Anytime Learning Schedules Overview

• Anytime learning schedules are adaptive policies for allocating computational effort that deliver improving performance irrespective of the training horizon.
• They employ methodologies like horizon-free step sizes, latent dynamical system schedulers, and RL-based controllers to ensure optimal outcomes under dynamic resource constraints.
• Empirical evaluations demonstrate that techniques such as constant or 1/√t learning rates achieve state-of-the-art performance in deep learning and resource-constrained environments.

Anytime Learning Schedules

Anytime learning schedules designate adaptive policies for allocating computational or optimization effort such that performance—measured as solution quality, accuracy, regret, or task completion—improves monotonically or nearly monotonically with elapsed computation, and crucially, at every possible interruption point. In machine learning, this framework encompasses both how model parameters are updated (e.g., via learning rate schedules, curriculum selection, or replay frequency) and how compute is dynamically distributed across tasks or resources. A defining property is independence from the total training horizon: at any stop-point, the learner is guaranteed to deliver the highest-possible performance given prior resource usage, without horizon-dependent pre-tuning.

1. Formal Definitions and Theoretical Foundations

The central object in anytime scheduling is a schedule or control policy that can be invoked or queried at arbitrary, potentially unknown, points in time or training. Classic anytime algorithms yield solutions whose quality is a non-decreasing function of compute. The formal cost function for multi-process anytime scheduling under shared computational resources is given by

$$E_u = \int_0^\infty \prod_{i=1}^n [1-F_i(\tau_i(t))]\,dt,$$

where $F_i$ is the cumulative “time-to-solution” profile of process $i$, and $\tau_i(t)$ is its subjective compute allocation (Finkelstein et al., 2011). The schedule is called anytime if, at every possible interruption, the set of completed steps maximizes expected utility or solution quality under resource constraints.

In convex learning, an anytime learning rate schedule is a sequence $(\eta_t)_{t\geq 1}$ or adaptive control law $\eta^*_t = f(\mathcal{H}_{\leq t})$ (where $\mathcal{H}_{\leq t}$ is the observed data or optimization trajectory) with the property that for all $T$,

$$R(T) = \sum_{t=1}^T \left[ \ell_t(\theta_t) - \ell_t(\theta^*_t) \right]$$

is (nearly) minimized with respect to unknown $T$ or nonstationary data distributions (Fahrbach et al., 2023).

2. Algorithms and Methodologies for Anytime Scheduling

2.1. Horizon-Free Step Size Control

A major focus is on learning rate schedules that do not require advance knowledge of the training horizon:

• Constant and polynomially decaying step sizes: Schedules of the form $\eta_t = C t^{-\gamma}$, with $\gamma$ tuned according to the spectral and source conditions of the regression or learning problem, can be proven minimax optimal for overparameterized models under weight averaging (Meterez et al., 3 Feb 2026). The “1/$\sqrt{t}$” schedule ($\gamma=0.5$) is particularly robust.
• Warmup-Stable-Decay (WSD): A composite schedule with linear warmup, extended plateau, and final linear decay is nearly horizon-free; the schedule can be checkpointed and restarted as more compute budget becomes available (Meterez et al., 3 Feb 2026).

2.2. Latent Dynamical System Schedulers

Generative models such as Latent ODEs are trained over historical training trajectories, encoding the joint evolution of loss, validation accuracy, and learning rate: $$z'(t) = f_\theta(z(t), t),\quad x(t) = (\ell(t), \nu(t), \eta(t))$$ The scheduler can be queried at any $(t^*)$, producing a future block of learning rates via an ensemble of ODE rollouts from the inferred latent state; the process is iterative and horizon-agnostic, constituting a true anytime learning-rate generator (Sampson et al., 27 Sep 2025).

2.3. RL-Based Adaptive Schedulers

Reinforcement learning controllers (e.g., actor-critic PPO) parameterize the schedule as a policy $\pi_\theta(a_t | s_t)$, where $s_t$ encodes current training statistics and $a_t$ decides learning-rate adjustment. Such controllers can be updated online and can generalize or transfer across tasks; the action can be invoked at any optimization step, conferring the anytime property (Xu et al., 2019, Klasson et al., 2022).

2.4. Rolling Horizon and Evolutionary Curriculum Scheduling

In rolling-horizon evolutionary algorithms (e.g., RHEA CL for curriculum learning), a population of curricula is iteratively optimized, and at each epoch only the leading segment is executed, after which the curriculum is re-optimized with updated state. This guarantees monotonic, elitist improvement at any interruption point (Jiwatode et al., 2024).

3. Key Domains and Empirical Evaluations

3.1. Deep Learning Optimization

For neural network training, anytime learning-rate schedules outperform parametric or reactive schedules by permitting arbitrary interruption and adaptation:

• Latent ODE-based schedules improved test accuracy by 1–3% over fixed baselines and located wider, flatter minima, as quantified by Hessian eigenvalues (e.g., 0.62 for LODE vs. 1.2–13.2 for other schedules) (Sampson et al., 27 Sep 2025).
• For LLM pretraining, constant or 1/$\sqrt{t}$ step sizes with EMA weight averaging closely tracked—often within $10^{-3}$ to $10^{-2}$ loss—an oracle-tuned horizon-dependent cosine schedule, with no advance budget knowledge (Meterez et al., 3 Feb 2026).

