We analyze a variant of stochastic gradient descent with initial regularization (SGDIR) and derive dimension-free upper bounds on its expected excess risk for the squared loss. In the noiseless case, we obtain new bounds for both averaged and non-averaged SGDIR under moment, source, and capacity assumptions. For a particular value of the source parameter, these bounds are of order $m^{-2}\log^{2}m$, where the number of training samples is of order $m$. For another value of the source parameter, we obtain, for any $ε>0$, bounds of order $m^{-3+ε}$, provided that the capacity parameter exceeds $ε^{-1}$. We also establish a lower bound that matches our upper bounds in certain regimes up to a polylogarithmic factor. In the noisy case, we provide an instance-based comparison between SGDIR and ridge regression. Under general assumptions and a mild lower bound on the regularization parameter, we show that the expected excess risk of SGDIR is no larger than that of ridge regression, up to a polylogarithmic factor. Numerical experiments on synthetic and real data are consistent with our theoretical findings.
Training error is what we can observe on a training set; test error is the quantity we actually care about. We study linear regression with squared-error in a deterministic $(d+1)$-dimensional single-spike model. Each stylized training vector has the same informative spike coordinate, of amplitude $\sqrtγ$ with $γ>1$. The remaining directions are nuisance, and the nuisance components of distinct training vectors all have equal norm and are mutually orthogonal. The training labels are all $1$. Fresh test points are drawn from $\vec{x}_{\rm test} \sim \mathcal{N}(\vec{0},\operatorname{diag}(γ,1,\ldots,1))$, with the noise-free test labels being the normalized spike coordinate $x_{\rm test}[1]/\sqrtγ$. We focus on linear predictors in the span of the training vectors, the class naturally reached by zero-initialized linear gradient methods. We exhibit a range of training-set sizes $n$ in which every span predictor that generalizes well must fit the training data \emph{worse} than the zero predictor. We call this regime \emph{benign misfitting}, or the fourth quadrant. The best span predictor begins to generalize when $n\gg d/γ^2$, while interpolation does not generalize until the later threshold $n\gg d/γ$. In the window $d/γ^2 \ll n \ll d/γ$, useful prediction within the linear span lies beyond interpolation: predictions on the training points overshoot the labels. We show that one-pass stochastic gradient descent (SGD), with a large constant learning rate, reaches small test error throughout this window---matching the best span predictor up to a logarithmic factor. We also verify directly that it indeed has \emph{large} empirical training error (despite the descent premise in its name). Finally, we show that the unavoidable nuisance component responsible for the training misfit also controls the predictor's adversarial sensitivity.
Long-range learning is hard for recurrent networks trained with stochastic gradient descent, because the influence of a past input fades with the lag $\ell$, and if it fades too fast the dependence cannot be learned from finite data. This fade is captured by an envelope $f(\ell)$. An exponential fade makes the data needed to learn a lag-$\ell$ dependence grow exponentially, putting long horizons out of reach; a power-law fade keeps the cost polynomial. We show that the asymptotic decay behavior of $f(\ell)$ is not fixed by the architecture. Instead, it emerges from the coupling between the state dynamics and parameter dynamics, settling into either a collapsed regime (fast, exponential forgetting) or an extended, anti-collapsed regime (slow, power-law forgetting). The intuition is a competition within these coupled dynamics. Training drives the network's effective time scales toward short ones, while rare, heavy-tailed fluctuations of the learning dynamics push a few of them to very long values. Along the route studied here, the extended regime survives only when these heavy-tailed pushes are strong enough to balance the pull. We make this mathematically precise with a coarse-grained stochastic process and derive an explicit threshold at which this route to the extended regime becomes available. A single exponent, the spectral exponent~$β$, then governs both the spread of time scales and how slowly the network forgets. Realizing the regime in practice needs one more ingredient: the joint action of the architecture and the optimizer must be able to hold such a broad spread. A network whose capacity to generate broad time-scale spectra is severely constrained still collapses, even when supplied with strong heavy-tailed forcing. Heavy-tailed fluctuations thus act not as noise to be suppressed, but as the mechanism that sustains long-range learning.
In this paper, we develop a fractional stochastic neural network with residual dynamics driven by fractional Brownian motion. By introducing a discrete stochastic maximum principle for the network, we construct the corresponding adjoint recursion. For deterministic network parameters, we prove mean square convergence of projected samplewise stochastic gradient descent. Numerical experiments include a closed form convergence test, noisy regression with uncertainty quantification, long memory time series generation and image classification under structured perturbations. The results identify settings in which fractional drivers improve long memory recovery or robustness relative to Brownian and deterministic baselines.
This work addresses the problem of variance in stochastic gradient estimation for machine learning optimization. Deep learning relies on mini-batch methods such as stochastic gradient descent, which approximate full gradients but introduce noise, creating trade-offs between convergence stability, speed, and generalization. Existing methods, including variance reduction techniques (e.g., SVRG and SAG) and adaptive optimizers, aim to mitigate gradient noise but may introduce additional computational overhead. We propose a model-assisted sampling framework that interprets mini-batch gradients through survey sampling theory, treating the dataset as a fixed finite population and gradients as sample-based estimates. Our aim is to bridge machine learning optimization and survey sampling theory by combining their perspectives on sample-based estimation and variance reduction. By incorporating auxiliary gradient-prediction models, we construct more efficient gradient estimators, with uniform sampling arising as a special case when no auxiliary information is used. Our approach integrates easily with existing optimizers, improving efficiency without altering their dynamics. Empirical results on synthetic and six benchmark datasets show performance gains in 71-86% of the experiments, particularly for medium-sized input spaces in our benchmarks. Notably, with momentum-based optimizers such as AdamW, the proposed estimator achieves clearly better generalization in roughly half the training epochs compared to baseline estimator.
Farhad Pashakhanloo, Jacob A. Zavatone-Vethq-bio.NC cs.LG
What can representational similarity matrices (RSMs) tell us about a neural code? As the popularity of these summary statistics grows, so too does the need for a more complete characterization of their properties. Here, we show that symmetries in network inputs can confound RSM-based analyses. Stimulus symmetries render many representations functionally equivalent, but these different configurations can lead to different RSMs. These different RSMs reflect qualitatively different representational geometries. We show that stochastic gradient descent or energetic regularization can generate sparse, drifting codes, leading in turn to drifting RSMs. Moreover, we demonstrate that these phenomena are present in networks trained to encode image data, where the symmetry is latent. Our results illustrate the challenges inherent in comparing nonlinear neural codes, when functionally-equivalent representations are not related by a simple rotation.