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Implicit Regularization in Matrix Factorization

Suriya Gunasekar, Blake Woodworth, Srinadh Bhojanapalli, Behnam Neyshabur, Nathan Srebro

TL;DR

The paper investigates implicit regularization in underdetermined matrix regression by optimizing a full-dimensional factorization $X=UU^T$ via gradient descent on $U$. It derives gradient-flow dynamics and conjectures that, under small steps and near-origin initialization, the limit solution attains the minimum nuclear-norm subject to $A(X)=y$, effectively biasing toward the simplest enriched representation. Theoretical results establish the conjecture in the commuting case, while non-commuting measurement matrices pose substantial analytical challenges, complemented by extensive empirical evidence across synthetic and real data showing a bias toward low nuclear norm even when reconstruction is not guaranteed. These findings suggest that optimization dynamics themselves can act as a powerful implicit regularizer, with implications for generalization in non-convex matrix factorization and related architectures.

Abstract

We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix $X$ with gradient descent on a factorization of $X$. We conjecture and provide empirical and theoretical evidence that with small enough step sizes and initialization close enough to the origin, gradient descent on a full dimensional factorization converges to the minimum nuclear norm solution.

Implicit Regularization in Matrix Factorization

TL;DR

The paper investigates implicit regularization in underdetermined matrix regression by optimizing a full-dimensional factorization via gradient descent on . It derives gradient-flow dynamics and conjectures that, under small steps and near-origin initialization, the limit solution attains the minimum nuclear-norm subject to , effectively biasing toward the simplest enriched representation. Theoretical results establish the conjecture in the commuting case, while non-commuting measurement matrices pose substantial analytical challenges, complemented by extensive empirical evidence across synthetic and real data showing a bias toward low nuclear norm even when reconstruction is not guaranteed. These findings suggest that optimization dynamics themselves can act as a powerful implicit regularizer, with implications for generalization in non-convex matrix factorization and related architectures.

Abstract

We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix with gradient descent on a factorization of . We conjecture and provide empirical and theoretical evidence that with small enough step sizes and initialization close enough to the origin, gradient descent on a full dimensional factorization converges to the minimum nuclear norm solution.

Paper Structure

This paper contains 10 sections, 2 theorems, 12 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Factorized Gradient Descent for Matrix Regression
  3. Gradient Flow and Main Conjecture
  4. Theoretical Analysis
  5. Warm up:
  6. Getting started: $\mathbf{m=1}$
  7. What we can prove: Commutative $\mathbf{\left\{A_i\right\}_{i\in[m]}}$
  8. The plot thickens: Non-commutative $\mathbf{\left\{A_i\right\}_{i\in[m]}}$
  9. Empirical Evidence
  10. Discussion

Key Result

Theorem 1

In the case where matrices $\left\{A_i\right\}_{i=1}^m$ commute, if $\widehat{X} = \lim_{\alpha\to 0}X_\infty(\alpha I)$ exists and is a global optimum for eq:lstsq with $\mathcal{A}(\widehat{X}) = y$, then $\widehat{X} \in \mathop{\mathrm{argmin}}\limits_{X\succeq0} \norm{X}_*\ \textrm{s.t.}\ \math

Figures (4)

  • Figure 1: Reconstruction error of the solutions for the planted $50 \times 50$ matrix reconstruction problem. In $(a)$$X^*$ is of rank $r=2$ and $m=3nr$, in $(b)$$X^*$ has a spectrum decaying as $O(1/k^{1.5})$ normalized to have $\|X^*\|_*=\sqrt{r}\norm{X^*}_F$ for $r=2$ and $m=3nr$, and in $(c)$ we look at a non-reconstructable setting where the number of measurements $m=nr/4$ is much smaller than the requirement to reconstruct a rank $r=2$ matrix. The plots compare the reconstruction error of gradient descent on $U$ for different choices initialization $U_0$ and step size $\eta$, including fixed step-size and exact line search clipped for stability ($\eta_{\overline{ELS}}$). Additonally, the orange dashed reference line represents the performance of $X_{gd}$ --- a rank unconstrained global optima obtained by projected gradient descent on $X$ space for \ref{['eq:lstsq']}, and 'SVD-Initialization' is an example of an alternate rank $d$ global optima, where initialization $U_0$ is picked based on SVD of $X_{gd}$ and gradient descent with small stepsize is run on factor space. The results are averaged across $3$ random initialization and (nearly zero) errorbars indicate the standard deviation.
  • Figure 2: Nuclear norm of the solutions from Figure \ref{['fig:test_err_gauss']}. In addition to the reference of $X_{gd}$ from Figure \ref{['fig:test_err_gauss']}, the magenta dashed line (almost overlapped by the plot of $\|U\|_F=10^{-4},\eta=10^{-3}$) is added as a reference for the (rank unconstrained) minimum nuclear norm global optima. The error bars indicate the standard deviation across $3$ random initializations. We have dropped the plot for $\norm{U}_F=1,\eta=10^{-3}$ to reduce clutter.
  • Figure 3: Additional matrix reconstruction experiments
  • Figure 4: Histogram of relative sub-optimality of nuclear norm of $X_\infty$ in grid search experiments. In this figure, we plot the histogram of $\Delta(X_\infty)=\frac{\norm{X_\infty}_*-\norm{X_\text{min}}_*}{\norm{X_\text{min}}_*}$, where $\norm{X_\text{min}}_*=\underset{\mathcal{A}(X)=y}{\min}\norm{X}_*$. The three panels correspond to different values of norm of initialization $\bar{\alpha}=\|U_0\|_F$. In $(a)$$\bar{\alpha}=10^{-5}$, in $(a)$$\bar{\alpha}=10^{-3}$, and in $(c)$$\bar{\alpha}=1$.

Theorems & Definitions (4)

  • Conjecture
  • Theorem 1
  • proof
  • Corollary 2