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Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced

Simon S. Du, Wei Hu, Jason D. Lee

arXiv:1806.00900v2cs.LGmath.OCstat.ML

TL;DR

The paper asks how first-order optimization behaves on homogeneous models, where equivalent parameterizations can be severely unbalanced and difficult to analyze. It proves invariance properties for gradient flow across several neural-network architectures and analyzes discretized gradient descent for asymmetric matrix factorization. The results show automatic balancing and global convergence under specified step-size conditions, while generic positive-step-size analysis remains an open direction.

  • Problem

    Homogeneity permits unbalanced parameterizations and makes convergence analysis difficult because small loss can coexist with unbounded factors and non-smooth behavior near optima.

  • Method

    The paper analyzes invariants of gradient-flow trajectories for homogeneous neural networks and uses discretization arguments to study unregularized gradient descent for asymmetric matrix factorization.

  • Results

    Gradient flow preserves differences between squared layer norms across broad homogeneous networks; decreasing-step-size gradient descent balances matrix factors and converges globally, while rank-1 constant-step-size descent has a globally linear rate.

  • Takeaways & Limitations

    First-order algorithms can implicitly regularize homogeneous models by maintaining layer balancedness, providing a basis for analyzing optimization in deep models.

  • Takeaways & Limitations

    The paper does not provide a generic analysis of discretization from gradient flow to positive-step-size gradient descent and leaves other first-order methods for future work.

Abstract

from arXiv · show

We study the implicit regularization imposed by gradient descent for learning multi-layer homogeneous functions including feed-forward fully connected and convolutional deep neural networks with linear, ReLU or Leaky ReLU activation. We rigorously prove that gradient flow (i.e. gradient descent with infinitesimal step size) effectively enforces the differences between squared norms across different layers to remain invariant without any explicit regularization. This result implies that if the weights are initially small, gradient flow automatically balances the magnitudes of all layers. Using a discretization argument, we analyze gradient descent with positive step size for the non-convex low-rank asymmetric matrix factorization problem without any regularization. Inspired by our findings for gradient flow, we prove that gradient descent with step sizes $η_t = O\left(t^{-\left( \frac12+δ\right)} \right)$ ($0<δ\le\frac12$) automatically balances two low-rank factors and converges to a bounded global optimum. Furthermore, for rank-$1$ asymmetric matrix factorization we give a finer analysis showing gradient descent with constant step size converges to the global minimum at a globally linear rate. We believe that the idea of examining the invariance imposed by first order algorithms in learning homogeneous models could serve as a fundamental building block for studying optimization for learning deep models.

1 Introduction

Homogeneity makes deep models and asymmetric matrix factorization difficult to analyze because equivalent parameterizations can be arbitrarily unbalanced and non-coercive. The paper studies how gradient-based methods implicitly preserve balancedness and uses this property to establish convergence results.

  • Motivation: Homogeneous networks can preserve the same prediction while scaling one layer up and another down, allowing small loss with unbounded parameters.This unbalancedness undermines standard convergence analyses for first-order methods.
  • Motivation: In asymmetric matrix factorization, homogeneity makes the unregularized objective non-smooth even near a globally optimal point.A small perturbation to one factor can cause a dramatic change in its gradient when the factors are highly unbalanced.
  • Related work: Prior analyses add a balancing regularizer, which removes the homogeneity issue and yields favorable optimization-landscape properties.For the modified objective, every local minimum is global, saddle points are strict, and the objective is smooth.
  • Empirical motivation: Experiments show that unregularized gradient descent can converge to a global minimum while the ratio between the Frobenius norms of the two factors remains constant.This motivates the question of why gradient descent balances layers without explicit regularization.
  • Contributions: Gradient flow preserves differences between squared layer norms for a broad class of homogeneous models, including fully connected and convolutional networks with linear, ReLU, or Leaky ReLU activations.Small initial norm differences therefore remain small throughout the trajectory, producing automatic layer balancing.
  • Contributions: For asymmetric matrix factorization, decreasing-step-size gradient descent keeps factor imbalance small, bounds the iterates, and converges to a globally optimal solution.For rank-1 factorization, constant-step-size gradient descent converges to the global minimum at a globally linear rate.

2 The Auto-Balancing Properties in Deep Neural Networks

Gradient flow imposes auto-balancing invariants in deep homogeneous networks, preserving squared-norm differences locally and across layers. These results extend to linear, ReLU, Leaky ReLU, sparse, shared-weight, and convolutional architectures.

