Vatsal Sharan

Julian Asilis, Siddartha Devic, Shaddin Dughmi et al.

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.

Annie Marsden, Vatsal Sharan, Aaron Sidford et al.

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.

Siddartha Devic, David Kempe, Vatsal Sharan et al.

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.

Deqing Fu, Tian-Qi Chen, Robin Jia et al.

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.

Bhavya Vasudeva, Kameron Shahabi, Vatsal Sharan

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.

Bhavya Vasudeva, Puneesh Deora, Alberto Bietti et al.

Transformer-based language models excel at in-context learning (ICL), where they can adapt to new tasks based on contextual examples, without parameter updates. In a specific form of ICL, which we refer to as \textit{contextual recall}, models pretrained on open-ended text leverage pairwise examples to recall specific facts in novel prompt formats. We investigate whether contextual recall emerges from pretraining alone, what finetuning is required, and what mechanisms drive the necessary representations. For this, we introduce a controlled synthetic framework where pretraining sequences consist of subject-grammar-attribute tuples, with attribute types tied to grammar statistics. We demonstrate that while such pretraining successfully yields factual knowledge, it is insufficient for contextual recall: models fail to implicitly infer attribute types when the grammar statistics are removed in ICL prompts. However, we show that finetuning on tasks requiring implicit inference, distinct from the ICL evaluation, using a subset of subjects, triggers the emergence of contextual recall across all subjects. This transition is accompanied by the formation of low-dimensional latent encodings of the shared attribute type. For mechanistic insight, we derive a construction for an attention-only transformer that replicates the transition from factual to contextual recall, corroborated by empirical validation.

Lunjia Hu, Haipeng Luo, Spandan Senapati et al.

Multicalibration [HJKRR18] is an algorithmic fairness perspective that demands that the predictions of a predictor are correct conditional on themselves and membership in a collection of potentially overlapping subgroups of a population. The work of [NR23] established a surprising connection between multicalibration for an arbitrary property $Γ$ (e.g., mean or median) and property elicitation: a property $Γ$ can be multicalibrated if and only if it is elicitable, where elicitability is the notion that the true property value of a distribution can be obtained by solving a regression problem over the distribution. In the online setting, [NR23] proposed an inefficient algorithm that achieves $\sqrt T$ $\ell_2$-multicalibration error for a hypothesis class of group membership functions and an elicitable property $Γ$, after $T$ rounds of interaction between a forecaster and adversary. In this paper, we generalize multicalibration for an elicitable property $Γ$ from group membership functions to arbitrary bounded hypothesis classes and introduce a stronger notion -- swap multicalibration, following [GKR23]. Subsequently, we propose an oracle-efficient algorithm which, when given access to an online agnostic learner, achieves $T^{1/(r+1)}$ $\ell_r$-swap multicalibration error with high probability (for $r\ge2$) for a hypothesis class with bounded sequential Rademacher complexity and an elicitable property $Γ$. For the special case of $r=2$, this implies an oracle-efficient algorithm that achieves $T^{1/3}$ $\ell_2$-swap multicalibration error, which significantly improves on the previously established bounds for the problem [NR23, GMS25, LSS25a], and completely resolves an open question raised in [GJRR24] on the possibility of an oracle-efficient algorithm that achieves $\sqrt{T}$ $\ell_2$-mean multicalibration error by answering it in a strongly affirmative sense.

Bhavya Vasudeva, Deqing Fu, Tianyi Zhou et al.

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.

Deqing Fu, Tianyi Zhou, Mikhail Belkin et al.

