Logistic Regression Regret: What's the Catch?
This work provides theoretical insights into regret bounds for online logistic regression, which is important for machine learning practitioners dealing with high-dimensional online classification problems, though it appears to be an incremental theoretical analysis building on prior work.
The paper addresses achievable regret rates for online logistic regression, deriving lower bounds dominated by d/2 log T under L1, L2, and L∞ constraints and showing achievability for d=o(T^{1/3}) with Bayesian methods up to a d/2 log d term, while revealing different behaviors for larger dimensionality such as Ω(d log T) regret for d=Ω(√T) under L∞ constraints.
We address the problem of the achievable regret rates with online logistic regression. We derive lower bounds with logarithmic regret under $L_1$, $L_2$, and $L_\infty$ constraints on the parameter values. The bounds are dominated by $d/2 \log T$, where $T$ is the horizon and $d$ is the dimensionality of the parameter space. We show their achievability for $d=o(T^{1/3})$ in all these cases with Bayesian methods, that achieve them up to a $d/2 \log d$ term. Interesting different behaviors are shown for larger dimensionality. Specifically, on the negative side, if $d = Ω(\sqrt{T})$, any algorithm is guaranteed regret of $Ω(d \log T)$ (greater than $Ω(\sqrt{T})$) under $L_\infty$ constraints on the parameters (and the example features). On the positive side, under $L_1$ constraints on the parameters, there exist algorithms that can achieve regret that is sub-linear in $d$ for the asymptotically larger values of $d$. For $L_2$ constraints, it is shown that for large enough $d$, the regret remains linear in $d$ but no longer logarithmic in $T$. Adapting the redundancy-capacity theorem from information theory, we demonstrate a principled methodology based on grids of parameters to derive lower bounds. Grids are also utilized to derive some upper bounds. Our results strengthen results by Kakade and Ng (2005) and Foster et al. (2018) for upper bounds for this problem, introduce novel lower bounds, and adapt a methodology that can be used to obtain such bounds for other related problems. They also give a novel characterization of the asymptotic behavior when the dimension of the parameter space is allowed to grow with $T$. They additionally establish connections to the information theory literature, demonstrating that the actual regret for logistic regression depends on the richness of the parameter class, where even within this problem, richer classes lead to greater regret.