  • 2.1 Fully Connected Neural Networks: Gradient flow automatically balances incoming and outgoing weights at every neuron in fully connected deep networks.Theorem 2.1 establishes an invariance for each hidden neuron.
  • 2.1 Fully Connected Neural Networks: The difference between squared ℓ2-norms of incoming and outgoing weights remains exactly invariant over time.The incoming vector is the row of one layer, and the outgoing vector is the corresponding column of the next layer.
  • 2.1 Fully Connected Neural Networks: Summing the neuron-wise invariants shows that gradient flow preserves differences between squared Frobenius norms of consecutive weight matrices.With small initialization, these differences remain small throughout training, yielding balanced layer magnitudes.
  • 2.1 Fully Connected Neural Networks: For linear activation, a stronger layer-wise matrix invariance holds even when only two consecutive layers lack nonlinear activations.This extends a result previously known for entirely linear networks.
  • 2.2 Convolutional Neural Networks: The same conservation principle extends to arbitrary sparsity patterns and weight sharing, including convolutional layers, so gradient flow balances layer magnitudes there as well.The parameterization explicitly represents missing and shared weights.
  • 2.1 Fully Connected Neural Networks: The proofs use back-propagation through the chain rule together with homogeneity of activations such as ReLU, Leaky ReLU, and linear activation.For nondifferentiable activations, the analysis uses sub-differentials and differential inclusions.

3 Gradient Descent Converges to Global Minimum for Asymmetric Matrix Factorization

The paper analyzes unregularized gradient descent for asymmetric matrix factorization, extending gradient-flow balancedness to positive step sizes. With suitable diminishing steps, the method converges to a bounded solution near the global minimum; rank-1 factorization also admits constant-step linear convergence.

  • General rank-r case: Gradient descent directly solves unregularized asymmetric matrix factorization from random small initialization, according to the paper’s general-rank result.The authors describe this as the first such result for the unregularized problem.
  • General rank-r case: The analysis uses gradient-flow invariance as inspiration: small initial differences between U^T U and V^T V remain controlled under discretized updates.Positive step sizes no longer preserve exact invariance, so the proof analyzes the update errors.
  • General rank-r case: The objective decreases along the iterates, while the factors remain bounded with high probability over initialization.The lemma records monotone objective decrease and explicit Frobenius-norm bounds.
  • Rank-1 case: For rank 1, constant-step gradient descent keeps the two layer norms within a constant factor and converges linearly to a global minimum.The result relies on approximate balancedness in the signal space; the general rank-r constant-step questions remain open.

4 Empirical Verification

Experiments on a three-layer ReLU network test whether gradient descent preserves and improves layer balancedness. With small step size and initialization, layer squared-norm differences stay small and norm ratios approach one.

  • Empirical verification: Small-step gradient descent with small initialization keeps differences between any two layers’ squared Frobenius norms small throughout training.The experiment explicitly tests this auto-balancing property in a ReLU network.
  • Empirical verification: Norm ratios approach 1 during training, including after roughly 1,000 iterations in the reported figure.The paper reports this behavior for the plotted layer norms.
  • Experimental setup: The experiment uses a 3-layer fully connected ReLU network with input dimension 1,000, hidden widths 100 and 100, and output dimension 10.It trains on 1,000 data points with quadratic loss.
  • Balanced initialization: After 10,000 iterations of balanced initialization, ||W1||_F^2 = 43.76 and ||W3||_F^2 = 43.68.The corresponding layer-norm differences remain bounded by 0.14.
  • Unbalanced initialization: Under the reported unbalanced initialization, after 10,000 iterations ||W1||_F^2 = 55.50, while the supplied passage begins the corresponding ||W2||_F^2 report.The passage does not provide the complete numerical comparison.

5 Conclusion and Future Work

The paper characterizes invariance imposed by first-order optimization and connects gradient-flow auto-balancing with global convergence results for asymmetric matrix factorization. It identifies generic discretization analysis and extensions beyond gradient descent as future directions.

  • Conclusion: Gradient flow automatically balances layer magnitudes in deep neural networks with homogeneous activations.The conclusion frames this as a central finding of the paper.
  • Conclusion: For asymmetric matrix factorization, balancedness supports a result that gradient descent converges to a global minimum.This conclusion concerns the concrete matrix-factorization model analyzed in the paper.
  • Future work: Characterizing invariance for acceleration and adaptive first-order methods remains a future direction.The paper states that its analysis focuses on gradient descent.
  • Future work: A generic analysis of how positive-step discretization affects gradient-flow invariance remains open.The paper uses specialized techniques for asymmetric matrix factorization rather than a general discretization theory.

A Proofs for Section 2

The appendix proves the gradient-flow invariance results using chain-rule calculations. It relates derivatives of layer weights to layer activations, then combines the resulting norm dynamics to establish invariance.

