In transfer learning, the learner leverages auxiliary data to improve generalization on a main task. However, the precise theoretical understanding of when and how auxiliary data help remains incomplete. We provide new insights on this issue in two canonical linear settings: ordinary least squares regression and under-parameterized linear neural networks. For linear regression, we derive exact closed-form expressions for the expected generalization error with bias-variance decomposition, yielding necessary and sufficient conditions for auxiliary tasks to improve generalization on the main task. We also derive globally optimal task weights as outputs of solvable optimization programs, with consistency guarantees for empirical estimates. For linear neural networks with shared representations of width q≤K, where K is the number of auxiliary tasks, we derive a non-asymptotic expectation bound on the generalization error, yielding the first non-vacuous sufficient condition for beneficial auxiliary learning in this setting, as well as principled directions for task weight curation. We achieve this by proving a new column-wise low-rank perturbation bound for random matrices, which improves upon existing bounds by preserving fine-grained column structures. Our results are verified on synthetic data simulated with controlled parameters.
Figures & tables
Source
Object
Assumption
Bound
Possible to be smaller than ∥Z∥ or ∥Zej∥ ?
EYM ( Tran et al., 2025 )
∥[M]q−[S]q∥
-
2(σq+1(S)+∥Z∥)
No
TVV ( Tran et al., 2025 )
∥[M]q−[S]q∥
r≥4
r−26r∥Z∥(δq(S)σq(S)+logδq(S)6σ1(S))
No
Ours ( \cref th:4.2)
∥([M]q−[S]q)ej∥
r>1
r−11(∥Sej∥+∥Zej∥)+∥PUZej∥
Yes
Table 1: Comparison of low-rank perturbation bounds. Here, δq(S)=σq(S)−σq+1(S) , r=δq(S)/∥Z∥ , and PU=U:,qU:,q⊤ , where U:,q contains the top q left singular vectors of S .
(a) Simulation results of for both linear regression and linear neural networks.
Auxiliary learning is an optimization paradigm in which a neural network's performance on a target task is improved by jointly training it on additional tasks. However, the mechanisms behind this improvement remain poorly understood. We study this problem using a teacher-student framework and derive a closed system of differential equations describing the dynamics of online stochastic gradient descent in the large-input limit. For linear networks, we obtain a closed-form expression for the generalization error to leading order in the learning rate, quantifying how task correlations and label noise determine the benefit of auxiliary learning. For non-linear activation functions, we develop a fluctuation-dissipation analytical theory that establishes a general relation linking the main and auxiliary errors to the corresponding single-task error. Numerical experiments support the theoretical predictions and show how auxiliary tasks improve generalization by balancing the forcing dynamics towards the optimal solution with gradient noise.
Department of Physics and INFN University of Turin Via Giuria 1, 10125 Turin, Italy · Donders Centre for Neuroscience Radboud University Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
We consider nonlinear networks as perturbations of linear ones. Based on this approach, we present novel generalization bounds that become non-vacuous for networks that are close to being linear. The main advantage over the previous works which propose non-vacuous generalization bounds is that our bounds are a-priori: performing the actual training is not required for evaluating the bounds. To the best of our knowledge, they are the first non-vacuous generalization bounds for neural nets possessing this property.
Eugene Golikov
Chair of Statistical Field Theory École Polytechnique Fédérale de Lausanne (EPFL)
We study transfer learning for a linear regression task using several least-squares pretrained models that can be overparameterized. We formulate the target learning task as optimization that minimizes squared errors on the target dataset with penalty on the distance of the learned model from the pretrained models. We analytically formulate the test error of the learned target model and provide the corresponding empirical evaluations. Our results elucidate when using more pretrained models can improve transfer learning. Specifically, if the pretrained models are overparameterized, using sufficiently many of them is important for beneficial transfer learning. However, the learning may be compromised by overparameterization bias of pretrained models, i.e., the minimum ℓ2-norm solution's restriction to a small subspace spanned by the training examples in the high-dimensional parameter space. We propose a simple debiasing via multiplicative correction factor that can reduce the overparameterization bias and leverage more pretrained models to learn a target predictor.
Daniel Boharon, Yehuda Dar
Faculty of Computer and Information Science, Ben-Gurion University.