arXiv subjects

Vatsal Sharan

Publications and source records attributed to Vatsal Sharan.

At least 37 records · Page 2Linked to original sources

arXiv2025

Improved Bounds for Swap Multicalibration and Swap Omniprediction

In this paper, we consider the related problems of multicalibration -- a multigroup fairness notion and omniprediction -- a simultaneous loss minimization paradigm, both in the distributional and online settings. The recent work of Garg et al. (2024) raised the open problem of whether it is possible to efficiently achieve $O(\sqrt{T})$ $\ell_{2}$-multicalibration error against bounded linear functions. In this paper, we answer this question in a strongly affirmative sense. We propose an efficient algorithm that achieves $O(T^{\frac{1}{3}})$ $\ell_{2}$-swap multicalibration error (both in high probability and expectation). On propagating this bound onward, we obtain significantly improved rates for $\ell_{1}$-swap multicalibration and swap omniprediction for a loss class of convex Lipschitz functions. In particular, we show that our algorithm achieves $O(T^{\frac{2}{3}})$ $\ell_{1}$-swap multicalibration and swap omniprediction errors, thereby improving upon the previous best-known bound of $O(T^{\frac{7}{8}})$. As a consequence of our improved online results, we further obtain several improved sample complexity rates in the distributional setting. In particular, we establish a $O(\varepsilon ^ {-3})$ sample complexity of efficiently learning an $\varepsilon$-swap omnipredictor for the class of convex and Lipschitz functions, $O(\varepsilon ^{-2.5})$ sample complexity of efficiently learning an $\varepsilon$-swap agnostic learner for the squared loss, and $O(\varepsilon ^ {-5}), O(\varepsilon ^ {-2.5})$ sample complexities of learning $\ell_{1}, \ell_{2}$-swap multicalibrated predictors against linear functions, all of which significantly improve on the previous best-known bounds.

cs.LG↗

arXiv2025

Textual Steering Vectors Can Improve Visual Understanding in Multimodal Large Language Models

Steering methods have emerged as effective and targeted tools for guiding large language models' (LLMs) behavior without modifying their parameters. Multimodal large language models (MLLMs), however, do not currently enjoy the same suite of techniques, due in part to their recency and architectural diversity. Inspired by this gap, we investigate whether MLLMs can be steered using vectors derived from their text-only LLM backbone, via sparse autoencoders (SAEs), mean shift, and linear probing. We find that text-derived steering consistently enhances multimodal accuracy across diverse MLLM architectures and visual tasks. In particular, mean shift boosts spatial relationship accuracy on CV-Bench by up to +7.3% and counting accuracy by up to +3.3%, outperforming prompting and exhibiting strong generalization to out-of-distribution datasets. These results highlight textual steering vectors as a powerful, efficient mechanism for enhancing grounding in MLLMs with minimal additional data collection and computational overhead.

cs.LG↗

arXiv2025

Transformers Learn Low Sensitivity Functions: Investigations and Implications

Transformers achieve state-of-the-art accuracy and robustness across many tasks, but an understanding of their inductive biases and how those biases differ from other neural network architectures remains elusive. In this work, we identify the sensitivity of the model to token-wise random perturbations in the input as a unified metric which explains the inductive bias of transformers across different data modalities and distinguishes them from other architectures. We show that transformers have lower sensitivity than MLPs, CNNs, ConvMixers and LSTMs, across both vision and language tasks. We also show that this low-sensitivity bias has important implications: i) lower sensitivity correlates with improved robustness; it can also be used as an efficient intervention to further improve the robustness of transformers; ii) it corresponds to flatter minima in the loss landscape; and iii) it can serve as a progress measure for grokking. We support these findings with theoretical results showing (weak) spectral bias of transformers in the NTK regime, and improved robustness due to the lower sensitivity. The code is available at https://github.com/estija/sensitivity.

cs.LG↗

arXiv2024

Transformers Learn to Achieve Second-Order Convergence Rates for In-Context Linear Regression

Transformers excel at in-context learning (ICL) -- learning from demonstrations without parameter updates -- but how they do so remains a mystery. Recent work suggests that Transformers may internally run Gradient Descent (GD), a first-order optimization method, to perform ICL. In this paper, we instead demonstrate that Transformers learn to approximate second-order optimization methods for ICL. For in-context linear regression, Transformers share a similar convergence rate as Iterative Newton's Method, both exponentially faster than GD. Empirically, predictions from successive Transformer layers closely match different iterations of Newton's Method linearly, with each middle layer roughly computing 3 iterations; thus, Transformers and Newton's method converge at roughly the same rate. In contrast, Gradient Descent converges exponentially more slowly. We also show that Transformers can learn in-context on ill-conditioned data, a setting where Gradient Descent struggles but Iterative Newton succeeds. Finally, to corroborate our empirical findings, we prove that Transformers can implement $k$ iterations of Newton's method with $k + \mathcal{O}(1)$ layers.

cs.LG↗

arXiv2024

When is Multicalibration Post-Processing Necessary?

