Scalable Online Kernel Learning

• Scalable online kernel learning frameworks are algorithmic designs that use finite-dimensional approximations and budget mechanisms to efficiently process streaming data with sublinear regret guarantees.
• They employ techniques such as random feature linearization, Nyström methods, and support vector budgeting to reduce computational and memory costs while maintaining competitive accuracy.
• Distributed and federated protocols enhance these frameworks by coordinating kernel updates across multiple devices, ensuring scalability, reduced communication overhead, and robust performance.

A scalable online kernel learning framework refers to any algorithmic and systems-level solution for online learning with kernel methods that is specifically engineered to enable efficient operation as the number of data points, feature dimensions, or distributed participants scales to very large regimes. Such frameworks address the classic bottleneck of kernel methods—quadratic or worse memory and computation due to nonparametric representation—by leveraging principled finite-dimensional approximations, budget mechanisms, stochastic algorithmics, and system-level orchestration to achieve sublinear regret and competitive predictive performance at large scale.

1. Fundamental Principles and Online Protocols

The defining features of scalable online kernel learning frameworks are: (i) the capacity to process streaming or sequentially arriving data with bounded per-round memory and computation, and (ii) the preservation of theoretical guarantees, such as sublinear regret, relative to the best function in the underlying reproducing kernel Hilbert space (RKHS).

Common underlying protocols include:

• Classic online kernel ridge regression (Kernel-AWV): Regret-optimal forecasters, but O($n^2$) per round due to explicit Gram matrix operations (Jézéquel et al., 2019).
• Random feature (RF) linearization: Replace the infinite-dimensional mapping with a D-dimensional explicit embedding, reducing kernel evaluations to O(D) per sample (Ghari et al., 2023, Ghari et al., 2021, Shen et al., 2017, Chae et al., 2021).
• Nyström or low-rank subspaces: Either before data arrival (e.g., Taylor features) or adaptively (leverage-score sampling), project to m-dimensional subspaces, reducing per-round updates to O($m^2$) while controlling approximation error (Jézéquel et al., 2019, Sheikholeslami et al., 2016).
• Support vector budget mechanisms: Limit the number of active support vectors via uniform or importance-weighted eviction, enabling O(B) time and memory per round (Lu et al., 2015, Zhao et al., 2012, Sheikholeslami et al., 2016).
• Decentralized/parallel computation: Federated networks, peer-to-peer diffusion, and ADMM-style distributed protocols to distribute workload and storage, addressing both scale-out and privacy requirements (Ghari et al., 2023, Chae et al., 2021, Xu et al., 2022).

The broad goal is to ensure that, for any $T$, the cumulative regret $$R_T = \sum_{t=1}^T \ell(f_t(x_t), y_t) - \min_{f^*\in\mathcal{F}} \sum_{t=1}^T \ell(f^*(x_t), y_t)$$ is $O(\sqrt{T})$ (or sometimes $O(\log T)$ under curvature), while all algorithmic quantities are bounded by polylogarithms or low-order polynomials in $T$ and in fixed parameters $d$, $J$ (kernels), $K$ (clients), $B$ (budget), or $D$ (random feature/Gaussian Taylor basis size) (Ghari et al., 2023, Jézéquel et al., 2019, Sheikholeslami et al., 2016, Zhao et al., 2012).

2. Core Algorithmic Techniques

Explicit Random Features and Linearization

Bochner's theorem enables the approximation of any shift-invariant $k(x, x')$ by a Monte Carlo sum: $$k(x, x') \approx \phi(x)^T \phi(x'),~~ \phi(x) = \frac{1}{\sqrt{D}}[\sin(\rho_1^T x), \dots, \cos(\rho_D^T x)]^T,$$ with $\rho_j$ sampled from the kernel's spectral density (Ghari et al., 2023, Ghari et al., 2021, Shen et al., 2017). This approach turns the kernel problem into an explicit linear problem in fixed $D$ dimensions, supporting plain stochastic gradient descent and facilitating downstream parallelization.

