Conceptual Overfitting in Machine Learning

• Conceptual overfitting is a phenomenon where models over-specialize on representational assumptions, supervisory targets, and optimization geometry rather than just parameter counts.
• The concept highlights that overfitting arises from a mismatch between the model’s learned structure and the underlying data-generating process, affecting transferability.
• Empirical studies in self-supervised, reinforcement, and diffusion models reveal that overfitting manifests through sharp supervisory signals, fragile decision boundaries, and overly sample-specific information.

Editor’s term “conceptual overfitting” is not a uniformly standardized expression across machine-learning theory, but a plausible synthesis is that it denotes overfitting understood at the level of representational assumptions, supervisory targets, relational objectives, and optimization geometry, rather than only as a rise in test loss at high parameter count. In this sense, overfitting may appear as optimistic bias of training risk relative to population risk, excessive information transfer from dataset to learned model, pressure to fit overly crisp labels to intrinsically ambiguous observations, non-transferable notions of similarity in self-supervision, memorization of training environments in reinforcement learning, or convergence toward empirical attractors that differ from the population optimum (Schmidt, 2023, Bashir et al., 2020, Korhan et al., 2023, Rabin et al., 2024, Zhang et al., 2018, Maleknia et al., 2 Apr 2026).

1. Formal scope of the concept

A first formalization treats overfitting as a discrepancy between empirical and expected risk for a data-dependent hypothesis. If $S\sim_{iid} P_{X\times Y}$, empirical risk minimization selects

$$y_S \in \arg\min_{\tilde y\in H} e_S(\tilde y), \qquad e_S(\tilde y)=\frac{1}{|S|}\sum_{(x,y)\in S} c_{\tilde y}(x,y).$$

The key observation is that $e_S(\tilde y)\approx E(c_{\tilde y})$ is straightforward only for a fixed $\tilde y$; the learned model $y_S$ depends on the same sample used to evaluate it. On that basis, a trained model $y_S$ is said to $\epsilon$-overfit when

$e_S(y_S) < E(c_{y_S}) - \epsilon,$

to $\epsilon$-underfit when

$e_S(y_S) > E(c_{y_S}) + \epsilon,$

and to $$y_S \in \arg\min_{\tilde y\in H} e_S(\tilde y), \qquad e_S(\tilde y)=\frac{1}{|S|}\sum_{(x,y)\in S} c_{\tilde y}(x,y).$$0-generalize otherwise. This framing makes overfitting a problem of adaptivity and dependence, not merely of low training error (Schmidt, 2023).

The same work proposes a hypothesis test that compares training and holdout empirical risks through a concentration argument. For bounded loss $$y_S \in \arg\min_{\tilde y\in H} e_S(\tilde y), \qquad e_S(\tilde y)=\frac{1}{|S|}\sum_{(x,y)\in S} c_{\tilde y}(x,y).$$1 and an independent holdout sample $$y_S \in \arg\min_{\tilde y\in H} e_S(\tilde y), \qquad e_S(\tilde y)=\frac{1}{|S|}\sum_{(x,y)\in S} c_{\tilde y}(x,y).$$2, if $$y_S \in \arg\min_{\tilde y\in H} e_S(\tilde y), \qquad e_S(\tilde y)=\frac{1}{|S|}\sum_{(x,y)\in S} c_{\tilde y}(x,y).$$3 $$y_S \in \arg\min_{\tilde y\in H} e_S(\tilde y), \qquad e_S(\tilde y)=\frac{1}{|S|}\sum_{(x,y)\in S} c_{\tilde y}(x,y).$$4-generalizes, then

$$y_S \in \arg\min_{\tilde y\in H} e_S(\tilde y), \qquad e_S(\tilde y)=\frac{1}{|S|}\sum_{(x,y)\in S} c_{\tilde y}(x,y).$$5

This is important because it reframes the familiar train–validation gap as a statistically calibrated observable, while also making explicit that the same discrepancy can reflect overfitting, underfitting, or distributional shift (Schmidt, 2023).

