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What Happens after SGD Reaches Zero Loss? --A Mathematical Framework

Zhiyuan Li, Tianhao Wang, Sanjeev Arora

arXiv:2110.06914v4cs.LGstat.ML

TL;DR

The paper addresses how to characterize SGD’s implicit bias near manifolds of loss minimizers, a question tied to generalization in overparametrized models. It develops a framework for limiting SGD dynamics that accommodates arbitrary noise covariance and analyzes the dynamics over Θ(η^-2) steps. For sparse overparametrized linear models, it establishes an O(κ ln d) sample bound for label-noise SGD, compared with Ω(d) for kernel-regime GD.

  • Problem

    The implicit bias that may underlie generalization in overparametrized models remains difficult to characterize mathematically.

    Source: §1 · p. 1

  • Method

    The paper develops a framework, inspired by Katzenberger (1991), for characterizing limiting SGD dynamics near manifolds of local minimizers with arbitrary noise covariance.

    Source: §7 · p. 12

  • Results

    For κ-sparse overparametrized linear models, label-noise SGD achieves O(κ ln d) sample complexity, while kernel-regime GD requires Ω(d) samples.

    Source: §1.2 · p. 4, §6.3 · p. 11

  • Takeaways & Limitations

    The results establish a sample-complexity gap between label-noise SGD and kernel-regime GD for the studied overparametrized linear model.

    Source: §1.2 · p. 4, §7 · p. 12

    The framework assumes a C3 loss and a C2 manifold of local minimizers, and one theorem’s convergence argument assumes convergence of (Yn, Zn) for brevity.

    Source: §3.1 · p. 5, §4.1 · p. 6

Abstract

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Understanding the implicit bias of Stochastic Gradient Descent (SGD) is one of the key challenges in deep learning, especially for overparametrized models, where the local minimizers of the loss function $L$ can form a manifold. Intuitively, with a sufficiently small learning rate $\eta$, SGD tracks Gradient Descent (GD) until it gets close to such manifold, where the gradient noise prevents further convergence. In such a regime, Blanc et al. (2020) proved that SGD with label noise locally decreases a regularizer-like term, the sharpness of loss, $\mathrm{tr}[\nabla^2 L]$. The current paper gives a general framework for such analysis by adapting ideas from Katzenberger (1991). It allows in principle a complete characterization for the regularization effect of SGD around such manifold -- i.e., the "implicit bias" -- using a stochastic differential equation (SDE) describing the limiting dynamics of the parameters, which is determined jointly by the loss function and the noise covariance. This yields some new results: (1) a global analysis of the implicit bias valid for $\eta^{-2}$ steps, in contrast to the local analysis of Blanc et al. (2020) that is only valid for $\eta^{-1.6}$ steps and (2) allowing arbitrary noise covariance. As an application, we show with arbitrary large initialization, label noise SGD can always escape the kernel regime and only requires $O(\kappa\ln d)$ samples for learning an $\kappa$-sparse overparametrized linear model in $\mathbb{R}^d$ (Woodworth et al., 2020), while GD initialized in the kernel regime requires $\Omega(d)$ samples. This upper bound is minimax optimal and improves the previous $\tilde{O}(\kappa^2)$ upper bound (HaoChen et al., 2020).

1 Introduction

The paper studies SGD’s implicit bias near manifolds of loss minimizers, where noise can induce regularizing drift even at small learning rates. It develops a global, arbitrary-covariance framework for that drift and applies it to sparse overparametrized linear models.

1 Introduction

  • The paper frames implicit bias as a mechanism for better generalization within a single basin, despite small learning rates limiting jumps between basins.The authors contrast this setting with claims that large learning rates or noise favor flatter basins.Source: §1 · pp. 1–2

1.1 Intuitive explanation of regularization effect due to SGD

  • Near a minimizer manifold, fast normal-space noise fluctuations induce slow tangent drift that decreases tr[∇²L].Locally, the normal dynamics are approximated by an Ornstein–Uhlenbeck process; averaging its fluctuations yields the tangent drift.Source: §1.1 · pp. 2–3
  • The earlier sharpness analysis is local to O(η^-1.6) time, whereas nontrivial limiting dynamics require global analysis on the η^-2 time scale.The local analysis yields only O(η^0.4) total movement as η tends to zero.Source: §1 · p. 2, §1.1 · p. 3

