Online Learning via Learned Latent Bayesian Tracking

• Figure 1 contrasts AURA’s enhanced error–complexity trade-off with approaches that alter the learning algorithm in parameter space.Source: Figure 1 · p. 2
• Direct Bayesian filtering scales at least quadratically, and often cubically, with parameter dimension, while existing approximations can limit adaptation performance and remain costly.Source: §1 · p. 2
• AURA performs a single-step EKF update in a learned low-dimensional latent space, then reconstructs full model parameters through a lifting map.Source: §1 · p. 2
• The framework meta-learns its latent dynamics and lifting map from offline non-stationary trajectories to tailor parameter-space geometry for online adaptation.Source: §1 · p. 2
• AURA is reported to enable accurate adaptation with a single low-complexity update per sample on wireless receivers and non-stationary image classification.Source: §1 · p. 3

• Streaming methods include online gradient methods, adaptive optimizers, and proximal methods, while continual learning and test-time adaptation commonly use batches or multiple samples.Source: §2 · p. 3
• Bayesian-filtering approaches track parameters as evolving states, but deep-model applications use restrictive covariance approximations or manually predefined adaptation subspaces.Source: §2 · p. 3
• AURA instead learns a low-dimensional latent state-space and its mapping to parameter space for online Kalman-based updates.Source: §2 · p. 3

• At each time step, an input-label pair is drawn from a time-dependent distribution that may change continuously, through gradual drift or more abrupt shifts.Source: §3 · p. 3
• As the distribution evolves, fixed model parameters may become suboptimal, motivating adaptation to maintain accurate predictions.Source: §3 · p. 3
• The learner observes labeled samples sequentially and updates parameters through an online operator that may depend on the previous parameterization.Source: §3 · p. 4
• The formulation requires sequential adaptation from limited data and an update rule that is computationally efficient and adapts rapidly.Source: §3 · p. 4
• AURA’s adaptation mechanism is designed using an auxiliary labeled dataset of non-stationary trajectories; deployment instead uses sequential streaming samples.Source: §3 · p. 4

4 Proposed Method: AURA

• AURA replaces direct high-dimensional Bayesian filtering with a learned low-dimensional latent state, tracked by a single-step EKF and lifted to full model parameters.Source: §4 · p. 4

4.1 Latent Bayesian Online Adaptation

• The latent state represents adaptation-relevant degrees of freedom; a stochastic linear dynamical system models its evolution, and the EKF updates it from streaming labeled observations.Source: §4.1 · pp. 4–5
• The EKF tracks a Gaussian approximation of the latent-state posterior, and the lifting map reconstructs the adapted predictor parameters after each sample.Source: §4.1 · p. 5

4.2 Offline Meta-Learning of the Latent Adaptation Space

• AURA meta-learns the lifting map and latent dynamics from non-stationary trajectories by minimizing cumulative online loss and differentiating through the adaptation trajectory.Source: §4.2 · pp. 5–6

4.3 Discussion and Practical Considerations

• Per-step complexity is O(n + m^2), compared with at least quadratic scaling in the full parameter dimension d for direct Bayesian filtering.Source: §4.3 · p. 6
• Larger latent spaces and lifting maps can increase flexibility and computational cost, while AURA’s Bayesian posterior concerns the latent state rather than full model weights.Source: §4.3 · pp. 6–7

5.1 General-Purpose Online Adaptation for Non-stationary Image Classification

• Image streams gradually vary mixtures of corruption types across MNIST-C, CIFAR-10-C, and CIFAR-100-C, with supervised streaming methods receiving up to 15 labeled samples per step.The experiments also compare no adaptation, online gradient descent, EKF-FC, and batch-wise test-time adaptation methods.Source: §5.1 · pp. 7–8
• AURA achieves the lowest overall classification error on MNIST-C and CIFAR-10-C, and ranks second on CIFAR-100-C, where streaming-incompatible ROID leads.On the first two benchmarks it outperforms every batch-wise baseline while using as few as five adaptation samples per frame; on CIFAR-100-C it improves substantially over streaming Online GD and EKF-FC.Source: §5.1 · p. 8, Table 1 · p. 8
• In image classification, AURA dominates parameter-space EKF-FC in the complexity–accuracy plane, while increasing latent dimension initially improves performance before gains saturate or slightly deteriorate.Selected operating points lie near the empirical Pareto frontier for each dataset.Source: Figure 2 · p. 8, §5.1 · p. 9

