Goal-Baseline Regularization

• Goal-baseline regularization is a technique that anchors models to a reference solution by adding an explicit penalty for deviation, balancing exploration and stability.
• It applies in diverse settings—contextual bandits, reinforcement learning, continual learning, and Bayesian modeling—by integrating baseline constraints into the loss function.
• Adaptive regularization strategies enhance sample efficiency and reduce overfitting, ensuring robust performance and improved task-specific outcomes.

Goal-baseline regularization is a class of regularization techniques designed to guide an estimator, policy, or predictive function toward a reference solution or baseline, balancing exploration, plasticity, and stability constraints across a variety of learning frameworks. Central instances occur in contextual bandits, goal-conditioned reinforcement learning, continual and lifelong learning, domain generalization, and Bayesian modeling—each deploying regularization to preserve, imitate, or anchor learned solutions near an empirically or theoretically effective baseline while enabling further optimization toward a task-specific goal.

1. Conceptual Foundations and Definitions

Goal-baseline regularization operates by incorporating an explicit term or constraint in the learning objective that penalizes deviation from a baseline, reference, or prior solution. The baseline may be:

• a previously validated policy or parameter vector (contextual bandits, RL),
• empirical statistics from source domains (domain generalization),
• initialization or pre-trained weights (continual/lifelong learning),
• prior parameterizations of a function (Bayesian semi-parametric models),
• self-generated historical solutions (hindsight regularizations in RL).

Mathematically, this is realized as an additive or multiplicative regularization term to the objective:

$\mathcal{L}(w) + \lambda\, \rho(w, \tilde{w})$

where $\mathcal{L}(w)$ is the base loss, $\tilde{w}$ is the baseline or prior, and $\rho(\cdot,\cdot)$ is a convex or strongly convex penalty (e.g., KL divergence, Euclidean distance, or entropy).

A key property is that the regularizer is designed not just for overfitting control but for principled anchoring to a performance baseline, with the goal and baseline explicitly encoded in the regularization design.

2. Instantiations in Contextual Bandits and RL

In contextual bandit settings (Fontaine et al., 2018), goal-baseline regularization anchors learned policies $p(x)$ toward a baseline $q$ by augmenting the loss function:

$$L(p) = \int_{x} \big[\mu(x)\cdot p(x) + \lambda(x)\,\rho(p(x))\big]\,dx$$

Here, $\rho(p(x))$ can be KL divergence, $\ell^2$-distance, or negative entropy with respect to baseline $q$. The regularization weight $\lambda(x)$ can be spatially or contextually varying. Under nonparametric binning, the context space is split, and regularized multi-armed bandit instances are solved locally, yielding explicit convergence rates that interpolate between slow, fast, and intermediate regimes under conditions on smoothness, strong convexity, and a new margin parameter $\alpha$ controlling proximity to the simplex boundary.

In goal-conditioned RL (Laezza et al., 15 Mar 2024, Lei et al., 8 Aug 2025, Hiraoka, 2023), regularization strategies:

• Use behavior cloning regularization (BC) to keep policies close to observed data, mitigating extrapolation errors.
• Employ hindsight self-imitation regularization (HSR) and hindsight goal-conditioned regularization (HGR), which generate regularization priors based on intermediate or achieved goals along trajectories, improving sample efficiency and action coverage.
• Utilize ensemble averaging, layer normalization, and bounded target computations to address overestimation and variance in Q-values, especially under high replay ratios and sparse rewards.

Offline RL frameworks, such as TD3+BC (Laezza et al., 15 Mar 2024), use regularization terms directly constraining policy-updates to remain near offline data, with the regularization strength $\alpha$ carefully tuned. HGR constructs action support priors using all intermediate goals, regularizing the policy via KL divergence against this hindsight mixture.

3. Continual, Lifelong, and Online Learning Dynamics

In continual linear regression (Levinstein et al., 6 Jun 2025), explicit isotropic $\ell_2$ regularization (or implicit regularization via a finite gradient step budget) provably closes the gap with sequential learning lower bounds. The parameter schedule controlling regularization strength ($\lambda_t$ per task) is fundamental: a gradually increasing regularization coefficient over tasks yields an $O(1/k)$ convergence rate for worst-case expected loss after $k$ learning iterations, matching the information-theoretic lower bound. The schedule is:

$$\lambda_t = \frac{13 R^2}{3}\cdot \frac{k+1}{k-t+2}$$

where $R$ is the data matrix radius; early tasks allow more plasticity, later tasks more stability.