3.2. Online, Continual, and Streaming Learning

• In online settings with data arriving in large chunks, the optimal update frequency is typically neither greedy (after every chunk) nor maximally tardy (only once, at the end), but at a moderate interval $w^*$ yielding a lower cumulative error rate. Empirical guidance suggests $w^* \approx 0.1 B$ for $B$ chunks (Caccia et al., 2021).
• Task replay in continual learning benefits from adaptive replay schedules (found via MCTS or RL), yielding 3–5% absolute accuracy improvement over naive equal-task replay under tight memory constraints (Klasson et al., 2022).

3.3. Anytime Scheduling in Resource-Constrained Systems

• For assigning anytime algorithmic tasks to heterogenous servers, adaptive quality control—setting target quality per task or globally via bisection—enables the scheduler to maintain zero averaged lateness while maximizing the attained solution quality, even as overload fluctuates (Módos et al., 2018).

3.4. Theoretical Insights: Regret and Optimality

• For SGD with distribution shift, an SDE-based closed-form control for the optimal schedule,

$$\zeta^*(\tau) = \min \left\{1, \frac{v_\tau}{\tfrac{d+1}{\nu(\tau)}v_\tau + \tfrac{d\sigma^2}{\nu(\tau)}}\right\}$$

achieves regret bounds of the form $O(d\sigma^2\ln T + \Gamma^2 T)$ with no horizon knowledge (Fahrbach et al., 2023).

4. Properties, Guarantees, and Practical Recommendations

• Monotonicity and Robustness: Many anytime schedules (e.g., rolling elitist EA, weight averaging + horizon-free step sizes, SDE-controlled learning rates) guarantee monotonic non-degradation if interrupted.
• Optimizer-agnosticism: Generative (LODE) and averaging-based schedules do not require changes to the underlying optimizer nor task-specific re-tuning (Sampson et al., 27 Sep 2025, Meterez et al., 3 Feb 2026).
• Empirical Best Practices: In open-ended pretraining, constant or 1/$\sqrt{t}$ step size plus EMA is “fire and forget,” requires minimal tuning, and delivers state-of-the-art performance for variable compute (Meterez et al., 3 Feb 2026). For curriculum learning or task allocation, rolling-horizon selection of next steps enables rapid early progress and high quality upon early stop (Jiwatode et al., 2024).

5. Extensions and Connections

Anytime learning principles generalize to:

• Distributed and parallel settings, where optimal interleaving across multiple anytime processes depends on hazard function properties; for monotonic hazard, simple sequential or round-robin schedules are optimal, but heavy-tailed cases may require intricate suspend-resume strategies (Finkelstein et al., 2011).
• Replay selection in CL, curriculum scheduling in RL, and compute allocation in server environments—all of which benefit from data-driven or learned anytime policies (Klasson et al., 2022, Jiwatode et al., 2024, Módos et al., 2018).
• Direct schedule generation for complex, nonstationary or open-ended scenarios, where model-based or RL-based schedules adaptively optimize over unbounded horizons (Sampson et al., 27 Sep 2025, Fahrbach et al., 2023, Xu et al., 2019).

6. Limitations, Open Questions, and Future Directions

• Estimation of performance/time profiles: In resource-allocation and combinatorial settings, accurate estimation of task time-quality profiles is essential but remains challenging for high variability tasks (Módos et al., 2018).
• Adaptation to nonstationary environments: While online convex optimization admits regret-optimal anytime schedules, deep nonconvex learning is fundamentally harder. Empirical gap persists versus batch training, especially in streaming regimes (Caccia et al., 2021).
• Learning schedule transfer: RL-based and generative schedule models show promise for cross-task transferability, but generalization across drastic domain shifts requires further study (Xu et al., 2019, Klasson et al., 2022).
• Integration with memory/computation trade-offs: Optimal checkpointing, replay, and lazy-update policies remain open for general nonconvex or heterogeneous systems.

7. Summary Table: Representative Anytime Schedule Types

Method/Class	Anytime Guarantee	Key Setting/Domain
Constant/1/√t lr + EMA (Meterez et al., 3 Feb 2026)	Yes (minimax with avg.)	Large-scale LLM pretraining
Latent ODE scheduler (Sampson et al., 27 Sep 2025)	Yes (any step t*)	DL optimization, all models
RL-based controller (Xu et al., 2019, Klasson et al., 2022)	Yes (online stepwise)	Learning rate/replay in CL
Rolling horizon EAs (Jiwatode et al., 2024)	Yes (segment-wise)	Curriculum learning in RL
Bisection/individual ctrl. (Módos et al., 2018)	Yes (per scheduler call)	Task-server allocation

Each approach embodies the core principle that high-performance is delivered at any interruption, with the schedule being horizon-independent, and improvements monotonic or non-degrading under reasonable assumptions. Empirically and theoretically, anytime learning schedules are essential for continuous, open-ended, and resource-constrained learning paradigms.