Estimating stationary mass, frequency by frequency

Suppose we observe a trajectory of length $n$ from an $\alpha$-mixing stochastic process over a finite but potentially large state space. We consider the problem of estimating the probability mass placed by the stationary distribution of any such process on elements that occur with a certain frequency in the observed sequence. We estimate this vector of probabilities in total variation distance, showing universal consistency in $n$ and recovering known results for i.i.d. sequences as special cases. Our proposed methodology carefully combines the plug-in (or empirical) estimator with a recently-proposed modification of the Good--Turing estimator called \textsc{WingIt}, which was originally developed for Markovian sequences. En route to controlling the error of our estimator, we develop new performance bounds on \textsc{WingIt} and the plug-in estimator for $\alpha$-mixing stochastic processes. Importantly, the extensively used method of Poissonization can no longer be applied in our non i.i.d. setting, and so we develop complementary tools -- including concentration inequalities for a natural self-normalized statistic of mixing sequences -- that may prove independently useful in the design and analysis of estimators for related problems.

Task Shift: From Classification to Regression in Overparameterized Linear Models

Modern machine learning methods have recently demonstrated remarkable capability to generalize under task shift, where latent knowledge is transferred to a different, often more difficult, task under a similar data distribution. We investigate this phenomenon in an overparameterized linear regression setting where the task shifts from classification during training to regression during evaluation. In the zero-shot case, wherein no regression data is available, we prove that task shift is impossible in both sparse signal and random signal models for any Gaussian covariate distribution. In the few-shot case, wherein limited regression data is available, we propose a simple postprocessing algorithm which asymptotically recovers the ground-truth predictor. Our analysis leverages a fine-grained characterization of individual parameters arising from minimum-norm interpolation which may be of independent interest. Our results show that while minimum-norm interpolators for classification cannot transfer to regression a priori, they experience surprisingly structured attenuation which enables successful task shift with limited additional data.

Multi-Agent Performative Prediction Beyond the Insensitivity Assumption: A Case Study for Mortgage Competition

Performative prediction models account for feedback loops in decision-making processes where predictions influence future data distributions. While existing work largely assumes insensitivity of data distributions to small strategy changes, this assumption usually fails in real-world competitive (i.e. multi-agent) settings. For example, in Bertrand-type competitions, a small reduction in one firm's price can lead that firm to capture the entire demand, while all others sharply lose all of their customers. We study a representative setting of multi-agent performative prediction in which insensitivity assumptions do not hold, and investigate the convergence of natural dynamics. To do so, we focus on a specific game that we call the ''Bank Game'', where two lenders compete over interest rates and credit score thresholds. Consumers act similarly as to in a Bertrand Competition, with each consumer selecting the firm with the lowest interest rate that they are eligible for based on the firms' credit thresholds. Our analysis characterizes the equilibria of this game and demonstrates that when both firms use a common and natural no-regret learning dynamic -- exponential weights -- with proper initialization, the dynamics always converge to stable outcomes despite the general-sum structure. Notably, our setting admits multiple stable equilibria, with convergence dependent on initial conditions. We also provide theoretical convergence results in the stochastic case when the utility matrix is not fully known, but each learner can observe sufficiently many samples of consumers at each time step to estimate it, showing robustness to slight mis-specifications. Finally, we provide experimental results that validate our theoretical findings.

The Group Robustness is in the Details: Revisiting Finetuning under Spurious Correlations

Modern machine learning models are prone to over-reliance on spurious correlations, which can often lead to poor performance on minority groups. In this paper, we identify surprising and nuanced behavior of finetuned models on worst-group accuracy via comprehensive experiments on four well-established benchmarks across vision and language tasks. We first show that the commonly used class-balancing techniques of mini-batch upsampling and loss upweighting can induce a decrease in worst-group accuracy (WGA) with training epochs, leading to performance no better than without class-balancing. While in some scenarios, removing data to create a class-balanced subset is more effective, we show this depends on group structure and propose a mixture method which can outperform both techniques. Next, we show that scaling pretrained models is generally beneficial for worst-group accuracy, but only in conjuction with appropriate class-balancing. Finally, we identify spectral imbalance in finetuning features as a potential source of group disparities -- minority group covariance matrices incur a larger spectral norm than majority groups once conditioned on the classes. Our results show more nuanced interactions of modern finetuned models with group robustness than was previously known. Our code is available at https://github.com/tmlabonte/revisiting-finetuning.

Precise asymptotics of reweighted least-squares algorithms for linear diagonal networks

The classical iteratively reweighted least-squares (IRLS) algorithm aims to recover an unknown signal from linear measurements by performing a sequence of weighted least squares problems, where the weights are recursively updated at each step. Varieties of this algorithm have been shown to achieve favorable empirical performance and theoretical guarantees for sparse recovery and $\ell_p$-norm minimization. Recently, some preliminary connections have also been made between IRLS and certain types of non-convex linear neural network architectures that are observed to exploit low-dimensional structure in high-dimensional linear models. In this work, we provide a unified asymptotic analysis for a family of algorithms that encompasses IRLS, the recently proposed lin-RFM algorithm (which was motivated by feature learning in neural networks), and the alternating minimization algorithm on linear diagonal neural networks. Our analysis operates in a "batched" setting with i.i.d. Gaussian covariates and shows that, with appropriately chosen reweighting policy, the algorithm can achieve favorable performance in only a handful of iterations. We also extend our results to the case of group-sparse recovery and show that leveraging this structure in the reweighting scheme provably improves test error compared to coordinate-wise reweighting.

