Optimizers as Inductive Bias

• Optimizers as inductive bias are defined as the implicit preferences introduced by training dynamics that steer models toward simpler, more generalizable solutions.
• They actively influence convergence behavior and determine solution properties, affecting fairness, robustness, and the qualitative expressivity of neural networks.
• Adaptive and meta-learned optimizers tailor inductive biases to specific tasks, enabling enhanced performance and better alignment with real-world applications.

Optimizers as inductive bias constitute a central theme in modern machine learning theory and practice, particularly in deep learning. While model architecture classically embodies explicit inductive biases, the choice and behavior of optimization algorithms—through the dynamics of training—impart significant, sometimes dominant, implicit biases. These biases can affect not only the efficiency of convergence but also the generalization properties, solution types, fairness, and even qualitative expressivity of trained models.

1. Theoretical Foundations: Optimizers as a Vehicle for Inductive Bias

Inductive bias refers to the set of assumptions or constraints that guide a learning algorithm toward selecting particular solutions from the hypothesis space, especially when many models fit the training data equally well. In Baxter’s framework, this is formalized as the learner’s selection of a hypothesis space from a family $\mathcal{H}$, such that the chosen space balances richness (containing good solutions) and parsimony (ensuring reliable generalization from limited data) (1106.0245). This process, fundamentally achieved through optimization, is not restricted to parameter tuning for a single task. Rather, it generalizes over families of tasks, as in meta-learning, where the optimizer refines the bias by considering empirical performance across an environment of related problems. Explicit sample complexity bounds and covering number arguments formalize how optimization "learns the bias." In neural networks, this often manifests as the optimizer adjusting internal representations (features) for maximal transfer across tasks, as captured by formulas like:

$$\inf erz(H_w) = \frac{1}{n} \sum_{i=1}^n \inf_{a_0, \ldots, a_k} \frac{1}{m} \sum_{j=1}^m \left( \sigma\left(\sum_{l=1}^k a_l \cdot \Phi_{w,i}(I_{ij}) + a_0\right) - y_{ij} \right)^2$$

hence directly linking feature optimization to bias generalization.

2. Implicit Regularization: Optimizer Dynamics Over Architecture Size

Deep learning empirical evidence demonstrates that the traditional view—where architecture capacity limits solely determine generalization—is incomplete. In highly over-parameterized regimes, optimizers such as stochastic gradient descent (SGD) and its variants exhibit implicit regularization: they systematically select low-complexity solutions, as measured by weight norms or function simplicity, independent of network size (1412.6614). This phenomenon is supported by experiments where test error continues to decrease as hidden units increase well beyond the interpolation threshold.

A theoretical explanation recasts network training as analogous to matrix factorization with norm constraints. Notably, imposing global $\ell_2$ regularization is provably equivalent to constraining hidden weights and imposing an $\ell_1$ norm on output weights (for ReLU activations):

$$\sum_{t=1}^n L(y_t, \sum_{h=1}^H v_h [\langle u_h, x_t \rangle]_+) + \frac{\lambda}{2} \sum_{h=1}^H ( \|u_h\|^2 + |v_h|^2 )$$

$\Longleftrightarrow$

$$\sum_{t=1}^n L(y_t, \sum_{h=1}^H v_h [\langle u_h, x_t \rangle]_+) + \lambda \sum_{h=1}^H |v_h|,\quad \text{subject to}\ \|u_h\| \leq 1$$

This equivalence illustrates that the optimization path—induced by the optimizer—imposes a preference for solutions lying in lower norm (and thus lower complexity) parts of the hypothesis space, providing a direct mechanism for implicit inductive bias.

3. Qualitative Solution Properties: Optimizer-Dependent Expressivity

Unlike convex optimization, where all algorithms eventually converge to the same minimizer, in deep neural networks the optimizer fundamentally influences the outcome. Due to nonconvex landscapes, different optimizers may visit distinct basins of attraction, converging to minima associated with qualitatively different representation properties (Pascanu et al., 16 Jul 2025). For example, preconditioned second-order methods like Shampoo yield hidden representations with lower effective rank and higher localization compared to first-order methods like SGD or AdamW. Such observed differences reveal that the optimizer’s dynamics restrict the set of reachable functions—an implicit form of inductive bias which interacts with and sometimes outweighs the architecture’s expressive power.

This observation extends to the domain of regularization: for deep matrix factorization problems, optimizers that implicitly or explicitly minimize the sharpness of the loss (e.g., SGD with noise or explicit Hessian trace penalties) induce bias toward solutions with low nuclear norm (Schatten 1-norm), which tend to generalize better:

$F(M) \approx L (d_0 d_L)^{1/L} \|M\|_*^{2(L-1)/L}$

where $M$ is the end-to-end weight matrix (Gatmiry et al., 2023). The optimizer, by pushing for flat minima, effectively reduces the functional complexity of the solution regardless of parameter redundancy.

