Self-Guided Training for Autoregressive Models

• Self-Guided Training for Autoregressive Models (ST-AR) is a suite of techniques that incorporate self-supervision to improve prediction and generalization under noise and adversarial conditions.
• It leverages extended prediction horizons and algebraic parameterization to emulate ARMA effects, ensuring stability and performance across different data types.
• ST-AR has been successfully applied in time series forecasting, image generation, reinforcement learning, and density modeling, demonstrating significant practical improvements.

Self-Guided Training for Autoregressive Models (ST-AR) encompasses a suite of techniques that augment classical autoregressive (AR) and ARMA models with principled self-supervision and optimization strategies. The central concept is that the training process itself is guided by intrinsic structures, properties, or objectives—enabling robust learning and generalization even in the presence of unobserved noise, adversarial conditions, or insufficient labeled data. Recent research articulates varied instantiations of ST-AR across time series, image generation, reinforcement learning, and density modeling. The following sections synthesize the main methodologies, mathematical formulations, and applications underpinning ST-AR as surveyed in contemporary literature.

1. Foundations: Autoregressive Models and Self-Guided Training Principles

Autoregressive models define joint or conditional distributions by chaining conditionals over ordered variables. The canonical ARMA formulation for time series is:

$$X_t = \sum_{i=1}^k \alpha_i X_{t-i} + \sum_{j=1}^q \beta_j \epsilon_{t-j} + \epsilon_t$$

Self-guided training refers to parameter estimation or latent representation learning using intrinsic signals (statistical averages, conditional structure, contrastive learning) in lieu of full supervision or explicit noise modeling. This often means deploying improper learning choices—e.g., predicting with augmented AR models that embed unknown MA effects, using self-supervised objectives that align representations, or regularizing forecasting with scheduled autoregression.

Critical ST-AR elements include:

• Extended prediction horizons (embedding ARMA within AR) (Anava et al., 2013)
• Algebraic parameter selection via long-term statistics (Harlim et al., 2014)
• Self-supervised objectives for representation alignment (Yue et al., 18 Sep 2025)
• Iterative denoising under heavy-tailed noise (Banerjee et al., 18 Aug 2025)
• Contrastive and masked modeling for semantic consistency (Yue et al., 18 Sep 2025)

These designs yield predictors that adaptively compensate for unknown dynamics, correlate representations across token, view, or step, and dynamically expand their context windows—haLLMarks of self-guided learning.

2. Online Learning and Improper ARMA Embedding

One major contribution to ST-AR is the use of improper AR models to mimic ARMA prediction without requiring explicit estimation of noise terms (Anava et al., 2013). The strategy augments the AR horizon by $m$ lags, approximating MA effects:

$$\tilde{X}_t(\gamma) = \sum_{i=1}^{m+k} \gamma_i X_{t-i}$$

Key online algorithms include:

• ARMA-ONS (Online Newton Step): Second-order updates for exp-concave losses, with sublinear regret $O((GD + 1/\lambda)\log T)$.
• ARMA-OGD (Online Gradient Descent): First-order updates for generic convex losses, regret $O(GD\sqrt{T})$.

The improper learning setup obviates direct estimation of noise coefficients ($\beta$), instead embedding their effect into autoregressive weights. Selecting $m$ to satisfy error decay allows these methods to closely match the performance of best ARMA models in hindsight, even under weak noise assumptions.

3. Algebraic Self-Guidance from Long-Time Statistics

For nonlinear and chaotic signals, stability and consistency can be ensured via algebraic construction of AR filters solely from descriptive long-term statistics (Harlim et al., 2014). The approach imposes stability (roots of characteristic polynomial within unit circle) and multistep consistency (matching Adams-Bashforth constraints), yielding coefficient parametric forms such as:

$$\begin{cases} a_1 = (s - 3/2)\lambda \delta t \ a_2 = -(2s - 5/2)\lambda \delta t \ a_3 = s \lambda \delta t \end{cases}$$

where $s \in \mathbb{C}$ is chosen to optimize admissible discretization steps $\delta t$ for which the filter remains stable and consistent. This “self-guided” algebraic parameterization enables robust construction of SCAR models without regression over training data, and is especially beneficial for turbulent systems with available equilibrium statistics but sparse observation.