Low-rank and Budget Subspace Techniques

• Taylor expansion (for Gaussian kernel): Construct a fixed basis $G_M$ of multivariate orthonormal polynomials, with $D=O((\log n)^d)$ features, guaranteeing kernel approximation error decaying super-polynomially in $M$ (Jézéquel et al., 2019).
• Nyström adaptive dictionary: Grow a dictionary of active basis vectors using leverage-score sampling. The dictionary size $m$ is chosen for given effective dimension, delivering error bounds matching full kernel learning as long as $m=O(d_{\text{eff}} \log^2 n)$ (Jézéquel et al., 2019, Sheikholeslami et al., 2016).
• Support-vector budget schemes: Limit model capacity by tightly controlling the number of stored kernel atoms using uniform or non-uniform (importance-weighted) sampling, with regret guarantees and efficient update procedures (Zhao et al., 2012, Lu et al., 2015).

Federated and Distributed Protocols

• Federated multi-kernel learning: Each client maintains local kernel weights and confidence vectors, updating a personal combination of global model components, with subset-of-kernel communication for bandwidth control (Ghari et al., 2023).
• Decentralized ADMM/Hedge: Each agent updates kernel parameters with local data and broadcasts or diffuses updates amongst neighbors, achieving consensus and sublinear regret through network-wide synchronization (Chae et al., 2021, Xu et al., 2022).
• Communication efficiency techniques: Randomized subset selection (sending updates for only $M \ll J$ kernels), quantization/bandwidth censoring, and event-triggered communication (Ghari et al., 2023, Xu et al., 2022).

3. Regret Analysis and Theoretical Guarantees

Scalable online kernel learning frameworks are analyzed using adversarial or stochastic regret bounds, depending on underlying assumptions:

• Projected Kernel-AWV (PKAWV):

$$R_n(f) \leq \lambda\|f\|^2 + B^2 \sum_{j=1}^n \log(1+\lambda_j(K_n)/\lambda) + \text{approx. error},$$

with the approximation error controlled by the dimension of the subspace (Taylor/Nyström) and can be made negligible with suitable choices (Jézéquel et al., 2019).

• Random feature-based methods:

$$R_T = O(\sqrt{T}) + O(\varepsilon T)~ \text{with}~ \varepsilon = O(1/\sqrt{D}),$$

choosing $D$ large enough so that the random feature error term is subdominant (Ghari et al., 2023, Shen et al., 2017).

• Federated/Distributed algorithms: Client and server regrets are bounded as $O(\sqrt{T})$, and consensus errors decay as $O(\sqrt{T})$ as well, with explicit dependence on communication/computation budgets $M, J, K$ (Ghari et al., 2023, Chae et al., 2021).
• Budgeted OGD (BOGD):

$$\mathbb{E}\left[\sum_{t=1}^T \ell(y_t f_t(x_t)) - \sum_{t=1}^T \ell(y_t f(x_t))\right] = O\left(\sqrt{T} + \frac{T}{B}\right),$$

offering consistency as long as $B\gg 1$ (Zhao et al., 2012).

• Pairwise/metric-learning with random Fourier features and dynamic averaging: Regret bound $O(\sqrt{T})$ for kernelized AUC and similar objectives, with $D=O(\sqrt{T}\log T)$ (AlQuabeh et al., 2 Feb 2024).

4. Communication and Computational Complexity

The frameworks deploy several approaches to control computational and communication demands:

Framework/Technique	Per-round Computation	Communication per Round
Random feature-based	$O(JD)$ (per client/node)	$O(M D)$ (if communicating top-$M$ kernels)
Federated MKL (e.g., POF-MKL)	$O(J D)$ (prediction), $O(M D)$ (update)	$O(M D)$ floats per client-server
Budgeted OGD (BOGD/BOMKL)	$O(M B)$	N/A (single-node or multi-kernel)
Decentralized ADMM	$O(P M^2)$ (per node)	$O(P M)$ floats (per link per iteration)
Graph-aided OMKL	$O(M D)$	N/A (all feature maps kept local)

• Explicit feature methods require only maintaining $D$-dimensional vectors per kernel, with $J$ kernels, and can batch or subset updates/communications to further control scaling.
• Subset selection and quantized communication can reduce per-round bits to $O(M D b)$, where $b$ is the number of bits in quantization (Ghari et al., 2023, Xu et al., 2022).
• Frameworks such as FORKS deliver $O(\log T)$ regret with linear time per budget unit via incremental sketching and decomposition, outperforming previous $O(B^2)$ second-order approaches (Wen et al., 15 Oct 2024).