Language models trained on natural text learn to represent numbers using periodic features with dominant periods at $T=2, 5, 10$. In this paper, we identify a two-tiered hierarchy of these features: while Transformers, Linear RNNs, LSTMs, and classical word embeddings trained in different ways all learn features that have period-$T$ spikes in the Fourier domain, only some learn geometrically separable features that can be used to linearly classify a number mod-$T$. To explain this incongruity, we prove that Fourier domain sparsity is necessary but not sufficient for mod-$T$ geometric separability. Empirically, we investigate when model training yields geometrically separable features, finding that the data, architecture, optimizer, and tokenizer all play key roles. In particular, we identify two different routes through which models can acquire geometrically separable features: they can learn them from complementary co-occurrence signals in general language data, including text-number co-occurrence and cross-number interaction, or from multi-token (but not single-token) addition problems. Overall, our results highlight the phenomenon of convergent evolution in feature learning: A diverse range of models learn similar features from different training signals.

Siddartha Devic, Aleksandra Korolova, David Kempe et al.

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.

Haipeng Luo, Spandan Senapati, Vatsal Sharan

Calibration is a fundamental concept that aims at ensuring the reliability of probabilistic predictions by aligning them with real-world outcomes. There is a surge of studies on new calibration measures that are easier to optimize compared to the classical $\ell_1$-Calibration while still having strong implications for downstream applications. One recent such example is the work by Fishelson et al. (2025) who show that it is possible to achieve $O(T^{1/3})$ pseudo $\ell_2$-Calibration error via minimizing pseudo swap regret of the squared loss, which in fact implies the same bound for all bounded proper losses with a smooth univariate form. In this work, we significantly generalize their result in the following ways: (a) in addition to smooth univariate forms, our algorithm also simultaneously achieves $O(T^{1/3})$ swap regret for any proper loss with a twice continuously differentiable univariate form (such as Tsallis entropy); (b) our bounds hold not only for pseudo swap regret that measures losses using the forecaster's distributions on predictions, but also hold for the actual swap regret that measures losses using the forecaster's actual realized predictions. We achieve so by introducing a new stronger notion of calibration called (pseudo) KL-Calibration, which we show is equivalent to the (pseudo) swap regret for log loss. We prove that there exists an algorithm that achieves $O(T^{1/3})$ KL-Calibration error and provide an explicit algorithm that achieves $O(T^{1/3})$ pseudo KL-Calibration error. Moreover, we show that the same algorithm achieves $O(T^{1/3}(\log T)^{-1/3}\log(T/δ))$ swap regret w.p. $\ge 1-δ$ for any proper loss with a smooth univariate form, which implies $O(T^{1/3})$ $\ell_2$-Calibration error. A technical contribution of our work is a new randomized rounding procedure and a non-uniform discretization scheme to minimize the swap regret for log loss.

Siddartha Devic, Tejas Srinivasan, Jesse Thomason et al.

Large Language Models (LLMs) are increasingly assisting users in the real world, yet their reliability remains a concern. Uncertainty quantification (UQ) has been heralded as a tool to enhance human-LLM collaboration by enabling users to know when to trust LLM predictions. We argue that current practices for uncertainty quantification in LLMs are not optimal for developing useful UQ for human users making decisions in real-world tasks. Through an analysis of 40 LLM UQ methods, we identify three prevalent practices hindering the community's progress toward its goal of benefiting downstream users: 1) evaluating on benchmarks with low ecological validity; 2) considering only epistemic uncertainty; and 3) optimizing metrics that are not necessarily indicative of downstream utility. For each issue, we propose concrete user-centric practices and research directions that LLM UQ researchers should consider. Instead of hill-climbing on unrepresentative tasks using imperfect metrics, we argue that the community should adopt a more human-centered approach to LLM uncertainty quantification.

Bhavya Vasudeva, Jung Whan Lee, Vatsal Sharan et al.

