All Random Features Representations are Equivalent (2406.18802v2)

Abstract: Random features are a powerful technique for rewriting positive-definite kernels as linear products. They bring linear tools to bear in important nonlinear domains like KNNs and attention. Unfortunately, practical implementations require approximating an expectation, usually via sampling. This has led to the development of increasingly elaborate representations with ever lower sample error. We resolve this arms race by deriving an optimal sampling policy. Under this policy all random features representations have the same approximation error, which we show is the lowest possible. This means that we are free to choose whatever representation we please, provided we sample optimally.

• The paper demonstrates that the optimal importance sampling strategy minimizes sample variance for kernel approximations.
• It shows that under this strategy, any random feature representation yields equivalent approximation errors.
• The work provides theoretical bounds and rigorous analysis to simplify kernel estimation in large-scale learning.

Equivalence of All Random Features Representations in Kernel Approximations

The paper "All Random Features Representations are Equivalent" presents a comprehensive examination of random features in the context of approximating positive-definite kernels. The authors, Luke Sernau, Silvano Bonacina, and Rif A. Saurous from Google DeepMind and Google Research, provide a novel analysis and a resolution to ongoing pursuits in the optimization of random feature approximations.

Overview and Contributions

Kernel methods, despite their historical significance, face practical limitations when dealing with large datasets due to their linear scaling with dataset size. To mitigate this issue, random feature representations have been used to approximate kernels more efficiently. These representations allow kernels to be rewritten as infinite-dimensional dot products, and thus, their expectations can be estimated using sampling. This paper addresses the competition in choosing optimal random feature representations by deriving an optimal sampling policy that equalizes the approximation error across all random feature representations.

The paramount contributions of the paper are:

1. Optimal Sampling Policy:
   • The authors derive an optimal importance sampling strategy which minimizes the sample variance of kernel estimates. This sampling strategy is shown to be independent of the specific random feature representation.
2. Equivalence of Representations:
   • It is demonstrated that, under this optimal sampling policy, all random feature representations yield the same approximation error. Therefore, the choice of random feature representation is rendered inconsequential provided the sampling is optimal.
3. Theoretical Bounds and Global Optimality:
   • The paper establishes a lower bound on the sample variance that holds universally across all random feature representations.

Theoretical Insights

The theoretical backbone of the paper lies in the application of importance sampling to minimize variance in kernel approximation.

Importance Sampling

Importance sampling is a method to reduce variance in expectation estimates by sampling more frequently from regions with higher impact on the estimate. The authors apply this method to the problem of estimating kernel functions with random features. Formally, if $K(x_1, x_2)$ can be expressed as an expectation of $\phi(x, \omega)$ functions sampled over a distribution $\Omega$, the expectation can equally be approximated by sampling from a distribution $\Psi$ and rescaling, provided that $\Psi$ and $\Omega$ share the same support.

The variance of such an estimate can be minimized by carefully choosing $\Psi$ such that the impact of the high-variance terms is mitigated. The authors derive that the optimal choice of $\Psi$ is proportional to $p_\Omega(\omega) q_\phi(\omega)$, where $q_\phi(\omega)$ is a function of the second moments of $\phi(x, \omega)$.

Optimal Variance and Representation Equivalence

The optimal sample variance, as derived, does not depend on the specific choice of $\phi$. This fundamental result implies that once the optimal sampling distribution is employed, all random feature representations will produce equivalent variances. This is a significant finding as it obviates the need for complex evaluations of different representations, thereby simplifying the computational processes involved in kernel approximation.

Practical Implications and Future Work

Practically, the sampling procedure so designed is of theoretical interest, but its empirical and computational tractability remains to be fully explored. The adoption of Markov Chain Monte Carlo (MCMC) samplers within machine learning frameworks like TensorFlow and Jax provides a practical route for implementing these optimal sampling strategies. Additionally, in practice, resampling can be amortized across multiple training steps, which may reduce computational overhead.

Future work could delve into more empirical studies to validate the theoretical findings and to explore the potential simplifications in practical implementations. There is also a prospect of further theoretical exploration into the structure of the optimal sampling distribution, which may yield even more efficient methods for kernel approximation in larger-scale machine learning tasks.

Conclusion

This paper advances the understanding of random feature representations in kernel methods by proving that all such representations are equivalent under an optimal sampling policy. This finding simplifies decision-making in computational implementations and suggests a unified approach to kernel approximation using random features. Both the theoretical and practical implications of this work promise to influence future methodologies in machine learning involving kernel methods.