1.2 Our Approach: Separating the Slow from the Fast

  • To separate slow manifold motion from fast dynamics, the approach tracks a projection onto the minimizer manifold that is invariant under gradient flow.The projection is chosen as the limiting point of gradient flow from the current parameter.Source: §1.2 · p. 3
  • The framework characterizes limiting SGD dynamics over Θ(η^-2) steps as η tends to zero, and allows arbitrary noise covariance.For label noise, it recovers the regularization effect associated with tr[∇²L].Source: §1.2 · p. 4
  • For κ-sparse overparametrized linear models, label-noise SGD achieves O(κ ln d) sample complexity, while kernel-regime GD requires Ω(d) samples.The SGD bound is minimax optimal and improves the earlier eO(κ^2) upper bound; its implicit regularizer is a data-dependent weighted ℓ1 norm.Source: §1.2 · p. 4

2 Related Works

Related work connects overparametrization with manifolds of minimizers and studies how optimization dynamics select among solutions. Prior accounts examine implicit bias from gradient methods and continuous-time SDE models of stochastic optimization.

Loss Landscape of Overparametrized Models

  • For overparametrized models, local minimizers can form low-dimensional manifolds, and SGD convergence to such manifolds has been studied from nearby initial points.Mode connectivity describes simple paths connecting local minimizers.Source: §2 · p. 4

Implicit Bias in Overparametrized Models

  • Prior work links implicit bias to noise shape and parameter dependence, while gradient methods on quadratically overparametrized linear models can favor ℓ1-regularized solutions.In the kernel regime, gradient descent behaves like a kernel method and selects solutions minimizing the corresponding RKHS norm.Source: §2 · pp. 4–5

Modelling Stochastic First-Order Methods with Itô SDE

  • SDE approximations of SGD have been used to study scaling rules, intrinsic learning rates, flat minima, and relations between implicit regularization and training speed.The cited work also studies invariant distributions collapsing to minimizer manifolds.Source: §2 · p. 5

3 Notation and Preliminaries

The preliminaries define gradient flow and its limiting map, then specify assumptions on a smooth loss and a manifold of local minimizers. They also describe a condition under which interpolating global minimizers form such a manifold.

3 Notation and Preliminaries

  • The gradient-flow map φ follows the flow generated by −∇L, and Φ(x) denotes its limiting point when that limit exists.The text also defines 1_ξ as the one-hot vector for coordinate ξ and 1 as the all-ones vector.Source: §3 · p. 5

3.1 Manifold of local minimizers

  • The analysis assumes a C3 loss and a C2 manifold Γ of local minimizers, with Hessian rank M on Γ.It also assumes an open neighborhood U from which gradient flow converges to Γ.Source: §3.1 · p. 5
  • If all global minimizers interpolate the data and the Jacobian of the model outputs has rank n on Γ, the interpolating set is a manifold of dimension D − n.The rank condition is equivalent to full rank of the NTK at points on Γ.Source: §3.1 · p. 5

4 Limiting Diffusion of SGD

The paper adapts Katzenberger’s framework to characterize SGD’s limiting dynamics near a manifold of local minimizers. It derives the required geometric quantities and shows that, under stated assumptions, SGD iterates converge to the limiting process over the η^-2 timescale.

4.1 Recap of Katzenberger’s Theorem

  • The framework adapts Katzenberger’s theorem with a user-friendly convergence result, computes derivatives of Φ from L, and rewrites SGD as a Katzenberger process.The transformation of the process makes its limit continuous at time zero; the paper also identifies obstacles involving derivative expressions and convergence.Source: §4.1 · p. 6

4.2 Closed-Form expression of the limiting diffusion

  • On Γ, ∂Φ is the orthogonal projection onto the tangent space, and the noise covariance is decomposed into tangent, normal, and cross-space components.The paper introduces a Lyapunov operator to express second-order derivatives compactly.Source: §4.2 · p. 7

4.3 Main Result

  • Under Assumptions 3.1 and 3.2, SGD iterates converge in distribution to the limiting process at time T after ⌊T/η²⌋ steps, provided the process stays in U.A stronger result gives path convergence; a stopped version applies when the limiting process stays in a compact subset of Γ with high probability.Source: §4.3 · pp. 7–8
  • The limiting dynamics preserve tangent noise and compensate to remain on Γ, while normal-space noise induces noise and mixed regularization through its second-order effect.The tangent-noise compensation depends on Γ itself, while the second-order terms correspond to mixed and normal noise covariance.Source: §4.3 · p. 7

5 Implications and Examples

The paper derives limiting dynamics for isotropic and label noise within its SGD framework. For isotropic noise, the limit combines Brownian motion on the manifold with a Riemannian gradient flow; for label noise, the regularization comes from normal-space noise.

