Embedding Inequalities for Barron-type Spaces

One of the fundamental problems in deep learning theory is understanding the approximation and generalization properties of two-layer neural networks in high dimensions. In order to tackle this issue, researchers have introduced the Barron space $\mathcal{B}_s(\Omega)$ and the spectral Barron space $\mathcal{F}_s(\Omega)$, where the index $s$ characterizes the smoothness of functions within these spaces and $\Omega\subset\mathbb{R}^d$ represents the input domain. However, it is still not clear what is the relationship between the two types of Barron spaces. In this paper, we establish continuous embeddings between these spaces as implied by the following inequality: for any $\delta\in (0,1), s\in \mathbb{N}^{+}$ and $f: \Omega \mapsto\mathbb{R}$, it holds that \[ \delta\gamma^{\delta-s}_{\Omega}\|f\|_{\mathcal{F}_{s-\delta}(\Omega)}\lesssim_s \|f\|_{\mathcal{B}_s(\Omega)}\lesssim_s \|f\|_{\mathcal{F}_{s+1}(\Omega)}, \] where $\gamma_{\Omega}=\sup_{\|v\|_2=1,x\in\Omega}|v^Tx|$ and notably, the hidden constants depend solely on the value of $s$. Furthermore, we provide examples to demonstrate that the lower bound is tight.

Theoretical Analysis of Inductive Biases in Deep Convolutional Networks

In this paper, we study the inductive biases in convolutional neural networks (CNNs), which are believed to be vital drivers behind CNNs' exceptional performance on vision-like tasks. We first analyze the universality of CNNs, i.e., the ability to approximate continuous functions. We prove that a depth of $\mathcal{O}(\log d)$ is sufficient for achieving universality, where $d$ is the input dimension. This is a significant improvement over existing results that required a depth of $\Omega(d)$. We also prove that learning sparse functions with CNNs needs only $\tilde{\mathcal{O}}(\log^2d)$ samples, indicating that deep CNNs can efficiently capture long-range sparse correlations. Note that all these are achieved through a novel combination of increased network depth and the utilization of multichanneling and downsampling. Lastly, we study the inductive biases of weight sharing and locality through the lens of symmetry. To separate two biases, we introduce locally-connected networks (LCNs), which can be viewed as CNNs without weight sharing. Specifically, we compare the performance of CNNs, LCNs, and fully-connected networks (FCNs) on a simple regression task. We prove that LCNs require ${\Omega}(d)$ samples while CNNs need only $\tilde{\mathcal{O}}(\log^2d)$ samples, which highlights the cruciality of weight sharing. We also prove that FCNs require $\Omega(d^2)$ samples while LCNs need only $\tilde{\mathcal{O}}(d)$ samples, demonstrating the importance of locality. These provable separations quantify the difference between the two biases, and our major observation behind is that weight sharing and locality break different symmetries in the learning process.

A duality framework for generalization analysis of random feature models and two-layer neural networks

We consider the problem of learning functions in the $\mathcal{F}_{p,\pi}$ and Barron spaces, which are natural function spaces that arise in the high-dimensional analysis of random feature models (RFMs) and two-layer neural networks. Through a duality analysis, we reveal that the approximation and estimation of these spaces can be considered equivalent in a certain sense. This enables us to focus on the easier problem of approximation and estimation when studying the generalization of both models. The dual equivalence is established by defining an information-based complexity that can effectively control estimation errors. Additionally, we demonstrate the flexibility of our duality framework through comprehensive analyses of two concrete applications. The first application is to study learning functions in $\mathcal{F}_{p,\pi}$ with RFMs. We prove that the learning does not suffer from the curse of dimensionality as long as $p>1$, implying RFMs can work beyond the kernel regime. Our analysis extends existing results [CMM21] to the noisy case and removes the requirement of overparameterization. The second application is to investigate the learnability of reproducing kernel Hilbert space (RKHS) under the $L^\infty$ metric. We derive both lower and upper bounds of the minimax estimation error by using the spectrum of the associated kernel. We then apply these bounds to dot-product kernels and analyze how they scale with the input dimension. Our results suggest that learning with ReLU (random) features is generally intractable in terms of reaching high uniform accuracy.

