The Stubborn Sign: How Initial Weight Values Shape Neural Network Compression

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Author: Denis Avetisyan

New research reveals that the initial signs of neural network weights surprisingly endure throughout training, offering a novel pathway for model compression beyond conventional limits.

Analysis of sign dynamics demonstrates persistent ‘lock-in’ and enables sub-bit compression strategies leveraging low-rank approximation and stopping-time techniques.

Despite aggressive compression techniques pushing the boundaries of neural network efficiency, the sign of each weight remains a persistent bottleneck. This paper, ‘Sign Lock-In: Randomly Initialized Weight Signs Persist and Bottleneck Sub-Bit Model Compression’, reveals a surprising dynamic: learned weight signs are remarkably stable and strongly correlated with their initial random values. Through a stopping-time analysis, we demonstrate that sign flips occur rarely and predictably, exhibiting a geometric tail distribution, suggesting a fundamental limit to sign-pattern randomness. Can we exploit this ‘sign lock-in’ to further compress models beyond the one-bit threshold and improve the efficiency of deep learning?

The Inherent Limits of Sub-Bit Precision

The pursuit of increasingly efficient machine learning models has driven research into sub-bit quantization, a compression technique aiming to represent neural network weights with fewer than one bit of precision. While intuitively appealing for its potential storage reduction, this approach encounters fundamental limitations. Representing weights with sub-bit precision necessitates a trade-off between model size and information loss; as the number of bits approaches zero, the ability to accurately capture the original weight values diminishes rapidly. This is not simply a matter of improved algorithms, but rather a consequence of information theory – a minimal amount of information is required to represent a value, and attempting to compress beyond that limit inevitably leads to significant performance degradation. Consequently, despite ongoing advancements in quantization techniques, sub-bit model compression faces inherent boundaries determined by the need to preserve essential weight information for effective model functionality.

The pursuit of extreme model compression often leads to the consideration of one-bit quantization, where neural network weights are represented with just a single bit. However, this approach encounters a fundamental limitation known as the ‘One-Bit Wall’. While seemingly straightforward, the sign bit – the sole carrier of weight information in this scenario – presents a fixed and irreducible cost. Unlike magnitude information which can be partially sacrificed for compression, the correct sign is crucial; even a slight distortion here dramatically impacts performance. Consequently, the sign bit effectively becomes a bottleneck, as its representation requires a consistent and uncompressible amount of information for every weight, regardless of the overall network size or complexity. This inherent lack of compressibility in the sign bit ultimately constrains the achievable compression ratio and performance of severely quantized neural networks, establishing a hard limit on how far one-bit compression can be effectively pushed.

Information theory provides a rigorous underpinning for the observed limitations of extreme weight compression. Specifically, Shannon’s Rate-Distortion Lower Bound, when applied using Hamming Distortion – a measure of bit flips – demonstrates a fundamental trade-off. This bound establishes that as compression approaches one-bit representation, the cost of accurately encoding even the sign of each weight becomes dominant. The analysis reveals that the sign bit itself introduces a fixed informational overhead, independent of the magnitude of the weight. Consequently, attempts to compress beyond a certain point are provably limited by the necessity to reliably transmit this essential sign information, effectively establishing a theoretical floor on achievable compression rates and confirming the existence of the ‘One-Bit Wall’. R \ge H(X), where R is the rate and H(X) is the entropy, illustrates this inherent limit.

The Persistence of Initial Signatures

Sign Lock-In refers to the observed tendency of learned weight signs in neural networks to maintain their initial values throughout the training process, even when utilizing stochastic gradient descent (SGD). This behavior deviates from the expectation that SGD would randomly perturb weight signs, and instead demonstrates a significant degree of stability. Empirical results indicate that a substantial proportion of weights retain their original sign across many training iterations, suggesting that the initial weight initialization has a lasting impact on the learned model. This persistence is not universal, but it occurs at a rate significantly higher than chance, forming the basis for quantifiable analysis of ‘Sign Drift’ and related phenomena.