Adam is the de facto optimization algorithm for several deep learning applications, but an understanding of its implicit bias and how it differs from other algorithms, particularly standard first-order methods such as (stochastic) gradient descent (GD), remains limited. In practice, neural networks (NNs) trained with SGD are known to exhibit simplicity bias -- a tendency to find simple solutions. In contrast, we show that Adam is more resistant to such simplicity bias. First, we investigate the differences in the implicit biases of Adam and GD when training two-layer ReLU NNs on a binary classification task with Gaussian data. We find that GD exhibits a simplicity bias, resulting in a linear decision boundary with a suboptimal margin, whereas Adam leads to much richer and more diverse features, producing a nonlinear boundary that is closer to the Bayes' optimal predictor. This richer decision boundary also allows Adam to achieve higher test accuracy both in-distribution and under certain distribution shifts. We theoretically prove these results by analyzing the population gradients. Next, to corroborate our theoretical findings, we present extensive empirical results showing that this property of Adam leads to superior generalization across various datasets with spurious correlations where NNs trained with SGD are known to show simplicity bias and do not generalize well under certain distributional shifts.

Woody Haosheng Gan, Deqing Fu, Julian Asilis et al.

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.

Tianyi Zhou, Deqing Fu, Mahdi Soltanolkotabi et al.

Large Language Models (LLMs) typically represent numbers using multiple tokens, which requires the model to aggregate these tokens to interpret numerical values. This fragmentation makes both training and inference less efficient and adversely affects the model's performance on number-related tasks. Inspired by the observation that pre-trained LLMs internally learn Fourier-like features for number tokens, we propose Fourier Number Embedding (FoNE), a novel method that directly maps numbers into the embedding space with their Fourier features. FoNE encodes each number as a single token with only two embedding dimensions per digit, effectively capturing numerical values without fragmentation. This compact representation accelerates both training and inference. Compared to traditional subword and digit-wise embeddings, FoNE not only reduces computational overhead but also achieves higher accuracy across various numerical tasks including addition, subtraction and multiplication. On 6-digit decimal addition, FoNE requires 64$\times$ less data to achieve 99% accuracy than subword and digit-wise embeddings while using 3$\times$ and 6$\times$ fewer tokens per number, respectively. Furthermore, FoNE is the only method that yields 100% accuracy on over 100,000 test examples for addition, subtraction, and multiplication. The codes and visualization are available at https://fouriernumber.github.io/.

Guan Zhe Hong, Bhavya Vasudeva, Vatsal Sharan et al.

When large language models (LLMs) use in-context learning (ICL) to solve a new task, they must infer latent concepts from demonstration examples. This raises the question of whether and how transformers represent latent structures as part of their computation. Our work experiments with several controlled tasks, studying this question using mechanistic interpretability. First, we show that in transitive reasoning tasks with a latent, discrete concept, the model successfully identifies the latent concept and does step-by-step concept composition. This builds upon prior work that analyzes single-step reasoning. Then, we consider tasks parameterized by a latent numerical concept. We discover low-dimensional subspaces in the model's representation space, where the geometry cleanly reflects the underlying parameterization. Overall, we show that small and large models can indeed disentangle and utilize latent concepts that they learn in-context from a handful of abbreviated demonstrations.

Julian Asilis, Siddartha Devic, Shaddin Dughmi et al.

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.

Haipeng Luo, Spandan Senapati, Vatsal Sharan

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.

Bhavya Vasudeva, Puneesh Deora, Yize Zhao et al.

The growing adoption of spectrum-aware matrix-valued optimizers such as Muon and Shampoo in deep learning motivates a systematic study of their generalization properties and, in particular, when they might outperform competitive algorithms. We approach this question by introducing appropriate simplifying abstractions as follows: First, we use imbalanced data as a testbed. Second, we study the canonical form of such optimizers, which is Spectral Gradient Descent (SpecGD) -- each update step is $UV^T$ where $UΣV^T$ is the truncated SVD of the gradient. Third, within this framework we identify a canonical setting for which we precisely quantify when SpecGD outperforms vanilla Euclidean GD. For a Gaussian mixture data model and both linear and bilinear models, we show that unlike GD, which prioritizes learning dominant principal components of the data first, SpecGD learns all principal components of the data at equal rates. We demonstrate how this translates to a growing gap in balanced accuracy favoring SpecGD early in training and further show that the gap remains consistent even when the GD counterpart uses adaptive step-sizes via normalization. By extending the analysis to deep linear models, we show that depth amplifies these effects. We empirically verify our theoretical findings on a variety of imbalanced datasets. Our experiments compare practical variants of spectral methods, like Muon and Shampoo, against their Euclidean counterparts and Adam. The results validate our findings that these spectral optimizers achieve superior generalization by promoting a more balanced learning of the data's underlying components.

