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Implicit Regularization of Discrete Gradient Dynamics in Linear Neural Networks
Gauthier Gidel, Francis Bach, Simon Lacoste-Julien
TL;DR
Over-parameterized models can reach zero training error through many parameter choices, motivating the study of how optimization induces implicit regularization. This paper analyzes discrete gradient dynamics in two-layer linear networks and shows that, with vanishing initialization and a sufficiently small step size, time-rescaled training sequentially learns reduced-rank regression solutions with increasing rank.
Problem
The paper examines how optimization choices select among multiple zero-training-error solutions and addresses limited understanding of generalization in over-parameterized linear models.
Method
The paper studies discrete and continuous least-squares gradient dynamics for two-layer linear networks using vanishing initialization, small step sizes, time rescaling, and perturbation analysis.
Results
Two-layer dynamics sequentially learns increasingly ranked hierarchical components corresponding to gradually less regularized reduced-rank regression solutions, unlike linear-model dynamics that learn components simultaneously.
Takeaways & Limitations
The gradient-dynamics path provides an implicit regularization that may yield better generalization properties, with experiments observing improved matrix reconstruction over a linear model.
Takeaways & Limitations
The analysis is restricted to over-parameterized two-layer linear networks and relies on assumptions including vanishing initialization, sufficiently small step size, and the stated over-parameterization regime.
Abstract
from arXiv · showhide
When optimizing over-parameterized models, such as deep neural networks, a large set of parameters can achieve zero training error. In such cases, the choice of the optimization algorithm and its respective hyper-parameters introduces biases that will lead to convergence to specific minimizers of the objective. Consequently, this choice can be considered as an implicit regularization for the training of over-parametrized models. In this work, we push this idea further by studying the discrete gradient dynamics of the training of a two-layer linear network with the least-squares loss. Using a time rescaling, we show that, with a vanishing initialization and a small enough step size, this dynamics sequentially learns the solutions of a reduced-rank regression with a gradually increasing rank.
1 Introduction
The paper studies implicit regularization in two-layer linear networks, extending continuous-dynamics analysis to discrete gradient updates. Under small-step and vanishing-initialization conditions, the dynamics sequentially learns increasingly ranked reduced-rank solutions, with experiments indicating improved matrix-reconstruction generalization over linear models.
- Motivation: The work frames optimization choices and hyperparameters as implicit regularization because they bias which zero-training-error minimizer is reached.Different minimizers can have different test errors and generalization properties.
- Contribution: The paper analyzes discrete gradient updates for a two-layer linear network trained with least-squares loss, extending prior work on continuous gradient dynamics.The setting uses multi-dimensional-output regression and compares two-layer linear networks with linear prediction models.
- Contribution: With vanishing initialization and a small enough step size, two-layer dynamics sequentially learns components organized by a hierarchical structure.The corresponding linear model learns the components simultaneously and does not capture this hierarchy.
- Contribution: The network’s optimization path successively solves the regression problem under a growing low-rank constraint, corresponding to reduced-rank regression.This path concerns training dynamics, distinct from paths describing information propagation inside a fixed network.
- Contribution: Under stated assumptions, both discrete and continuous dynamics sequentially learn solutions of gradually less regularized reduced-rank regression.The discrete-dynamics result is presented as novel, while the continuous analysis weakens a standard commutativity assumption through perturbation analysis.
- Experiments: Experiments observe improved matrix-reconstruction generalization for the two-layer linear network compared with the linear model.The experiments also assess the reasonableness of the paper’s assumption.
2 A Simple Deep Linear Model
The paper formulates multi-output least-squares regression with deep linear networks and explains how depth and bottleneck widths alter the learning problem. Over-parameterization preserves linear expressivity, while narrower hidden layers induce reduced-rank constraints.
- Model formulation: The model predicts multi-dimensional outputs from finite inputs using a deep linear network trained with least-squares loss.Inputs are x_i ∈ R^d and outputs are y_i ∈ R^p, represented by design matrices X and Y.