A more structural formalization treats overfitting as a mismatch between the information a learning algorithm can transfer from data into its output and the complexity of the dataset. In that view, algorithm capacity is

$$y_S \in \arg\min_{\tilde y\in H} e_S(\tilde y), \qquad e_S(\tilde y)=\frac{1}{|S|}\sum_{(x,y)\in S} c_{\tilde y}(x,y).$$6

with distributional version $$y_S \in \arg\min_{\tilde y\in H} e_S(\tilde y), \qquad e_S(\tilde y)=\frac{1}{|S|}\sum_{(x,y)\in S} c_{\tilde y}(x,y).$$7. Overfitting is then defined by

$$y_S \in \arg\min_{\tilde y\in H} e_S(\tilde y), \qquad e_S(\tilde y)=\frac{1}{|S|}\sum_{(x,y)\in S} c_{\tilde y}(x,y).$$8

and underfitting by the reverse mismatch for a time-indexed capacity $$y_S \in \arg\min_{\tilde y\in H} e_S(\tilde y), \qquad e_S(\tilde y)=\frac{1}{|S|}\sum_{(x,y)\in S} c_{\tilde y}(x,y).$$9. This formulation shifts the explanatory burden from “large hypothesis class” to “too much sample-specific information retained by the dataset-to-model channel.” The same paper also proves the undecidability of determining, for an arbitrary classification algorithm and dataset, whether the algorithm will overfit, which strengthens the case for structural signatures rather than universal deciders (Bashir et al., 2020).

2. Alignment between hypothesis class and data-generating structure

One of the clearest demonstrations that overfitting depends on structural alignment rather than parameter count alone comes from matrix product state learning. In that setting, each scalar input is mapped by

$e_S(\tilde y)\approx E(c_{\tilde y})$0

the full lifted representation is a tensor product $e_S(\tilde y)\approx E(c_{\tilde y})$1, and the predictor is linear in that lifted space, with model capacity controlled by the bond dimension $e_S(\tilde y)\approx E(c_{\tilde y})$2. The regression loss is

$e_S(\tilde y)\approx E(c_{\tilde y})$3

Artificial labels are generated exactly by a target MPS,

$e_S(\tilde y)\approx E(c_{\tilde y})$4

so approximation error can be made arbitrarily small when $e_S(\tilde y)\approx E(c_{\tilde y})$5 is large enough. In this setting, classical overfitting appears strongly for effectively one-dimensional data: training loss keeps improving, but test loss reaches an optimum at a finite $e_S(\tilde y)\approx E(c_{\tilde y})$6 and then worsens. By contrast, on more complex artificial data and on MNIST, test performance is monotone or saturating rather than sharply U-shaped. The paper’s central interpretation is that overfitting is strongest when the model class is well aligned with the true generative structure and expressive enough to fit it closely; when the architecture is mismatched to the task, increasing capacity mainly reduces approximation error instead of entering a variance-dominated regime (Strashko et al., 2022).

This alignment-based account is closely related to a sample-size view developed for kernel ridge regression and two-layer ReLU networks trained by gradient flow. There the focus is not exact interpolation, but what the paper calls “almost benign overfitting”: for every $e_S(\tilde y)\approx E(c_{\tilde y})$7, there exists $e_S(\tilde y)\approx E(c_{\tilde y})$8 such that, with probability at least $e_S(\tilde y)\approx E(c_{\tilde y})$9,

$\tilde y$0

The conceptual claim is that the classical U-shaped excess-risk curve is not rejected, but incomplete if one reasons at fixed sample size. As $\tilde y$1 grows, the location of the optimal complexity regime can move “down and to the right,” so larger models can achieve both very small training error and very small excess risk. In that sense, overfitting is again relational: it depends on complexity relative to sample size, not on complexity in isolation (Park et al., 16 May 2025).