Qilin Ye, Deqing Fu, Robin Jia et al.

Transformers often fail to learn generalizable algorithms, instead relying on brittle heuristics. Using graph connectivity as a testbed, we explain this phenomenon both theoretically and empirically. We consider a simplified Transformer architecture, the disentangled Transformer, and prove that an $L$-layer model has capacity to solve for graphs with diameters up to exactly $3^L$, implementing an algorithm equivalent to computing powers of the adjacency matrix. We analyze the training-dynamics, and show that the learned strategy hinges on whether most training instances are within this model capacity. Within-capacity graphs (diameter $\leq 3^L$) drive the learning of a correct algorithmic solution while beyond-capacity graphs drive the learning of a simple heuristic based on node degrees. Finally, we empirically demonstrate that restricting training data within a model's capacity leads to both standard and disentangled transformers learning the exact algorithm rather than the degree-based heuristic.

Rina Panigrahy, Vatsal Sharan

Safety, trust and Artificial General Intelligence (AGI) are aspirational goals in artificial intelligence (AI) systems, and there are several informal interpretations of these notions. In this paper, we propose strict, mathematical definitions of safety, trust, and AGI, and demonstrate a fundamental incompatibility between them. We define safety of a system as the property that it never makes any false claims, trust as the assumption that the system is safe, and AGI as the property of an AI system always matching or exceeding human capability. Our core finding is that -- for our formal definitions of these notions -- a safe and trusted AI system cannot be an AGI system: for such a safe, trusted system there are task instances which are easily and provably solvable by a human but not by the system. We note that we consider strict mathematical definitions of safety and trust, and it is possible for real-world deployments to instead rely on alternate, practical interpretations of these notions. We show our results for program verification, planning, and graph reachability. Our proofs draw parallels to Gödel's incompleteness theorems and Turing's proof of the undecidability of the halting problem, and can be regarded as interpretations of Gödel's and Turing's results.

Nathan Derhake, Siddartha Devic, Dutch Hansen et al.

Calibration is a critical property for establishing the trustworthiness of predictors that provide uncertainty estimates. Multicalibration is a strengthening of calibration which requires that predictors be calibrated on a potentially overlapping collection of subsets of the domain. As multicalibration grows in popularity with practitioners, an essential question is: how do we measure how multicalibrated a predictor is? Błasiok et al. (2023) considered this question for standard calibration by introducing the distance to calibration framework (dCE) to understand how calibration metrics relate to each other and the ground truth. Building on the dCE framework, we consider the auditability of the distance to multicalibration of a predictor $f$. We begin by considering two natural generalizations of dCE to multiple subgroups: worst group dCE (wdMC), and distance to multicalibration (dMC). We argue that there are two essential properties of any multicalibration error metric: 1) the metric should capture how much $f$ would need to be modified in order to be perfectly multicalibrated; and 2) the metric should be auditable in an information theoretic sense. We show that wdMC and dMC each fail to satisfy one of these two properties, and that similar barriers arise when considering the auditability of general distance to multigroup fairness notions. We then propose two (equivalent) multicalibration metrics which do satisfy these requirements: 1) a continuized variant of dMC; and 2) a distance to intersection multicalibration, which leans on intersectional fairness desiderata. Along the way, we shed light on the loss-landscape of distance to multicalibration and the geometry of the set of perfectly multicalibrated predictors. Our findings may have implications for the development of stronger multicalibration algorithms as well as multigroup auditing more generally.

