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1. SGD Noise and Implicit Low-Rank Bias in Deep Neural Networks

   Galanti, Tomer; Poggio, Tomaso (March 2022, Center for Brains, Minds and Machines (CBMM))

   We analyze deep ReLU neural networks trained with mini-batch stochastic gradient decent and weight decay. We prove that the source of the SGD noise is an implicit low rank constraint across all of the weight matrices within the network. Furthermore, we show, both theoretically and empirically, that when training a neural network using Stochastic Gradient Descent (SGD) with a small batch size, the resulting weight matrices are expected to be of small rank. Our analysis relies on a minimal set of assumptions and the neural networks may include convolutional layers, residual connections, as well as batch normalization layers. 

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2. The Janus effects of SGD vs GD: high noise and low rank

   Xu, Mengjia; Galanti, Tomer; Rosasco, Lorenzo; Rangamani, Akshay; Pinto, Andrea; Poggio, Tomaso (February 2024, Center for Brains, Minds and Machines (CBMM))

   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. 

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3. Sub-quadratic Algorithms for Kernel Matrices via Kernel Density Estimation

   Bakshi, Ainesh; Kacham, Praneeth; Indyk, Piotr; Silwal, Sandeep; Zhou, Samson (January 2023, International Conference on Learning Representations)

   Kernel matrices, as well as weighted graphs represented by them, are ubiquitous objects in machine learning, statistics and other related fields. The main drawback of using kernel methods (learning and inference using kernel matrices) is efficiency – given n input points, most kernel-based algorithms need to materialize the full n × n kernel matrix before performing any subsequent computation, thus incurring Ω(n^2) runtime. Breaking this quadratic barrier for various problems has therefore, been a subject of extensive research efforts. We break the quadratic barrier and obtain subquadratic time algorithms for several fundamental linear-algebraic and graph processing primitives, including approximating the top eigenvalue and eigenvector, spectral sparsification, solving lin- ear systems, local clustering, low-rank approximation, arboricity estimation and counting weighted triangles. We build on the recently developed Kernel Density Estimation framework, which (after preprocessing in time subquadratic in n) can return estimates of row/column sums of the kernel matrix. In particular, we de- velop efficient reductions from weighted vertex and weighted edge sampling on kernel graphs, simulating random walks on kernel graphs, and importance sampling on matrices to Kernel Density Estimation and show that we can generate samples from these distributions in sublinear (in the support of the distribution) time. Our reductions are the central ingredient in each of our applications and we believe they may be of independent interest. We empirically demonstrate the efficacy of our algorithms on low-rank approximation (LRA) and spectral sparsi- fication, where we observe a 9x decrease in the number of kernel evaluations over baselines for LRA and a 41x reduction in the graph size for spectral sparsification. 

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4. Nonlinear spiked covariance matrices and signal propagation in deep neural networks

   Wang, Zhichao; Wu, Denny; Fan, Zhou (June 2024, Proceedings of Machine Learning Research)

   Many recent works have studied the eigenvalue spectrum of the Conjugate Kernel (CK) defined by the nonlinear feature map of a feedforward neural network. However, existing results only establish weak convergence of the empirical eigenvalue distribution, and fall short of providing precise quantitative characterizations of the “spike” eigenvalues and eigenvectors that often capture the low-dimensional signal structure of the learning problem. In this work, we characterize these signal eigenvalues and eigenvectors for a nonlinear version of the spiked covariance model, including the CK as a special case. Using this general result, we give a quantitative description of how spiked eigenstructure in the input data propagates through the hidden layers of a neural network with random weights. As a second application, we study a simple regime of representation learning where the weight matrix develops a rank-one signal component over training and characterize the alignment of the target function with the spike eigenvector of the CK on test data. 

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5. NOLA: Compressing LoRA using Linear Combination of Random Basis

   Koohpayegani, Soroush A; Navaneet, K L; Nooralinejad, Parsa; Kolouri, Soheil; Pirsiavash, Hamed (May 2024, International Conference on Learning Representations)

   ine-tuning Large Language Models (LLMs) and storing them for each downstream task or domain is impractical because of the massive model size (e.g., 350GB in GPT-3). Current literature, such as LoRA, showcases the potential of low-rank modifications to the original weights of an LLM, enabling efficient adaptation and storage for task-specific models. These methods can reduce the number of parameters needed to fine-tune an LLM by several orders of magnitude. Yet, these methods face two primary limitations: (1) the parameter count is lower-bounded by the rank one decomposition, and (2) the extent of reduction is heavily influenced by both the model architecture and the chosen rank. We introduce NOLA, which overcomes the rank one lower bound present in LoRA. It achieves this by re-parameterizing the low-rank matrices in LoRA using linear combinations of randomly generated matrices (basis) and optimizing the linear mixture coefficients only. This approach allows us to decouple the number of trainable parameters from both the choice of rank and the network architecture. We present adaptation results using GPT-2, LLaMA-2, and ViT in natural language and computer vision tasks. NOLA performs as well as LoRA models with much fewer number of parameters compared to LoRA with rank one, the best compression LoRA can archive. Particularly, on LLaMA-2 70B, our method is almost 20 times more compact than the most compressed LoRA without degradation in accuracy. Our code is available here: https://github.com/UCDvision/NOLA 

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