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While previous optimization results have suggested that deep neural networks tend to favour low-rank weight matrices, the implications of this inductive bias on generalization bounds remain underexplored. In this paper, we apply a chain rule for Gaussian complexity (Maurer, 2016a) to analyze how low-rank layers in deep networks can prevent the accumulation of rank and dimensionality factors that typically multiply across layers. This approach yields generalization bounds for rank and spectral norm constrained networks. We compare our results to prior generalization bounds for deep networks, highlighting how deep networks with low-rank layers can achieve better generalization than those with full-rank layers. Additionally, we discuss how this framework provides new perspectives on the generalization capabilities of deep networks exhibiting neural collapse. Keywords: Gaussian complexity, Generalization bounds, Neural collapse, Low rank layers 

Over the last four decades, the amazing success of deep learning has been driven by the use of Stochastic Gradient Descent (SGD) as the main optimization technique. The default implementation for the computation of the gradient for SGD is backpropagation, which, with its variations, is used to this day in almost all computer implementations. From the perspective of neuroscientists, however, the consensus is that backpropagation is unlikely to be used by the brain. Though several alternatives have been discussed, none is so far supported by experimental evidence. Here we propose a circuit for updating the weights in a network that is biologically plausible, works as well as backpropagation, and leads to verifiable predictions about the anatomy and the physiology of a characteristic motif of four plastic synapses between ascending and descending cortical streams. A key prediction of our proposal is a surprising property of self-assembly of the basic circuit, emerging from initial random connectivity and heterosynaptic plasticity rules. 

Understanding neural representations will help open the black box of neural networks and advance our scientific understanding of modern AI systems. However, how complex, structured, and transferable representations emerge in modern neural networks has remained a mystery. Building on previous results, we propose the Canonical Representation Hypothesis (CRH), which posits a set of six alignment relations to universally govern the formation of representations in most hidden layers of a neural network. Under the CRH, the latent representations (R), weights (W), and neuron gradients (G) become mutually aligned during training. This alignment implies that neural networks naturally learn compact representations, where neurons and weights are invariant to task-irrelevant transformations. We then show that the breaking of CRH leads to the emergence of reciprocal power-law relations between R, W, and G, which we refer to as the Polynomial Alignment Hypothesis (PAH). We present a minimal-assumption theory demonstrating that the balance between gradient noise and regularization is crucial for the emergence the canonical representation. The CRH and PAH lead to an exciting possibility of unifying major key deep learning phenomena, including neural collapse and the neural feature ansatz, in a single framework. 

Originally proposed for handling time series data, Auto-regressive Decision Trees (ARDTs) have not yet been explored for language modeling. This paper delves into both the theoretical and practical applications of ARDTs in this new context. We theoretically demonstrate that ARDTs can compute complex functions, such as simulating automata, Turing machines, and sparse circuits, by leveraging "chain-of-thought" computations. Our analysis provides bounds on the size, depth, and computational efficiency of ARDTs, highlighting their surprising computational power. Empirically, we train ARDTs on simple language generation tasks, showing that they can learn to generate coherent and grammatically correct text on par with a smaller Transformer model. Additionally, we show that ARDTs can be used on top of transformer representations to solve complex reasoning tasks. This research reveals the unique computational abilities of ARDTs, aiming to broaden the architectural diversity in language model development. 

The Average Gradient Outer Product (AGOP) provides a novel approach to feature learning in neural networks. We applied both AGOP and Gradient Descent to learn the matrix M in the Hyper Basis Function Network (HyperBF) and observed very similar performance. We show formally that AGOP is a greedy approximation of gradient descent. 

Neural networks have demonstrated impressive success in various domains, raising the question of what fundamental principles underlie the effectiveness of the best AI systems and quite possibly of human intelligence. This perspective argues that compositional sparsity, or the property that a compositional function have “few” constituent functions, each depending on only a small subset of inputs, is a key principle underlying successful learning architectures. Surprisingly, all functions that are efficiently Turing computable have a compositional sparse representation. Furthermore, deep networks that are also sparse can exploit this general property to avoid the “curse of dimensionality”. This framework suggests interesting implications about the role that machine learning may play in mathematics. 

It was always obvious that SGD with small minibatch size yields for neural networks much higher asymptotic fluctuations in the updates of the weight matrices than GD. It has also been often reported that SGD in deep RELU networks shows empirically a low-rank bias in the weight matrices. A recent theoretical analysis derived a bound on the rank and linked it to the size of the SGD fluctuations [25]. In this paper, we provide an empirical and theoretical analysis of the convergence of SGD vs GD, first for deep RELU networks and then for the case of linear regression, where sharper estimates can be obtained and which is of independent interest. In the linear case, we prove that the component $$W^\perp$$ of the matrix $$W$$ corresponding to the null space of the data matrix $$X$$ converges to zero for both SGD and GD, provided the regularization term is non-zero. Because of the larger number of updates required to go through all the training data, the convergence rate {\it per epoch} of these components is much faster for SGD than for GD. In practice, SGD has a much stronger bias than GD towards solutions for weight matrices $$W$$ with high fluctuations -- even when the choice of mini batches is deterministic -- and low rank, provided the initialization is from a random matrix. Thus SGD with non-zero regularization, shows the coupled phenomenon of asymptotic noise and a low-rank bias-- unlike GD. 

In this paper, we derive norm-based generalization bounds for sparse ReLU neural networks, including convolutional neural networks. These bounds differ from previous ones because they consider the sparse structure of the neural network architecture and the norms of the convolutional filters, rather than the norms of the (Toeplitz) matrices associated with the convolutional layers. Theoretically, we demonstrate that these bounds are significantly tighter than standard norm-based generalization bounds. Empirically, they offer relatively tight estimations of generalization for various simple classification problems. Collectively, these findings suggest that the sparsity of the underlying target function and the model’s architecture plays a crucial role in the success of deep learning. 

While the Transformer architecture has made a substantial impact in the field of machine learning, it is unclear what purpose each component serves in the overall architecture. Heterogeneous nonlinear circuits such as multi-layer RELU networks are interleaved with layers of soft-max units. We introduce here a homogeneous architecture based on Hyper Radial Basis Function (HyperBF) units. Evalua- tions on CIFAR10, CIFAR100, and Tiny ImageNet demonstrate a performance comparable to standard vision transformers. 

Artificial neural networks are being proposed as models of parts of the brain. The networks are compared to recordings of biological neurons, and good performance in reproducing neural responses is considered to support the model’s validity. A key question is how much this system identification approach tells us about brain computation. Does it validate one model architecture over another? We evaluate the most commonly used comparison techniques, such as a linear encoding model and centered kernel alignment, to correctly identify a model by replacing brain recordings with known ground truth models. System identification performance is quite variable; it also depends significantly on factors independent of the ground truth architecture, such as stimuli images. In addition, we show the limitations of using functional similarity scores in identifying higher-level architectural motifs.