5. Empirical Benchmarks and Application Domains

Evaluation of scalable online kernel learning frameworks spans:

• Data sets: High-dimensional regression/forecasting (Naval Propulsion, UJI WiFi, Air Quality, Wave Energy), real-world federated tasks (multi-site, non-IID splits), large-scale classification (SUSY, codrna, w8a, a9a), pairwise/AUC maximization, causal structure learning (Ghari et al., 2023, Lu et al., 2015, Tanaka, 7 Nov 2025, AlQuabeh et al., 2 Feb 2024).
• Key results:
  • Federated RF-based POF-MKL: Achieves up to 2$\times$ lower MSE than best single-kernel approaches with dramatically reduced communication (down to 1/51st for $M=1$, $J=51$), and retains sublinear regret under non-IID conditions (Ghari et al., 2023).
  • Budgeted methods (BOGD/BOMKL): Outperform both unbudgeted OMKC and other compressed/budgeted baselines over millions of samples, with order-of-magnitude reductions in memory and walltime (Lu et al., 2015, Zhao et al., 2012).
  • Graph-aided OMKL: Gains up to $10^3\times$ speedup over naive OMKL/OMKR, with superior MSE, due to dynamic pruning of irrelevant kernels per round (Ghari et al., 2021).
  • Online subspace tracking (OK-FE): Outperforms Nyström-based reductions in time-varying data, matches full-kernel downstream accuracy, and adapts to data drift (Sheikholeslami et al., 2016).
  • Causal inference (SEM-Kernel): Achieves near-zero bias and root-$n$ rate for bidirectional causal effect estimation, with $\sim$linear scaling vs. data, outperforming polynomial expansions and naive single-equation methods (Tanaka, 7 Nov 2025).
  • Pairwise learning (LM-OGD): O($\sqrt{T}$) regret and O($\sqrt{T}\log T$) cost via low-rank Fourier features, matching or exceeding more expensive offline and buffer-based AUC maximization baselines (AlQuabeh et al., 2 Feb 2024).

6. Personalization, Heterogeneity, and Dynamic Adaptation

Personalization and heterogeneity resilience are enabled through:

• Per-client kernel-confidence vectors and multiplicative weights (Hedge), so each client/node effectively learns its own convex combination of kernels, adapting to local data (Ghari et al., 2023, Lu et al., 2015, Chae et al., 2021).
• Randomized and adaptive subset selection—either binning kernels per weight profile at each client or employing feedback graphs for global kernel selection—balancing exploration (probing new or less-tested kernels) and exploitation (focusing on high-performing kernels) (Ghari et al., 2023, Ghari et al., 2021).
• Dynamic adaptivity in AdaRaker: Interval-based meta-learning schemes for nonstationary or time-varying environments, with dynamic regret bounds $O(\widetilde{T}^{2/3}V_T^{1/3})$ for total variation $V_T$ (Shen et al., 2017).

7. Limitations and Open Directions

While scalable online kernel learning frameworks represent a significant advance, several open issues persist:

• Hyperparameter tuning (kernel bandwidths, $D$, $B$, learning rates, communication budgets) remains problem-dependent and can impact empirical outcomes.
• Budget methods exhibit trade-off between bias and variance: smaller budgets degrade accuracy, non-uniform importance sampling mitigates this but requires additional computation (Lu et al., 2015, Zhao et al., 2012).
• Communication cost may still be high when the kernel dictionary is large or network connectivity is dense (Ghari et al., 2023, Chae et al., 2021, Xu et al., 2022).
• Real-time adaptation to adversarial concept drift benefits from dynamic, meta-expert layers but may require careful resource management as instances or intervals proliferate (Shen et al., 2017).
• Frameworks for domain-specific kernels or highly-structured input spaces (graphs, sequences) largely remain to be adapted to these scalable online paradigms.

In sum, scalable online kernel learning frameworks constitute an active and technically rich area offering a spectrum of algorithmic strategies (random features, budgeted subspaces, distributed consensus, dynamic kernel combinations) to realize the statistical power of kernel methods at the scale and complexity of contemporary data streams and distributed systems.