Self-supervised multimodal neuroimaging yields predictive representations for a spectrum of Alzheimer's phenotypes

Recent neuroimaging studies that focus on predicting brain disorders via modern machine learning approaches commonly include a single modality and rely on supervised over-parameterized models.However, a single modality provides only a limited view of the highly complex brain. Critically, supervised models in clinical settings lack accurate diagnostic labels for training. Coarse labels do not capture the long-tailed spectrum of brain disorder phenotypes, which leads to a loss of generalizability of the model that makes them less useful in diagnostic settings. This work presents a novel multi-scale coordinated framework for learning multiple representations from multimodal neuroimaging data. We propose a general taxonomy of informative inductive biases to capture unique and joint information in multimodal self-supervised fusion. The taxonomy forms a family of decoder-free models with reduced computational complexity and a propensity to capture multi-scale relationships between local and global representations of the multimodal inputs. We conduct a comprehensive evaluation of the taxonomy using functional and structural magnetic resonance imaging (MRI) data across a spectrum of Alzheimer's disease phenotypes and show that self-supervised models reveal disorder-relevant brain regions and multimodal links without access to the labels during pre-training. The proposed multimodal self-supervised learning yields representations with improved classification performance for both modalities. The concomitant rich and flexible unsupervised deep learning framework captures complex multimodal relationships and provides predictive performance that meets or exceeds that of a more narrow supervised classification analysis. We present elaborate quantitative evidence of how this framework can significantly advance our search for missing links in complex brain disorders.

Improving Electricity Market Economy via Closed-Loop Predict-and-Optimize

The electricity market clearing is usually implemented via an open-loop predict-then-optimize (O-PO) process: it first predicts the available power of renewable energy sources (RES) and the system reserve requirements; then, given the predictions, the markets are cleared via optimization models, i.e., unit commitment (UC) and economic dispatch (ED), to pursue the optimal electricity market economy. However, the market economy could suffer from the open-loop process because its predictions may be overly myopic to the optimizations, i.e., the predictions seek to improve the immediate statistical forecasting errors instead of the ultimate market economy. To this end, this paper proposes a closed-loop predict-and-optimize (C-PO) framework based on the tri-level mixed-integer programming, which trains economy-oriented predictors tailored for the market-clearing optimization to improve the ultimate market economy. Specifically, the upper level trains the economy-oriented RES and reserve predictors according to their induced market economy; the middle and lower levels, with given predictions, mimic the market-clearing process and feed the induced market economy results back to the upper level. The trained economy-oriented predictors are then embedded into the UC model, forming a prescriptive UC model that can simultaneously provide RES-reserve predictions and UC decisions with enhanced market economy. Numerical case studies on an IEEE 118-bus system illustrate potential economic and practical advantages of C-PO over O-PO, robust UC, and stochastic UC.

REZCR: A Zero-shot Character Recognition Method via Radical Extraction

The long-tail effect is a common issue that limits the performance of deep learning models on real-world datasets. Character image dataset development is also affected by such unbalanced data distribution due to differences in character usage frequency. Thus, current character recognition methods are limited when applying to real-world datasets, in particular to the character categories in the tail which are lacking training samples, e.g., uncommon characters, or characters from historical documents. In this paper, we propose a zero-shot character recognition framework via radical extraction, i.e., REZCR, to improve the recognition performance of few-sample character categories, in which we exploit information on radicals, the graphical units of characters, by decomposing and reconstructing characters following orthography. REZCR consists of an attention-based radical information extractor (RIE) and a knowledge graph-based character reasoner (KGR). The RIE aims to recognize candidate radicals and their possible structural relations from character images. The results will be fed into KGR to recognize the target character by reasoning with a pre-designed character knowledge graph. We validate our method on multiple datasets, REZCR shows promising experimental results, especially for few-sample character datasets.

