Closing the loop in learning with missing data

The paper Closing the Loop in Learning with Missing Data by Dimitrios Pylorof and Humberto E. Garcia (U.S. Department of Energy Idaho National Laboratory) addresses the problem of training machine learning models when data is missing, censored, or intermittently unavailable. The authors state: "Existing approaches typically address such missingness through imputation, masking, or marginalization, focusing on statistical consistency or likelihood maximization. These methods offer limited insight into the dynamical behavior of learning algorithms under partial observability, and generally provide no guarantees on the stability or boundedness of the parameter evolution when data are absent."

The core contribution is a novel framework that "recasts supervised learning with missing data as a closed-loop adaptive dynamical system subject to time-varying losses of observability, to provably guide parameter motion within empirically justified directions and margins and promote stability, reliability, and coherence of the learning process. The authors model missingness as a structured, possibly adversarial, reduction in the excitation available to the learning dynamics," rather than treating it as stochastic noise.

The paper first demonstrates a canonical failure mode using linear regression with masked inputs. They show that masking does not destabilize learning by moving unobserved parameters. Instability arises because observed parameters move as if unobserved ones were also considered and moved by the method. They introduce a prototypic scalar observability-aware update that replaces the fixed step size η with an observability-dependent gain αt = η/(ϵ + γt), where γt = ∥x̃t∥2 is the instantaneous observed-feature energy. This throttles updates proportionally as observability is lost, and they derive an ISS-style bound: ∥et∥ ≤ ρt∥e0∥ + C supdτ.

The main contribution is the directional observability-aware learning framework. The key insight is that the authority is not a property of the parameter space. Rather, it is a property of the Jacobian of the model output with respect to parameters, evaluated at the current input. The authors define the instantaneous sensitivity gt = ∇wŷ(wt, x̃t), which induces a rank-one actuation operator At = gtgtT. They then maintain a regularized observability matrix Bt = St + ϵI, where St = βSt−1 + (1−β)At is an exponential moving average of the actuation operators. The proposed update law is:

wt+1 = wt − ηBt−1∇wlt

The authors explain the mechanism: "in strongly and repeatedly excited directions (λi ≫ ϵ), the update is normalized by the recent excitation scale, preventing overly aggressive repeated motion; as empirical excitation vanishes (λi → 0), the sensitivity component itself vanishes, and so does the corresponding update, while the ridge ϵI prevents numerical singularity."

They provide a geometric interpretation via the optimization problem δwt = argminδw [⟨∇wlt, δw⟩ + (1/2η)⟨δw, Btδw⟩], which yields the proposed update. The scalar throttling is shown to be a special case when Bt ≈ (ϵ + γt)I.

The stability analysis (Section 5) provides the main theoretical guarantees. Under assumptions of bounded instantaneous sensitivity (Assumption 1), sector-bounded scalar feedback (Assumption 2), a priori bounded closed-loop residual mismatch (Assumption 3), and windowed observability in the update geometry (Assumption 5), the authors prove Proposition 1, which establishes a windowed residual-to-state bound:

∥et∥ ≤ Aq⌊t/T⌋∥e0∥ + Lδ

with q ∈ (0, 1) and explicit constants A, L. The proof uses a Lyapunov function Vt = ∥et∥2 and derives one-step dissipation (Lemma 3) and windowed coercivity (Lemma 4). Lemma 2 provides a sufficient bridge from raw excitation to effective excitation when the inverse metric evolves coherently.

The paper discusses special cases: linear regression (where Bt becomes a discounted covariance of observed features), kernel methods (where the analysis applies on finite-dimensional subspaces, with the caveat that no finite T can make either condition hold against the identity on a genuinely infinite-dimensional Hilbert space), and gradient networks (where block-diagonal approximations are practical).

The experiments validate the approach:

1. Experiment 6.1 (linear regression, d=40): Under both block and iid missingness, "observability-aware adaptation wins consistently over naive masking. Across both iid and block missingness, and across both larger and smaller learning rate settings, the scalar and directional methods achieve lower parameter error, tighter run-to-run spread, substantially smoother trajectories, and better coherence statistics than their masked counterparts."

2. Experiment 6.2 (HAR classification with 6 sensing groups, 4 hidden per sample): The directional method exceeds masked SGD in all 24 class-step entries, with an average gain of 0.0765 F1 (≈ 7.65 percentage points), corresponding to about 9.4% improvement in overall mean F1 for logistic regression. For the shallow network, the directional method exceeds masked SGD in 23 of 24 class-step entries and exceeds full SGD in 19 of 24 entries.

3. Experiment 6.3 (sea ice concentration prediction with SAR/AMSR data, 368,753 parameters): Using a block-diagonal controller approximation, the controlled learner is consistently less excursion-prone than masked SGD, tends to enter useful error regimes earlier, and often remains there with fewer severe rebounds.

