The classical bias–variance decomposition is a foundational concept in statistical learning. In low-dimensional settings, the expected prediction error of an estimator can be decomposed into three components: bias squared, variance, and irreducible error (noise). The bias measures the systematic error introduced by the estimator, while the variance reflects the sensitivity to fluctuations in the training data. Traditionally, increasing model complexity reduces bias but increases variance, and vice versa, leading to the familiar U-shaped test error curve as a function of model complexity. This decomposition guides how you select model complexity to balance underfitting and overfitting.

However, in high-dimensional regimes — where the number of parameters rivals or exceeds the number of observations — this classical framework faces serious limitations. The variance term can become explosively large, sometimes dominating the error regardless of the bias. In these settings, classical estimators such as the ordinary least squares (OLS) estimator can have extremely high variance or even become undefined if the design matrix is not full rank. As a result, the simple bias–variance trade-off picture breaks down, and new techniques are needed to achieve meaningful generalization.

Regularization fundamentally alters the bias–variance balance, especially in high-dimensional and sparse settings. Regularization techniques such as ridge regression or lasso introduce a penalty term to the loss function, which constrains the solution space. This penalty explicitly increases the bias of the estimator by shrinking coefficients toward zero or enforcing sparsity, but it can dramatically reduce variance by stabilizing the solution.

When dimensionality is high, the variance of unregularized estimators can overwhelm any potential gains from low bias. Regularization shifts the bias–variance equilibrium: the optimal estimator is intentionally biased to suppress variance. In sparse models — where only a small subset of features are expected to be truly relevant — regularization methods like lasso exploit this sparsity by setting many coefficients exactly to zero. This not only reduces variance but also enhances interpretability.

The amount of regularization required depends on the interplay between sparsity and dimensionality. In extremely high-dimensional settings, stronger regularization is often necessary. The optimal regularization regime is found by balancing the increased bias from the penalty with the reduction in variance, typically using techniques such as cross-validation to select the penalty parameter. In summary, in high dimensions, you must accept increased bias as the price for any hope of controlling variance and achieving stable predictions.

High-dimensional geometry provides an intuitive lens to understand why variance is so problematic and why bias becomes indispensable. In high dimensions, data points become sparse and are often nearly orthogonal to each other. This means that small perturbations in the data can lead to wild fluctuations in the fitted model — variance is amplified by the geometry of the space. Imagine fitting a hyperplane in a space with thousands of dimensions but only a few data points: there are infinitely many hyperplanes that perfectly interpolate the data, but their predictions outside the observed points can be erratic.

Regularization acts like a geometric constraint, shrinking the solution toward a lower-dimensional subspace or toward the origin. This geometric "pull" introduces bias but tames the wild variance induced by the high-dimensional geometry. In sparse settings, regularization can focus on a small subset of directions, further stabilizing the estimator. The key geometric insight is that in high dimensions, a little bias can go a long way toward making learning feasible and robust.

Choosing the optimal regularization regime in high-dimensional, sparse models involves a delicate balance. Too little regularization leaves the model at the mercy of high variance, while too much regularization can wash out meaningful signal by introducing excessive bias. In practice, the best trade-off is often achieved by data-driven methods such as cross-validation, which directly estimate predictive performance.

However, there are fundamental limits to the bias–variance trade-off in high-dimensional statistics. If the signal is not sufficiently sparse relative to the dimensionality, no amount of regularization can fully overcome the curse of dimensionality — variance will dominate, and bias cannot compensate without sacrificing all useful signal. Conversely, in settings where strong sparsity holds, carefully tuned regularization can yield estimators that approach the minimax optimal error rates, achieving both low bias and low variance within the constraints of the model. Understanding these limits is essential for effective modeling in modern high-dimensional applications.