When does SGD favor flat minima? A quantitative characterization via linear stability

The observation that stochastic gradient descent (SGD) favors flat minima has played a fundamental role in understanding implicit regularization of SGD and guiding the tuning of hyperparameters. In this paper, we provide a quantitative explanation of this striking phenomenon by relating the particular noise structure of SGD to its \emph{linear stability} (Wu et al., 2018). Specifically, we consider training over-parameterized models with square loss. We prove that if a global minimum $\theta^*$ is linearly stable for SGD, then it must satisfy $\|H(\theta^*)\|_F\leq O(\sqrt{B}/\eta)$, where $\|H(\theta^*)\|_F, B,\eta$ denote the Frobenius norm of Hessian at $\theta^*$, batch size, and learning rate, respectively. Otherwise, SGD will escape from that minimum \emph{exponentially} fast. Hence, for minima accessible to SGD, the flatness -- as measured by the Frobenius norm of the Hessian -- is bounded independently of the model size and sample size. The key to obtaining these results is exploiting the particular geometry awareness of SGD noise: 1) the noise magnitude is proportional to loss value; 2) the noise directions concentrate in the sharp directions of local landscape. This property of SGD noise provably holds for linear networks and random feature models (RFMs) and is empirically verified for nonlinear networks. Moreover, the validity and practical relevance of our theoretical findings are justified by extensive numerical experiments.

HoVer-Trans: Anatomy-aware HoVer-Transformer for ROI-free Breast Cancer Diagnosis in Ultrasound Images

Ultrasonography is an important routine examination for breast cancer diagnosis, due to its non-invasive, radiation-free and low-cost properties. However, it is still not the first-line screening test for breast cancer due to its inherent limitations. It would be a tremendous success if we can precisely diagnose breast cancer by breast ultrasound images (BUS). Many learning-based computer-aided diagnostic methods have been proposed to achieve breast cancer diagnosis/lesion classification. However, most of them require a pre-define ROI and then classify the lesion inside the ROI. Conventional classification backbones, such as VGG16 and ResNet50, can achieve promising classification results with no ROI requirement. But these models lack interpretability, thus restricting their use in clinical practice. In this study, we propose a novel ROI-free model for breast cancer diagnosis in ultrasound images with interpretable feature representations. We leverage the anatomical prior knowledge that malignant and benign tumors have different spatial relationships between different tissue layers, and propose a HoVer-Transformer to formulate this prior knowledge. The proposed HoVer-Trans block extracts the inter- and intra-layer spatial information horizontally and vertically. We conduct and release an open dataset GDPH&GYFYY for breast cancer diagnosis in BUS. The proposed model is evaluated in three datasets by comparing with four CNN-based models and two vision transformer models via a five-fold cross validation. It achieves state-of-the-art classification performance with the best model interpretability.

The Multiscale Structure of Neural Network Loss Functions: The Effect on Optimization and Origin

Local quadratic approximation has been extensively used to study the optimization of neural network loss functions around the minimum. Though, it usually holds in a very small neighborhood of the minimum, and cannot explain many phenomena observed during the optimization process. In this work, we study the structure of neural network loss functions and its implication on optimization in a region beyond the reach of good quadratic approximation. Numerically, we observe that neural network loss functions possesses a multiscale structure, manifested in two ways: (1) in a neighborhood of minima, the loss mixes a continuum of scales and grows subquadratically, and (2) in a larger region, the loss shows several separate scales clearly. Using the subquadratic growth, we are able to explain the Edge of Stability phenomenon[4] observed for gradient descent (GD) method. Using the separate scales, we explain the working mechanism of learning rate decay by simple examples. Finally, we study the origin of the multiscale structure and propose that the non-uniformity of training data is one of its cause. By constructing a two-layer neural network problem we show that training data with different magnitudes give rise to different scales of the loss function, producing subquadratic growth or multiple separate scales.

Exploiting the Potential of Datasets: A Data-Centric Approach for Model Robustness

Robustness of deep neural networks (DNNs) to malicious perturbations is a hot topic in trustworthy AI. Existing techniques obtain robust models given fixed datasets, either by modifying model structures, or by optimizing the process of inference or training. While significant improvements have been made, the possibility of constructing a high-quality dataset for model robustness remain unexplored. Follow the campaign of data-centric AI launched by Andrew Ng, we propose a novel algorithm for dataset enhancement that works well for many existing DNN models to improve robustness. Transferable adversarial examples and 14 kinds of common corruptions are included in our optimized dataset. In the data-centric robust learning competition hosted by Alibaba Group and Tsinghua University, our algorithm came third out of more than 3000 competitors in the first stage while we ranked fourth in the second stage. Our code is available at \url{https://github.com/hncszyq/tianchi_challenge}.