5.2 Rapid Online Adaptation for Wireless Receivers

• Wireless evaluation uses a three-user, five-antenna DeepSIC receiver on 150-frame channel sequences, with six labeled pilots per frame during tracking.The setup evaluates linear and nonlinear QuaDRiGa channels and compares AURA with online gradient descent and full-parameter Bayesian filtering variants.Source: §5.2 · p. 9
• Across linear and nonlinear QuaDRiGa channels, AURA achieves the lowest BER across the SNR sweep and more stable temporal tracking than the considered baselines.The gains are especially pronounced relative to full-parameter Bayesian filtering methods.Source: §5.2 · p. 9

6 Conclusion

• AURA learns a low-dimensional state-space representation tailored to online adaptation, enabling an efficient single-step Bayesian update and strong performance across domains under computational constraints.The conclusion frames the framework as combining Kalman-based filtering with learned representations.Source: §6 · p. 9

A Additional Related Work

• Unlike online continual learning methods that preserve performance on past distributions, AURA tracks predictors aligned with the current distribution and does not use replay or explicit consolidation.This distinction suits settings where past operating conditions become obsolete rather than tasks that must later be revisited.Source: A Additional Related Work · p. 17
• AURA uses differentiable Bayesian filtering as a meta-learning mechanism for an online adaptation rule, rather than learning a Kalman filter for physical-state estimation.Its tracked latent state represents a learned low-dimensional parameterization of evolving optimal model parameters.Source: A Additional Related Work · p. 17
• AURA differs from parameter-efficient fine-tuning by targeting streaming adaptation with few labeled samples and strict latency constraints, rather than fixed-task adaptation with substantial data and multiple optimization iterations.It learns a latent adaptation geometry for real-time Bayesian tracking of drifting optima.Source: A Additional Related Work · p. 17

B Theoretical Motivation for the Latent State-Space Formulation

• The theoretical motivation treats evolving optimal parameters as trackable in a learned low-dimensional representation with smooth dynamics, and outlines conditions for noisy observations and an ELBO interpretation of latent Kalman tracking.The appendix says proofs are provided at its end.Source: B Theoretical Motivation for the… · pp. 17–18

B.1 Existence of Low-Dimensional Model Representations

• Under strong quasi-convexity and bounded-loss conditions, the proposition supports a qualitative link between distribution shift and latent variation; the text cautions that these assumptions may not hold exactly in practical deep learning.The discussion motivates modeling latent representations as a smoothly evolving dynamical system rather than prescribing a specific representation.Source: §B.1 · p. 18

B.2 Observations Model as an Additive Noise Channel

• With sufficiently expressive predictors and cross-entropy loss, the observation error is zero-mean and uncorrelated with model output, motivating the assumed additive-noise observation model.The appendix also states that, under a first-order linearized Gaussian approximation, EKF updates coincide with Gaussian-prior ELBO natural-gradient minimization.Source: §B.2 · p. 18, §B.3 · p. 19

B.4.2 Proof of Lemma B.2

• Under the expressivity assumption, a parameter vector realizing the true posterior probabilities minimizes instantaneous cross-entropy risk.The proof identifies this optimizer with the instantaneous optimum.Source: §B.4.2 · pp. 20–21
• The observation error is zero-mean and orthogonal to the model output.The proof establishes these properties after showing the true-posterior parameter is an instantaneous-risk optimizer.Source: §B.4.2 · pp. 20–21

B.4.3 Proof of Lemma B.3

• With Gaussian process and observation noise and a first-order observation-model linearization, one natural-gradient step minimizing the ELBO is equivalent to a Kalman-style adaptation update.The derivation uses a Gaussian variational posterior for the latent state.Source: §B.4.3 · pp. 21–23