Devansh Gupta, Meisam Razaviyayn, Vatsal Sharan

Differentially private zeroth-order optimization methods have recently gained popularity in private fine tuning of machine learning models due to their reduced memory requirements. Current approaches for privatizing zeroth-order methods rely on adding Gaussian noise to the estimated zeroth-order gradients. However, since the search direction in the zeroth-order methods is inherently random, researchers including Tang et al. (2024) and Zhang et al. (2024a) have raised an important question: is the inherent noise in zeroth-order estimators sufficient to ensure the overall differential privacy of the algorithm? This work settles this question for a class of oracle-based optimization algorithms where the oracle returns zeroth-order gradient estimates. In particular, we show that for a fixed initialization, there exist strongly convex objective functions such that running (Projected) Zeroth-Order Gradient Descent (ZO-GD) is not differentially private. Furthermore, we show that even with random initialization and without revealing (initial and) intermediate iterates, the privacy loss in ZO-GD can grow superlinearly with the number of iterations when minimizing convex objective functions.

Julian Asilis, Siddartha Devic, Shaddin Dughmi et al.

Proper learning refers to the setting in which learners must emit predictors in the underlying hypothesis class $H$, and often leads to learners with simple algorithmic forms (e.g. empirical risk minimization (ERM), structural risk minimization (SRM)). The limitation of proper learning, however, is that there exist problems which can only be learned improperly, e.g. in multiclass classification. Thus, we ask: Under what assumptions on the hypothesis class or the information provided to the learner is a problem properly learnable? We first demonstrate that when the unlabeled data distribution is given, there always exists an optimal proper learner governed by distributional regularization, a randomized generalization of regularization. We refer to this setting as the distribution-fixed PAC model, and continue to evaluate the learner on its worst-case performance over all distributions. Our result holds for all metric loss functions and any finite learning problem (with no dependence on its size). Further, we demonstrate that sample complexities in the distribution-fixed PAC model can shrink by only a logarithmic factor from the classic PAC model, strongly refuting the role of unlabeled data in PAC learning (from a worst-case perspective). We complement this with impossibility results which obstruct any characterization of proper learnability in the realizable PAC model. First, we observe that there are problems whose proper learnability is logically undecidable, i.e., independent of the ZFC axioms. We then show that proper learnability is not a monotone property of the underlying hypothesis class, and that it is not a local property (in a precise sense). Our impossibility results all hold even for the fundamental setting of multiclass classification, and go through a reduction of EMX learning (Ben-David et al., 2019) to proper classification which may be of independent interest.

Omar Salemohamed, Laurent Charlin, Shivam Garg et al.

We propose a general framework for end-to-end learning of data structures. Our framework adapts to the underlying data distribution and provides fine-grained control over query and space complexity. Crucially, the data structure is learned from scratch, and does not require careful initialization or seeding with candidate data structures/algorithms. We first apply this framework to the problem of nearest neighbor search. In several settings, we are able to reverse-engineer the learned data structures and query algorithms. For 1D nearest neighbor search, the model discovers optimal distribution (in)dependent algorithms such as binary search and variants of interpolation search. In higher dimensions, the model learns solutions that resemble k-d trees in some regimes, while in others, they have elements of locality-sensitive hashing. The model can also learn useful representations of high-dimensional data and exploit them to design effective data structures. We also adapt our framework to the problem of estimating frequencies over a data stream, and believe it could also be a powerful discovery tool for new problems.

Dutch Hansen, Siddartha Devic, Preetum Nakkiran et al.

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'.

Tianyi Zhou, Deqing Fu, Vatsal Sharan et al.

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.

Parikshit Gopalan, Nina Narodytska, Omer Reingold et al.