Taken together, these results suggest that conceptual overfitting is often best understood through the geometry of the hypothesis class relative to the data-generating process. A model can overfit most classically when it speaks the “right language” for the data and has enough capacity for that language to become highly sample-specific. When alignment is weak, the same increase in capacity may simply continue reducing bias (Strashko et al., 2022, Park et al., 16 May 2025).

3. Supervisory rigidity and concept-level regularization

A different line of work locates overfitting in the sharpness of supervision itself. The “negotiated representations” approach argues that much overfitting occurs in “reconciling sharply defined membership ratios to specific classes.” Standard supervised learning is written as

$\tilde y$2

whereas negotiated training introduces a negotiation rate $\tilde y$3 and uses

$\tilde y$4

The prose specifies that negotiated labels are a weighted average of the predicted labels and the original labels, that original labels are switched with negotiated labels after each negotiation phase, and that the negotiation rate increases linearly after each epoch. The conceptual thesis is that one-hot targets are often epistemically too sharp for ambiguous, noisy, or weakly observed samples, and that forcing exact class membership distorts the representation so that the model encodes “exceptions,” “wrong labels,” “outliers,” and “individual identities.” Empirically, test accuracy improved on all four reported datasets: MNIST from $\tilde y$5 to $\tilde y$6, Fashion-MNIST from $\tilde y$7 to $\tilde y$8, CIFAR-10 from $\tilde y$9 to $y_S$0, and CIFAR-100 from $y_S$1 to $y_S$2 (Korhan et al., 2023).

Few-shot adaptation of vision-LLMs yields a related notion of concept-level overfitting. In Conceptual Codebook Learning, CLIP encoders remain frozen while a learnable conceptual codebook

$y_S$3

retrieves visual concept keys and associated conceptual prompt values. The classifier is trained with

$y_S$4

where $y_S$5 regularizes learned prompt-induced text features toward handcrafted concept-based prompts derived from a concept cache. The setting is mainly 16-shot adaptation, and the main hyperparameters are $y_S$6, $y_S$7, $y_S$8, prompt length $y_S$9, and codebook size $y_S$0. The paper’s argument is that prompt tuning overfits when free prompt parameters absorb class-specific or sample-specific idiosyncrasies, while concept-indexed retrieval and concept-grounded regularization preserve transferability across base-to-novel, cross-dataset, and domain-generalization settings (Zhang et al., 2024).

These two works converge on a common point. In one case, overfitting is conceptual rigidity of labels; in the other, it is over-specialization of prompts to base classes and few-shot support sets. In both cases, mitigation comes from softening or constraining the supervision space so that learning operates through reusable structure rather than through exact sample-to-label or sample-to-prompt matching (Korhan et al., 2023, Zhang et al., 2024).

4. Relational overfitting in self-supervised and sequential settings

In unsupervised contrastive learning, overfitting cannot be defined through label error, so the relevant object becomes the self-supervised objective itself. Using SimCLR with a ResNet-18 backbone on a four-class subset of CIFAR-10, trained with Adam at learning rate $y_S$1 for up to $y_S$2 epochs, overfitting is identified when training contrastive loss continues decreasing while validation contrastive loss begins increasing. Around epoch $y_S$3, the validation loss bottoms out and then worsens. Decomposing the SimCLR loss,

$y_S$4

the paper finds that the negative similarity term continues improving on validation, while the positive similarity term deteriorates. The interpretation is that the model becomes increasingly specialized to training-set-specific positive correspondences and loses the ability to align unseen positive pairs. Conceptually, this is overfitting of the learned notion of “same underlying instance under augmentation,” not of class labels (Rabin et al., 2024).

The recommended intervention in that work is early stopping based on validation positive similarity, because the positive term reveals overfitting earlier than the full contrastive loss. At the same time, the paper does not study downstream linear probing or transfer accuracy, so its strongest claim remains pretext-task overfitting rather than fully semantic failure (Rabin et al., 2024).