C.1 Non-stationary Wireless Receiver Adaptation

• Wireless tests use a DeepSIC receiver on time-varying MIMO channels, with both linear and tanh-distorted observations.Evaluation uses 150-frame sequences: four synchronization frames with 64 labeled pilots per frame, followed by 146 tracking frames with six labeled pilots each; each block detector has 868 trainable parameters. Methods include Online GD and full, diagonal, and DLR parameter-space EKF baselines.Source: §C.1 · pp. 23–24

C.2 Non-stationary Image Classification

• Image tests model gradual distribution shift as mixtures of six corruption types and evaluate adaptation on MNIST-C, CIFAR-10-C, and CIFAR-100-C.Each run has 1,000 time steps with batches of 100 images; the benchmark includes source, gradient-based, normalization-based, self-supervised, and Bayesian baselines.Source: §C.2 · pp. 24–25

D Additional Experiments

• The supplementary studies examine latent-dimension sensitivity, train-test mismatch, and dependence on offline meta-learning in both application domains.The appendix frames these experiments as analyses of AURA's behavior, robustness, and design trade-offs.Source: D Additional Experiments · p. 25

D.1.1 Temporal Tracking Behavior

• AURA maintains stable BER tracking across the channel evolution for linear and nonlinear QuaDRiGa channels.Figure 5 plots BER against snapshot index.Source: §D.1.1 · p. 25, Figure 5 · p. 26

D.1.2 Effect of Latent Compression and Dynamics Parameterization

• Across tested compression levels and both dynamics models, AURA outperforms corresponding full-parameter Bayesian baselines; larger latent dimensions improve BER, while learned dynamics generally perform better when capacity is sufficient.The qualitative compression trends are similar for OU and learned linear dynamics; the reported comparisons cover aggregate BER across SNR and temporal BER.Source: §D.1.2 · p. 26, Figure 6 · p. 26, Figure 7 · p. 26

D.1.3 Cross-Dataset Generalization

• Under QuaDRiGa-to-COST2100 transfer, AURA’s BER is broadly comparable to leading full-parameter Bayesian baselines while using a compressed latent space, without a uniform BER advantage.The result indicates transfer of adaptation structure beyond the offline simulator.Source: Figure 11 · p. 27, Figure 10 · p. 27, §D.1.3 · p. 27

D.1.4 Effect of Offline Meta-Learning.

• Offline meta-learning reduces BER by orders of magnitude versus cold-start variants across prediction models and latent dimensions under the same online budget.The comparison holds latent dimensionality and prediction model fixed within each warm-start/cold-start pair.Source: §D.1.4 · p. 27, Table 2 · p. 28

D.1.5 Computational Cost versus Adaptation Accuracy.

• At SNR = 12 dB, AURA achieves lower error than full-space BONG with fewer FLOPs; larger latent dimensions trade increased computation for improved adaptation quality.Among AURA configurations, diagonal-F has the lowest error at a moderate cost increase relative to OU dynamics.Source: Figure 12 · p. 28, §D.1.5 · p. 28

D.1.6 Intermittent Supervision

• Across all 15 tested supervision and pilot-budget combinations, AURA has the lowest BER, although performance degrades as supervision becomes less frequent or pilots decrease.The results support moderate intermittent supervision, not stability over arbitrarily long periods without feedback.Source: Table 3 · p. 29, §D.1.6 · p. 29

D.1.7 Full Transition Matrix Ablation.

• Full-F dynamics can track strongly at sufficiently large latent dimensions, but dense covariance prediction costs O(m^3), making the variant runtime-attractive only around m ≤250 in this implementation.The main experiments therefore focus on structured OU and diagonal-F dynamics.Source: §D.1.7 · pp. 29–30, Figure 13 · p. 30

D.2.1 Lifting Map Architecture

• Across CIFAR-10-C and CIFAR-100-C and all tested supervision budgets, nonlinear lifting improves accuracy over affine lifting, with greater expressiveness increasing online computation.The lifting map requires forward evaluation and differentiation for the observation Jacobian during EKF correction.Source: Table 4 · p. 30, §D.2.1 · pp. 30–31

