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Life sciences · Preprint
arXiv · September 10, 2026
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This is a theoretical paper in convex optimization extending the CONES framework to time-varying loss functions. The authors derive algorithmic regret and movement cost bounds for projected proximal methods under convex and strongly convex loss settings, along with lower bounds for weakly adaptive and anytime algorithms. No empirical validation, clinical application, or real-world test is provided.
Preprint.
Projected proximal algorithm achieves O(T^(1−β)) regret and O(T^β) movement cost simultaneously for any β ∈ [0,1) under convex losses Any weakly adaptive online algorithm with O(T^β) regret has movement cost lower bound of Ω(T^((1−β)/2)) for any β ∈ [0,1) Under strongly convex loss functions, projected proximal algorithm achieves O(1) regret and O(log T) movement cost simultaneously
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This is a theoretical computer science paper presenting algorithmic results and complexity bounds for an optimization problem, not clinical evidence or empirical validation in any applied domain.
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Convex Optimization with Nested Evolving Feasible Sets (CONES)} was introduced in \cite{CONESVaze} where the objective function \(f\) remains fixed but the feasible region evolves over time as a nested sequence \(S_1 \supseteq S_2 \supseteq \cdots \supseteq S_T\). The goal of an online algorithm is to simultaneously minimize the regret with respect to hindsight static optimal benchmark and the total movement cost $M_\cA(T)$ while ensuring feasibility at all times. CONES is an optimization-oriented generalization of the well-known \emph{nested convex body chasing} (NCBC). In this paper, we extend CONES to allow for loss functions $f_t'$s to also change over time. When all loss functions are convex, we show that the projected proximal algorithm achieves $O(T^{1-β}), O(T^β)$ simultaneous regret and movement cost, respectively, for any $β\in [0,1)$, over a time horizon of $T$. We also show that any {\it weakly adaptive} online algorithm with $O(T^β)$ regret has a movement cost of $Ω\left(T^{\frac{1-β}{2}}\right)$ for any $β\in [0,1)$. When all loss functions are strongly convex, we show that the projected proximal algorithm simultaneously achieves $O(1)$ regret and a movement cost of $O(\log T)$. To complement this, we show that any online algorithm with sublinear {\it anytime} regret has a movement cost of $Ω\left(\log T\right)$.
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