Deep reinforcement learning exhibits an analogous phenomenon at the level of task structure rather than pairwise similarity. Zhang, Vinyals, Munos, and Bengio define overfitting by training on one set of environments and evaluating on held-out environments sampled from the same family. The standard objective remains

$y_S$5

but generalization is measured by the gap between training and test returns across distinct MDP instances. In maze-like tasks, agents trained on a small number of random seeds can achieve near-optimal training return while generalizing poorly to unseen seeds. The same work shows memorization-like behavior under randomized reward assignments and emphasizes that stochasticity such as random starts or action noise does not reliably expose overfitting. The central conceptual point is that robustness to perturbations of familiar environments is not the same as learning task structure that transfers to new environments from the same distribution (Zhang et al., 2018).

Both cases generalize the classical notion of overfitting. In contrastive learning, the non-generalizing object is a relation of sameness; in reinforcement learning, it is a policy’s latent conception of the task family. Neither failure is adequately described by counting parameters alone (Rabin et al., 2024, Zhang et al., 2018).

5. Geometry, dynamics, and local decision structure

Several recent theories connect overfitting to specific geometric and dynamical mechanisms. In diffusion models trained by score matching, the central claim is that benign overfitting does not occur in the practically relevant regime. The population target is learned through explicit score matching,

$y_S$6

while training uses the empirical denoising score matching objective. The paper proves impossibility results showing that, unless sample size grows exponentially with intrinsic dimension, one cannot make empirical and population score-matching losses simultaneously small at low noise scales. In a linear random-feature analysis, population loss is U-shaped in model complexity rather than double-descent-like, because score matching lacks the favorable target–covariance alignment that can make overparameterized regression benign. Time-smoothness across diffusion times and early stopping are identified as the regularizing mechanisms that actually prevent harmful overfitting (Farghly et al., 2 Jul 2026).

A physics-based account explains overfitting control through noisy optimization and free-energy selection. For stochastic gradient Langevin dynamics,

$y_S$7

the stationary distribution is Gibbs, and transition rates between wells are governed by the Eyring factor

$y_S$8

Wide minima have larger entropy and therefore lower free energy, so noisy dynamics preferentially retains them; the paper interprets those wide minima as algorithmically stable and therefore less overfitting. In GANs, an analogous heuristic is proposed through a predator–prey model, where adversarial competition pushes the discriminator away from narrow likelihood maxima and toward broader ones (Kozyrev et al., 2024).

A more explicitly dynamical description is given for one-hidden-layer tanh MLPs trained on finite noisy data. With empirical loss

$y_S$9

and population error

$\epsilon$0

the exact realizers of the target,

$\epsilon$1

cease to be critical points of the empirical loss as soon as observational noise is present, because

$\epsilon$2

is almost surely nonzero for $\epsilon$3. The overfitting region is the empirical minimizer set

$\epsilon$4

and the paper argues that trajectories pass through plateau regions and near-optimal saddle-like regions before converging to $\epsilon$5. Under boundedness and a dimension/sample-size condition, the overfitting region collapses to a single attractor modulo permutation and sign symmetries with high probability (Maleknia et al., 2 Apr 2026).

Local decision geometry provides yet another perspective. The counterfactual-regularization work argues that the easier it is to find a valid counterfactual

$\epsilon$6

the more overfit the model is likely to be, because highly convoluted boundaries place points closer to prediction-flipping perturbations. The proposed objective

$\epsilon$7

pushes points farther from their counterfactuals and improves test accuracy in most reported settings. The paper is careful, however, that these counterfactuals are primarily geometric rather than semantic (Giorgi et al., 13 Feb 2025).

High-dimensional imbalanced classification reveals a more distributional geometric distortion. For linearly separable Gaussian-mixture data, test logits asymptotically remain Gaussian, but training logits become rectified: $\epsilon$8 The formal statement is that the empirical logit distribution is a margin-rectified version of the testing logit distribution, which explains why training data look easier than unseen data and why the minority class is more severely affected. Margin rebalancing corrects this by shifting the intercept; at the optimum, $\epsilon$9 and minority and majority class errors are equalized (Lyu et al., 17 Feb 2025).

