Technical Summary: The Implicit Bias of Depth: From Neural Collapse to Softmax Codes

Problem Statement

Deep neural network classifiers trained beyond interpolation often converge to a structured geometric arrangement known as Neural Collapse (NC), where class means form a simplex equiangular tight frame (ETF) and features align with this frame. While NC is well-understood in shallow models and regularized deep settings, its emergence in unregularized deep architectures remains theoretically ambiguous.

Previous theoretical work on the Unconstrained Features Model (UFM) has largely relied on explicit $L_2$ regularization to induce NC. However, recent findings suggest that in deep UFMs, $L_2$ regularization actually induces a low-rank bias that suppresses NC, making NC suboptimal. A critical gap exists in understanding the implicit biases of unregularized deep training: Does depth alone, via gradient descent dynamics, favor NC, or does it drive the system toward alternative geometric structures? Specifically, how do depth, optimization dynamics, and initialization interact to determine the final geometric configuration in the absence of explicit regularization?

Methodology

The authors analyze the Deep Unconstrained Features Model (UFM) trained with Cross-Entropy (CE) loss without explicit regularization. This model treats final-layer features as free optimization variables, effectively abstracting a deep linear network with orthogonal inputs. The study employs two complementary analytical perspectives:

1. Asymptotic Analysis: The authors examine the limit of gradient flow (GF) as $t \to \infty$. They utilize the framework of Lyu & Li (2019) to show that GF on CE loss implicitly performs non-convex margin maximization. They characterize the landscape of the corresponding constrained optimization problem to identify global and local minima.
2. Dynamical Analysis: To understand finite-time behavior, the authors employ Hadamard initialization (specifically Sylvester Hadamard matrices). This initialization preserves the singular vectors of parameter matrices throughout training, reducing the complex matrix dynamics to a tractable system of differential equations governing only the singular values. This allows for a precise characterization of stability and trajectory evolution near the origin and during the transition to the large-norm regime.

Key Contributions and Results

1. Depth Breaks the Benign Landscape

The paper demonstrates that the "benign" landscape of the shallow UFM (where NC is the unique global minimum) does not persist in deep settings ($L > 1$).

• Theorem 3.1: For $L=2$ with $K \ge 6$ classes (or $L \ge 3$ with $K \ge 4$), NC is not globally optimal. While NC remains a local optimum, the landscape admits multiple distinct, stable local minima.
• Mechanism: The suboptimality of NC arises because low-rank structures propagate norms more efficiently through successive matrix multiplications. In a deep network, low-rank matrices allow for larger logit norms (and thus larger Euclidean margins) for a fixed parameter norm compared to the full-rank NC configuration. This is a purely geometric effect of depth, independent of explicit regularization.

2. Characterization of Optimal Low-Rank Structures: Softmax Codes

The authors identify the specific geometric structures that replace NC in the deep limit.

• Theorem 3.2: In the limit of infinite depth ($L \to \infty$), the global optimum corresponds to a rank-2 solution where class means are arranged as vertices of a regular $K$-gon on a circle.
• Connection to Softmax Codes: This structure coincides with the optimal solution of a single-layer UFM under an extreme width bottleneck ($d=2$), known as a softmax code (Jiang et al., 2023). The paper argues that depth acts as an implicit rank constraint, selecting solutions that would otherwise require explicit architectural bottlenecks. Empirical results suggest this connection extends to finite depths, where networks converge to rank-$r$ softmax codes.

3. Dynamical Instability and the "Rich-Get-Richer" Effect

The paper analyzes the trajectory of gradient flow to explain how the system avoids NC.

• Instability of NC: Unlike the shallow case where NC is stable throughout the trajectory, Theorem 4.3 proves that for $L > 1$, the NC direction is unstable near the origin (small parameter norms). The region of instability grows with the number of layers.
• Mechanism: The instability is driven by a "rich-get-richer" effect (Theorem 4.4). In the linearized regime near initialization, larger singular values grow faster than smaller ones ($L > 1$), amplifying initial disparities and driving the system toward low-rank structures before it exits the linearized regime.
• Divergence of KL Divergence: The paper shows that the KL divergence to NC, which serves as a Lyapunov function in shallow models, can diverge in deep models, confirming that trajectories can move away from NC entirely.

4. The Role of Initialization and Width

Despite the theoretical preference for low-rank structures, NC is often observed in practice. The authors explain this via initialization effects.

• Theorem 4.5: Under random initialization with sufficiently large network width ($d$), the initial velocity of the logits is biased toward alignment with the simplex ETF (the signature of NC).
• Concentration of Measure: Large width induces a concentration of measure in the initial eigenvalues, keeping singular values nearly equal. This counteracts the "rich-get-richer" effect, allowing the system to enter the large-norm regime with high effective rank, where NC is locally stable. This provides a dynamical justification for why NC persists in wide, randomly initialized networks despite the implicit low-rank bias of depth.

5. Empirical Verification

The authors validate their theory on deep UFMs and ResNet-20 architectures trained on MNIST and CIFAR-10.

• Depth vs. Rank: Increasing the depth of the network head systematically decreases the effective rank of the mean logit matrix, confirming the depth-induced low-rank bias.
• Width vs. Rank: Increasing the width generally increases the effective rank, supporting the theoretical claim that width biases training toward higher-rank (NC-like) solutions.
• Generalization: Lower effective rank (associated with low-rank alternatives) is generally correlated with poorer generalization compared to NC solutions, suggesting that the implicit bias of depth toward low-rank structures may come at the cost of reduced margin and generalization performance.

Significance and Claims

The paper claims to provide the first comprehensive characterization of how implicit biases from depth, optimization dynamics, and initialization interact to determine the emergence of Neural Collapse in unregularized deep networks.

• Theoretical Novelty: It establishes that depth alone is sufficient to induce a low-rank bias that makes NC suboptimal, challenging the view that NC is the inevitable outcome of deep learning or that it requires explicit regularization to emerge.
• Mechanism Identification: It identifies the "rich-get-richer" dynamic in the linearized regime as the driver for low-rank emergence and characterizes the resulting optima as softmax codes.
• Practical Insight: The work offers a dynamical explanation for the ubiquity of NC in practice: while depth theoretically favors low-rank structures, large width and random initialization can steer optimization back toward NC.
• Limitations: The authors acknowledge that their analysis relies on the UFM abstraction and specific initialization schemes (Hadamard) for tractable dynamics, though they provide evidence that these phenomena persist under small random initialization. They note that the theory is asymptotic and that finite-time rates remain an open direction.

In summary, the paper reframes the understanding of deep learning geometry, positing that depth introduces an implicit low-rank bias that competes with the high-rank structure of Neural Collapse, with the final outcome determined by the interplay between network depth, width, and initialization.