Streaming Complexity of SVMs

We study the space complexity of solving the bias-regularized SVM problem in the streaming model. In particular, given a data set (x_i,y_i) ∈ ℝ^d× {-1,+1}, the objective function is F_λ(θ,b) = λ/2‖(θ,b)‖₂² + 1/n∑_{i=1}ⁿ max{0,1-y_i(θ^Tx_i+b)} and the goal is to find the parameters that (approximately) minimize this objective. This is a classic supervised learning problem that has drawn lots of attention, including for developing fast algorithms for solving the problem approximately: i.e., for finding (θ,b) such that F_λ(θ,b) ≤ min_{(θ',b')} F_λ(θ',b')+ε. One of the most widely used algorithms for approximately optimizing the SVM objective is Stochastic Gradient Descent (SGD), which requires only O(1/λε) random samples, and which immediately yields a streaming algorithm that uses O(d/λε) space. For related problems, better streaming algorithms are only known for smooth functions, unlike the SVM objective that we focus on in this work. We initiate an investigation of the space complexity for both finding an approximate optimum of this objective, and for the related "point estimation" problem of sketching the data set to evaluate the function value F_λ on any query (θ, b). We show that, for both problems, for dimensions d = 1,2, one can obtain streaming algorithms with space polynomially smaller than 1/λε, which is the complexity of SGD for strongly convex functions like the bias-regularized SVM [Shalev-Shwartz et al., 2007], and which is known to be tight in general, even for d = 1 [Agarwal et al., 2009]. We also prove polynomial lower bounds for both point estimation and optimization. In particular, for point estimation we obtain a tight bound of Θ(1/√{ε}) for d = 1 and a nearly tight lower bound of Ω̃(d/{ε}²) for d = Ω(log(1/ε)). Finally, for optimization, we prove a Ω(1/√{ε}) lower bound for d = Ω(log(1/ε)), and show similar bounds when d is constant.