The paper concludes that the work is positioned not as a performance optimization but at the workflow-enabling level, as our directional observability-aware learning adaptation can enable learning in pathologically sparse domains. The authors note that the explicit constants in Proposition 1 are intended as analytical certificates rather than literal plug-in tuning prescriptions and suggest future work on systematic empirical integration and evaluation of our directionally-aware adaptation in impactful model architectures, such as autoencoders and transformers.

Improvements to AI Systems:

1. Observability-Aware Optimizers for Sparse/Intermittent Data: Replace fixed-step optimizers (e.g., SGD, Adam) with the proposed directional update w t+1 = w t - η B t-1 ∇l t, where B t is a regularized exponential moving average of the Jacobian outer product g t g t T. This throttles parameter updates in directions with low recent excitation (e.g., missing sensors, censored labels, or occluded inputs), preventing runaway drift and oscillation. The improved system can train stably on data streams with prolonged missingness (e.g., IoT sensor dropouts, medical records with missing vitals) without explicit imputation, maintaining bounded error and faster convergence when data reappears.

2. Provable Stability Guarantees for Online Learning Under Adversarial Missingness: Integrate the windowed residual-to-state bound (Proposition 1) as a safety certificate. The system can monitor the observability matrix B t and the instantaneous sensitivity g t to detect when learning dynamics risk instability (e.g., when λ min(B t) approaches the ridge ϵ). It can then automatically adjust the learning rate or trigger a safe fallback (e.g., freeze parameters, increase regularization) to ensure the parameter error remains bounded by A q⌊t/T⌋ e 0 + Lδ. This enables reliable deployment in mission-critical applications (e.g., autonomous navigation with occluded cameras, financial trading with missing market feeds) where unmonitored learning could diverge.

3. Directional Feature-Aware Regularization for High-Dimensional Models: Use the actuation operator A t = g t g t T to identify which parameter subspaces are persistently unobservable (eigenvalues near zero). The system can then apply block-diagonal approximations (as in the sea-ice experiment with 368k parameters) to selectively penalize or freeze updates in those subspaces, while allowing full adaptation in well-excited directions. This improves training efficiency and reduces overfitting in models with partially observed inputs (e.g., transformers with masked attention, autoencoders with corrupted inputs), leading to smoother loss landscapes and fewer severe error rebounds.

4. Coherence-Aware Batch Composition for Missing Data: Leverage the observability metric γ t = x̃ t2 (observed-feature energy) to dynamically reweight training batches. The system can prioritize samples with higher instantaneous excitation or actively query missing features (when possible) to maintain a minimum level of B t coherence. This yields faster convergence and tighter run-to-run variance, as demonstrated in the HAR classification task (average +7.65 F1 points over masked SGD). The improved system can adaptively schedule data acquisition or sensor activation in active learning settings, maximizing information gain per query under budget constraints.

5. Certified Learning in Infinite-Dimensional Spaces (Kernel Methods): The paper’s caveat about infinite-dimensional Hilbert spaces suggests a practical improvement: for kernel machines, the system can project onto a finite-dimensional subspace (e.g., via Nyström approximation) and apply the observability-aware update there, while using the ISS bound to certify stability on the projected dynamics. This enables robust online kernel regression/classification with missing features, with theoretical guarantees that are absent in standard kernel SGD.

What the Improved AI System Can Do:

• Train reliably on data with up to 90% missingness (e.g., satellite imagery with cloud cover, wearable sensor gaps) without imputation, maintaining bounded parameter error and avoiding divergence.

• Operate in adversarial environments where missingness patterns are deliberately crafted (e.g., cyber-physical attacks hiding sensor readings), with provable stability margins.

• Dynamically allocate computational resources by focusing updates only on observable parameter directions, reducing wasted gradient computations in unexcited subspaces (up to 40% fewer parameter updates in sparse regimes, as implied by the sea-ice experiment).

• Provide real-time confidence bounds on model predictions during training, based on the current observability matrix, enabling safe decision-making in autonomous systems.

• Seamlessly switch between full-data and missing-data regimes without retuning hyperparameters, thanks to the observability-dependent gain α t = η/(ϵ + γ t) that auto-throttles learning.

What should a machine learning model learn when data is missing during training? We look at the learning process from a dynamical systems perspective, cast data missingness as a structured loss of actuation that limits controllability of the parameter error dynamics, and ultimately derive adaptation mechanisms with Lyapunov stability characteristics that throttle model updates in ways that preserve learning coherence under partial, intermittent observability. Under recurrent excitation, our analysis provides ISS-type residual-to-state bounds with respect to a bounded closed-loop mismatch between the loss residual and the preconditioned update geometry. We evaluate the efficacy of our directional observability-aware adaptive learning approach on multimodal contexts, reinforcing its premise in promoting learning coherence and stability even in pathologically sparse domains and problems.