4. Adaptive and Task-Specific Optimization: Bias via Learning Algorithms

Optimizers themselves can be meta-learned to encode domain-specific or task-specific biases. Learned optimizers incorporate structure—such as coordinate-wise updates, root-mean-square scaling, or recurrent state—that mirrors the inductive bias found in manually designed algorithms (Metz et al., 2018, Lan et al., 2023). When trained across diverse tasks, such optimizers (for instance, Optim4RL in RL domains) use forms like:

$$\Delta\theta = -\alpha \frac{m}{\sqrt{v + \epsilon}}$$

where $m$ and $v$ are moving averages learned or computed by internal recurrent models, enforcing an inductive bias corresponding to adaptivity and resilience to nonstationarity (Lan et al., 2023). This structure is crucial in RL, where optimization must contend with highly non-i.i.d., high-variance agent gradients, and stochastic agent-environment interactions.

Furthermore, learned optimizers can be tuned for desirable meta-properties such as stability, robustness, or fair parameter updates. For example, including nominal descent terms in learned optimizers’ architectures ensures eigenvalues of the update matrix lie within the unit circle, conferring both stability and a bias toward descent directions (Harrison et al., 2022).

5. Beyond Generalization: Inductive Bias and Fairness, Representation, and Flexibility

The optimizer’s role in encoding inductive bias extends to domains beyond accuracy or generalization. For group fairness, the paper demonstrates that adaptive optimizers (e.g., RMSProp, Adam) intrinsically promote fairer outcomes under group imbalance, compared to plain SGD, as established through stochastic differential equation analysis and extensive empirical validation (Kolahdouzi et al., 21 Apr 2025). The adaptive scaling inherent in such algorithms shrinks bias-inducing updates and leads to improved measures of demographic parity and equalized odds.

Similarly, optimizers impact the representational alignment and transfer of architectural priors. By incorporating representational alignment losses (such as centered kernel alignment, CKA), even "untrainable" architectures can absorb inductive biases from guide networks and dramatically improve performance, offering a practical handle to inject architectural priors through the optimization process (Subramaniam et al., 26 Oct 2024):

$$\mathcal{L}(\theta^T) = \mathcal{L}_t(\theta^T) + \sum_{i \in I} [1 - \text{CKA}(\text{Activations}^{T}_i, \text{Activations}^{G}_i)]$$

This tool provides a means to make inductive biases more flexible and potentially turn architectural bias into a continuous, optimizable parameter.

6. Inductive Bias Measurement, Adaptation, and Design Guidelines

Recent work introduces systematic ways to measure the inductive bias encoded by an entire optimizer–architecture combination. For example, the information-theoretic approach developed in (Boopathy et al., 22 Jun 2024) estimates the inductive bias by quantifying the information required to encode well-generalizing solutions, comparing distributions over hypotheses (for example, via divergence or approximation error bounds):

• Explicit formulas such as $\xi_h \leq |\mu_p - \mu_q|$ for Laplace-distributed errors
• Sampling-based methods for high- or infinite-dimensional spaces

Empirical results show that as the complexity (dimensionality) of tasks increases, models must encode correspondingly stronger inductive biases to generalize well. This quantification provides practical guidance for the design and adaptation of optimization strategies, aligning optimization bias with task requirements.

Furthermore, several studies emphasize that the community should not focus solely on convergence speed or efficiency when evaluating optimizers. Instead, understanding and leveraging the qualitative, solution-determining properties of optimizers is essential, suggesting new research trajectories to synthesize optimizers that expressly encode desired characteristics—whether simplicity, sparsity, fairness, or stability (Pascanu et al., 16 Jul 2025).

7. Implications for Future Research and Practice

Optimizers as inductive bias elevate the role of the learning algorithm from a mere minimizer of loss to an agent that fundamentally determines which solutions are accessible—and hence what generalization, fairness, or robustness properties a model will possess. The evidence across multiple domains and tasks recommends a paradigm where optimizer design is recognized as a primary lever in shaping model outcomes, to be wielded alongside architecture and data. Ongoing and future work is probing how to systematically characterize, analyze, and design optimizers for target bias profiles, as well as developing metrics and methods for quantifying and adapting inductive bias to diverse problem classes.

This view has led to methods such as explicit adaptation of inductive bias via progressive reparameterization (e.g., interpolating between convolution and attention mechanisms during training (Lee et al., 2022)), meta-tailoring (optimizing unsupervised objectives at prediction time (Alet et al., 2020)), and meta-learning of activation functions to adjust the complexity bias of networks for task-specific needs (Teney et al., 13 Mar 2025), all coordinated by the choice and behavior of the optimizer.

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In summary, the optimizer is not a passive participant in the learning workflow but a principal determiner of inductive bias in modern machine learning. Its direct and indirect effects influence not only which solutions are reached, but also their qualitative nature—enabling or suppressing generalization, fairness, sparsity, robustness, and more. Recognizing, measuring, and engineering these biases, both through empirical paper and mathematical characterization, is a vital and active research direction.