Sharp analysis of out-of-distribution error for "importance-weighted" estimators in the overparameterized regime

Overparameterized models that achieve zero training error are observed to generalize well on average, but degrade in performance when faced with data that is under-represented in the training sample. In this work, we study an overparameterized Gaussian mixture model imbued with a spurious feature, and sharply analyze the in-distribution and out-of-distribution test error of a cost-sensitive interpolating solution that incorporates "importance weights". Compared to recent work Wang et al. (2021), Behnia et al. (2022), our analysis is sharp with matching upper and lower bounds, and significantly weakens required assumptions on data dimensionality. Our error characterizations also apply to any choice of importance weights and unveil a novel tradeoff between worst-case robustness to distribution shift and average accuracy as a function of the importance weight magnitude.

Just Wing It: Optimal Estimation of Missing Mass in a Markovian Sequence

We study the problem of estimating the stationary mass -- also called the unigram mass -- that is missing from a single trajectory of a discrete-time, ergodic Markov chain. This problem has several applications -- for example, estimating the stationary missing mass is critical for accurately smoothing probability estimates in sequence models. While the classical Good--Turing estimator from the 1950s has appealing properties for i.i.d. data, it is known to be biased in the Markov setting, and other heuristic estimators do not come equipped with guarantees. Operating in the general setting in which the size of the state space may be much larger than the length $n$ of the trajectory, we develop a linear-runtime estimator called \emph{Windowed Good--Turing} (\textsc{WingIt}) and show that its risk decays as $\widetilde{\mathcal{O}}(\mathsf{T_{mix}}/n)$, where $\mathsf{T_{mix}}$ denotes the mixing time of the chain in total variation distance. Notably, this rate is independent of the size of the state space and minimax-optimal up to a logarithmic factor in $n / \mathsf{T_{mix}}$. We also present a bound on the variance of the missing mass random variable, which may be of independent interest. We extend our estimator to approximate the stationary mass placed on elements occurring with small frequency in $X^n$. Finally, we demonstrate the efficacy of our estimators both in simulations on canonical chains and on sequences constructed from a popular natural language corpus.

Balanced Data, Imbalanced Spectra: Unveiling Class Disparities with Spectral Imbalance

Classification models are expected to perform equally well for different classes, yet in practice, there are often large gaps in their performance. This issue of class bias is widely studied in cases of datasets with sample imbalance, but is relatively overlooked in balanced datasets. In this work, we introduce the concept of spectral imbalance in features as a potential source for class disparities and study the connections between spectral imbalance and class bias in both theory and practice. To build the connection between spectral imbalance and class gap, we develop a theoretical framework for studying class disparities and derive exact expressions for the per-class error in a high-dimensional mixture model setting. We then study this phenomenon in 11 different state-of-the-art pretrained encoders and show how our proposed framework can be used to compare the quality of encoders, as well as evaluate and combine data augmentation strategies to mitigate the issue. Our work sheds light on the class-dependent effects of learning, and provides new insights into how state-of-the-art pretrained features may have unknown biases that can be diagnosed through their spectra.

Towards Last-layer Retraining for Group Robustness with Fewer Annotations

Empirical risk minimization (ERM) of neural networks is prone to over-reliance on spurious correlations and poor generalization on minority groups. The recent deep feature reweighting (DFR) technique achieves state-of-the-art group robustness via simple last-layer retraining, but it requires held-out group and class annotations to construct a group-balanced reweighting dataset. In this work, we examine this impractical requirement and find that last-layer retraining can be surprisingly effective with no group annotations (other than for model selection) and only a handful of class annotations. We first show that last-layer retraining can greatly improve worst-group accuracy even when the reweighting dataset has only a small proportion of worst-group data. This implies a "free lunch" where holding out a subset of training data to retrain the last layer can substantially outperform ERM on the entire dataset with no additional data or annotations. To further improve group robustness, we introduce a lightweight method called selective last-layer finetuning (SELF), which constructs the reweighting dataset using misclassifications or disagreements. Our empirical and theoretical results present the first evidence that model disagreement upsamples worst-group data, enabling SELF to nearly match DFR on four well-established benchmarks across vision and language tasks with no group annotations and less than 3% of the held-out class annotations. Our code is available at https://github.com/tmlabonte/last-layer-retraining.

Faster Margin Maximization Rates for Generic Optimization Methods

First-order optimization methods tend to inherently favor certain solutions over others when minimizing a given training objective with multiple local optima. This phenomenon, known as implicit bias, plays a critical role in understanding the generalization capabilities of optimization algorithms. Recent research has revealed that gradient-descent-based methods exhibit an implicit bias for the $\ell_2$-maximal margin classifier in the context of separable binary classification. In contrast, generic optimization methods, such as mirror descent and steepest descent, have been shown to converge to maximal margin classifiers defined by alternative geometries. However, while gradient-descent-based algorithms demonstrate fast implicit bias rates, the implicit bias rates of generic optimization methods have been relatively slow. To address this limitation, in this paper, we present a series of state-of-the-art implicit bias rates for mirror descent and steepest descent algorithms. Our primary technique involves transforming a generic optimization algorithm into an online learning dynamic that solves a regularized bilinear game, providing a unified framework for analyzing the implicit bias of various optimization methods. The accelerated rates are derived leveraging the regret bounds of online learning algorithms within this game framework.