Type I: Isotropic Noise

  • With isotropic noise, the limiting diffusion is Brownian motion plus Riemannian gradient flow with respect to the pseudo-determinant of ∇²L.The pseudo-determinant is the product of the nonzero eigenvalues of ∇²L, equivalently the exponential of their summed logarithms.Source: §5 · p. 8

Implications and Examples

  • The framework gives a global Θ(η^-2)-step analysis with simpler derivations than prior discrete-time analyses, enabling a generalization upper bound for overparametrized linear models.The paper says the framework avoids manual Taylor expansion and careful joint control of infinitesimal terms.Source: §5 · p. 8

Type II: Label Noise

  • When Σ=c∇²L on Γ for c>0, the label-noise SDE simplifies to a limiting flow whose regularization comes from normal-space noise.The section fixes the expected loss and noise distribution and takes η to zero when deriving these noise-type examples.Source: §5 · pp. 8–9

6 Provable Generalization Benefit with Label Noise

The section proves that label-noise SGD can learn sparse overparametrized linear models with far fewer samples than GD initialized in the kernel regime. Its analysis connects convergence of a limiting flow to recovery by regularizer minimization.

6 Provable Generalization Benefit with Label Noise

  • O(κ ln d) samples suffice for label-noise SGD to learn a κ-sparse overparametrized linear model, while kernel-regime GD requires Ω(d).The SGD guarantee holds for Gaussian or Boolean data and arbitrary initialization outside a zero-measure set; the GD lower bound holds for all learning-rate schedules and fixed step counts.Source: §6 · p. 9, §6.3 · p. 11

6.1 Limiting Flow Converges to Minimizers of Regularizer

  • The limiting flow from any point on Γ converges to the regularizer's optimum, despite noncompactness and boundary stationary points.The proof establishes a Polyak–Łojasiewicz condition for convergence and uses a potential function to rule out suboptimal stationary points.Source: §6.1 · pp. 10–11

6.2 Minimizer of the Regularizer Recovers the Sparse Groundtruth

  • With n ≥ Cκ ln d Gaussian or Boolean samples, the regularizer's unique optimum up to coordinate sign flips recovers the sparse groundtruth.For Boolean data, minimizing the regularizer on Γ is equivalent to finding a minimum-ℓ1-norm solution; the Gaussian case is established using techniques from Tropp (2015).Source: §6.2 · p. 11

6.3 Lower Bound for Gradient Descent in the Kernel Regime

  • Kernel-regime GD needs at least Ω(d) samples, regardless of its learning-rate schedule or number of steps.The lower-bound analysis considers the infinite-initialization limit and a fixed neural tangent kernel.Source: §6.3 · p. 11

7 Conclusion and Future Work

The conclusion presents a framework for characterizing SGD's implicit bias through limiting dynamics, including for arbitrary noise covariance. It highlights a sparse-learning application and identifies extensions to more complex models and other optimization algorithms as future work.

7 Conclusion and Future Work

  • For infinitesimal learning rates, the framework characterizes SGD's implicit bias through a limiting diffusion on a manifold of local minimizers, allowing arbitrary noise covariance.The stated convergence scale is Θ(η^-2) SGD steps as η approaches zero.Source: §7 · p. 12
  • For specific noise types, the framework recovers and strengthens prior implicit-bias results with simpler analysis.The conclusion says the framework may be applied to more complex models and extended to algorithms such as ADAM and SGD with momentum.Source: §7 · p. 12

A Preliminaries on Stochastic Processes

  • The preliminaries establish notation and review stochastic-process basics, referring readers to standard texts for systematic derivations.They define conventions for derivatives, tangent and normal spaces, and Banach-space norms.Source: A Preliminaries on Stochastic Processes · p. 18

A.1 C`adl`ag Function and Metric

This appendix develops stochastic-process, time-scaling, and convergence tools for deriving SGD’s limiting diffusion and verifying conditions in Katzenberger’s framework, including compactness and limit-point properties and the equivalence of uniform and Skorohod convergence in distribution.

