Scaling hyperparameter optimisation to very large datasets remains an open problem in the Gaussian process community.

There is a pressing need for scalable machine learning approaches to extract rich statistical structure from large datasets.

Scaling hyperparameter optimisation to very large datasets remains an open problem in the Gaussian process community.

This paper studies theoretical lower bounds for estimating the trace of a matrix function, $\text{tr}(f(A))$, focusing on methods that use Hutchinson's method along with Block Krylov techniques. These methods work by approximating matrix-vector products like $f(A)V$ using a Block Krylov subspace. This is closely related to approximating functions with polynomials. We derive theoretical upper bounds on how many Krylov steps are needed for functions such as $A^{-1/2}$ and $A^{-1}$ by analyzing the upper bounds from the polynomial approximation of their scalar equivalent. In addition, we also develop lower limits on the number of queries needed for trace estimation, specifically for $\text{tr}(W^{-p})$ where $W$ is a Wishart matrix. Our study clarifies the connection between the number of steps in Block Krylov methods and the degree of the polynomial used for approximation. This links the total cost of trace estimation to basic limits in polynomial approximation and how much information is needed for the computation.

There have been attempts to utilize linear probe for detoxification, with existing studies relying on a single toxicity probe vector to reduce toxicity. However, toxicity can be fine-grained into various subcategories, making it difficult to remove certain types of toxicity by using a single toxicity probe vector. To address this limitation, we propose a category-specific toxicity probe vector approach. First, we train multiple toxicity probe vectors for different toxicity categories. During generation, we dynamically select the most relevant toxicity probe vector based on the current context. Finally, the selected vector is dynamically scaled and subtracted from model. Our method successfully mitigated toxicity from categories that the single probe vector approach failed to detoxify. Experiments demonstrate that our approach achieves up to a 78.52% reduction in toxicity on the evaluation dataset, while fluency remains nearly unchanged, with only a 0.052% drop compared to the unsteered model.

We propose novel O(n) approaches to estimating these quantities from only fast matrix vector multiplications (MVMs). These stochastic approximations are based on Chebyshev, Lanczos, and surrogate models, and converge quickly even for kernel matrices that have challenging spectra. We leverage these approximations to develop a scalable Gaussian process approach to kernel learning. We find that Lanczos is generally superior to Chebyshev for kernel learning, and that a surrogate approach can be highly efficient and accurate with popular kernels.

For applications as varied as Bayesian neural networks, determinantal point processes, elliptical graphical models, and kernel learning for Gaussian processes (GPs), one must compute a log determinant of an n by n positive definite matrix, and its derivatives---leading to prohibitive O(n^3) computations. We propose novel O(n) approaches to estimating these quantities from only fast matrix vector multiplications (MVMs). These stochastic approximations are based on Chebyshev, Lanczos, and surrogate models, and converge quickly even for kernel matrices that have challenging spectra. We leverage these approximations to develop a scalable Gaussian process approach to kernel learning. We find that Lanczos is generally superior to Chebyshev for kernel learning, and that a surrogate approach can be highly efficient and accurate with popular kernels.