-overlapping packing of these rectangles into a minimum number of rounds such that all rectangles lie completely below the capacity profile of the edges. We show that in contrast to BIN PACKING, both the problems do not admit an asymptotic polynomial-time approximation scheme (APTAS), even when all edge capacities are equal. However, for this setting, we obtain asymptotic $(2+\varepsilon)$-approximations for both problems. For the general case, we obtain an $O(\log\log n)$-approximation algorithm and an $O(\log\log\frac{1}{\delta})$-approximation under $(1+\delta)$-resource augmentation for both problems. For the intermediate setting of the no bottleneck assumption (i.e., the maximum task demand is at most the minimum edge capacity), we obtain absolute $12$- and asymptotic $(16+\varepsilon)$-approximation algorithms for ROUND-UFP and ROUND-SAP, respectively.
Binary neural networks (BNNs) represent original full-precision weights and activations into 1-bit with sign function. Since the gradient of the conventional sign function is almost zero everywhere which cannot be used for back-propagation, several attempts have been proposed to alleviate the optimization difficulty by using approximate gradient. However, those approximations corrupt the main direction of factual gradient. To this end, we propose to estimate the gradient of sign function in the Fourier frequency domain using the combination of sine functions for training BNNs, namely frequency domain approximation (FDA). The proposed approach does not affect the low-frequency information of the original sign function which occupies most of the overall energy, and high-frequency coefficients will be ignored to avoid the huge computational overhead. In addition, we embed a noise adaptation module into the training phase to compensate the approximation error. The experiments on several benchmark datasets and neural architectures illustrate that the binary network learned using our method achieves the state-of-the-art accuracy. Code will be available at \textit{//gitee.com/mindspore/models/tree/master/research/cv/FDA-BNN}.
Despite the considerable success of neural networks in security settings such as malware detection, such models have proved vulnerable to evasion attacks, in which attackers make slight changes to inputs (e.g., malware) to bypass detection. We propose a novel approach, \emph{Fourier stabilization}, for designing evasion-robust neural networks with binary inputs. This approach, which is complementary to other forms of defense, replaces the weights of individual neurons with robust analogs derived using Fourier analytic tools. The choice of which neurons to stabilize in a neural network is then a combinatorial optimization problem, and we propose several methods for approximately solving it. We provide a formal bound on the per-neuron drop in accuracy due to Fourier stabilization, and experimentally demonstrate the effectiveness of the proposed approach in boosting robustness of neural networks in several detection settings. Moreover, we show that our approach effectively composes with adversarial training.
One of the central problems in machine learning is domain adaptation. Unlike past theoretical work, we consider a new model for subpopulation shift in the input or representation space. In this work, we propose a provably effective framework for domain adaptation based on label propagation. In our analysis, we use a simple but realistic ``expansion'' assumption, proposed in \citet{wei2021theoretical}. Using a teacher classifier trained on the source domain, our algorithm not only propagates to the target domain but also improves upon the teacher. By leveraging existing generalization bounds, we also obtain end-to-end finite-sample guarantees on the entire algorithm. In addition, we extend our theoretical framework to a more general setting of source-to-target transfer based on a third unlabeled dataset, which can be easily applied in various learning scenarios.
Self-training algorithms, which train a model to fit pseudolabels predicted by another previously-learned model, have been very successful for learning with unlabeled data using neural networks. However, the current theoretical understanding of self-training only applies to linear models. This work provides a unified theoretical analysis of self-training with deep networks for semi-supervised learning, unsupervised domain adaptation, and unsupervised learning. At the core of our analysis is a simple but realistic ``expansion'' assumption, which states that a low-probability subset of the data must expand to a neighborhood with large probability relative to the subset. We also assume that neighborhoods of examples in different classes have minimal overlap. We prove that under these assumptions, the minimizers of population objectives based on self-training and input-consistency regularization will achieve high accuracy with respect to ground-truth labels. By using off-the-shelf generalization bounds, we immediately convert this result to sample complexity guarantees for neural nets that are polynomial in the margin and Lipschitzness. Our results help explain the empirical successes of recently proposed self-training algorithms which use input consistency regularization.
Contrastive learning (CL) is a popular technique for self-supervised learning (SSL) of visual representations. It uses pairs of augmentations of unlabeled training examples to define a classification task for pretext learning of a deep embedding. Despite extensive works in augmentation procedures, prior works do not address the selection of challenging negative pairs, as images within a sampled batch are treated independently. This paper addresses the problem, by introducing a new family of adversarial examples for constrastive learning and using these examples to define a new adversarial training algorithm for SSL, denoted as CLAE. When compared to standard CL, the use of adversarial examples creates more challenging positive pairs and adversarial training produces harder negative pairs by accounting for all images in a batch during the optimization. CLAE is compatible with many CL methods in the literature. Experiments show that it improves the performance of several existing CL baselines on multiple datasets.
Interpretation of Deep Neural Networks (DNNs) training as an optimal control problem with nonlinear dynamical systems has received considerable attention recently, yet the algorithmic development remains relatively limited. In this work, we make an attempt along this line by reformulating the training procedure from the trajectory optimization perspective. We first show that most widely-used algorithms for training DNNs can be linked to the Differential Dynamic Programming (DDP), a celebrated second-order trajectory optimization algorithm rooted in the Approximate Dynamic Programming. In this vein, we propose a new variant of DDP that can accept batch optimization for training feedforward networks, while integrating naturally with the recent progress in curvature approximation. The resulting algorithm features layer-wise feedback policies which improve convergence rate and reduce sensitivity to hyper-parameter over existing methods. We show that the algorithm is competitive against state-ofthe-art first and second order methods. Our work opens up new avenues for principled algorithmic design built upon the optimal control theory.
Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.
It is known that the current graph neural networks (GNNs) are difficult to make themselves deep due to the problem known as \textit{over-smoothing}. Multi-scale GNNs are a promising approach for mitigating the over-smoothing problem. However, there is little explanation of why it works empirically from the viewpoint of learning theory. In this study, we derive the optimization and generalization guarantees of transductive learning algorithms that include multi-scale GNNs. Using the boosting theory, we prove the convergence of the training error under weak learning-type conditions. By combining it with generalization gap bounds in terms of transductive Rademacher complexity, we show that a test error bound of a specific type of multi-scale GNNs that decreases corresponding to the depth under the conditions. Our results offer theoretical explanations for the effectiveness of the multi-scale structure against the over-smoothing problem. We apply boosting algorithms to the training of multi-scale GNNs for real-world node prediction tasks. We confirm that its performance is comparable to existing GNNs, and the practical behaviors are consistent with theoretical observations. Code is available at //github.com/delta2323/GB-GNN
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