Estimating the Kullback-Leibler (KL) divergence between two distributions given samples from them is well-studied in machine learning and information theory. Motivated by considerations of multi-group fairness, we seek KL divergence estimates that accurately reflect the contributions of sub-populations to the overall divergence. We model the sub-populations coming from a rich (possibly infinite) family $\mathcal{C}$ of overlapping subsets of the domain. We propose the notion of multi-group attribution for $\mathcal{C}$, which requires that the estimated divergence conditioned on every sub-population in $\mathcal{C}$ satisfies some natural accuracy and fairness desiderata, such as ensuring that sub-populations where the model predicts significant divergence do diverge significantly in the two distributions. Our main technical contribution is to show that multi-group attribution can be derived from the recently introduced notion of multi-calibration for importance weights [HKRR18, GRSW21]. We provide experimental evidence to support our theoretical results, and show that multi-group attribution provides better KL divergence estimates when conditioned on sub-populations than other popular algorithms.

Brian Axelrod, Shivam Garg, Yanjun Han et al.

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.

Jonathan Kelner, Annie Marsden, Vatsal Sharan et al.

We provide new gradient-based methods for efficiently solving a broad class of ill-conditioned optimization problems. We consider the problem of minimizing a function $f : \mathbb{R}^d \rightarrow \mathbb{R}$ which is implicitly decomposable as the sum of $m$ unknown non-interacting smooth, strongly convex functions and provide a method which solves this problem with a number of gradient evaluations that scales (up to logarithmic factors) as the product of the square-root of the condition numbers of the components. This complexity bound (which we prove is nearly optimal) can improve almost exponentially on that of accelerated gradient methods, which grow as the square root of the condition number of $f$. Additionally, we provide efficient methods for solving stochastic, quadratic variants of this multiscale optimization problem. Rather than learn the decomposition of $f$ (which would be prohibitively expensive), our methods apply a clean recursive "Big-Step-Little-Step" interleaving of standard methods. The resulting algorithms use $\tilde{\mathcal{O}}(d m)$ space, are numerically stable, and open the door to a more fine-grained understanding of the complexity of convex optimization beyond condition number.

Parikshit Gopalan, Adam Tauman Kalai, Omer Reingold et al.

Loss minimization is a dominant paradigm in machine learning, where a predictor is trained to minimize some loss function that depends on an uncertain event (e.g., "will it rain tomorrow?''). Different loss functions imply different learning algorithms and, at times, very different predictors. While widespread and appealing, a clear drawback of this approach is that the loss function may not be known at the time of learning, requiring the algorithm to use a best-guess loss function. We suggest a rigorous new paradigm for loss minimization in machine learning where the loss function can be ignored at the time of learning and only be taken into account when deciding an action. We introduce the notion of an (${\mathcal{L}},\mathcal{C}$)-omnipredictor, which could be used to optimize any loss in a family ${\mathcal{L}}$. Once the loss function is set, the outputs of the predictor can be post-processed (a simple univariate data-independent transformation of individual predictions) to do well compared with any hypothesis from the class $\mathcal{C}$. The post processing is essentially what one would perform if the outputs of the predictor were true probabilities of the uncertain events. In a sense, omnipredictors extract all the predictive power from the class $\mathcal{C}$, irrespective of the loss function in $\mathcal{L}$. We show that such "loss-oblivious'' learning is feasible through a connection to multicalibration, a notion introduced in the context of algorithmic fairness. In addition, we show how multicalibration can be viewed as a solution concept for agnostic boosting, shedding new light on past results. Finally, we transfer our insights back to the context of algorithmic fairness by providing omnipredictors for multi-group loss minimization.

Atish Agarwala, Abhimanyu Das, Brendan Juba et al.

Can deep learning solve multiple tasks simultaneously, even when they are unrelated and very different? We investigate how the representations of the underlying tasks affect the ability of a single neural network to learn them jointly. We present theoretical and empirical findings that a single neural network is capable of simultaneously learning multiple tasks from a combined data set, for a variety of methods for representing tasks -- for example, when the distinct tasks are encoded by well-separated clusters or decision trees over certain task-code attributes. More concretely, we present a novel analysis that shows that families of simple programming-like constructs for the codes encoding the tasks are learnable by two-layer neural networks with standard training. We study more generally how the complexity of learning such combined tasks grows with the complexity of the task codes; we find that combining many tasks may incur a sample complexity penalty, even though the individual tasks are easy to learn. We provide empirical support for the usefulness of the learning bounds by training networks on clusters, decision trees, and SQL-style aggregation.

