Scaling and renormalization in high-dimensional regression

• URL: http://arxiv.org/abs/2405.00592v4
• Date: Mon, 30 Jun 2025 15:11:57 GMT
• Title: Scaling and renormalization in high-dimensional regression
• Authors: Alexander Atanasov, Jacob A. Zavatone-Veth, Cengiz Pehlevan,
• Abstract summary: We present a unifying perspective on recent results on ridge regression.<n>We use the basic tools of random matrix theory and free probability, aimed at readers with backgrounds in physics and deep learning.<n>Our results extend and provide a unifying perspective on earlier models of scaling laws.
• Score: 72.59731158970894
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: From benign overfitting in overparameterized models to rich power-law scalings in performance, simple ridge regression displays surprising behaviors sometimes thought to be limited to deep neural networks. This balance of phenomenological richness with analytical tractability makes ridge regression the model system of choice in high-dimensional machine learning. In this paper, we present a unifying perspective on recent results on ridge regression using the basic tools of random matrix theory and free probability, aimed at readers with backgrounds in physics and deep learning. We highlight the fact that statistical fluctuations in empirical covariance matrices can be absorbed into a renormalization of the ridge parameter. This `deterministic equivalence' allows us to obtain analytic formulas for the training and generalization errors in a few lines of algebra by leveraging the properties of the $S$-transform of free probability. From these precise asymptotics, we can easily identify sources of power-law scaling in model performance. In all models, the $S$-transform corresponds to the train-test generalization gap, and yields an analogue of the generalized-cross-validation estimator. Using these techniques, we derive fine-grained bias-variance decompositions for a very general class of random feature models with structured covariates. This allows us to discover a scaling regime for random feature models where the variance due to the features limits performance in the overparameterized setting. We also demonstrate how anisotropic weight structure in random feature models can limit performance and lead to nontrivial exponents for finite-width corrections in the overparameterized setting. Our results extend and provide a unifying perspective on earlier models of neural scaling laws.