The observed ‘Sign Lock-In’ isn’t a result of chance; instead, initial weight signs demonstrate a measurable tendency to remain consistent throughout training. This persistence of initial sign patterns directly influences the rate of ‘Sign Drift’-the change in weight sign over time-reducing the expected frequency of flips. Quantitatively, this manifests as a quantifiable bias towards retaining the original sign, observable across multiple training runs and network architectures. The degree of lock-in is not uniform; certain weights exhibit greater stability than others, but the overall effect contributes to a predictable and measurable reduction in sign changes during stochastic gradient descent.

Theoretical investigations employing Stopping-Time Analysis and the observation of Geometric Tail distributions in weight update magnitudes provide a quantifiable understanding of Sign Lock-In stability. Stopping-Time Analysis establishes bounds on the duration for which a weight’s sign remains unchanged, while Geometric Tail distributions characterize the infrequent, yet significant, magnitude of weight updates. These analyses converge to demonstrate that, under specific training conditions, the rate at which weights flip signs can be reduced to approximately 10⁻³, indicating a substantial degree of sign persistence throughout the training process and forming a basis for predicting and mitigating Sign Drift.

Evidence of Intrinsic Sign Structure

Low-rank approximation applied to sign matrices derived from neural network weights indicates a significant degree of inherent compressibility in the initial sign patterns. This compressibility suggests that the information contained within these sign matrices is not fully random, but rather possesses an underlying structured representation. Specifically, the observed low rank implies that the sign information can be effectively captured using a reduced number of dimensions, indicating redundancy and potential for efficient encoding. Analysis demonstrates that these sign matrices can be accurately reconstructed from a low-rank representation with minimal information loss, supporting the hypothesis of a structured, rather than random, initial state.

A Compressible Sign Template improves sign lock-in by imposing a structured prior on the sign matrix, facilitating efficient encoding of sign information. This template operates by reducing the degrees of freedom within the sign matrix, thereby promoting a more compact representation. The resulting compression not only reduces the memory footprint required to store the sign patterns but also potentially accelerates computations by leveraging the structured representation. Empirical results demonstrate that utilizing this template leads to a measurable increase in sign consistency without significant performance degradation, as evidenced by a minimal increase in perplexity-approximately one point-when compared to models trained without such a template.

Evaluations conducted within a Transformer architecture consistently demonstrate that the observed ‘sign lock-in’ – the tendency of model weights to converge to a stable sign pattern – is a repeatable and quantifiable effect. These experiments move beyond initial observations by establishing that sign stability is not simply a consequence of specific initialization or training parameters. Reproducible results across multiple training runs, and variations in hyperparameters, confirm the robustness of this phenomenon. Quantitative analysis reveals that the stabilized sign patterns persist throughout training, indicating that they are not transient effects, and are demonstrably different from randomly generated sign patterns, as evidenced by comparisons to a Rademacher baseline.

Analysis of learned sign patterns within the Transformer architecture demonstrates non-randomness when contrasted against a Rademacher baseline, which generates random sign assignments. This comparison reveals a statistically significant structure in the learned signs, indicating they are not simply noise. Importantly, achieving this level of sign consistency – a structured, non-random sign pattern – results in a negligible increase in perplexity, approximately one point, suggesting minimal performance degradation from leveraging this inherent structure in model optimization.

Harnessing Sign Stability for Model Efficiency

Gap initialization represents a deliberate strategy to foster sign stability within neural networks. This technique moves initial weights away from the zero-value, effectively creating a ‘gap’ that discourages early sign changes during training. By establishing this margin, the network is less susceptible to fluctuations that might otherwise cause a weight to flip between positive and negative values prematurely. This proactive approach doesn’t merely address sign flips as they occur, but actively minimizes their probability from the outset, contributing to more robust and reliable models that maintain consistent sign patterns throughout the learning process.

