Memory-Sample Tradeoffs for Linear Regression with Small Error
This addresses memory-sample tradeoffs in streaming regression, providing the first nontrivial lower bounds for super-linear memory, which could impact continuous optimization algorithms.
The paper tackles the problem of linear regression over a stream of data under memory constraints, showing that algorithms with subquadratic memory require more samples to achieve small error compared to those with more memory. Specifically, it proves a lower bound of Ω(d log log(1/ε)) samples for memory d²/4 bits, while O(d² log(1/ε)) memory allows recovery with d examples.
We consider the problem of performing linear regression over a stream of $d$-dimensional examples, and show that any algorithm that uses a subquadratic amount of memory exhibits a slower rate of convergence than can be achieved without memory constraints. Specifically, consider a sequence of labeled examples $(a_1,b_1), (a_2,b_2)\ldots,$ with $a_i$ drawn independently from a $d$-dimensional isotropic Gaussian, and where $b_i = \langle a_i, x\rangle + η_i,$ for a fixed $x \in \mathbb{R}^d$ with $\|x\|_2 = 1$ and with independent noise $η_i$ drawn uniformly from the interval $[-2^{-d/5},2^{-d/5}].$ We show that any algorithm with at most $d^2/4$ bits of memory requires at least $Ω(d \log \log \frac{1}ε)$ samples to approximate $x$ to $\ell_2$ error $ε$ with probability of success at least $2/3$, for $ε$ sufficiently small as a function of $d$. In contrast, for such $ε$, $x$ can be recovered to error $ε$ with probability $1-o(1)$ with memory $O\left(d^2 \log(1/ε)\right)$ using $d$ examples. This represents the first nontrivial lower bounds for regression with super-linear memory, and may open the door for strong memory/sample tradeoffs for continuous optimization.