Calibration is a well-studied property of predictors which guarantees meaningful uncertainty estimates. Multicalibration is a related notion -- originating in algorithmic fairness -- which requires predictors to be simultaneously calibrated over a potentially complex and overlapping collection of protected subpopulations (such as groups defined by ethnicity, race, or income). We conduct the first comprehensive study evaluating the usefulness of multicalibration post-processing across a broad set of tabular, image, and language datasets for models spanning from simple decision trees to 90 million parameter fine-tuned LLMs. Our findings can be summarized as follows: (1) models which are calibrated out of the box tend to be relatively multicalibrated without any additional post-processing; (2) multicalibration post-processing can help inherently uncalibrated models and large vision and language models; and (3) traditional calibration measures may sometimes provide multicalibration implicitly. More generally, we also distill many independent observations which may be useful for practical and effective applications of multicalibration post-processing in real-world contexts. We also release a python package implementing multicalibration algorithms, available via `pip install multicalibration'.

cs.LG↗

arXiv2024

Transductive Learning Is Compact

We demonstrate a compactness result holding broadly across supervised learning with a general class of loss functions: Any hypothesis class $H$ is learnable with transductive sample complexity $m$ precisely when all of its finite projections are learnable with sample complexity $m$. We prove that this exact form of compactness holds for realizable and agnostic learning with respect to any proper metric loss function (e.g., any norm on $\mathbb{R}^d$) and any continuous loss on a compact space (e.g., cross-entropy, squared loss). For realizable learning with improper metric losses, we show that exact compactness of sample complexity can fail, and provide matching upper and lower bounds of a factor of 2 on the extent to which such sample complexities can differ. We conjecture that larger gaps are possible for the agnostic case. Furthermore, invoking the equivalence between sample complexities in the PAC and transductive models (up to lower order factors, in the realizable case) permits us to directly port our results to the PAC model, revealing an almost-exact form of compactness holding broadly in PAC learning.

cs.LG↗

arXiv2024

On the Statistical Complexity of Sample Amplification

The ``sample amplification'' problem formalizes the following question: Given $n$ i.i.d. samples drawn from an unknown distribution $P$, when is it possible to produce a larger set of $n+m$ samples which cannot be distinguished from $n+m$ i.i.d. samples drawn from $P$? In this work, we provide a firm statistical foundation for this problem by deriving generally applicable amplification procedures, lower bound techniques and connections to existing statistical notions. Our techniques apply to a large class of distributions including the exponential family, and establish a rigorous connection between sample amplification and distribution learning.

math.ST↗

arXiv2024

A Spectral View of Adversarially Robust Features

Given the apparent difficulty of learning models that are robust to adversarial perturbations, we propose tackling the simpler problem of developing adversarially robust features. Specifically, given a dataset and metric of interest, the goal is to return a function (or multiple functions) that 1) is robust to adversarial perturbations, and 2) has significant variation across the datapoints. We establish strong connections between adversarially robust features and a natural spectral property of the geometry of the dataset and metric of interest. This connection can be leveraged to provide both robust features, and a lower bound on the robustness of any function that has significant variance across the dataset. Finally, we provide empirical evidence that the adversarially robust features given by this spectral approach can be fruitfully leveraged to learn a robust (and accurate) model.

cs.LG↗

arXiv2024

Sample Amplification: Increasing Dataset Size even when Learning is Impossible

Given data drawn from an unknown distribution, $D$, to what extent is it possible to ``amplify'' this dataset and output an even larger set of samples that appear to have been drawn from $D$? We formalize this question as follows: an $(n,m)$ $\text{amplification procedure}$ takes as input $n$ independent draws from an unknown distribution $D$, and outputs a set of $m > n$ ``samples''. An amplification procedure is valid if no algorithm can distinguish the set of $m$ samples produced by the amplifier from a set of $m$ independent draws from $D$, with probability greater than $2/3$. Perhaps surprisingly, in many settings, a valid amplification procedure exists, even when the size of the input dataset, $n$, is significantly less than what would be necessary to learn $D$ to non-trivial accuracy. Specifically we consider two fundamental settings: the case where $D$ is an arbitrary discrete distribution supported on $\le k$ elements, and the case where $D$ is a $d$-dimensional Gaussian with unknown mean, and fixed covariance. In the first case, we show that an $\left(n, n + Θ(\frac{n}{\sqrt{k}})\right)$ amplifier exists. In particular, given $n=O(\sqrt{k})$ samples from $D$, one can output a set of $m=n+1$ datapoints, whose total variation distance from the distribution of $m$ i.i.d. draws from $D$ is a small constant, despite the fact that one would need quadratically more data, $n=Θ(k)$, to learn $D$ up to small constant total variation distance. In the Gaussian case, we show that an $\left(n,n+Θ(\frac{n}{\sqrt{d}} )\right)$ amplifier exists, even though learning the distribution to small constant total variation distance requires $Θ(d)$ samples. In both the discrete and Gaussian settings, we show that these results are tight, to constant factors. Beyond these results, we formalize a number of curious directions for future research along this vein.

