Real-world time series inherently intertwine global periodic structures with localized non-stationary variations. Existing approaches process these heterogeneous dynamics within a single computational domain, incurring fundamental limitations: time-domain models suffer from periodic misalignment over long horizons, while frequency-domain models over-smooth transient spikes. We argue that the optimal computational domain is not a property of the model, but of the data itself. Based on this principle, we propose RouteTS, a unified forecasting framework that partitions the frequency spectrum via amplitude routing and delegates components to their mathematically optimal domains. Dominant frequencies are processed by a complex-valued linear predictor in the frequency domain to preserve periodic structure, while residual spectral energy is reverted to the time domain and modeled by a lightweight MLP for local variations. Extensive experiments demonstrate that RouteTS achieves competitive prediction accuracy across diverse real-world datasets, with routing decisions guided by the underlying spectral signature. Furthermore, the lightweight design of RouteTS provides significant computational efficiency advantages, offering a principled solution to the longstanding dilemma between global periodicity and local transience.
A routing decision can be revised at the next transaction, but a latched source exclusion persists across later decisions. We ask what evidence should authorize these unequal-persistence actions when finite-population auditing and learning share a budget. ALIVE (Action-Layered Intervention via Evidence) is an auditable control layer: one randomized without-replacement prefix supplies cached evidence, heuristic warnings drive non-latching floor-bounded routing, and only two fresh simultaneous certificate separations may latch an exclusion request subject to capacity-feasible activation. Conditional on fixed support and labels under an ideal uniform audit permutation, any predictable controller preserving this interface inherits an anytime familywise bound of δon acting against a source that fails the pre-fixed absolute or relative strict-majority-disagreement predicate. With a published known-size, all-strict-majority PPR engine, median evidence count fell from 304 to 96 identities in e40 and from 171 to 62 in e60, while both engines used 48 in e80. In the matched CIFAR controller, the persistent-action layer added +0.1935 accuracy-AUBC percentage points over routing-only in all ten paired seed clusters. The +0.1954-point full-system contrast against CBR was also positive but did not meet the predeclared multiplicity-adjusted criterion (conditional Holm-adjusted sign-flip reference value =.097656). On a fixed natural panel, exploratory PPR used a median closure prefix of 95 rather than 105 for exploratory Serfling/FPC, but still exposed 88.0% of the panel and had no downstream task. Together these results map a restraint--power--cost--utility boundary: the action contract controls a defined persistent decision, while net value depends on evidence margin, audit cost, and budget regime.
A stochastic Gumbel-Top-K router defines, for every token of a mixture-of-experts (MoE) model, a routing law: a distribution over ordered expert lists and mixture weights. We ask which joint distributions over the routing choices of different tokens are reachable while every individual token's complete routing law is held exactly fixed. We give a two-sided construction, Hierarchical Copula-Gumbel-Top-K (CGA). Within a group of related tokens, an exchangeable Gaussian copula positively correlates the Gumbel perturbations at each expert coordinate, which can increase within-group expert-set coherence. Across disjoint pairs of groups, a tunable antithetic construction introduces a selectable amount of negative dependence. We prove that both operations leave each token's ordered Top-K sample, mixture weights, and inclusion probabilities identical in distribution to independent routing at a routing layer conditioned on its pre-routing logits; conditional expected expert traffic is preserved as a consequence. We characterize the resulting trade-off: positive within-group coupling can only inflate the variance of realized expert loads relative to independent routing, while nonnegative cross-group opposition can only reduce it relative to flat coupling at the same within-group strength. Coherence and load dispersion are thus controlled by two complementary dependence dials on the invariance constraint surface. Because the base model is untouched, the dials can be driven by a small controller over frozen features, trainable with a score-function estimator: the frozen network is evaluated only in the forward direction, and gradients are confined to the controller. An initial small-scale pilot validates the mechanism and the training route, but does not establish task-level fine-tuning gains.
Zijian Wang, Pengfei Li, Guangyu Yang +1cs.LG cs.AI cs.DC
Routing-prediction federated learning has emerged as a new paradigm that reframes inter-client heterogeneity as a resource for system-level intelligence: at inference time, the server routes each external query to the best-matched client for prediction. Existing approaches, however, typically treat each client as internally homogeneous, overlooking latent subpopulations within local data. For example, patients with the same diagnosis at one hospital may exhibit morphologically distinct disease subtypes. The coexistence of inter-client and intra-client heterogeneity, which we call dual heterogeneity, can impair both routing and prediction. To address this challenge, we propose FedSPM, a routing-enabled semiparametric mixture framework that represents each client using client-specific latent components. Each component combines a predictive distribution for classification with a feature distribution for routing. To flexibly model feature distributions while effectively sharing information across clients, FedSPM models their density ratios relative to a common nonparametric measure estimated via empirical likelihood. We develop a federated expectation-maximization algorithm that optimizes a tractable surrogate and prove convergence of the exact profiled objective at the standard $\mathcal{O}(1/\sqrt{T})$ rate when the surrogate errors are properly controlled. Experiments on controlled benchmarks and real-world medical data demonstrate consistent improvements in routing and prediction under dual heterogeneity. Code is available at https://github.com/zijianwang0510/FedSPM.