A.1 C`adl`ag Function and Metric

  • The Skorokhod framework defines c`adl`ag paths as right-continuous with left limits and equips them with a metric for comparing paths.The appendix also defines continuity moduli and jumps for these paths.Source: §A.1 · pp. 18–19

A.2 Stochastic Processes and Stochastic Integral

  • The appendix defines cross variation, martingales, local martingales, and semimartingales as foundational stochastic-process concepts.A semimartingale is decomposed into a local martingale and an adapted c`adl`ag process of bounded total variation.Source: §A.2 · p. 19
  • Itô’s stochastic integral is defined through left-endpoint sums; when the processes do not jump together, it coincides with the Riemann–Stieltjes integral.The integral’s defining limit is in probability as the partition mesh goes to zero.Source: §A.2 · pp. 19–20

A.3 Weak Convergence for Stochastic Processes

  • Weak convergence is defined through bounded continuous test functions, and convergence in Skorokhod metric implies uniform-metric convergence when the limit paths are continuous.The Skorokhod representation theorem also gives almost-sure convergence on a suitable probability space for weakly convergent measures.Source: §A.3 · pp. 20–21

B Limiting Diffusion of SGD

  • The appendix derives the limiting diffusion of SGD, verifies that Φ is C2 under Assumption 3.2, and outlines the supporting analysis.The supporting appendices address time scalings, conditions for applying Katzenberger’s results, and a user-friendly statement of the theorem.Source: B Limiting Diffusion of SGD · p. 21

B.1 Approximating SGD by SDE

  • Rescaling the SGD-approximating SDE by η^-2 yields a process whose constant-time evolution approximates Ω(1/η^2) SGD steps.The appendix also derives the SDE corresponding to the original time scaling.Source: §B.1 · pp. 21–22

B.2 Necessary Conditions

  • For SGD iterates defined in Lemma 4.2, the integrator and noise sequences satisfy Conditions B.2–B.5 required by the framework.The verification uses the noise process’s independent increments and martingale properties.Source: §B.2 · p. 23

B.3 Katzenberger’s Theorem for Asymptotically Continuous Case

  • Katzenberger’s full theorem requires asymptotically continuous integrators, encompassing SDE (3) and SGD as η tends to zero.The theorem’s stated setup also assumes Assumptions 3.1 and 3.2 and Conditions B.2–B.5.Source: §B.3 · p. 24
  • Any limit point is a continuous semimartingale whose Y component remains in Γ, with µ at least the first exit time from the interior of K.The sequence is relatively compact under the Skorohod metric.Source: §B.3 · p. 25
  • Convergence in distribution under the Skorohod metric is equivalent to convergence under the uniform metric, which the paper uses thereafter.The stated equivalence also applies to Prohorov metrics and δ-Prohorov distances.Source: §B.3 · p. 25

B.4 A User-friendly Interface for Katzenberger’s Theorem

This section makes Katzenberger’s limiting-diffusion result easier to apply and characterizes SGD’s two-phase behavior through derivative calculations involving the manifold projection, loss Hessian, and Lyapunov operator. It also shows that tangent-noise compensation depends only on the manifold, while normal-noise dynamics can cycle rather than stop.

B.4 A User-friendly Interface for Katzenberger’s Theorem

  • For small learning rate η, stopped SGD paths can be coupled to the stopped limiting diffusion within uniform distance ϵ with probability at least 1 − 2δ.Theorem B.9 gives this approximation when δ bounds the probability that the limiting process exits the compact set K by time T.Source: §B.4 · pp. 25–26
  • If the limiting diffusion is global and remains in U, then for each fixed T, the SGD process converges in distribution to it as η → 0.The limiting process stays on Γ almost surely.Source: §B.4 · pp. 26, 28
  • SGD first moves toward the minimizer manifold Γ along gradient flow over Θ(1/η) steps, then stays close to Γ and diffuses approximately according to (10) over Θ(1/η^2) steps.Theorem 4.6 reflects only the second phase because the first phase becomes negligible on that time scale as η → 0.Source: §B.4 · p. 28

