Stochastic Rounding Implicitly Regularizes Tall-and-Thin Matrices (2403.12278v3)

Abstract: Motivated by the popularity of stochastic rounding in the context of machine learning and the training of large-scale deep neural network models, we consider stochastic nearness rounding of real matrices $\mathbf{A}$ with many more rows than columns. We provide novel theoretical evidence, supported by extensive experimental evaluation that, with high probability, the smallest singular value of a stochastically rounded matrix is well bounded away from zero -- regardless of how close $\mathbf{A}$ is to being rank deficient and even if $\mathbf{A}$ is rank-deficient. In other words, stochastic rounding \textit{implicitly regularizes} tall and skinny matrices $\mathbf{A}$ so that the rounded version has full column rank. Our proofs leverage powerful results in random matrix theory, and the idea that stochastic rounding errors do not concentrate in low-dimensional column spaces.

• The paper demonstrates that stochastic rounding prevents the smallest singular value from vanishing, thereby implicitly regularizing tall-and-thin matrices.
• The study leverages random matrix theory and experiments to validate improvements in numerical conditioning through SR.
• Results indicate that SR-induced regularization enhances stability in model training, with significant implications for machine learning applications.

Singular Value Bounds for Stochastic Rounding in Tall-and-Thin Matrices

Introduction to Stochastic Rounding

Stochastic rounding (SR) has sprouted new interest in digital computation, especially within machine learning applications and deep neural network (DNN) model trains. Unlike deterministic rounding methods that may consistently bias towards the floor or the ceiling values, SR introduces a probabilistic element to rounding. This randomness can potentially benefit numerical stability, precision, and even model accuracy in large-scale computations.

The Role of SR in Regularizing Matrices

Our research explores the effects of SR on the singular values of real, tall-and-thin matrices, where the number of rows notably surpasses the columns ($n \gg d$). We primarily focus on how SR can influence the smallest singular value of such matrices, potentially acting as an implicit form of regularization. This implicit regularization could offer substantial benefits, including improved conditioning for inversion or system solving and, by extension, enhanced stability in downstream machine learning models.

Theoretical Insights into SR and Singular Values

Our main contributions draw from Random Matrix Theory (RMT) and hinge on a novel finding: With high probability, SR not only prevents the smallest singular value of a tall-and-thin matrix from vanishing but in fact guarantees it is bounded away from zero. This significant result holds under the assumption that the SR process introduces sufficient randomness (as characterized by the normalized column-wise variance $\nu$) and that the matrix dimensions satisfy a certain tall-and-thin criteria.

Formally, our findings can be summarized as follows:

For a tall-and-thin matrix $A \in \mathbb{R}^{n \times d}$ subject to SR, the smallest singular value, $ma_d(Atil)$, satisfies with high probability

$$ma_d(Atil) \geq \beta^{1-p}\sqrt{n}(\sqrt{\nu} - \epsilon_{n,d}),$$

where $\beta$ is the basis, $p$ the working precision of the floating point representation, and $\epsilon_{n,d}$ a term that vanishes as the matrix becomes 'taller' and 'thinner'.

Experimental Validation

Our experimental investigations align well with the theoretical predictions. We examined various scenarios across different precisions and singular value configurations, including rank-deficient and full-rank matrices. It was observed consistently that SR leads to an increase in the smallest singular value, reinforcing the notion that SR implicitly regularizes tall-and-thin matrices.

Implications and Future Directions

These findings underscore the potential of SR to act as an implicit regularizer in numerical computations, particularly for machine learning models where matrix conditioning plays a critical role. Future research will focus on relaxing our model’s assumptions, better understanding the necessary conditions for SR-induced regularization, and exploring the practical impacts in real-world machine learning implementations.

Conclusion

Our work provides a rigorous foundation for the benefits of stochastic rounding on the regularization of tall-and-thin matrices. By increasing the smallest singular value, SR implicitly improves the conditioning of these matrices, offering a potent, yet underexplored, tool in numerical computing and model training. Further explorations in this direction could pave the way for more stable and accurate computational methods in data science and artificial intelligence.