D.2.2 Cross-Dataset Generalization

• With offline training and deployment on disjoint corruption families, AURA remains competitive with the best baseline and outperforms the remaining alternatives.The result supports transfer across corruption families and temporal drift patterns beyond those seen during offline training.Source: Table 5 · p. 31, §D.2.2 · p. 31

Comparison with LoRA

• Both AURA lifting variants outperform LoRA under supervised streaming, with nonlinear lifting providing the largest gain.The comparison indicates that a compact parameterization alone does not account for AURA’s performance; learning the representation from non-stationary trajectories adds benefit.Source: §D.2.3 · pp. 31–32

Comparison with Efficient TTA

• EcoTTA and ELaTTA use no labeled feedback and have lower latency but higher classification error; the comparison is not supervision-matched because AURA uses five labeled samples per update.EcoTTA remains competitive on CIFAR-10-C but does not improve on the CIFAR-100-C source model, while ELaTTA does not improve on the source model in either setting.Source: Table 7 · p. 32, §D.2.3 · p. 32

Robustness to Abrupt Distribution Shifts

• On CIFAR-10-C at severity 5, both lifting variants reduce initial post-shift error by approximately 58% after 40 updates, with nonlinear lifting lower throughout recovery.The evaluation uses five abrupt corruption shifts and 15 labeled adaptation samples per update.Source: §D.2.4 · p. 32
• After abrupt shifts, innovation-based EKF corrections can progressively move the latent state toward the new regime despite mismatch with offline training dynamics.The findings support recovery from abrupt changes, while gradual variation remains the setting most naturally matched to AURA.Source: §D.2.4 · p. 32

E Proof-of-Concept: Adaptive Deep Software-Defined Radio

• The proof-of-concept implements AURA-style online adaptation in a real-time over-the-air IEEE 802.11a communication system using commodity RF platforms.The demonstration tests whether latent Bayesian adaptation translates into practical online adaptation under hardware and latency constraints.Source: E Proof-of-Concept: Adaptive Deep… · pp. 32–33

E.1 Hardware Setup and Adaptive Receiver Architecture

• The prototype uses PlutoSDR+ devices, a lightweight 1.8K-parameter CNN equalizer, and known WiFi preamble symbols for online adaptation before payload processing.The neural receiver runs inline beside a classical LMMSE receiver, with both processing the same packets; adaptation runs in NumPy on CPU without GPU acceleration.Source: Figure 14 · p. 33, §E.1 · p. 33

E.2 Hardware Performance and Runtime Characterization

• Across SNR and CFO conditions, the adaptive neural receiver enlarges the reliable operating regime, particularly at low SNR and high CFO where the classical receiver degrades.The measured bit-error-rate comparison is shown in Figure 15.Source: §E.2 · p. 33, Figure 15 · p. 34
• CPU-only measurements show full adaptation for the lightweight receiver in tens of milliseconds, with larger models remaining deployable at increased but practical latency.The runtime evaluation reports per-packet inference and adaptation latency for multiple CNN architectures.Source: Table 9 · p. 34, §E.2 · p. 34

F Limitations

• Compressing very high-dimensional parameter spaces into one latent state creates memory and optimization bottlenecks, requiring a balance between compactness and reconstruction expressiveness.Source: F Limitations · p. 34
• Performance may degrade when deployment-time latent dynamics differ substantially from those observed during offline training, because inaccurate predictions can reduce adaptation quality and stability.New observations can compensate during the correction step.Source: F Limitations · p. 34
• The current method assumes labeled samples during online adaptation, which may limit applicability in fully unsupervised deployment scenarios.Source: F Limitations · p. 34
• The formulation relies on first-order linearization and Gaussian approximations; significant deviations from these assumptions may reduce Bayesian-update fidelity.Source: F Limitations · p. 35
• AURA is best suited to structured, moderately smooth non-stationarity with online supervision; abrupt shifts, unsupervised deployment, and larger models remain challenges.Robustness to abrupt shifts, unsupervised use, and larger-scale models remains future work.Source: F Limitations · p. 35