C.1 Explicit Expression of the Derivatives

  • On Γ, the loss Hessian annihilates tangent vectors, and ∂Φ is the orthogonal projection onto the tangent space; for all x, ∂Φ(x)∇L(x) = 0.This subsection computes derivatives of Φ by relating them to derivatives of L.Source: §C.1 · p. 29, C Explicit Formula of the Limiting… · p. 29
  • Using the Lyapunov Operator from Definition 4.4 simplifies Lemma C.5 into Lemma C.6.Lemma 4.5 then follows directly from Lemmas C.4 and C.6.Source: §C.1 · p. 32

C.2 Tangent Noise Compensation only Dependends on the Manifold Itself

  • The tangent-noise compensation term in the limiting dynamics depends only on Γ, not on which loss function locally defines that manifold.For any tangent vector, the specified Hessian-pseudoinverse derivative expression is equal for two loss functions defining Γ locally.Source: §C.2 · p. 32

C.4 Example: k-Phase Motor

  • Aligning noise magnitude with regularization strength and using at least three auxiliary dimensions yields an anti-clockwise normal-regularization force of constant magnitude.The construction requires k ≥ 3, equivalently D ≥ 5.Source: §C.4 · pp. 34–35
  • The limiting flow in the k-phase motor example moves anti-clockwise around a cycle at constant angular speed (D − 2)/2, never stopping.The example provides a rigorous instance of normal-noise implicit bias that cannot be characterized by a fixed regularizer; Damian et al. (2021) had previously verified this phenomenon experimentally.Source: §C.4 · pp. 33–34

C.3 Proof of results in Section 5

The appendix gives explicit limiting-diffusion specializations for isotropic and label noise. It also introduces proofs for the Section 6 overparametrized linear-model results.

C.3 Proof of results in Section 5

  • For isotropic noise, Corollary 5.1 gives a specialization of the limiting diffusion on Γ.The supplied passage names the corollary but does not include the specialized equation.Source: §C.3 · p. 33
  • The pseudo-determinant of the Hessian equals the sum of the logarithms of its nonzero eigenvalues.The passage defines it as the limit of |∇2L(Y) + αI_D| divided by α raised to D − rank(∇2L(Y)).Source: §C.3 · p. 33
  • For label noise with covariance Σ = c∇2L on Γ, the limiting diffusion simplifies to a flow whose regularization comes from normal-space noise.The stated condition is for a constant c > 0.Source: §C.3 · p. 33

D.1 Proof of Theorem 6.1

Theorem 6.1 connects the label-noise SGD guarantee to an optimal solution of a regularized flow for the overparametrized linear model. Its assumptions include a sparse ground truth and sufficient Gaussian or Boolean training data.

D.1 Proof of Theorem 6.1

  • With n ≥ Ω(κ ln d) Gaussian or Boolean samples, label-noise SGD returns an ε-optimal solution after ⌊T/η²⌋ steps with probability at least 1 − e^−Ω(n).The guarantee holds for any initialization outside a zero-measure set and sufficiently small learning rate η.Source: §D.1 · p. 36
  • The proof transfers an ε/2-optimal endpoint of the Riemannian gradient flow to an ε-optimal SGD iterate using Theorem 4.6.The transfer is made for sufficiently small η, with probability at least 1 − e^−Ω(n).Source: §D.1 · p. 36

D.2 Proof of Lemma 6.2

  • When rank(Z) = n, Γ is a smooth manifold of dimension D − n and the Hessian rank is n along Γ.The rank condition holds with probability 1 for Gaussian data and probability 1 − c^d for Boolean data, for some constant c in (0, 1).Source: §D.2 · p. 37

D.3 Proof of Lemma 6.3

The proof establishes that gradient flow from the specified neighborhood converges to the minimizer manifold Γ. It uses invariants along the flow and a lower bound involving the smallest eigenvalue of K.