- Model formulation: The network objective minimizes 1/(2n) times the squared norm of Y − XW1 · · · WL.The weight matrices are learned through the mean-squared-error least-squares formulation.
- Model formulation: For L = 2, the deep linear network has different learning dynamics from a single-layer linear model despite representing only linear transformations.Intermediate products W_l · · · W_1x are treated as hidden-layer representations.
- Rank constraint: When the minimum hidden-layer width r is below min(d, p), increasing depth can impose a low-rank constraint equivalent to reduced-rank regression.The rank threshold is r := min{r_l : 1 ≤ l ≤ L − 1}.
- Over-parameterization: When r > min(p, d), the model is over-parameterized and can represent every p-by-d linear transformation.This is the regime in which the paper studies implicit regularization.
3 Gradient Dynamics as a Regularizer
The paper analyzes continuous and discrete gradient dynamics in linear and two-layer linear networks, showing that factorization induces sequential, rank-ordered learning. Under weaker covariance assumptions, vanishing initialization and suitable step sizes lead to progressively less regularized reduced-rank regression solutions.
- Setup: The dynamics are studied for least-squares regression using both linear models and two-layer linear neural networks.The discrete updates use step size η, while the continuous dynamics arise as η → 0.
- 3.1 Continuous dynamics: Under a weaker assumption than Σx = Id, perturbation analysis extends the continuous two-layer solution beyond the commuting case.The covariance decompositions allow a small non-diagonal perturbation B, with Grönwall’s inequality providing the perturbation bound.
- 3.1 Continuous dynamics: After rescaling time as t → δt, vanishing initialization separates phase transitions at Ti := 1/σi, causing components to be learned sequentially.Components remain inactive before their transition and are learned in decreasing order of the singular values of Σxy.
- 3.2 Discrete dynamics: The two-layer gradient path acts as implicit regularization by successively solving reduced-rank regression problems with gradually increasing rank.Experiments show step-like trace-norm behavior and sequential learning for the two-layer network, unlike the linear model.
- 3.1 Continuous dynamics: Between transitions Tk and Tk+1, the limiting product W1W2 has rank k and contains the k components associated with the largest singular values of Σxy.The product therefore solves the k-low-rank constrained regression problem, with the corresponding minimum ℓ2-norm solution identified in Corollary 2.
- 3.2 Discrete dynamics: The discrete analysis requires a new proof technique because no closed-form solution is known for the discrete dynamics.With sufficiently small η, the discrete dynamics reproduce sequential component learning, while the linear formulation learns components simultaneously.
4 Experiments
Experiments assess the assumptions on classification datasets and compare reconstruction dynamics for linear and two-layer linear models. The two-layer network exhibits sequential component learning consistent with the theoretical prediction.
- 4.1 Assump. 1 for Classification Datasets: The normalized quantities Δx and Δxy assess violations of the assumptions across MNIST, CIFAR10, and down-sampled ImageNet.Values closer to 0 indicate that an assumption approximately holds, while values closer to 1 indicate greater violation.
- 4.1 Assump. 1 for Classification Datasets: The quantity associated with the paper’s Assumption 1 is two orders of magnitude smaller than the quantity associated with Saxe et al.’s assumption.This pattern is reported across the three datasets and is presented as indicating the relevance of the paper’s assumption.
- 4.2 Matrix Reconstruction: The autoencoder experiment uses p = d = 20, n = 1000, r = 5, and synthetic data with latent variances 4, 2, 1, 1/2, and 1/4.The experiment compares reconstruction properties of the linear predictor with the product of the two network layers.
- 4.2 Matrix Reconstruction: The two-layer linear network’s trace-norm trajectory is close to a step function, with each step corresponding to learning a new component sequentially.Its experimental behavior is reported as close to the theoretical behavior predicted by the continuous dynamics.
5 Discussion
The paper studies implicit regularization in a simple two-layer linear network and finds that discrete gradient dynamics sequentially learn hierarchical problem components. Its conclusions provide insight into generalization while remaining limited to a simplified setting.
- 5 Discussion: The discrete gradient dynamics visits implicitly regularized solutions of the initial linear regression problem.The paper frames this as an instance of implicit regularization from gradient descent on an over-parameterized model.
