HT-PAder is proposed, a parameter-free algorithm combining restarted AdaGrad experts over a geometric pool of block lengths with a pathwise meta-algorithm, which requires no moment conditions on meta-losses and provides the first parameter-free minimax universal dynamic regret guarantee.
Abstract
We study online convex optimization (OCO) in non-stationary environments under heavy-tailed noise, where the stochastic gradient oracle admits only a finite $p$-th central moment for some $p \in (1, 2]$. While static regret is well-understood, achieving universal dynamic regret in a parameter-free manner remains an open challenge. We resolve this by proposing \textbf{HT-PAder}, a parameter-free algorithm combining restarted AdaGrad experts over a geometric pool of block lengths with a pathwise meta-algorithm, \textbf{AdaGrad-Hedge}, which requires no moment conditions on meta-losses. For a domain of diameter $D$, Lipschitz constant $G$, noise level $\sigma$, and comparator path length $P_T$, HT-PAder achieves an expected universal dynamic regret of \[ \widetilde O\left( GD\sqrt{T(1+P_T/D)} + \sigma D T^{1/p}(1+P_T/D)^{(p-1)/p} \right). \] The algorithm does not require prior knowledge of any of these problem parameters. Even in the special case of finite variance ($p=2$), HT-PAder provides the first parameter-free minimax universal dynamic regret guarantee. We also prove a matching lower bound, establishing the optimality of the path-length exponent.
Heavy-tailed distributions arise naturally in sequential decision-making problems such as financial investment, online advertising, and network management, where rare but extreme outcomes can dominate performance. Heavy-tailed bandits model online decision-making in these settings by assuming only that rewards $X$ satisfy $\mathbb{E}[|X|^{1+\epsilon}]\leq u$, for some tail exponent $\epsilon\in(0,1]$ and moment bound $u<+\infty$. However, most existing regret minimization algorithms require these parameters to be known. This assumption is particularly restrictive in practice: $\epsilon$ and $u$ govern the frequency and magnitude of rare events and are therefore precisely the quantities that are hardest to infer reliably from limited observations. Motivated by an open problem posed by Genalti and Metelli at COLT 2025, we resolve the assumption-free adaptation problem for heavy-tailed bandits and characterize the price in the regret of not knowing the tail parameters. We first study adaptation to the moment bound $u$ for a fixed tail exponent $\epsilon$. We prove that every algorithm unaware of $u$, or of any upper bound on it, must obey a sharp trade-off between its distribution-dependent and distribution-free regret guarantees. We then introduce a scheduled-exploration algorithm that requires no knowledge of $u$ and matches the resulting adaptation frontier up to logarithmic factors. Finally, we show that the same algorithm can be instanced without knowing $\epsilon$ by calibrating its exploration schedule to the endpoint $\epsilon=1$. It achieves sublinear regret for every fixed $\epsilon>0$, while no algorithm can guarantee sublinear regret uniformly over all $\epsilon\in(0,1]$. Altogether, our results resolve the COLT open problem without additional distributional assumptions and provide a sharp characterization of the statistical cost of adapting to unknown heavy tails.
We establish the first convergence guarantees for the plain vector-form Adam optimizer under heavy-tailed stochastic noise. While several Adam variants are known to achieve optimal iteration complexity in bounded-variance nonsmooth nonconvex optimization, little is understood about their behavior when stochastic gradients admit only a bounded $p$-th central moment for some $p \in (1,2]$, a setting increasingly observed in modern deep learning. To address this gap, we generalize the recent online-to-nonconvex conversion framework to accommodate heavy-tailed martingale-difference noise. Building on this generalized framework, we develop a discounted regret analysis for Adam, without restrictive parameter coupling. Our results show that Adam converges to $(\rho,\epsilon)$-stationary points under heavy-tailed noise. However, it exhibits a suboptimal iteration complexity and $p$-dependent convergence, a suboptimality that persists even in the bounded-variance case ($p=2$). Specifically, the $\epsilon$-dominant term in the iteration complexity for reaching in-expectation stationarity is $T=\mathrm{O}\left(\Delta \rho^{1/2}(G+\sigma)^{\frac{5p}{3p-4}}\epsilon^{-\left(\frac{5p}{3p-4}+\frac{3}{2}\right)}\right)$ for $p\in(\frac{4}{3},2]$, which simplifies to $T=\mathrm{O}(\epsilon^{-13/2})$ when $p=2$. When the domain radius is known and used to control the online-learner output, a standard setup in related literature, the convergence rate improves to match the optimal complexity. In this case, the $\epsilon$-dominant iteration complexity is $T=\mathrm{O}\left(\Delta \rho^{1/2}(G+\sigma)^{\frac{p}{p-1}}\epsilon^{-\left(\frac{p}{p-1}+\frac{3}{2}\right)}\right)$ for $p\in(1,2]$, which simplifies to $T=\mathrm{O}(\epsilon^{-7/2})$ when $p=2$. These findings provide new theoretical insight into the robustness and limitations of Adam in heavy-tailed regimes.
