Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limit (2207.08799v3)

Abstract: There is mounting evidence of emergent phenomena in the capabilities of deep learning methods as we scale up datasets, model sizes, and training times. While there are some accounts of how these resources modulate statistical capacity, far less is known about their effect on the computational problem of model training. This work conducts such an exploration through the lens of learning a $k$-sparse parity of $n$ bits, a canonical discrete search problem which is statistically easy but computationally hard. Empirically, we find that a variety of neural networks successfully learn sparse parities, with discontinuous phase transitions in the training curves. On small instances, learning abruptly occurs at approximately $n^{O(k)}$ iterations; this nearly matches SQ lower bounds, despite the apparent lack of a sparse prior. Our theoretical analysis shows that these observations are not explained by a Langevin-like mechanism, whereby SGD "stumbles in the dark" until it finds the hidden set of features (a natural algorithm which also runs in $n^{O(k)}$ time). Instead, we show that SGD gradually amplifies the sparse solution via a Fourier gap in the population gradient, making continual progress that is invisible to loss and error metrics.

• The paper demonstrates that SGD efficiently learns sparse parity functions, with convergence times meeting theoretical lower bounds for Statistical Query algorithms.
• It uncovers that hidden progress in population gradients, highlighted by Fourier gap analysis, is crucial for detecting relevant features.
• Empirical and theoretical results refute a random search hypothesis, informing better initialization and tuning practices in neural network training.

Emergent Phenomena in Deep Learning: Analyzing Sparse Parity Learning Capabilities

Recent analyses have increasingly demonstrated the emergent capabilities observed in deep learning settings, particularly as the involved resources—datasets, model sizes, and training durations—are scaled. Traditional statistical lenses provide insights into how higher resource allocations enhance the expressive capacities of models. However, the computational intricacies underlying gradient-based optimization under scaled resources remain less scrutinized. The paper under review focuses on investigating this through the paper of sparse parity learning. This problem is statistically tractable yet computationally intensive, providing a robust framework to discern these phenomena.

Empirical Findings on Sparse Parity Learning with Various Architectures

The investigation primarily centers around the problem of learning a $k$-sparse parity function of $n$ bits using neural networks. The label in this supervised learning setting is defined as the parity (XOR) of $k \ll n$ bits in a random binary string of length $n$. This task poses a challenge to various algorithms, including those based on gradient methods and streaming approaches.

A variety of neural network architectures were assessed within this framework. These include multi-layer perceptrons (MLPs), specialized architectures like PolyNets, sinusoidal neurons, and Transformers. Across these different setups, neural networks demonstrated the capability to solve the sparse parity problem, exhibiting discontinuous phase transitions in their training curves. The convergence times scaled approximately as $n^{O(k)}$ iterations, aligning closely with the theoretical lower bounds for Statistical Query (SQ) algorithms.

Theoretical Insights and Mechanisms Beyond Random Search

The empirical observations suggest that the success of Stochastic Gradient Descent (SGD) in learning sparse parities is not purely attributable to a process akin to random search. Several factors undermine a random search hypothesis:

1. The convergence times adapt to the sparsity parameter $k$, indicating a more nuanced mechanism at work.
2. There is no significant early convergence across a large number of trials, which would be expected in random search scenarios.
3. Convergence times exhibit substantial sensitivity to initialization but are relatively stable across different SGD runs.
4. Scaling behavior changes with problem size, diminishing the simple power-law models observed in smaller instances.

SGD and Feature Learning: Fourier Gaps in Population Gradients

The underlying theoretical contribution highlights that successful SGD on this task can be attributed to a hidden progress measure in the gradient dynamics. The analysis posits that population gradients (theoretical expectations of gradients over the data distribution) at initialization can encode sufficient information to eventually identify the relevant $k$ coordinates among $n$. This identification process is facilitated by analyzing Fourier coefficients, where a significant gap (denoted as a Fourier gap) exists between the coordinates linked to correct parities and those that are not.

Moreover, evidence against the "stumbling in the dark" hypothesis is further supported by the empirical scaling laws and the specific behavior of training dynamics. The paper provides comprehensive analytical results for 2-layer MLPs and introduces an idealized architecture, the disjoint-PolyNet, which demonstrates even more predictable behavior aligning with theoretical predictions.

Implications and Prospects

The implications of these findings are multifaceted:

1. Theoretical Understanding: Beyond just empirical results, the theoretical foundation establishes a more intricate understanding of why gradient-based methods succeed in learning sparse parities. This sets a precedent for analyzing other combinatorial problems within neural network training frameworks.
2. Practical Techniques: For practitioners, these results suggest that sophisticated feature-learning dynamics are at play even in settings without explicit sparsity priors. This informs hyperparameter tuning and initialization strategies to enhance model performance on specific tasks.
3. Broader Impact on AI Developments: This paper’s insights contribute to the ongoing discourse on emergent properties in AI systems. By understanding scaling laws and hidden progress measures, researchers can better anticipate and harness these emergent capabilities.

Future Directions

The paper paves the way for several future research avenues:

1. Extension to Other Combinatorial Problems: Investigating similar mechanisms in other combinatorial optimization problems would further generalize the findings.
2. Advanced Architectures: Exploring the influence of more complex architecture choices, including deeper and more intricately connected networks, can provide deeper insights into the scalability and limitations of these methods.
3. Quantitative Metrics for Hidden Progress: Developing precise quantitative metrics for hidden progress and applying these metrics to other machine learning tasks can potentially transform model debugging and optimization processes.

Conclusion

The paper makes significant strides in explaining emergent phenomena in deep learning, particularly through the paper of sparse parity learning. It bridges empirical observations with theoretical analysis, proposing that SGD performs continual, albeit hidden, progress rather than mere stochastic searches. These findings not only advance our theoretical understanding of gradient-based optimization processes but also suggest practical avenues for improving model training strategies in various settings.