- 5 Discussion: The dynamics sequentially learns hierarchical components, which could help explain generalization.This consequence is stated within the paper’s linear-network setting.
- 5 Discussion: The setting omits nonlinearities, large values of L, and stochasticity from SGD, so it does not claim to solve generalization in deep neural networks.The paper presents the model as providing insights into possible sources of generalization in deep learning.
A Deep Linear Autoencoder Recovers PCA.
The deep linear autoencoder’s gradient dynamics are formulated through gradient and recurrence relations whose limiting solution recovers a PCA formulation.
- A Deep Linear Autoencoder Recovers PCA: The two-layer linear autoencoder is introduced as the formulation under study.The subsequent derivation uses its objective and gradient dynamics.
- A Deep Linear Autoencoder Recovers PCA: The gradients of the objective define the dynamics used to analyze the network parameters.The supplied passage introduces these gradients before the discrete recurrence.
- A Deep Linear Autoencoder Recovers PCA: The discrete dynamics are represented through a recurrence for the network weights.The recurrence is paired with a continuous-time characterization in the derivation.
- A Deep Linear Autoencoder Recovers PCA: The continuous dynamics imply a relation between the two layer weights, yielding a limiting solution.That limit is identified as a formulation of PCA.
B.1 Proof of Prop. 1
The proof analyzes continuous two-layer linear-network dynamics under commuting and perturbed non-commuting covariance structures. It establishes boundedness, convergence, and component-wise dynamics using transformations, uniqueness, and perturbation arguments.
- Commutative case: The proof diagonalizes the relevant covariance structure and reduces the dynamics to component-wise scalar equations.Only the first r diagonal components have non-trivial dynamics, with each governed by products of layer coefficients.
- Commutative case: Under symmetric layer initialization, the scalar dynamics admit an analytic solution whose uniqueness follows from the Cauchy-Lipschitz theorem.The proof first constructs the solution and then verifies that it satisfies the nonlinear differential equation.
- Non-commutative case: The non-commutative analysis compares the perturbed dynamics with a commuting reference system using Lipschitz bounds and Grönwall’s inequality.The perturbation scale is controlled through ∥S∥ = O(ϵ).
- Non-commutative case: The predictor W(t) converges to X†Y and is therefore bounded for t > 0.The argument uses the span of the gradients and the limiting interpolation condition XW(t) → Y.
- Non-commutative case: The layer norms satisfy ∥W1(t)∥2 = O(t) and ∥W2(t)∥2 = O(t) when the predictor remains bounded.These bounds support the perturbation analysis of the two-layer dynamics.
B.3 Proof of Thm. 2
The proof establishes the behavior of the quantities defined in (16) under vanishing initialization, with distinct regimes determined by time and σ_i.
- Under m_i(0) = e^−δ for 1 ≤ i ≤ r, the theorem analyzes the values w_i(t) defined in (16).
- For each i and t ≥ 0, the resulting behavior changes at the threshold t = 1/σ_i.
B.5 Proof of Thm. 3
The proof develops recurrence properties for diagonal sequences governing the discrete two-layer linear-network dynamics, under positive step size and stability assumptions.
- The discrete update scheme is analyzed through sequences m_i(t), n_i(t), and a_i(t) associated with the diagonal coefficients of the network weights.
- The argument uses the transformation from §B.2 and inductive bounds on the recurrence to establish the theorem’s properties over successive time steps.
- If the initial weights have only diagonal coefficients, the discrete dynamics preserve this diagonal structure for all t ∈ N.
- The proof assumes η > 0 and distinguishes cases based on whether σ_i and λ_i are positive or zero.
- The sequence a_i(t) is shown to be increasing under the theorem’s recurrence and initialization conditions.
- The theorem requires a step-size condition 2ησ_i < 1 for the relevant indices, while the corollary imposes additional bounds involving σ_i − σ_i+1.
- The corollary defines transition times T_j := 1/(σ_jη), with T_j = +∞ beyond the effective rank and T_0 = 0.