Sohom Bhattacharya

Anvit Garg, Sohom Bhattacharya, Pragya Sur

Model collapse occurs when generative models degrade after repeatedly training on their own synthetic outputs. We study this effect in overparameterized linear regression in a setting where each iteration mixes fresh real labels with synthetic labels drawn from the model fitted in the previous iteration. We derive precise generalization error formulae for minimum-$\ell_2$-norm interpolation and ridge regression under this iterative scheme. Our analysis reveals intriguing properties of the optimal mixing weight that minimizes long-term prediction error and provably prevents model collapse. For instance, in the case of min-$\ell_2$-norm interpolation, we establish that the optimal real-data proportion converges to the reciprocal of the golden ratio for fairly general classes of covariate distributions. Previously, this property was known only for ordinary least squares, and additionally in low dimensions. For ridge regression, we further analyze two popular model classes -- the random-effects model and the spiked covariance model -- demonstrating how spectral geometry governs optimal weighting. In both cases, as well as for isotropic features, we uncover that the optimal mixing ratio should be at least one-half, reflecting the necessity of favoring real-data over synthetic. We validate our theoretical results with extensive simulations.

Yanke Song, Sohom Bhattacharya, Pragya Sur

This paper establishes the generalization error of pooled min-$\ell_2$-norm interpolation in transfer learning where data from diverse distributions are available. Min-norm interpolators emerge naturally as implicit regularized limits of modern machine learning algorithms. Previous work characterized their out-of-distribution risk when samples from the test distribution are unavailable during training. However, in many applications, a limited amount of test data may be available during training, yet properties of min-norm interpolation in this setting are not well-understood. We address this gap by characterizing the bias and variance of pooled min-$\ell_2$-norm interpolation under covariate and model shifts. The pooled interpolator captures both early fusion and a form of intermediate fusion. Our results have several implications: under model shift, for low signal-to-noise ratio (SNR), adding data always hurts. For higher SNR, transfer learning helps as long as the shift-to-signal (SSR) ratio lies below a threshold that we characterize explicitly. By consistently estimating these ratios, we provide a data-driven method to determine: (i) when the pooled interpolator outperforms the target-based interpolator, and (ii) the optimal number of target samples that minimizes the generalization error. Under covariate shift, if the source sample size is small relative to the dimension, heterogeneity between between domains improves the risk, and vice versa. We establish a novel anisotropic local law to achieve these characterizations, which may be of independent interest in random matrix theory. We supplement our theoretical characterizations with comprehensive simulations that demonstrate the finite-sample efficacy of our results.