Neural collapse (NC) refers to the surprising structure of the last layer of deep neural networks in the terminal phase of gradient descent training. Recently, an increasing amount of experimental evidence has pointed to the propagation of NC to earlier layers of neural networks. However, while the NC in the last layer is well studied theoretically, much less is known about its multi-layered counterpart - deep neural collapse (DNC). In particular, existing work focuses either on linear layers or only on the last two layers at the price of an extra assumption. Our paper fills this gap by generalizing the established analytical framework for NC - the unconstrained features model - to multiple non-linear layers. Our key technical contribution is to show that, in a deep unconstrained features model, the unique global optimum for binary classification exhibits all the properties typical of DNC. This explains the existing experimental evidence of DNC. We also empirically show that (i) by optimizing deep unconstrained features models via gradient descent, the resulting solution agrees well with our theory, and (ii) trained networks recover the unconstrained features suitable for the occurrence of DNC, thus supporting the validity of this modeling principle.
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Deep learning is also known as hierarchical learning, where the learner _learns_ to represent a complicated target function by decomposing it into a sequence of simpler functions to reduce sample and time complexity. This paper formally analyzes how multi-layer neural networks can perform such hierarchical learning _efficiently_ and _automatically_ by SGD on the training objective. On the conceptual side, we present a theoretical characterizations of how certain types of deep (i.e. super-constant layer) neural networks can still be sample and time efficiently trained on some hierarchical tasks, when no existing algorithm (including layerwise training, kernel method, etc) is known to be efficient. We establish a new principle called "backward feature correction", where the errors in the lower-level features can be automatically corrected when training together with the higher-level layers. We believe this is a key behind how deep learning is performing deep (hierarchical) learning, as opposed to layerwise learning or simulating some non-hierarchical method. On the technical side, we show for every input dimension $d > 0$, there is a concept class of degree $\omega(1)$ multi-variate polynomials so that, using $\omega(1)$-layer neural networks as learners, SGD can learn any function from this class in $\mathsf{poly}(d)$ time to any $\frac{1}{\mathsf{poly}(d)}$ error, through learning to represent it as a composition of $\omega(1)$ layers of quadratic functions using "backward feature correction." In contrast, we do not know any other simpler algorithm (including layerwise training, applying kernel method sequentially, training a two-layer network, etc) that can learn this concept class in $\mathsf{poly}(d)$ time even to any $d^{-0.01}$ error. As a side result, we prove $d^{\omega(1)}$ lower bounds for several non-hierarchical learners, including any kernel methods.
Traditionally, data selection has been studied in settings where all samples from prospective sources are fully revealed to a machine learning developer. However, in practical data exchange scenarios, data providers often reveal only a limited subset of samples before an acquisition decision is made. Recently, there have been efforts to fit scaling laws that predict model performance at any size and data source composition using the limited available samples. However, these scaling functions are black-box, computationally expensive to fit, highly susceptible to overfitting, or/and difficult to optimize for data selection. This paper proposes a framework called <projektor>, which predicts model performance and supports data selection decisions based on partial samples of prospective data sources. Our approach distinguishes itself from existing work by introducing a novel *two-stage* performance inference process. In the first stage, we leverage the Optimal Transport distance to predict the model's performance for any data mixture ratio within the range of disclosed data sizes. In the second stage, we extrapolate the performance to larger undisclosed data sizes based on a novel parameter-free mapping technique inspired by neural scaling laws. We further derive an efficient gradient-based method to select data sources based on the projected model performance. Evaluation over a diverse range of applications demonstrates that <projektor> significantly improves existing performance scaling approaches in terms of both the accuracy of performance inference and the computation costs associated with constructing the performance predictor. Also, <projektor> outperforms by a wide margin in data selection effectiveness compared to a range of other off-the-shelf solutions.
Learning generic high-dimensional tasks is notably hard, as it requires a number of training data exponential in the dimension. Yet, deep convolutional neural networks (CNNs) have shown remarkable success in overcoming this challenge. A popular hypothesis is that learnable tasks are highly structured and that CNNs leverage this structure to build a low-dimensional representation of the data. However, little is known about how much training data they require, and how this number depends on the data structure. This paper answers this question for a simple classification task that seeks to capture relevant aspects of real data: the Random Hierarchy Model. In this model, each of the $n_c$ classes corresponds to $m$ synonymic compositions of high-level features, which are in turn composed of sub-features through an iterative process repeated $L$ times. We find that the number of training data $P^*$ required by deep CNNs to learn this task (i) grows asymptotically as $n_c m^L$, which is only polynomial in the input dimensionality; (ii) coincides with the training set size such that the representation of a trained network becomes invariant to exchanges of synonyms; (iii) corresponds to the number of data at which the correlations between low-level features and classes become detectable. Overall, our results indicate how deep CNNs can overcome the curse of dimensionality by building invariant representations, and provide an estimate of the number of data required to learn a task based on its hierarchically compositional structure.