Discounted adaptive online learning (Zhang et al., 5 Feb 2024) exploits FTRL-based regularization that “remembers” useful offline priors and adapts regularization instance-wise:

$$\operatorname{reg}^{\lambda}_T(l_{1:T}, u) \leq \widetilde{\mathcal{O}}(\|u\|\sqrt{V_T})$$

where $V_T$ is the discounted gradient variance. This confers better adaptivity and stability than non-adaptive constant-learning-rate gradient descent.

4. Goal-Baseline Regularization in Domain Generalization and Bayesian Estimation

The ERM++ framework (Teterwak et al., 2023) in domain generalization demonstrates goal-baseline regularization via improved training utilization, initializations tied to strong pre-trained characteristics, and weight-space regularizers such as model parameter averaging and warm starts. These strategies ensure maintenance of generalizable features (the baseline) while minimizing overfitting and catastrophic forgetting during fine-tuning.

In Bayesian semi-parametric Cox models (Lázaro et al., 31 Jan 2024), flexible baseline hazard specification is regularized toward stable estimations via prior distributions with correlated structures (martingale or random walk). Regularization ensures that highly flexible baseline functions, such as mixtures of piecewise constant or B-spline hazards, do not overfit. Correlated priors (PC3, PC4, PS3) enforce smooth transitions and penalize spurious oscillations, achieving robust prediction and inference while capturing non-monotonic risk profiles.

5. Mathematical Formalization and Theoretical Guarantees

The mathematical structure across instantiations exhibits regularization terms that link current estimates to baseline references, often with convexity or strong convexity ensuring uniqueness and stability:

• Contextual bandit: $$L(p) = \int [\mu(x)\cdot p(x) + \lambda(x)\rho(p(x), q)] dx$$
• RL policy regularization: $$\pi = \arg\max_{\pi} \mathbb{E}[\lambda Q(s, \pi(s,g), g) - (\pi(s,g)-a)^2]$$
• Continual learning update: $$w_t = \arg\min_w \{\frac{1}{2}\|X_{\tau_t}w-y_{\tau_t}\|^2 + (\lambda_t/2)\|w-w_{t-1}\|^2\}$$

Theoretical guarantees, as in (Fontaine et al., 2018, Levinstein et al., 6 Jun 2025), derive explicit convergence rates:

• Contextual bandits: $$R(T) = O((T/\log^2 T)^{-(\beta/(2\beta+d))(1+\alpha)})$$
• Continual learning: $\mathbb{E}\,\mathcal{L}(w_k) \leq O(1/k)$
• Goal-conditioned RL: sample efficiency increases up to $8\times$ by ensemble regularization and bounded Q-values.
• FTRL adaptive online learning: regret bound adapts to comparator norm and actual gradient variance.

6. Practical Implications and Empirical Results

Empirical studies demonstrate that goal-baseline regularization yields improved generalization, sample efficiency, reduced variance, and more stable learning dynamics:

• In RL and navigation (Gireesh et al., 2022, Lei et al., 8 Aug 2025), regularization via data augmentation and value-consistency under transformation produces substantially increased success rates and SPL compared to non-regularized baselines.
• In continual learning (Levinstein et al., 6 Jun 2025), increased regularization schedules preserve performance across sequential tasks, mitigating catastrophic forgetting.
• In Bayesian Cox models (Lázaro et al., 31 Jan 2024), flexible hazard modeling with correlated priors gives stable survival estimates and robustness to overfitting.
• In LLM RL (Hao et al., 29 May 2025), optimal reward baseline regularization minimizes gradient variance, maintains high output entropy, and achieves stable policy alignment.

7. Design Considerations and Future Directions

Selection of regularization type (explicit vs. implicit), its schedule, and its interaction with baseline solutions is highly application-dependent and can dramatically affect the stability-plasticity trade-off and task optimality. The use of adaptive, context-dependent, or data-driven regularization weights is especially important in nonstationary and continual learning environments.

Ongoing directions include integrating regret-adaptive regularization, expanding hindsight regularization strategies for compositional goal achievement in RL, developing finer-grained prior structures for Bayesian inference, and generalizing instance-dependent regularization frameworks, especially for scale-free and high-dimensional domains.

Goal-baseline regularization unifies principled regularization for safe, stable learning across sequential, goal-directed, and nonstationary tasks, with performance guarantees and flexible mathematical structures to match evolving theoretical and practical demands.