Across these works, overfitting is not treated as a single pathology. It is score-field memorization at small diffusion times, free-energy selection of sharp extrema, attraction toward empirical rather than population optima, proximity to fragile decision boundaries, or rectification of train-time logit distributions (Farghly et al., 2 Jul 2026, Kozyrev et al., 2024, Maleknia et al., 2 Apr 2026, Giorgi et al., 13 Feb 2025, Lyu et al., 17 Feb 2025).

6. Distributional limits, misconceptions, and open directions

A particularly strong distribution-centric theory argues that overfitting can be bounded independently of model size for symmetric algorithms whose outputs depend only on the histogram of the training data. For a bounded loss $e_S(y_S) < E(c_{y_S}) - \epsilon,$0 and a symmetric learning algorithm $e_S(y_S) < E(c_{y_S}) - \epsilon,$1, the generalization error is bounded in terms of

$e_S(y_S) < E(c_{y_S}) - \epsilon,$2

where $e_S(y_S) < E(c_{y_S}) - \epsilon,$3 is the Rényi entropy of the data-generating distribution. The main theorem gives, with high probability,

$e_S(y_S) < E(c_{y_S}) - \epsilon,$4

A corresponding sufficient sample-size theorem makes the dominant scale essentially $e_S(y_S) < E(c_{y_S}) - \epsilon,$5. The same paper proves that randomizing labels increases Rényi entropy by $e_S(y_S) < E(c_{y_S}) - \epsilon,$6, thereby worsening the required sample size multiplicatively, and gives a data-distribution-dependent no-free-lunch theorem showing that this entropy dependence is not merely sufficient but essentially necessary (Suzuki, 30 May 2025).

These results jointly undermine several persistent misconceptions. One is that overfitting is primarily a monotone function of parameter count. The MPS study shows that it can become strongest when the architecture is best aligned with the data-generating structure (Strashko et al., 2022). Negotiated representations argue that rigid one-hot supervision can itself be the conceptual source of overfitting (Korhan et al., 2023). Contrastive learning shows that a model can keep optimizing its training objective while learning a non-transferable notion of positive-pair similarity (Rabin et al., 2024). Deep RL shows that high training return can coexist with poor transfer across held-out environment seeds even under stochastic evaluation (Zhang et al., 2018). Almost benign overfitting emphasizes that the operative sweet spot moves with sample size (Park et al., 16 May 2025). Diffusion theory shows that interpolation and good generalization need not coexist at all for score matching (Farghly et al., 2 Jul 2026).

Several limitations remain open and define the current frontier. The negotiated-representation framework is conceptually provocative but mathematically under-specified in its update rule (Korhan et al., 2023). The contrastive-learning analysis diagnoses pretext-task overfitting but does not measure downstream semantic transfer directly (Rabin et al., 2024). The Rényi-entropy theory is model-independent but currently excludes stochastic methods such as SGD and Adam (Suzuki, 30 May 2025). The MLP dynamical theory is strongest in a one-hidden-layer tanh setting with explicit noise assumptions rather than in modern deep architectures (Maleknia et al., 2 Apr 2026). The diffusion results establish a sharp contrast with regression, but open questions remain about how broadly the mechanism extends across practical parameterizations and training heuristics (Farghly et al., 2 Jul 2026).

A coherent synthesis is therefore possible, but it should remain qualified. Conceptual overfitting is not a single theorem or benchmark signature. It is a family of explanations in which generalization failure arises because a learner becomes too specialized within a particular conceptualization of the task: the wrong supervisory sharpness, the wrong relational invariance, the wrong environment family, the wrong local geometry, or the wrong empirical surrogate for a population object. The literature increasingly suggests that understanding overfitting requires tracking not only how much a model can fit, but what kind of structure it is being induced to fit, how that structure is represented, and under which data and optimization regimes that structure ceases to transfer (Strashko et al., 2022, Korhan et al., 2023, Rabin et al., 2024, Zhang et al., 2018, Suzuki, 30 May 2025).