Sign Betting for Misspecified Regression: Variance Adaptivity and Optimal Approximation

We study finite-sample linear regression in the presence of varied and unknown label noise, focusing on the heteroskedastic and adaptive linear regression models. Heteroskedastic linear regression models settings where the labels are of varying quality. We receive $n$ pairs $(X_i,Y_i)$ with labels $Y_i=X_i^\topβ+\varepsilon_i$, where $\varepsilon_i\sim N(0,σ_i^2)$ and the variances are unknown to the estimator. One natural measurement of the difficulty of this problem is the number of samples $m$ for which $σ_i^2\le1$ (larger $m$ is easier). We obtain a polynomial-time estimator with rate $\tilde{O}((nd^3/m^4)^{1/6})$ when $m\gg d^{3/4}n^{1/4}$, as well as nearly-matching lower bounds. For $d=O(1)$, our estimator achieves error $o(1)$ when $m\gg n^{1/4}$, whereas $L_1$ regression and other traditional approaches require $m\gg n^{1/2}$. In adaptive linear regression, the errors are drawn i.i.d. from an unknown distribution $p$, and our goal is to design a generic estimator that performs nearly as well as the best custom estimator that knows $p$. We introduce a (computationally inefficient) adaptive estimator that, so long as $p$ is a mixture of $k$ symmetric log-concave densities, achieves error comparable with the optimal estimator that knows $p$ and has $\tildeΘ(n/k)$ samples. For $k=1$, we show that $L_q$ regression (with data-dependent $q$) gives a polynomial-time estimator. Finally, to study the computational limits of both problems, we introduce the planted linear regression problem, where $X_i\sim N(0,I_d)$, $m$ unknown samples are noiseless, and the rest have error $\varepsilon_i\sim N(0,1)$. We conjecture that recovering $β$ up to error $\ll\sqrt{d/n}$ (or exactly) may have an information-computation gap between $m=d+1$ and $m\sim d^{3/4}n^{1/4}$, as is suggested by our near-matching polynomial-time estimator and statistical query (SQ) lower bound.

Statistical inference with bandit data presents fundamental challenges owing to adaptive sampling, which violates the independence assumptions underlying classical asymptotic theory. Recent work has identified stability~\citep{laiwei82} as a sufficient condition for valid inference under adaptivity. This paper first provides a refined stability condition, stated in terms of the iterates of an online algorithm, and shows that a large class of regularized stochastic-mirror-descent-style algorithms satisfy it. This refined condition allows us to strengthen the asymptotic results of~\citet{laiwei82} in several ways. First, we derive a non-asymptotic Berry--Esseen bound for the empirical reward estimates under adaptive sampling. Second, we derive matching non-asymptotic upper and lower bounds on the regret of the proposed algorithm, yielding a precise characterization of its regret. Third, we show that these regularized algorithms preserve asymptotic normality and valid inference under a prescribed level of adversarial corruption. Finally, we show that regularization is necessary rather than incidental: Lai--Wei stability is incompatible with the optimal $O(\sqrt{T})$ regret rate -- the rate attained by unregularized algorithms such as EXP3 -- so that a controlled, polylogarithmic inflation in regret is the price of valid inference.

Consider betting against a sequence of $[0,1]$-valued observations using bets that are fair if the data have a conditional mean $m_0 \in (0,1)$. Cover's universal portfolio algorithm achieves a worst-case regret of $O(\ln n)$ compared to the best constant bet in hindsight, and this logarithmic order is unimprovable against adversarial data. We construct an explicit full-support Robbins mixture that combines insights from Robbins and Cover. It retains $O(\ln n)$ regret on every path while also eventually satisfying $R_n\le(1+o(1))\ln\ln V_n$ on every path with bounded mixture wealth and $V_n\to\infty$, where $V_n$ is an intrinsic variance process. It also attains the optimal asymptotic wealth growth rate under iid alternatives. Under the conditional-mean claim and almost-sure divergence of $V_n$, the sharper regret bound holds almost surely, and its leading constant $1$ is unimprovable. We also give explicit finite-time, path-wise regret bounds. A simple hedging principle shows how splitting initial wealth between two strategies achieves a best-of-both-worlds guarantee by retaining the smaller of their regrets and the larger of their growth rates. We contrast our results to those for unbounded data in Agrawal and Ramdas (2026). Their worst-case regret is unbounded. However, hedging yields almost-sure iterated-logarithmic regret together with optimal growth. Finally, our wealth process witnesses a sharp game-theoretic upper law of iterated logarithm, analogous to Shafer and Vovk (2005).