Parikshit Gopalan, Omer Reingold, Vatsal Sharan et al.

The ratio between the probability that two distributions $R$ and $P$ give to points $x$ are known as importance weights or propensity scores and play a fundamental role in many different fields, most notably, statistics and machine learning. Among its applications, importance weights are central to domain adaptation, anomaly detection, and estimations of various divergences such as the KL divergence. We consider the common setting where $R$ and $P$ are only given through samples from each distribution. The vast literature on estimating importance weights is either heuristic, or makes strong assumptions about $R$ and $P$ or on the importance weights themselves. In this paper, we explore a computational perspective to the estimation of importance weights, which factors in the limitations and possibilities obtainable with bounded computational resources. We significantly strengthen previous work that use the MaxEntropy approach, that define the importance weights based on a distribution $Q$ closest to $P$, that looks the same as $R$ on every set $C \in \mathcal{C}$, where $\mathcal{C}$ may be a huge collection of sets. We show that the MaxEntropy approach may fail to assign high average scores to sets $C \in \mathcal{C}$, even when the average of ground truth weights for the set is evidently large. We similarly show that it may overestimate the average scores to sets $C \in \mathcal{C}$. We therefore formulate Sandwiching bounds as a notion of set-wise accuracy for importance weights. We study these bounds to show that they capture natural completeness and soundness requirements from the weights. We present an efficient algorithm that under standard learnability assumptions computes weights which satisfy these bounds. Our techniques rely on a new notion of multicalibrated partitions of the domain of the distributions, which appear to be useful objects in their own right.

Parikshit Gopalan, Vatsal Sharan, Udi Wieder

We consider the problem of detecting anomalies in a large dataset. We propose a framework called Partial Identification which captures the intuition that anomalies are easy to distinguish from the overwhelming majority of points by relatively few attribute values. Formalizing this intuition, we propose a geometric anomaly measure for a point that we call PIDScore, which measures the minimum density of data points over all subcubes containing the point. We present PIDForest: a random forest based algorithm that finds anomalies based on this definition. We show that it performs favorably in comparison to several popular anomaly detection methods, across a broad range of benchmarks. PIDForest also provides a succinct explanation for why a point is labelled anomalous, by providing a set of features and ranges for them which are relatively uncommon in the dataset.

Brian Axelrod, Shivam Garg, Vatsal Sharan et al.

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.

Vatsal Sharan, Aaron Sidford, Gregory Valiant

We consider the problem of performing linear regression over a stream of $d$-dimensional examples, and show that any algorithm that uses a subquadratic amount of memory exhibits a slower rate of convergence than can be achieved without memory constraints. Specifically, consider a sequence of labeled examples $(a_1,b_1), (a_2,b_2)\ldots,$ with $a_i$ drawn independently from a $d$-dimensional isotropic Gaussian, and where $b_i = \langle a_i, x\rangle + η_i,$ for a fixed $x \in \mathbb{R}^d$ with $\|x\|_2 = 1$ and with independent noise $η_i$ drawn uniformly from the interval $[-2^{-d/5},2^{-d/5}].$ We show that any algorithm with at most $d^2/4$ bits of memory requires at least $Ω(d \log \log \frac{1}ε)$ samples to approximate $x$ to $\ell_2$ error $ε$ with probability of success at least $2/3$, for $ε$ sufficiently small as a function of $d$. In contrast, for such $ε$, $x$ can be recovered to error $ε$ with probability $1-o(1)$ with memory $O\left(d^2 \log(1/ε)\right)$ using $d$ examples. This represents the first nontrivial lower bounds for regression with super-linear memory, and may open the door for strong memory/sample tradeoffs for continuous optimization.

