Dynamic Mixture-of-Experts Serving allocates k replica GPUs among m experts as workloads change. At each round, the online algorithm sees the current workload, chooses integral replica counts, and pays bottleneck service cost plus replica movement. It does not know future workloads. Huang, Lou, and Xiao gave an O(sqrt(log k))-competitive randomized algorithm for this problem. We prove a deterministic O(1)-competitive algorithm. For every number of experts and every k>=1, the algorithm satisfies ALG_det <= 10 C_PB OPT + (5 C_PB + 8) k + 16, where C_PB is the absolute constant from Chasing Positive Bodies at resource augmentation one and covering sparsity two. Consequently, CR_det(k)<=10 C_PB for every k>=1, so CR_det(k)=Theta(1). The multiplicative factor does not depend on the number of experts, replica budget, horizon, or workload values. Thus randomization is not needed for the asymptotic guarantee. The proof has two layers. A finite tangent envelope, summable positive resets, and a nonexpansive balanced projection reduce reciprocal-max service costs to a deterministic exact-budget fractional path. A new deterministic rounding theorem converts every such path to integral allocations with service distortion three and movement bounded by the fractional movement plus 6k. The complete reduction, rounding theorem, causal composition, and quantified main theorem are machine-checked in Lean 4 relative to the positive-body result as the sole scientific source premise. The theorem concerns the allocation model above. It does not include network topology, shared-edge congestion, or routing decisions.
Thomas Kesselheim, Marco Molinaro, Kalen Patton +1cs.LG cs.DS
Competitive analysis is central to the study of online algorithms, but upper bounds are often highly problem-specific. We develop a more unifying methodology via the minimax viewpoint. Guided by Yao's principle, we reduce worst-case competitive analysis to Bayesian online design under an arbitrary correlated prior over arrival sequences. For such a prior, let $X^*$ be the hindsight-optimal fractional solution for the realized instance, and let $X^{(t)}=\mathbb E[X^*\mid \mathcal F_t]$ be its posterior process. Our guiding rule is posterior matching: at each time $t$, choose the feasible online action that tracks the current posterior $X^{(t)}$ as closely as the online constraints permit. We show that this single principle yields optimal or near-optimal guarantees for several classical online fractional problems, including set cover, load balancing, matching and more general resource-allocation problems, recovering or improving state-of-the-art bounds in these settings with norm/concave objectives. Via known rounding reductions, it also yields randomized integral guarantees for weighted paging, MTS on star metrics, and ski-rental. At a technical level, our analysis reduces competitive guarantees to key probabilistic inequalities for the vector martingales generated by the posterior of the offline optimum. The resulting framework gives a reusable route from Bayesian online design under arbitrary correlated priors to information-theoretic worst-case competitive guarantees.
Tianhang Lu, Runtian Ren, Shengcai Liu +1cs.LG cs.CC
This paper studies learning-augmented and randomized online aggregation with delays on a line metric. We consider advice given as online suggested service lengths, and evaluate the algorithms in terms of robustness and consistency. For each $λ\in (0,1]$, we first propose a deterministic learning-augmented \textsc{Balance} algorithm that is $(4/λ+1/λ^2)$-robust and $(4+λ)$-consistent. We also propose a randomized algorithm for the problem in the classical adversarial model, which is $(e+1)$-competitive against an oblivious adversary, improving over the deterministic $5$-competitive \textsc{Balance} benchmark~\cite{bienkowski2013chain}. Notably, this competitive ratio is even lower than the lower bound of $4$ for deterministic online algorithms. Moreover, we establish a lower bound of $e$ on the competitive ratio of randomized online algorithms, improving the previous lower bound of $e/(e-1)$. Besides, we combine the two ideas and obtain a randomized learning-augmented algorithm that is $(e/λ+1/λ^2)$-robust and $(e+λ)$-consistent. Finally, we conduct numerical experiments to complement our theoretical analysis and evaluate the empirical performance of our algorithms.
We present a learning-augmented online algorithm for the preemptive FIFO buffer management problem, where packets arrive online to a finite-capacity buffer, must be transmitted in FIFO order, and the algorithm may preemptively discard buffered packets to accommodate future arrivals. Our algorithm simultaneously achieves 1-consistency, η-smoothness, and asymptotic \sqrt{3}-robustness, where ηdenotes the prediction error. Specifically, it attains an optimal competitive ratio of 1 under perfect predictions, degrades smoothly as the prediction error increases, and maintains an asymptotic competitive ratio of \sqrt{3} under arbitrarily inaccurate predictions, matching the best-known worst-case guarantee for the classical online problem, established by Englert and Westermann in 2009 [Algorithmica 53(4): 523-548]. A key technical contribution of our work is the introduction of an \emph{output-based prediction error metric}. Because capacity constraints dictate that only a strictly bounded subset of arriving packets is ultimately transmitted, our metric assesses prediction quality over the resulting optimal schedules rather than the raw input sequences, avoiding artificial error penalties. To guarantee robustness, our algorithm dynamically monitors predictions and executes a \emph{buffer-clearing strategy} upon transitioning to a worst-case fallback mechanism. We prove that the competitive loss incurred by this clearing operation is bounded by an additive capacity constant that vanishes asymptotically. Finally, we show that our algorithm provides a generalized framework for learning-augmented buffer management: substituting the fallback module with any β-competitive online algorithm immediately yields asymptotic β-robustness.