cs.LG↗

arXiv2024

Efficient Convex Optimization Requires Superlinear Memory

We show that any memory-constrained, first-order algorithm which minimizes $d$-dimensional, $1$-Lipschitz convex functions over the unit ball to $1/\mathrm{poly}(d)$ accuracy using at most $d^{1.25 - δ}$ bits of memory must make at least $\tildeΩ(d^{1 + (4/3)δ})$ first-order queries (for any constant $δ\in [0, 1/4]$). Consequently, the performance of such memory-constrained algorithms are a polynomial factor worse than the optimal $\tilde{O}(d)$ query bound for this problem obtained by cutting plane methods that use $\tilde{O}(d^2)$ memory. This resolves a COLT 2019 open problem of Woodworth and Srebro.

cs.LG↗

arXiv2024

Regularization and Optimal Multiclass Learning

The quintessential learning algorithm of empirical risk minimization (ERM) is known to fail in various settings for which uniform convergence does not characterize learning. It is therefore unsurprising that the practice of machine learning is rife with considerably richer algorithmic techniques for successfully controlling model capacity. Nevertheless, no such technique or principle has broken away from the pack to characterize optimal learning in these more general settings. The purpose of this work is to characterize the role of regularization in perhaps the simplest setting for which ERM fails: multiclass learning with arbitrary label sets. Using one-inclusion graphs (OIGs), we exhibit optimal learning algorithms that dovetail with tried-and-true algorithmic principles: Occam's Razor as embodied by structural risk minimization (SRM), the principle of maximum entropy, and Bayesian reasoning. Most notably, we introduce an optimal learner which relaxes structural risk minimization on two dimensions: it allows the regularization function to be "local" to datapoints, and uses an unsupervised learning stage to learn this regularizer at the outset. We justify these relaxations by showing that they are necessary: removing either dimension fails to yield a near-optimal learner. We also extract from OIGs a combinatorial sequence we term the Hall complexity, which is the first to characterize a problem's transductive error rate exactly. Lastly, we introduce a generalization of OIGs and the transductive learning setting to the agnostic case, where we show that optimal orientations of Hamming graphs -- judged using nodes' outdegrees minus a system of node-dependent credits -- characterize optimal learners exactly. We demonstrate that an agnostic version of the Hall complexity again characterizes error rates exactly, and exhibit an optimal learner using maximum entropy programs.

cs.LG↗

arXiv2024

Pre-trained Large Language Models Use Fourier Features to Compute Addition

Pre-trained large language models (LLMs) exhibit impressive mathematical reasoning capabilities, yet how they compute basic arithmetic, such as addition, remains unclear. This paper shows that pre-trained LLMs add numbers using Fourier features -- dimensions in the hidden state that represent numbers via a set of features sparse in the frequency domain. Within the model, MLP and attention layers use Fourier features in complementary ways: MLP layers primarily approximate the magnitude of the answer using low-frequency features, while attention layers primarily perform modular addition (e.g., computing whether the answer is even or odd) using high-frequency features. Pre-training is crucial for this mechanism: models trained from scratch to add numbers only exploit low-frequency features, leading to lower accuracy. Introducing pre-trained token embeddings to a randomly initialized model rescues its performance. Overall, our analysis demonstrates that appropriate pre-trained representations (e.g., Fourier features) can unlock the ability of Transformers to learn precise mechanisms for algorithmic tasks.

cs.LG↗

arXiv2024

Optimal Multiclass U-Calibration Error and Beyond

We consider the problem of online multiclass U-calibration, where a forecaster aims to make sequential distributional predictions over $K$ classes with low U-calibration error, that is, low regret with respect to all bounded proper losses simultaneously. Kleinberg et al. (2023) developed an algorithm with U-calibration error $O(K\sqrt{T})$ after $T$ rounds and raised the open question of what the optimal bound is. We resolve this question by showing that the optimal U-calibration error is $Θ(\sqrt{KT})$ -- we start with a simple observation that the Follow-the-Perturbed-Leader algorithm of Daskalakis and Syrgkanis (2016) achieves this upper bound, followed by a matching lower bound constructed with a specific proper loss (which, as a side result, also proves the optimality of the algorithm of Daskalakis and Syrgkanis (2016) in the context of online learning against an adversary with finite choices). We also strengthen our results under natural assumptions on the loss functions, including $Θ(\log T)$ U-calibration error for Lipschitz proper losses, $O(\log T)$ U-calibration error for a certain class of decomposable proper losses, U-calibration error bounds for proper losses with a low covering number, and others.

