The Space Complexity of Approximating Logistic Loss

• URL: http://arxiv.org/abs/2412.02639v1
• Date: Tue, 03 Dec 2024 18:11:37 GMT
• Title: The Space Complexity of Approximating Logistic Loss
• Authors: Gregory Dexter, Petros Drineas, Rajiv Khanna,
• Abstract summary: We prove a general $tildeOmega(dcdot mu_mathbfy(mathbfX))$ space lower bound when $epsilon$ is constant. We also refute a prior conjecture that $mu_mathbfy(mathbfX)$ is hard to compute.
• Score: 11.338399194998933
• License:
• Abstract: We provide space complexity lower bounds for data structures that approximate logistic loss up to $\epsilon$-relative error on a logistic regression problem with data $\mathbf{X} \in \mathbb{R}^{n \times d}$ and labels $\mathbf{y} \in \{-1,1\}^d$. The space complexity of existing coreset constructions depend on a natural complexity measure $\mu_\mathbf{y}(\mathbf{X})$, first defined in (Munteanu, 2018). We give an $\tilde{\Omega}(\frac{d}{\epsilon^2})$ space complexity lower bound in the regime $\mu_\mathbf{y}(\mathbf{X}) = O(1)$ that shows existing coresets are optimal in this regime up to lower order factors. We also prove a general $\tilde{\Omega}(d\cdot \mu_\mathbf{y}(\mathbf{X}))$ space lower bound when $\epsilon$ is constant, showing that the dependency on $\mu_\mathbf{y}(\mathbf{X})$ is not an artifact of mergeable coresets. Finally, we refute a prior conjecture that $\mu_\mathbf{y}(\mathbf{X})$ is hard to compute by providing an efficient linear programming formulation, and we empirically compare our algorithm to prior approximate methods.

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