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1. A theory of capacity and sparse neural encoding

   Baldi, Pierre; Vershynin, Roman (January 2021, Neural networks)

   null (Ed.)

   Motivated by biological considerations, we study sparse neural maps from an input layer to a target layer with sparse activity, and specifically the problem of storing K input-target associations (x, y), or memories, when the target vectors y are sparse. We mathematically prove that K undergoes a phase transition and that in general, and some-what paradoxically, sparsity in the target layers increases the storage capacity of the map.The target vectors can be chosen arbitrarily, including in random fashion, and the memories can be both encoded and decoded by networks trained using local learning rules, including the simple Hebb rule. These results are robust under a variety of statistical assumptions on the data. The proofs rely on elegant properties of random polytopes and sub-gaussian random vector variables. Open problems and connections to capacity theories and polynomial threshold maps are discussed 

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2. Willump: A Statistically-Aware End-to-end Optimizer for Machine Learning Inference

   Kraft, Peter; Kang, Daniel; Narayanan, Deepak; Palkar, Shoumik; Bailis, Peter; and Zaharia, Matei. (April 2020, MLSys 2020)

   null (Ed.)

   Systems for ML inference are widely deployed today, but they typically optimize ML inference workloads using techniques designed for conventional data serving workloads and miss critical opportunities to leverage the statistical nature of ML. In this paper, we present WILLUMP, an optimizer for ML inference that introduces two statistically-motivated optimizations targeting ML applications whose performance bottleneck is feature computation. First, WILLUMP automatically cascades feature computation for classification queries: WILLUMP classifies most data inputs using only high-value, low-cost features selected through empirical observations of ML model performance, improving query performance by up to 5× without statistically significant accuracy loss. Second, WILLUMP accurately approximates ML top-K queries, discarding low-scoring inputs with an automatically constructed approximate model and then ranking the remainder with a more powerful model, improving query performance by up to 10× with minimal accuracy loss. WILLUMP automatically tunes these optimizations’ parameters to maximize query performance while meeting an accuracy target. Moreover, WILLUMP complements these statistical optimizations with compiler optimizations to automatically generate fast inference code for ML applications. We show that WILLUMP improves the end-to-end performance of real-world ML inference pipelines curated from major data science competitions by up to 16× without statistically significant loss of accuracy. 

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3. A Piecewise Lyapunov Analysis of Sub-quadratic SGD: Applications to Robust and Quantile Regression

   https://doi.org/10.1145/3744970.3727269  

   Zhang, Yixuan; Huo, Dongyan Lucy; Chen, Yudong; Xie, Qiaomin (June 2025, ACM SIGMETRICS Performance Evaluation Review)

   Motivated by robust and quantile regression problems, we investigate the stochastic gradient descent (SGD) algorithm for minimizing an objective functionfthat is locally strongly convex with a sub--quadratic tail. This setting covers many widely used online statistical methods. We introduce a novel piecewise Lyapunov function that enables us to handle functionsfwith only first-order differentiability, which includes a wide range of popular loss functions such as Huber loss. Leveraging our proposed Lyapunov function, we derive finite-time moment bounds under general diminishing stepsizes, as well as constant stepsizes. We further establish the weak convergence, central limit theorem and bias characterization under constant stepsize, providing the first geometrical convergence result for sub--quadratic SGD. Our results have wide applications, especially in online statistical methods. In particular, we discuss two applications of our results. 1) Online robust regression: We consider a corrupted linear model with sub--exponential covariates and heavy--tailed noise. Our analysis provides convergence rates comparable to those for corrupted models with Gaussian covariates and noise. 2) Online quantile regression: Importantly, our results relax the common assumption in prior work that the conditional density is continuous and provide a more fine-grained analysis for the moment bounds. 

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4. Pure and Spurious Critical Points: a Geometric Study of Linear Networks

   Trager, Matthew; Kohn, Kathlen; Bruna, Joan (April 2020, international conference on learning representations)

   The critical locus of the loss function of a neural network is determined by the geometry of the functional space and by the parameterization of this space by the network's weights. We introduce a natural distinction between pure critical points, which only depend on the functional space, and spurious critical points, which arise from the parameterization. We apply this perspective to revisit and extend the literature on the loss function of linear neural networks. For this type of network, the functional space is either the set of all linear maps from input to output space, or a determinantal variety, i.e., a set of linear maps with bounded rank. We use geometric properties of determinantal varieties to derive new results on the landscape of linear networks with different loss functions and different parameterizations. Our analysis clearly illustrates that the absence of "bad" local minima in the loss landscape of linear networks is due to two distinct phenomena that apply in different settings: it is true for arbitrary smooth convex losses in the case of architectures that can express all linear maps ("filling architectures") but it holds only for the quadratic loss when the functional space is a determinantal variety ("non-filling architectures"). Without any assumption on the architecture, smooth convex losses may lead to landscapes with many bad minima. 

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5. Demystifying Disagreement-on-the-Line in High Dimensions

   Donghwan Lee, Behrad Moniri (July 2023, ICML 2023)

   Evaluating the performance of machine learning models under distribution shifts is challenging, especially when we only have unlabeled data from the shifted (target) domain, along with labeled data from the original (source) domain. Recent work suggests that the notion of disagreement, the degree to which two models trained with different randomness differ on the same input, is a key to tackling this problem. Experimentally, disagreement and prediction error have been shown to be strongly connected, which has been used to estimate model performance. Experiments have led to the discovery of the disagreement-on-the-line phenomenon, whereby the classification error under the target domain is often a linear function of the classification error under the source domain; and whenever this property holds, disagreement under the source and target domain follow the same linear relation. In this work, we develop a theoretical foundation for analyzing disagreement in high-dimensional random features regression; and study under what conditions the disagreement-on-the-line phenomenon occurs in our setting. Experiments on CIFAR-10-C, Tiny ImageNet-C, and Camelyon17 are consistent with our theory and support the universality of the theoretical findings. 

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