D.3 Proof of Lemma 6.3

  • Along gradient flow, each product u_j(t)v_j(t) stays constant, so neither coordinate changes sign.The sign claim follows from continuity and invariance of the product.Source: §D.3 · p. 37
  • The smallest eigenvalue of K(x(t)) is bounded below by 8 min_i |u_i(0)v_i(0)| λ_min(ZZᵀ) along the flow.The passage uses this bound to complete the auxiliary lemma.Source: §D.3 · p. 38
  • Every stationary point in U is a global minimizer, so the stationary points there belong to Γ.The stated characterization is Γ = {x ∈ U | ∇L(x) = 0}.Source: §D.3 · p. 38
  • Gradient flow starting in U converges, and its limit lies in Γ.The proof notes that the flow converges to a stationary point in U, then applies the characterization of Γ.Source: §D.3 · p. 38

D.4 Proof of results in Section 6.2

The appendix proves recovery and uniqueness results for the parameterized optimization problem under Boolean and Gaussian designs. Its proof reduces the parameterized problem to a convex program and transfers uniqueness back through a canonical parametrization.

D.4 Proof of results in Section 6.2

  • When (35) has unique optimum w*, the optimizer of (18) is unique up to coordinate sign flips, with one solution given by the canonical parametrization ψ(w*).Source: §D.4 · p. 39
  • For Boolean designs, the optimum of (18) is unique up to coordinate sign flips and recovers the groundtruth.Source: §D.4 · p. 39
  • For Gaussian designs, n ≥ C(κ ln d) makes the minimum conic singular value positive with probability at least 1 − (2d + 1)e^−cn, yielding unique recovery.Source: §D.4 · p. 42
  • The Gaussian proof establishes uniqueness by showing the minimum conic singular value is positive for the convex program in w.Source: §D.4 · pp. 38, 40, 41, 42

D.5 Proof of Lemma 6.5

The proof analyzes the constrained flow by extending its vector field to the ambient space and establishing properties that support convergence. It shows that from any initialization in Γ, the flow converges to an optimal solution of (18), ruling out suboptimal limits with a tailored potential.

D.5 Proof of Lemma 6.5

  • The ambient field F is defined as the minimum-norm Lagrangian gradient, and it is orthogonal to every constraint gradient.The minimum exists uniquely because the candidate gradients form an affine subspace.Source: §D.5 · p. 43
  • If F(x) = 0, then each coordinate pair (u_j, v_j) has at least one zero component.The proof rules out both components being nonzero using identities that fail only on a probability-zero event in the Boolean and Gaussian cases.Source: §D.5 · pp. 43–44
  • F is continuous throughout the ambient space R^D.The proof handles points with full-rank constraint-gradient Gram matrices locally, then treats general points by reducing to independent coordinates.Source: §D.5 · pp. 44, 45, 47
  • For any initialization in Γ, the Riemannian gradient flow is defined for all t ≥ 0.The proof argues that finite-time termination would contradict the maintained positive value of each initialized coordinate product.Source: §D.5 · p. 47
  • Near any x* in Γ, F satisfies a local PL inequality on Γ: ∥F(x)∥_2^2 ≥ c max(R(x) − R(x*), 0) for some c > 0.The proof reduces the condition to points where F vanishes and uses the coordinate representation w = u⊙2 − v⊙2.Source: §D.5 · p. 49
  • The proof establishes finite trajectory length, so the flow converges to a limit where F(x(∞)) = 0.It uses the PL condition near a limit point to show finite length; the limit is therefore stationary for F and has F(x(∞)) = 0.Source: §D.5 · pp. 51–52
  • To exclude a suboptimal limit, the proof constructs a potential that increases locally near it while tending to −∞ at the limit, yielding a contradiction.The potential's derivative along the flow is shown to force this contradiction.Source: §D.5 · pp. 52, 55
  • Every trajectory of (17) initialized in Γ converges to an optimal solution of (18).Lemma 6.5 states that the limit exists and is optimal.Source: §D.5 · p. 52

D.6 Proof of Theorem 6.7

The proof establishes a lower bound on gradient descent’s expected test loss in the kernel regime. It uses the limited dimension of the gradient span and rotational symmetry of the data and initialization.

D.6 Proof of Theorem 6.7

  • Theorem 6.7 lower-bounds expected ℓ2 loss for any groundtruth, learning-rate schedule, and fixed number of steps.Source: §D.6 · p. 55
  • Rotational equivariance and rotation-invariant data and initialization make expected loss equal to that for a uniformly random groundtruth of the same norm.Source: §D.6 · pp. 55–56
  • GD iterates lie in a space spanned by initialization gradients with dimension at most n, so test loss is bounded below by the target’s squared distance from that space.Source: §D.6 · p. 56
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