Graph neural networks (GNNs) have become increasingly popular for classification tasks on graph-structured data. Yet, the interplay between graph topology and feature evolution in GNNs is not well understood. In this paper, we focus on node-wise classification, illustrated with community detection on stochastic block model graphs, and explore the feature evolution through the lens of the "Neural Collapse" (NC) phenomenon. When training instance-wise deep classifiers (e.g. for image classification) beyond the zero training error point, NC demonstrates a reduction in the deepest features' within-class variability and an increased alignment of their class means to certain symmetric structures. We start with an empirical study that shows that a decrease in within-class variability is also prevalent in the node-wise classification setting, however, not to the extent observed in the instance-wise case. Then, we theoretically study this distinction. Specifically, we show that even an "optimistic" mathematical model requires that the graphs obey a strict structural condition in order to possess a minimizer with exact collapse. Interestingly, this condition is viable also for heterophilic graphs and relates to recent empirical studies on settings with improved GNNs' generalization. Furthermore, by studying the gradient dynamics of the theoretical model, we provide reasoning for the partial collapse observed empirically. Finally, we present a study on the evolution of within- and between-class feature variability across layers of a well-trained GNN and contrast the behavior with spectral methods.
Synthetic time series are often used in practical applications to augment the historical time series dataset for better performance of machine learning algorithms, amplify the occurrence of rare events, and also create counterfactual scenarios described by the time series. Distributional-similarity (which we refer to as realism) as well as the satisfaction of certain numerical constraints are common requirements in counterfactual time series scenario generation requests. For instance, the US Federal Reserve publishes synthetic market stress scenarios given by the constrained time series for financial institutions to assess their performance in hypothetical recessions. Existing approaches for generating constrained time series usually penalize training loss to enforce constraints, and reject non-conforming samples. However, these approaches would require re-training if we change constraints, and rejection sampling can be computationally expensive, or impractical for complex constraints. In this paper, we propose a novel set of methods to tackle the constrained time series generation problem and provide efficient sampling while ensuring the realism of generated time series. In particular, we frame the problem using a constrained optimization framework and then we propose a set of generative methods including ``GuidedDiffTime'', a guided diffusion model to generate realistic time series. Empirically, we evaluate our work on several datasets for financial and energy data, where incorporating constraints is critical. We show that our approaches outperform existing work both qualitatively and quantitatively. Most importantly, we show that our ``GuidedDiffTime'' model is the only solution where re-training is not necessary for new constraints, resulting in a significant carbon footprint reduction.
Generalized logit dynamic defines a time-dependent integro-differential equation with which a Nash equilibrium of an iterative game in a bounded and continuous action space is expected to be approximated. We show that the use of the exponential logit function is essential for the approximability, which will not be necessarily satisfied with functions with convex exponential-like functions such as q-exponential ones. We computationally analyze this issue and discuss influences of the choice of the logit function through an application to a fishing tourism problem in Japan.
Noise is a part of data whether the data is from measurement, experiment or ... A few techniques are suggested for noise reduction to improve the data quality in recent years some of which are based on wavelet, orthogonalization and neural networks. The computational cost of existing methods are more than expected and that's why their application in some cases is not beneficial. In this paper, we suggest a low cost techniques based on special linear algebra structures (tridiagonal systems) to improve the signal quality. In this method, we suggest a tridiagonal model for the noise around the most noisy elements. To update the predicted noise, the algorithm is equipped with a learning/feedback approach. The details are described below and based on presented numerical results this algorithm is successful in computing the noise with lower MSE (mean squared error) in computation time specially when the data size is lower than 5000. Our algorithm is used for low-range noise while for high-range noise it is sufficient to use the presented algorithm in hybrid with moving average. The algorithm is implemented in MATLAB 2019b on a computer with Windows 11 having 8GB RAM. It is then tested over many randomly generated experiments. The numerical results confirm the efficiency of presented algorithm in most cases in comparison with existing methods.
Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.
We describe the new field of mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of learning theory. These questions concern: the outstanding generalization power of overparametrized neural networks, the role of depth in deep architectures, the apparent absence of the curse of dimensionality, the surprisingly successful optimization performance despite the non-convexity of the problem, understanding what features are learned, why deep architectures perform exceptionally well in physical problems, and which fine aspects of an architecture affect the behavior of a learning task in which way. We present an overview of modern approaches that yield partial answers to these questions. For selected approaches, we describe the main ideas in more detail.
Deep Convolutional Neural Networks (CNNs) are a special type of Neural Networks, which have shown state-of-the-art results on various competitive benchmarks. The powerful learning ability of deep CNN is largely achieved with the use of multiple non-linear feature extraction stages that can automatically learn hierarchical representation from the data. Availability of a large amount of data and improvements in the hardware processing units have accelerated the research in CNNs and recently very interesting deep CNN architectures are reported. The recent race in deep CNN architectures for achieving high performance on the challenging benchmarks has shown that the innovative architectural ideas, as well as parameter optimization, can improve the CNN performance on various vision-related tasks. In this regard, different ideas in the CNN design have been explored such as use of different activation and loss functions, parameter optimization, regularization, and restructuring of processing units. However, the major improvement in representational capacity is achieved by the restructuring of the processing units. Especially, the idea of using a block as a structural unit instead of a layer is gaining substantial appreciation. This survey thus focuses on the intrinsic taxonomy present in the recently reported CNN architectures and consequently, classifies the recent innovations in CNN architectures into seven different categories. These seven categories are based on spatial exploitation, depth, multi-path, width, feature map exploitation, channel boosting and attention. Additionally, it covers the elementary understanding of the CNN components and sheds light on the current challenges and applications of CNNs.
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