4. Self-Supervised Denoising under Additive and Impulsive Noise

Autoregressive signals contaminated by impulsive or infinite-variance noise are notoriously difficult to estimate. The Stable-Noise2Noise (Stable-N2N) paradigm employs a feedforward network trained via self-supervised pairs drawn from the noisy series (Banerjee et al., 18 Aug 2025). For AR models,

$Y_t = X_t + Z_t$

training minimizes:

$$\min_\phi \mathbb{E}\left\| \Phi(Y_{s+1}^{\langle B' \rangle},..., Y_{s+q}^{\langle B' \rangle}; \phi) - (Y_{s+q+N+1}, ..., Y_{s+2q+N}) \right\|_2^2$$

with signed-power input transformation for infinite-variance regimes ($B' < \min(\alpha_x, \alpha_z)/2$). The loss is theoretically justified under the independence and zero-mean assumption of $Z_t$. Parameter estimation and forecasting benefits are validated both in Gaussian and $\alpha$-stable noise scenarios, outperforming methods that require explicit noise knowledge.

5. Self-Supervised and Contrastive Training for Image Generation

Autoregressive image generation is impaired by local dependence, semantic inconsistency, and spatial invariance deficiency (Yue et al., 18 Sep 2025). The ST-AR framework addresses these via multiple targeted losses:

• Masked Image Modeling (MIM): Masking attention maps in transformer layers forces global context integration:

$$\text{Attn}(Q, K, V) = \text{Softmax}\left(\frac{QK^T}{\sqrt{d_k}} + M\right) V$$

• Inter-Step Contrastive Loss: Projects and aligns token features across generation steps within the same view.
• Inter-View Contrastive Loss: Aligns corresponding token representations across different augmented views for spatial invariance.

The overall loss combines next-token prediction with weighted MIM and contrastive components:

$$L_{ST\text{-}AR} = L_{AR} + \alpha L_{MIM} + \beta \frac{1}{2}(L_{step} + L_{view})$$

No changes are made to the autoregressive sampling at inference. Empirical results indicate significant improvements in both representation quality (e.g., linear probing accuracy from 21% to 55%) and generation realism (FID reduced by ~42% for LlamaGen-L and ~49% for LlamaGen-XL), all without dependence on pretrained representations.

6. Extensions to Reinforcement Learning and Density Modeling

Autoregressive stochastic processes can be integrated into reinforcement learning policies to enhance exploration (Korenkevych et al., 2019). Policies sample noise from AR processes with tunable temporal coherence parameter $\alpha$, forming action distributions with smoother trajectories—demonstrated to improve sample efficiency and safe operation, notably in continuous robot control settings.

In density modeling, Autoregressive Conditional Score Models (AR-CSM) replace normalized conditionals $q(x_d|x_{<d})$ with unnormalized score functions $$s_{\theta,d}(x_{<d}, x_d) = \frac{\partial}{\partial x_d}\log q_\theta(x_d|x_{<d})$$ (Meng et al., 2020). Composite Score Matching (CSM) decomposes training into efficient one-dimensional matching problems, enabling scalable and stable density estimation and image generation.

7. Broader Implications and Current Trends

Self-guided training frameworks provide robust alternatives to classical supervised and regression-based learning in autoregressive models. Importantly, they:

• Bypass stringent noise assumptions via extended AR embedding and self-supervised denoising.
• Achieve stability and consistency by algebraic means using long-term statistics, not raw sample paths.
• Harness contrastive and masked modeling to enhance representation learning in generative vision models.
• Generalize to domains such as reinforcement learning and implicit density modeling through flexible, modular training objectives.

Performance metrics and ablation studies consistently demonstrate marked improvements in predictive skill, semantic representation quality, and model robustness—especially under challenging, noisy, or out-of-distribution scenarios.

Self-guided training in autoregressive models is thus a convergent strategy, producing predictive systems that adapt context, mitigate noise, and prioritize semantic coherence, with empirical validation spanning time series, vision, and control applications.