How Good Can Linear Models Be for Time-Series Forecasting?
Authors: Lang Huang, Jinglue Xu, Luke Darlow
Organizations: 1Sakana AI, Tokyo, Japan · 2National Institute of Informatics, Japan
Abstract
Time-series forecasting research has been moving steadily toward larger architectures, from specialized transformers to general-purpose foundation models, on the assumption that capacity is what unlocks accuracy. We take the opposite position: most of the gap can be closed at far lower cost by tuning preprocessing rather than scaling models. We use Ridge regression as the testbed, since it has a closed-form solution and interpretable weights, which let the optimal hyperparameters be read off the search directly. We search over context length, local normalization, regularization, and augmentation on eight standard benchmarks and find three patterns. (1) Optimal lookback is strongly series-specific and often non-monotonic in forecast horizon, with fitted power-law exponents ranging from +0.46 on ETTm2 to −0.19 on Exchange and Traffic, challenging the convention that longer horizons need longer history. (2) Normalizing over a learned trailing fraction of the context, rather than its entirety, is almost universally preferred. (3) Series within the same dataset often disagree on hyperparameters; the optimal degree of cross-series sharing varies from fully shared to fully per-series. The resulting models beat prior linear forecasters on most dataset-horizon entries and exceed Transformer, MLP, and CNN baselines on six of eight benchmarks. The optimized hyperparameters also serve as a diagnostic on the data itself, revealing structures that larger models absorb silently into their learned parameters. We provide an accompanying interactive online demonstration and the code at https://sakanaai.github.io/SearchCast/.
Transformer architectures have been widely adopted for time series forecasting, yet whether the representational mechanisms that make them powerful in NLP actually engage on time series data remains unexplored. The persistent competitiveness of simple linear models such as DLinear has fueled ongoing debate, but no mechanistic explanation for this phenomenon has been offered. We address this gap by applying sparse autoencoders (SAEs), a tool from mechanistic interpretability, to probe the internal representations of PatchTST. We first establish that a single-layer, narrow-dimensional transformer matches the forecasting performance of deeper configurations across commonly used benchmarks. We then train SAEs on the post-GELU intermediate FFN activations with dictionary sizes ranging from 0.5x to 4.0x the native dimensionality. Expanding the dictionary yields negligible downstream performance change (average 0.214%), with large portions of overcomplete dictionaries remaining inactive. Targeted causal interventions on dominant latent features produce minimal forecast perturbation. Across all evaluated settings, we observe no empirical evidence that the analyzed FFN representations rely on strong superposition. Instead, the representations remain sparse, stable under aggressive dictionary expansion, and largely insensitive to latent interventions. These results demonstrate that superposition is not necessary for competitive performance on standard forecasting benchmarks, suggesting they may not demand the rich compositional representations that drive transformer success in language modeling, and helping explain the persistent competitiveness of simple linear models
Time Series Foundation Models (TSFMs) have borrowed the long context paradigm from natural language processing under the premise that feeding more history into the model improves forecast quality. But in stochastic domains, distant history is often just high-frequency noise, not signal. Hence, the proposed work tests whether this premise actually holds by running continuous context architectures (PatchTST included) through the ETTh1 benchmark. The obtained results contradict the premise: an inverse scaling law shows up clearly, with forecasting error rising as context gets longer. A 3,000-step window causes performance to drop by over 68%, evidence that attention mechanisms are poor at ignoring irrelevant historical volatility. Retrieval-Augmented Forecasting (RAFT) is evaluated as an alternative. RAFT achieves a mean squared error (MSE) of 0.379 with a fixed 720-step window and selective retrieval, outperforming both long-context configurations and zero-shot foundation models (Chronos, Moirai) despite requiring far less computation. In addition, the retrieval step injects only the most relevant historical segments as dynamic exogenous variables, which gives the model a context-informed inductive bias it cannot build on its own from raw sequences. Therefore, foundation models going forward need to shift architecturally toward selective retrieval.
A growing family of indices scores how predictable a series is from its spectrum. Practitioners increasingly read these scores as answering a different question: whether \emph{adding context}, a longer lookback, a retrieval plug-in, or a pretrained model, will help. These are not the same question. The value of context is a property of the operating point, not of the series. Any index built from the power spectrum is invariant under phase randomization, whereas the beyond-second-order value that retrieval and foundation models supply is not, because a phase-randomized series is asymptotically Gaussian. We state this as an impossibility result and isolate it with surrogate pairs that fix the spectrum and the marginal by construction. We then give a label-free, configuration-level diagnostic, the coverage deficit, whose principal term measures beyond-spectrum structure as the gain of analog over linear prediction. On seven benchmarks the prediction holds: window-keyed retrieval's value collapses across surrogate pairs (ECL median +33%→−35%, p<10−40) while every spectral index stays frozen; a foundation model's value splits into a surviving second-order part and a small beyond-linear margin that collapses; a longer linear window's value survives. Leave-one-dataset-out, the structure term predicts the sign of beyond-spectrum value where the spectral indices trail it, and the reverse holds for the second-order mechanism. We introduce no new forecaster; the contribution is the distinction, a controlled comparison, and a diagnostic for the deployment decision. Code: https://github.com/KurbanIntelligenceLab/SINE
Mert Onur Cakiroglu, Mehmet Dalkilic, Hasan Kurban