Outer-drift regularization functions as a stabilizing force within neural networks by discouraging the assignment of near-zero weights, thereby reinforcing established sign patterns. This technique operates on the principle that weights with small magnitudes are more susceptible to being flipped during training, introducing instability. Methods like the ‘Log-Barrier’ approach introduce a penalty proportional to the logarithm of the weight’s absolute value as it approaches zero, effectively pushing these weights away from the unstable region. By actively minimizing these small magnitudes, the network is encouraged to commit to stronger, more definitive sign assignments, resulting in a more robust and compressible model architecture and a significantly reduced rate of undesirable sign flips during the learning process.

The combined application of gap initialization and outer-drift regularization yields a synergistic effect on neural network training, resulting in models that are both remarkably robust and highly compressible. This approach doesn’t merely minimize the frequency of sign changes within the network’s weights, but actively locks them in, decreasing the effective sign flip rate to approximately 10⁻³. Such stabilization is crucial, as it allows for more aggressive pruning and quantization without significant performance degradation. Consequently, these techniques facilitate the creation of models that maintain accuracy while drastically reducing computational demands and memory footprint, representing a substantial advancement in efficient deep learning.

The pursuit of efficient model compression has historically centered on reducing the numerical precision of model weights. However, a novel approach focuses on sign lock-in, a phenomenon where weights are encouraged to maintain a consistent positive or negative value throughout training. This strategy represents a fundamental shift, moving beyond simply minimizing storage space by decreasing bit-width; instead, it actively sculpts the model’s representational capacity by prioritizing stable, sparse patterns. By harnessing sign lock-in, researchers are discovering that models can achieve substantial compression – and maintain, or even improve – performance by leveraging the inherent structure of these stabilized sign patterns, effectively creating more robust and compressible architectures that are less susceptible to overfitting and noise.

The persistence of initial conditions, as demonstrated in the study of ‘sign lock-in’, echoes a fundamental principle of mathematical systems. The observed tendency for weights to retain their initial signs, even under stochastic gradient descent, suggests an underlying determinism often obscured by the apparent chaos of training data. This aligns with Alan Turing’s assertion: “Sometimes it is the people who no one imagines anything of who do the things that no one can imagine.” The study reveals that seemingly random initialization harbors a surprising degree of control, akin to discovering a hidden order within complexity – a control which, as the paper details with its stopping-time analysis, can be leveraged to compress models beyond conventional limits, demonstrating that even in the realm of neural networks, mathematical discipline endures.

What Remains to be Proven?

The observation of ‘sign lock-in’ is not merely a curiosity; it reveals a fundamental constraint on the optimization landscape. The persistence of initial weight signs, despite stochastic gradient descent, suggests that the true minimum-or even a stable basin of attraction-may be far more constrained than current assumptions allow. The question isn’t simply whether these signs can be altered, but whether doing so reliably improves generalization-or if it merely introduces a form of brittle instability. Reproducibility, of course, remains paramount; a result observed across a single dataset, or even a handful, is insufficient to establish a general principle.

Future work must move beyond empirical demonstration and toward a more rigorous, geometric understanding of these dynamics. The connection to low-rank approximation is intriguing, but requires formalization. Can the ‘geometric tail’ be predicted analytically, rather than observed post-hoc? A provably convergent algorithm, exploiting sign constraints for compression, would be a far more compelling result than yet another heuristic. The current emphasis on simply achieving compression ratios misses the core issue: information preservation, and the deterministic guarantee of its recovery.

Ultimately, the field must confront the possibility that current optimization methods are, at best, approximating a highly constrained solution. The ‘one-bit wall’ may not be a limit of representational capacity, but a symptom of a deeper, mathematical necessity. To break through it requires not merely clever tricks, but a fundamentally more elegant, and provable, approach to learning.

Original article: https://arxiv.org/pdf/2602.17063.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-02-22 23:25