Zahraa S. Abdallah

Riku Green, Zahraa S. Abdallah, Telmo M Silva Filho

Multi-step time series forecasting (MSF) is commonly evaluated using point-wise error metrics such as mean squared error (MSE), implicitly treating the conditional mean as a sufficient target. We show that this can be misleading under conditional uncertainty, where the conditional expectation becomes unrepresentative of typical realized values at longer horizons. We formalize this effect through a conditional uncertainty gap and prove that whenever this gap is nonzero, no deterministic predictor can simultaneously minimize MSE and match the marginal distribution of realized futures. This establishes a fundamental, model-agnostic trade-off between point accuracy and marginal realism in MSF evaluation. Using controlled stochastic dynamical systems and nine real-world forecasting benchmarks, we empirically characterize the resulting accuracy--realism frontier and \textbf{quantify the practical cost of MSE-only model selection}. As conditional uncertainty increases with forecast horizon, the attainable set expands into a pronounced Pareto front, separating MSE-optimal but under-dispersed predictors from methods that trade accuracy for realistic marginal variability. \textbf{Across benchmarks, we find that small relaxations in MSE ($\boldsymbol{\le 5\%}$) frequently unlock disproportionate gains in marginal realism, with median improvements of $\mathbf{17.3\%}$ and gains exceeding $\mathbf{30\%}$ in some datasets.} We further show that common forecasting strategies systematically occupy different regions of this frontier: direct multi-output predictors concentrate near the accuracy-optimal extreme, while recursive strategies and sample-based inference favors marginal realism. Together, these results expose a structural failure mode of MSE-based evaluation in long-horizon forecasting and recast strategy and inference selection as navigation of an unavoidable accuracy--realism trade-off.

Riku Green, Zahraa S. Abdallah, Telmo M Silva Filho

In financial forecasting, predictive performance depends not only on which model is trained, but also on how the trained model is deployed. We study this issue in multi-horizon volatility forecasting. Our starting point is that a trained multi-output (MIMO) forecaster does not define a single deployable predictor: by changing the inference-time rollout rule, the same trained model induces a family of forecasts with different accuracy and cost profiles. Across 20 stock-volatility series, three forecast horizons, and architectures ranging from linear models to PatchTST, we find that non-default rollout rules often improve over standard MIMO deployment. However, the best fixed rule varies substantially across architectures and horizons, making any single static replacement unreliable. We therefore evaluate validation-based deployment policies over the induced rule family. Under the primary MSE objective, validation-selected singletons provide a low-cost improvement over default MIMO, while small rule subsets recover much of the benefit of larger ensembles at substantially lower inference cost. We also find that policy rankings are metric-sensitive: MSE-selected policies do not transfer uniformly to QLIKE, a finance-standard volatility loss. These results show that inference-time deployment is a meaningful source of adaptiveness in financial forecasting, and that trained volatility forecasters should be evaluated not only by their architecture, but also by their deployment policy.

Kairui Zhang, Ziwen Yu, Zahraa S. Abdallah et al.

Sparse autoencoders (SAEs) provide useful decompositions of Transformer residual streams, but their learned features are usually named post hoc rather than directly connected to the Transformer's token vocabulary. We introduce Vocabulary-Aligned Sparse Autoencoder (VASAE), a method that trains SAE features under vocabulary-aligned anchoring and assigns each feature an intrinsic token name: the token string whose embedding is nearest to that feature. Without reducing reconstruction quality compared with a standard SAE, VASAE produces dictionaries with vocabulary-aligned features. Using a 0.8 cutoff on the nearest-token alignment score, dictionaries trained on GPT-2-small post-residual streams align about 90% of features in layers 0--10. In Llama-3.1-8B, representative shallow and middle-layer dictionaries contain strongly aligned features, including 92.8% in the shallow layer, while the representative final-layer dictionary shows limited alignment. After subtracting the sentence-level mean sparse code, case studies show that many remaining intrinsic token names are relevant to nearby input tokens. These results suggest that vocabulary-aligned anchoring can connect learned features to intrinsic token names during training, complementing post hoc interpretation of learned dictionaries.

Riku Green, Zahraa S. Abdallah, Telmo M Silva Filho

Recursive multi-step forecasting is usually framed as distribution shift: models are trained on observed histories but deployed on their own predictions. We show this framing is incomplete by proving that, under partial observability or state truncation, recursive rollout is also an epistemic underidentification problem. Even with deterministic latent dynamics, one-step Bayes supervision identifies behavior only on observed contexts and need not identify the deployed recursive predictor once rollout queries self-generated induced states whose correct local targets are not determined by numeric state alone. We formalize this with induced states $Z$ and provenance variables $P$, and derive a decomposition of induced-state error into teacher-forcing/rollout mismatch, representation--class approximation, and provenance information gaps. Empirically, we show that rollout enters a distinct induced-state regime, that fixed induced states define a distinct local corrective task, and that closed-loop gains arise not only from local adaptation but also from changing the induced states visited during rollout. Using a simple binary provenance encoding, provenance-aware correction can further improve performance, though gains are conditional rather than uniform. These results recast exposure bias as reasoning under self-induced epistemic uncertainty.