stat.ML↗

arXiv2024

Stability and Multigroup Fairness in Ranking with Uncertain Predictions

Rankings are ubiquitous across many applications, from search engines to hiring committees. In practice, many rankings are derived from the output of predictors. However, when predictors trained for classification tasks have intrinsic uncertainty, it is not obvious how this uncertainty should be represented in the derived rankings. Our work considers ranking functions: maps from individual predictions for a classification task to distributions over rankings. We focus on two aspects of ranking functions: stability to perturbations in predictions and fairness towards both individuals and subgroups. Not only is stability an important requirement for its own sake, but -- as we show -- it composes harmoniously with individual fairness in the sense of Dwork et al. (2012). While deterministic ranking functions cannot be stable aside from trivial scenarios, we show that the recently proposed uncertainty aware (UA) ranking functions of Singh et al. (2021) are stable. Our main result is that UA rankings also achieve multigroup fairness through successful composition with multiaccurate or multicalibrated predictors. Our work demonstrates that UA rankings naturally interpolate between group and individual level fairness guarantees, while simultaneously satisfying stability guarantees important whenever machine-learned predictions are used.

cs.LG↗

arXiv2023

Mitigating Simplicity Bias in Deep Learning for Improved OOD Generalization and Robustness

Neural networks (NNs) are known to exhibit simplicity bias where they tend to prefer learning 'simple' features over more 'complex' ones, even when the latter may be more informative. Simplicity bias can lead to the model making biased predictions which have poor out-of-distribution (OOD) generalization. To address this, we propose a framework that encourages the model to use a more diverse set of features to make predictions. We first train a simple model, and then regularize the conditional mutual information with respect to it to obtain the final model. We demonstrate the effectiveness of this framework in various problem settings and real-world applications, showing that it effectively addresses simplicity bias and leads to more features being used, enhances OOD generalization, and improves subgroup robustness and fairness. We complement these results with theoretical analyses of the effect of the regularization and its OOD generalization properties.

cs.LG↗

arXiv2023

Recovery Guarantees for Quadratic Tensors with Sparse Observations

We consider the tensor completion problem of predicting the missing entries of a tensor. The commonly used CP model has a triple product form, but an alternate family of quadratic models, which are the sum of pairwise products instead of a triple product, have emerged from applications such as recommendation systems. Non-convex methods are the method of choice for learning quadratic models, and this work examines their sample complexity and error guarantee. Our main result is that with the number of samples being only linear in the dimension, all local minima of the mean squared error objective are global minima and recover the original tensor. We substantiate our theoretical results with experiments on synthetic and real-world data, showing that quadratic models have better performance than CP models where there are a limited amount of observations available.

cs.LG↗

arXiv2023

Fairness in Matching under Uncertainty

The prevalence and importance of algorithmic two-sided marketplaces has drawn attention to the issue of fairness in such settings. Algorithmic decisions are used in assigning students to schools, users to advertisers, and applicants to job interviews. These decisions should heed the preferences of individuals, and simultaneously be fair with respect to their merits (synonymous with fit, future performance, or need). Merits conditioned on observable features are always \emph{uncertain}, a fact that is exacerbated by the widespread use of machine learning algorithms to infer merit from the observables. As our key contribution, we carefully axiomatize a notion of individual fairness in the two-sided marketplace setting which respects the uncertainty in the merits; indeed, it simultaneously recognizes uncertainty as the primary potential cause of unfairness and an approach to address it. We design a linear programming framework to find fair utility-maximizing distributions over allocations, and we show that the linear program is robust to perturbations in the estimated parameters of the uncertain merit distributions, a key property in combining the approach with machine learning techniques.

cs.LG↗

arXiv2023

NeuroSketch: Fast and Approximate Evaluation of Range Aggregate Queries with Neural Networks

Range aggregate queries (RAQs) are an integral part of many real-world applications, where, often, fast and approximate answers for the queries are desired. Recent work has studied answering RAQs using machine learning (ML) models, where a model of the data is learned to answer the queries. However, there is no theoretical understanding of why and when the ML based approaches perform well. Furthermore, since the ML approaches model the data, they fail to capitalize on any query specific information to improve performance in practice. In this paper, we focus on modeling ``queries'' rather than data and train neural networks to learn the query answers. This change of focus allows us to theoretically study our ML approach to provide a distribution and query dependent error bound for neural networks when answering RAQs. We confirm our theoretical results by developing NeuroSketch, a neural network framework to answer RAQs in practice. Extensive experimental study on real-world, TPC-benchmark and synthetic datasets show that NeuroSketch answers RAQs multiple orders of magnitude faster than state-of-the-art and with better accuracy.

cs.DB↗