This paper studies the distributed online control problem over a network of linear time-invariant (LTI) systems in the presence of adversarial disturbances and time-varying convex costs. The network cost is characterized by the summation of local cost functions, where each local function is sequentially revealed only to the corresponding agent. The goal of each agent is to generate a control sequence, using only local observations and neighbor communication, that competes with the best {\it centralized} linear policy in hindsight. We extend the recently proposed Online Spectral Control framework from the centralized setting to the distributed setting. In particular, each agent applies a spectral controller obtained by convolving past disturbances with the leading eigenvectors of a Hankel matrix, while the controller parameters are updated through a distributed online gradient descent step over the local surrogate costs. We formulate this problem this problem as a {\it regret} minimization problem based on the spectral parameterization, and under standard assumptions, we establish a sublinear regret bound of $O(\frac{\sqrt{T}\text{poly}(\log T)}{γ^3})$, where $T$ is the time horizon and $γ$ denotes the stability margin. The resulting bound also captures the dependence on the network size and connectivity.
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 $(ρ,ε)$-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 $ε$-dominant term in the iteration complexity for reaching in-expectation stationarity is $T=\mathrm{O}\left(Δρ^{1/2}(G+σ)^{\frac{5p}{3p-4}}ε^{-\left(\frac{5p}{3p-4}+\frac{3}{2}\right)}\right)$ for $p\in(\frac{4}{3},2]$, which simplifies to $T=\mathrm{O}(ε^{-13/2})$ 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 $ε$-dominant iteration complexity is $T=\mathrm{O}\left(Δρ^{1/2}(G+σ)^{\frac{p}{p-1}}ε^{-\left(\frac{p}{p-1}+\frac{3}{2}\right)}\right)$ for $p\in(1,2]$, which simplifies to $T=\mathrm{O}(ε^{-7/2})$ when $p=2$. These findings provide new theoretical insight into the robustness and limitations of Adam in heavy-tailed regimes.