Nested Convex-Body Chasing for Online Optimization with Evolving Feasible Sets

We study online optimization with nested shrinking feasible regions in two settings: convex optimization with nested evolving feasible sets (CONES) and adversarial constrained online convex optimization (COCO). Our algorithms separate loss control from geometric movement: constrained minimizers and cumulative-loss tests preserve regret guarantees, while a deterministic resettable nested convex-body chaser limits movement. For CONES with a $G$-Lipschitz, $μ$-strongly convex objective on a diameter-$D$ domain, we chase intersections of the current feasible set with adaptive objective sublevel sets. Using the Euclidean chasing ratio $O(\sqrt{d\log(1+d)})$, we obtain nonpositive regret at every prefix and movement $O(\sqrt{d\log(1+d)\,GD\log(eT)/μ})$. The bound adapts to the increase in the constrained optimum value. In dimension two, with all other parameters fixed, every randomized algorithm with terminal expected regret $O(T^β)$, $β<1$, suffers $Ω(\sqrt{\log T})$ expected movement on some deterministic nested sequence, proving optimal horizon dependence. Under linear growth away from the constrained minimizer set, Steiner-point tracking yields movement independent of $T$. For general convex COCO, one-step-delayed chasing with regularized-leader resets gives regret $O(G_fD\sqrt{d\log(1+d)T})$ and cumulative constraint violation $O(G_gD\sqrt{d\log(1+d)T})$. For strongly convex losses, both are $O(d\log(1+d)\log(eT))$ when other parameters are fixed. These reductions replace the $O(d^{d/2})$ projection-path factor in prior analyses by the polynomial dimension dependence of Euclidean nested convex-body chasing.

Sharp Root Anti-Concentration via Projective Incidence and Ordered Root Laws

This paper answers the one-dimensional local root anti-concentration questions posed by Balcan, Pegden, and Sharma in the context of online optimization of piecewise-Lipschitz functions. For a homogeneous feature curve and coefficients whose density relative to the uniform law on a symmetric convex body $K$ is bounded by $A$, we show that the worst-case interval-hitting constant equals $A$ times a section-averaged projective incidence speed. For cube-supported coefficients, this speed is equivalent, up to universal constants, to the projective Lipschitz constant. This yields a sharp, dimension-free characterization and removes the previous $\sqrt N$ loss. For monic degree-$d$ polynomials under arbitrary coefficient laws, we prove that the interval-hitting constant is finite if and only if the ordered real-root laws have bounded densities, with a factor-$d$ comparison that is sharp. Conditional and joint coefficient-space area formulas, together with a two-chart certificate, make this criterion verifiable for dependent and singular coefficient laws. We also give two graph-learning applications that complete the transition-to-regret chain. A cost-sensitive Gaussian-RBF harmonic classifier uses the projective incidence theorem and achieves expected regret $\widetilde O((An^2D e^{BD}/\ell+1)\sqrt T)$. A common-offset polynomial-kernel model uses rigid translation of the ordered roots and achieves $\widetilde O((qn^2κ+1)\sqrt T)$ regret, even when the induced coefficient law is singular in the ambient coefficient space.

Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functions

We study decentralized online optimization for strongly geodesically convex (strongly g-convex) losses on Riemannian manifolds with bounded sectional curvature, including positively curved manifolds. In centralized Riemannian optimization, strong g-convexity tightens the optimal regret from $O(\sqrt{T})$ to $O(\log T)$, where $T$ is the time horizon; in the decentralized Riemannian setting, however, existing methods address only g-convex losses, leaving the strongly g-convex regime unexplored. One challenge is that the required decaying step size in the centralized regime is incompatible with existing network-error analyses, which typically assume a fixed step size. First, we provide a general network-error analysis for time-varying schedules. Next, we build on this analysis to establish the first $O(\log T)$ static regret bound for decentralized online Riemannian gradient descent, matching the minimax-optimal rate for strongly-convex Euclidean online optimization. Finally, we prove the same $O(\log T)$ regret bound for the two-point bandit feedback setting using novel strong subconvexity arguments for the smoothed versions of the loss functions.

Online Optimization of Difference-of-Convex Compositions with Smooth Mappings

We study online optimization for a broad class of structured non-convex non-smooth problems where each loss is a composition of a difference-of-convex function with a smooth mapping, and the feasible region is defined by constraint functions of the same kind. We propose a time-smoothed proximal linear algorithm and a local-regret measure based on a proximal residual mapping. We show that this residual is a proper stationarity measure for the original problem: its fixed-point condition implies first-order stationarity. Our analysis relies on a tangent-cone characterization for a feasible region described by composite difference-of-convex constraints, which is of independent interest and allows each update to be computed via a convex optimization oracle, despite the non-convexity of the problem. We establish a local-regret bound and a bound on the total number of inner convex subproblems. We also derive an error bound connecting the proximal residual to the distance to stationarity, providing a quantitative certificate of approximate stationarity.

On the Convergence of Adam, Revisited

We show that projected Adam for online optimization with arbitrary moment decay parameters $β_1,β_2\in[0,1)$ can have average regret bounded away from zero. A similar result of Reddi-Kale-Kumar from 2018 required $β_1<\sqrt{β_2}$. Similar to their result, we use a three-periodic sequence of linear functions on $[-1,1]$ with slopes $c,-1,-1$, though we use $c$ slightly larger than $2$. This nonzero average regret result extends to Adam variants such as AdamW, RMSProp, NAdam, Adan, AdaMax, Muon, and to an i.i.d. variant of the three-periodic sequence of slopes for Adam.

From Non-Convex to Strongly Convex: Curvature-Adaptive FTPL for Online Optimization

Curvature adaptivity is a classical theme in online optimization: for convex Lipschitz losses, adaptive methods interpolate between the optimal $O(\sqrt{T})$ regret for general convex losses and $O(\log T)$ regret under strong convexity. Recent work has shown that Follow-the-Perturbed-Leader (FTPL) achieves optimal $O(\sqrt{T})$ regret even for online non-convex Lipschitz losses, assuming access to an approximate offline-optimization oracle, but these guarantees do not exploit curvature. We show that FTPL can be made curvature-adaptive in the non-convex setting, without knowing in advance how curvature will accumulate over time. Our algorithm replaces the fixed perturbation scale of standard FTPL with a time-varying scale chosen using only past information. We give a simple follow-the-leader tuning rule for this scale and show that it competes, up to constants, with the best choice in hindsight. The resulting method achieves $O(\sqrt{T})$ regret for arbitrary non-convex Lipschitz losses and improves as cumulative curvature grows; with sufficiently accurate oracle calls, it achieves $O(\log T)$ regret when cumulative curvature grows linearly, which includes the classical strongly convex regime. We complement these upper bounds with matching lower bounds for prescribed cumulative-curvature sequences, already for one-dimensional convex losses, showing that the tradeoff between worst-case non-convex regret and curvature-driven fast rates is intrinsic.