  • Proof strategy: The proofs model the loss as a scalar function of the network output and introduce intermediate layer activations for chain-rule calculations.The appendix defines successive activations and applies the chain rule to layerwise derivatives.
  • Linear activation: For consecutive linear layers, the derivative expressions connect gradients with the adjacent weight matrices and intermediate activations.The displayed derivative identities are used to compare the dynamics of neighboring layer norms.
  • Invariance derivation: Combining the layerwise derivative identities yields the time derivative of the relevant norm difference.The appendix concludes the proof by comparing the resulting equations.

B Proof for Rank-r Matrix Factorization (Theorem 3.1)

This section proves smoothness of the asymmetric matrix-factorization objective over bounded factor sets by bounding its Hessian's largest eigenvalue.

  • Objective derivatives: The objective f(U, V) is analyzed through its gradient and Hessian with respect to the two factor matrices.The Hessian acts on vectorized matrices having the same shape as the concatenated factors.
  • Smoothness over bounded sets: For any c > 0, restricting both factors to a bounded set makes f smooth with parameter (6c + 2)||M*||_F.The proof establishes this by upper-bounding the maximum Hessian eigenvalue.

B.1 Proof of Lemma 3.1

The proof of Lemma 3.1 uses induction to establish descent, boundedness, and balancedness of the gradient-descent iterates for low-rank matrix factorization.

  • Inductive structure: The induction tracks three properties: objective descent, bounded iterates, and balanced Gram matrices across the two factors.Each property supplies the hypotheses needed to prove another at the next iteration.
  • Initialization: The initialization satisfies the three properties with high probability when the Gaussian variance is sufficiently small.The proof then assumes these properties hold and propagates them inductively.
  • Objective descent: Objective descent follows from smoothness on a bounded region and a sufficiently small step size.The argument bounds current and next-step factor norms before applying the standard gradient-descent decrease inequality.

B.2 Convergence to a Stationary Point

Under the theorem's conditions, bounded and descending iterates converge with high probability to a stationary point satisfying the balancedness property.

  • Convergence: With the balancedness and boundedness properties, the iterates converge to a stationary point of f.A descent result rules out divergence because all iterates are bounded.
  • Balancedness: The limiting stationary point also satisfies the balancedness condition inherited from Lemma 3.1.This follows directly from the first inductive property stated in that lemma.
  • Stationarity: The accumulated squared gradient norms are bounded, and smoothness then implies that the limiting point has zero gradient.The proof uses the step-size scaling and boundedness to establish stationarity.

B.3 Proof of Lemma 3.2

The proof shows that every stationary point is either close to a global minimum or has a direction of strictly negative curvature, making it a strict saddle.

  • Negative-curvature direction: The proof searches for a direction Δ whose Hessian quadratic form is negative unless the factorization is close to a global minimum.The construction is based on the imbalance between the two factors' Gram matrices.
  • Global minimizer construction: An SVD-based factorization of M* provides a global minimizer (U*, V*) with equal Gram matrices.Orthogonal alignment is used to define the comparison direction Δ.
  • Stationary-point analysis: At a stationary point, the first-order conditions set both factor gradients to zero.These conditions are used to simplify the Hessian analysis and bound its quadratic form.
  • Strict-saddle conclusion: The final alternative is ||UV^T − M*||_F ≤ ε or the Hessian has a direction with quadratic form at most −1/2 ε^2.In the latter case, the stationary point is a strict saddle.

B.4 Finishing the Proof of Theorem 3.1

Theorem 3.1 follows from earlier lemmas together with the result that gradient descent almost surely avoids strict saddle points.

  • Theorem 3.1 is a direct corollary of Lemmas B.2 and 3.2, plus gradient descent’s almost-sure avoidance of strict saddle points.

C Proof for Rank-1 Matrix Factorization (Theorem 3.2)

The proof analyzes rank-1 asymmetric matrix factorization through two stages: escaping the saddle point and converging to the global minimum. The dynamics grow signal components, keep layer strengths comparable, shrink complementary components, and achieve logarithmic iteration complexity.

  • The proof divides the dynamics into two stages: escaping from the saddle point and converging to the global minimum.
  • Stage 1: escaping from saddle point: During stage 1, the two layer strengths remain comparable, with |α_t − β_t| bounded by 99/101 of their sum.
  • Stage 1: escaping from saddle point: During stage 1, signal-space magnitudes grow exponentially while complementary-space strengths remain small.
  • Stage 2: convergence to global minimum: During stage 2, signal strengths stay non-vanishing, complementary magnitudes shrink geometrically, and the signal-space error contracts up to a diminishing complement-space term.
  • Stage 2: convergence to global minimum: The factorization loss reaches O(εσ_1) after O(log(d/ε)) iterations.
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