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1. Addressing Spectral Bias of Deep Neural Networks by Multi-Grade Deep Learning

   Fang, Ronglong; Xu, Yuesheng (December 2024, 38th Conference on Neural Information Processing Systems (NeurIPS 2024).)

   Deep neural networks (DNNs) have showcased their remarkable precision in approximating smooth functions. However, they suffer from the spectral bias, wherein DNNs typically exhibit a tendency to prioritize the learning of lower-frequency components of a function, struggling to effectively capture its high-frequency features. This paper is to address this issue. Notice that a function having only low frequency components may be well-represented by a shallow neural network (SNN), a network having only a few layers. By observing that composition of low frequency functions can effectively approximate a high-frequency function, we propose to learn a function containing high-frequency components by composing several SNNs, each of which learns certain low-frequency information from the given data. We implement the proposed idea by exploiting the multi-grade deep learning (MGDL) model, a recently introduced model that trains a DNN incrementally, grade by grade, a current grade learning from the residue of the previous grade only an SNN (with trainable parameters) composed with the SNNs (with fixed parameters) trained in the preceding grades as features. We apply MGDL to synthetic, manifold, colored images, and MNIST datasets, all characterized by presence of high-frequency features. Our study reveals that MGDL excels at representing functions containing high-frequency information. Specifically, the neural networks learned in each grade adeptly capture some low-frequency information, allowing their compositions with SNNs learned in the previous grades effectively representing the high-frequency features. Our experimental results underscore the efficacy of MGDL in addressing the spectral bias inherent in DNNs. By leveraging MGDL, we offer insights into overcoming spectral bias limitation of DNNs, thereby enhancing the performance and applicability of deep learning models in tasks requiring the representation of high-frequency information. This study confirms that the proposed method offers a promising solution to address the spectral bias of DNNs. The code is available on GitHub: Addressing Spectral Bias via MGDL. 

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2. Getting the Best Bang For Your Buck: Choosing What to Evaluate for Faster Bayesian Optimization

   Iqbal, Md Shahriar; Su, Jianhai; Kotthoff, Lars; Jamshidi, Pooyan (May 2022, Proceedings of the First International Conference on Automated Machine Learning, Late Break Abstracts, PMLR 188:6)

   Machine learning system design frequently necessitates balancing multiple objectives, such as prediction error and energy consumption for deep neural networks (DNNs). Typically, no single design performs well across all objectives; thus, finding Pareto-optimal designs is of interest. Measuring different objectives frequently incurs different costs; for example, measuring the prediction error of DNNs is significantly more expensive than measuring the energy consumption of a pre-trained DNN because it requires re-training the DNN. Current state-of-the-art methods do not account for this difference in objective evaluation cost, potentially wasting costly evaluations of objective functions for little information gain. To address this issue, we propose a novel cost-aware decoupled approach that weights the improvement of the hypervolume of the Pareto region by the measurement cost of each objective. We perform experiments on a of range of DNN applications for comprehensive evaluation of our approach. 

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3. Getting the Best Bang For Your Buck: Choosing What to Evaluate for Faster Bayesian Optimization

   Iqbal, Shahriar; Su, Jianhai; Kotthoff, Lars; Jamshidi, Pooyan (June 2022, 1st International Conference on Automated Machine Learning)

   Machine learning system design frequently necessitates balancing multiple objectives, such as prediction error and energy consumption, for deep neural networks (DNNs). Typically, no single design performs well across all objectives; thus, finding Pareto-optimal designs is of interest. Measuring different objectives frequently incurs different costs; for example, measuring the prediction error of DNNs is significantly more expensive than measuring the energy consumption of a pre-trained DNN because it requires re-training the DNN. Current state-of-the-art methods do not account for this difference in objective evaluation cost, potentially wasting costly evaluations of objective functions for little information gain. To address this issue, we propose a novel cost-aware decoupled approach that weights the improvement of the hypervolume of the Pareto region by the measurement cost of each objective. To evaluate our approach, we perform experiments on several machine learning systems deployed on energy constraints environments. 

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4. Multi-grade Deep Learning

   https://doi.org/10.1007/s42967-024-00474-y  

   Xu, Yuesheng (March 2025, Communications on Applied Mathematics and Computation)

   Abstract Deep learning requires solving a nonconvex optimization problem of a large size to learn a deep neural network (DNN). The current deep learning model is of asingle-grade, that is, it trains a DNN end-to-end, by solving a single nonconvex optimization problem. When the layer number of the neural network is large, it is computationally challenging to carry out such a task efficiently. The complexity of the task comes from learning all weight matrices and bias vectors from one single nonconvex optimization problem of a large size. Inspired by the human education process which arranges learning in grades, we propose a multi-grade learning model: instead of solving one single optimization problem of a large size, we successively solve a number of optimization problems of small sizes, which are organized in grades, to learn a shallow neural network (a network having a few hidden layers) for each grade. Specifically, the current grade is to learn the leftover from the previous grade. In each of the grades, we learn a shallow neural network stacked on the top of the neural network, learned in the previous grades, whose parameters remain unchanged in training of the current and future grades. By dividing the task of learning a DDN into learning several shallow neural networks, one can alleviate the severity of the nonconvexity of the original optimization problem of a large size. When all grades of the learning are completed, the final neural network learned is astair-shapeneural network, which is thesuperpositionof networks learned from all grades. Such a model enables us to learn a DDN much more effectively and efficiently. Moreover, multi-grade learning naturally leads to adaptive learning. We prove that in the context of function approximation if the neural network generated by a new grade is nontrivial, the optimal error of a new grade is strictly reduced from the optimal error of the previous grade. Furthermore, we provide numerical examples which confirm that the proposed multi-grade model outperforms significantly the standard single-grade model and is much more robust to noise than the single-grade model. They include three proof-of-concept examples, classification on two benchmark data sets MNIST and Fashion MNIST with two noise rates, which is to find classifiers, functions of 784 dimensions, and as well as numerical solutions of the one-dimensional Helmholtz equation. 

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5. DNN Verification, Reachability, and the Exponential Function Problem

   Isac, Omri; Zohar, Yoni; Barrett, Clark; Katz, Guy (September 2023, Proceedings of the 34th International Conference on Concurrency Theory (CONCUR 2023))

   Pérez, Guillermo A.; Raskin, Jean-François (Ed.)

   Deep neural networks (DNNs) are increasingly being deployed to perform safety-critical tasks. The opacity of DNNs, which prevents humans from reasoning about them, presents new safety and security challenges. To address these challenges, the verification community has begun developing techniques for rigorously analyzing DNNs, with numerous verification algorithms proposed in recent years. While a significant amount of work has gone into developing these verification algorithms, little work has been devoted to rigorously studying the computability and complexity of the underlying theoretical problems. Here, we seek to contribute to the bridging of this gap. We focus on two kinds of DNNs: those that employ piecewise-linear activation functions (e.g., ReLU), and those that employ piecewise-smooth activation functions (e.g., Sigmoids). We prove the two following theorems: 1. The decidability of verifying DNNs with piecewise-smooth activation functions is equivalent to a well-known, open problem formulated by Tarski; and 2. The DNN verification problem for any quantifier-free linear arithmetic specification can be reduced to the DNN reachability problem, whose approximation is NP-complete. These results answer two fundamental questions about the computability and complexity of DNN verification, and the ways it is affected by the network’s activation functions and error tolerance; and could help guide future efforts in developing DNN verification tools. 

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