Related papers

• Scaling Law for Stochastic Gradient Descent in Quadratically Parameterized Linear Regression [5.801904710149222]
  In machine learning, the scaling law describes how the model performance improves with the model and data size scaling up.<n>This paper studies the scaling law over a linear regression with the model being quadratically parameterized.<n>As a result, in the canonical linear regression, we provide explicit separations for curves between generalization with and without feature learning, and the information-theoretical lower bound that is to parametrization method and the algorithm.
  arXiv  Detail & Related papers  (2025-02-13T09:29:04Z)
• Distribution learning via neural differential equations: a nonparametric statistical perspective [1.4436965372953483]
  This work establishes the first general statistical convergence analysis for distribution learning via ODE models trained through likelihood transformations. We show that the latter can be quantified via the $C1$-metric entropy of the class $mathcal F$. We then apply this general framework to the setting of $Ck$-smooth target densities, and establish nearly minimax-optimal convergence rates for two relevant velocity field classes $mathcal F$: $Ck$ functions and neural networks.
  arXiv  Detail & Related papers  (2023-09-03T00:21:37Z)
• Structured Radial Basis Function Network: Modelling Diversity for Multiple Hypotheses Prediction [51.82628081279621]
  Multi-modal regression is important in forecasting nonstationary processes or with a complex mixture of distributions. A Structured Radial Basis Function Network is presented as an ensemble of multiple hypotheses predictors for regression problems. It is proved that this structured model can efficiently interpolate this tessellation and approximate the multiple hypotheses target distribution.
  arXiv  Detail & Related papers  (2023-09-02T01:27:53Z)
• Probabilistic Unrolling: Scalable, Inverse-Free Maximum Likelihood Estimation for Latent Gaussian Models [69.22568644711113]
  We introduce probabilistic unrolling, a method that combines Monte Carlo sampling with iterative linear solvers to circumvent matrix inversions. Our theoretical analyses reveal that unrolling and backpropagation through the iterations of the solver can accelerate gradient estimation for maximum likelihood estimation. In experiments on simulated and real data, we demonstrate that probabilistic unrolling learns latent Gaussian models up to an order of magnitude faster than gradient EM, with minimal losses in model performance.
  arXiv  Detail & Related papers  (2023-06-05T21:08:34Z)
• A Solvable Model of Neural Scaling Laws [72.8349503901712]
  Large language models with a huge number of parameters, when trained on near internet-sized number of tokens, have been empirically shown to obey neural scaling laws. We propose a statistical model -- a joint generative data model and random feature model -- that captures this neural scaling phenomenology. Key findings are the manner in which the power laws that occur in the statistics of natural datasets are extended by nonlinear random feature maps.
  arXiv  Detail & Related papers  (2022-10-30T15:13:18Z)
• On the Generalization and Adaption Performance of Causal Models [99.64022680811281]
  Differentiable causal discovery has proposed to factorize the data generating process into a set of modules. We study the generalization and adaption performance of such modular neural causal models. Our analysis shows that the modular neural causal models outperform other models on both zero and few-shot adaptation in low data regimes.
  arXiv  Detail & Related papers  (2022-06-09T17:12:32Z)
• Bias-variance decomposition of overparameterized regression with random linear features [0.0]
  "Over parameterized models" avoid overfitting even when the number of fit parameters is large enough to perfectly fit the training data. We show how each transition arises due to small nonzero eigenvalues in the Hessian matrix. We compare and contrast the phase diagram of the random linear features model to the random nonlinear features model and ordinary regression.
  arXiv  Detail & Related papers  (2022-03-10T16:09:21Z)
• A generalization gap estimation for overparameterized models via the Langevin functional variance [6.231304401179968]
  We show that a functional variance characterizes the generalization gap even in overparameterized settings. We propose an efficient approximation of the function variance, the Langevin approximation of the functional variance (Langevin FV)
  arXiv  Detail & Related papers  (2021-12-07T12:43:05Z)
• Hessian Eigenspectra of More Realistic Nonlinear Models [73.31363313577941]
  We make a emphprecise characterization of the Hessian eigenspectra for a broad family of nonlinear models. Our analysis takes a step forward to identify the origin of many striking features observed in more complex machine learning models.
  arXiv  Detail & Related papers  (2021-03-02T06:59:52Z)
• Memorizing without overfitting: Bias, variance, and interpolation in over-parameterized models [0.0]
  The bias-variance trade-off is a central concept in supervised learning. Modern Deep Learning methods flout this dogma, achieving state-of-the-art performance.
  arXiv  Detail & Related papers  (2020-10-26T22:31:04Z)
• Goal-directed Generation of Discrete Structures with Conditional Generative Models [85.51463588099556]
  We introduce a novel approach to directly optimize a reinforcement learning objective, maximizing an expected reward. We test our methodology on two tasks: generating molecules with user-defined properties and identifying short python expressions which evaluate to a given target value.
  arXiv  Detail & Related papers  (2020-10-05T20:03:13Z)
• The Neural Tangent Kernel in High Dimensions: Triple Descent and a Multi-Scale Theory of Generalization [34.235007566913396]
  Modern deep learning models employ considerably more parameters than required to fit the training data. Whereas conventional statistical wisdom suggests such models should drastically overfit, in practice these models generalize remarkably well. An emerging paradigm for describing this unexpected behavior is in terms of a emphdouble descent curve. We provide a precise high-dimensional analysis of generalization with the Neural Tangent Kernel, which characterizes the behavior of wide neural networks with gradient descent.
  arXiv  Detail & Related papers  (2020-08-15T20:55:40Z)
• Slice Sampling for General Completely Random Measures [74.24975039689893]
  We present a novel Markov chain Monte Carlo algorithm for posterior inference that adaptively sets the truncation level using auxiliary slice variables. The efficacy of the proposed algorithm is evaluated on several popular nonparametric models.
  arXiv  Detail & Related papers  (2020-06-24T17:53:53Z)
• Multiplicative noise and heavy tails in stochastic optimization [62.993432503309485]
  empirical optimization is central to modern machine learning, but its role in its success is still unclear. We show that it commonly arises in parameters of discrete multiplicative noise due to variance. A detailed analysis is conducted in which we describe on key factors, including recent step size, and data, all exhibit similar results on state-of-the-art neural network models.
  arXiv  Detail & Related papers  (2020-06-11T09:58:01Z)
• Dimension Independent Generalization Error by Stochastic Gradient Descent [12.474236773219067]
  We present a theory on the generalization error of descent (SGD) solutions for both and locally convex loss functions. We show that the generalization error does not depend on the $p$ dimension or depends on the low effective $p$logarithmic factor.
  arXiv  Detail & Related papers  (2020-03-25T03:08:41Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.