We study whether stochastic nonconvex optimization can be reduced to ordinary static regret minimization in online convex optimization in a black-box manner. For smooth nonconvex objectives, our reduction maintains a predictable gradient tracker, while a black-box online learner selects a preconditioner that determines how this tracker is transformed into the update direction. The learner receives linear convex losses and is evaluated against a single fixed comparator over one undiscounted online game. For a $\beta$-smooth objective with range bounded by $M$ and an unbiased stochastic-gradient oracle with variance bounded by \(\sigma^2\), we establish $$\frac{1}{T}\sum_{t=1}^T \mathbb E\!\left[\|\nabla f(x_t)\|_2^2\right] \lesssim \frac{\sigma\sqrt{M\beta}}{\sqrt T} + \frac{\sqrt{M\beta}\, \mathscr R_T(\mathcal A,I_d)}{T} + \frac{M\beta}{T}.$$ Consequently, any black-box OCO algorithm with $\mathscr R_T(\mathcal A,I_d)=O(\sqrt T)$ recovers the classical $O(\frac{1}{\sqrt{T}})$ convergence rate. We further show that the same black-box framework extends beyond the smooth setting to Lipschitz nonconvex objectives without Lipschitz continuous gradients. Importantly, this extension continues to rely only on an ordinary static-regret guarantee and requires no stronger notion of online regret. When the OCO oracle admits square-root static regret, the resulting conversion achieves the optimal $O(T^{-2/7})$ convergence rate for the corresponding Goldstein stationary point. These results resolve the open problem posed by Chen and Hazan (2024). More broadly, our framework separates optimizer design into gradient prediction and online preconditioner selection, providing a principled perspective on how adaptive optimization methods may be understood through static regret and applied in nonconvex optimization.
In this paper we first study the problem of generalized linear bandit (GLB) under heavy-tailed noise. The characteristics of heavy-tailed distributions are widely observed in real-world applications such as personalized recommendation, financial markets, and medical treatments. Based on the online mirror descent (OMD) method, we propose an algorithm EHM that extends the adaptive Huber loss method (Wang et al., 2025) with one-pass update ($\mathcal{O}(1)$ computational complexity with respect to current round $t$ and the time horizon $T$), which simultaneously achieves an almost optimal regret of $\widetilde{\mathcal{O}}(T^{\frac{1}{1+\epsilon}})$ where $T$ is the time horizon. In addition, by utilizing a special property of some link function (Sawarni et al., 2025), our algorithm eliminates the need to know a commonly used parameter. Next, we study the GLB problem under the case when contextual characteristic becomes piecewise constant, and we slightly revised former algorithm to obtain the PGLB-EHM algorithm. After theoretical analysis, we prove that the regret upper bound order stays the same. Furthermore, we look deeper into a special case of nonlinear bandit (NB) and present the NB-EHM algorithm with bisection method and special restriction. Eventually we utilize the affine lifting approach and show that the general NB problem can be applied with NB-EHM to achieve a sublinear regret bound.
Tianshuo Zheng, Ting Wu, Zhi-Hua Zhou et al.· 0 citations
We study Bayesian optimization in a time-varying environment where the unknown reward function evolves according to a Gaussian process drift model. Existing GP-UCB analyses in this setting typically require the exploration parameter to grow with the horizon to maintain uniform confidence bounds. Using per-round local confidence events, we show that GP-UCB can instead be run with a constant exploration parameter and obtain an expected-regret bound whose coefficient depends on the drift rate. We also derive a sharper time-varying maximum-information-gain bound. For the squared exponential kernel, it yields $\tilde\gamma_T/T=\widetilde{\mathcal O}(\epsilon^{1/2})$ and expected average regret $\widetilde{\mathcal O}(\epsilon^{1/4})$ in the persistent-drift regime. The same constant-exploration analysis also yields realized-regret guarantees. Simulations support the predicted logarithmic dependence of the bound-suggested exploration parameter on $1/\epsilon$.
In this paper, we study alternating regret in online convex optimization (OCO), motivated by the success of alternating learning dynamics in two-player games. Although previous works have shown that $o(\sqrt{T})$ alternating regret is achievable under various assumptions on the loss functions and feasible domains, the minimax regret rate has remained open even for the expert problem. In this paper, we resolve this question by showing matching lower and upper bounds for both the expert problem and general OCO. Somewhat surprisingly, for the $d$-expert problem, we show that the minimax alternating regret is $\Theta(\log d)$, independent of the horizon $T$. This significantly improves upon the best-known $\mathcal{O}(T^{1/3}\log^{2/3} d)$ established by Hait et al. [2025]. We further extend our results to general OCO over a $d$-dimensional compact convex set and prove that the worst-case minimax alternating regret is $\Theta\left(d\log \left(1+\frac{T}{d}\right)\right)$, also significantly improving upon the best-known $\mathcal{O}((d\log T)^{2/3}T^{1/3})$ upper bound and resolving the open problem posed by Cevher et al. [2023], Hait et al. [2025]. Technically, our upper bound for the expert problem is achieved by a corrected variant of Hedge, in which carefully designed correction terms cancel the unfavorable curvature arising in the alternating-regret analysis. We extend the same corrected-potential argument to continuous action sets to obtain the optimal alternating-regret rate for OCO. For the lower bounds, the expert construction repeatedly eliminates half of the candidate experts, while the OCO lower bound instance construction replaces this discrete elimination by a more involved multiscale construction on the unit disk.