Shivam Garg, Vatsal Sharan, Brian Hu Zhang et al.

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.

Hongyang R. Zhang, Vatsal Sharan, Moses Charikar et al.

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.

Vatsal Sharan, Parikshit Gopalan, Udi Wieder

We consider the problem of finding anomalies in high-dimensional data using popular PCA based anomaly scores. The naive algorithms for computing these scores explicitly compute the PCA of the covariance matrix which uses space quadratic in the dimensionality of the data. We give the first streaming algorithms that use space that is linear or sublinear in the dimension. We prove general results showing that \emph{any} sketch of a matrix that satisfies a certain operator norm guarantee can be used to approximate these scores. We instantiate these results with powerful matrix sketching techniques such as Frequent Directions and random projections to derive efficient and practical algorithms for these problems, which we validate over real-world data sets. Our main technical contribution is to prove matrix perturbation inequalities for operators arising in the computation of these measures.

Vatsal Sharan, Sham Kakade, Percy Liang et al.

We study the problem of learning overcomplete HMMs---those that have many hidden states but a small output alphabet. Despite having significant practical importance, such HMMs are poorly understood with no known positive or negative results for efficient learning. In this paper, we present several new results---both positive and negative---which help define the boundaries between the tractable and intractable settings. Specifically, we show positive results for a large subclass of HMMs whose transition matrices are sparse, well-conditioned, and have small probability mass on short cycles. On the other hand, we show that learning is impossible given only a polynomial number of samples for HMMs with a small output alphabet and whose transition matrices are random regular graphs with large degree. We also discuss these results in the context of learning HMMs which can capture long-term dependencies.

Kai Sheng Tai, Vatsal Sharan, Peter Bailis et al.

We introduce a new sub-linear space sketch---the Weight-Median Sketch---for learning compressed linear classifiers over data streams while supporting the efficient recovery of large-magnitude weights in the model. This enables memory-limited execution of several statistical analyses over streams, including online feature selection, streaming data explanation, relative deltoid detection, and streaming estimation of pointwise mutual information. Unlike related sketches that capture the most frequently-occurring features (or items) in a data stream, the Weight-Median Sketch captures the features that are most discriminative of one stream (or class) compared to another. The Weight-Median Sketch adopts the core data structure used in the Count-Sketch, but, instead of sketching counts, it captures sketched gradient updates to the model parameters. We provide a theoretical analysis that establishes recovery guarantees for batch and online learning, and demonstrate empirical improvements in memory-accuracy trade-offs over alternative memory-budgeted methods, including count-based sketches and feature hashing.

Vatsal Sharan, Kai Sheng Tai, Peter Bailis et al.

What learning algorithms can be run directly on compressively-sensed data? In this work, we consider the question of accurately and efficiently computing low-rank matrix or tensor factorizations given data compressed via random projections. We examine the approach of first performing factorization in the compressed domain, and then reconstructing the original high-dimensional factors from the recovered (compressed) factors. In both the matrix and tensor settings, we establish conditions under which this natural approach will provably recover the original factors. While it is well-known that random projections preserve a number of geometric properties of a dataset, our work can be viewed as showing that they can also preserve certain solutions of non-convex, NP-Hard problems like non-negative matrix factorization. We support these theoretical results with experiments on synthetic data and demonstrate the practical applicability of compressed factorization on real-world gene expression and EEG time series datasets.

Vatsal Sharan, Gregory Valiant

The popular Alternating Least Squares (ALS) algorithm for tensor decomposition is efficient and easy to implement, but often converges to poor local optima---particularly when the weights of the factors are non-uniform. We propose a modification of the ALS approach that is as efficient as standard ALS, but provably recovers the true factors with random initialization under standard incoherence assumptions on the factors of the tensor. We demonstrate the significant practical superiority of our approach over traditional ALS for a variety of tasks on synthetic data---including tensor factorization on exact, noisy and over-complete tensors, as well as tensor completion---and for computing word embeddings from a third-order word tri-occurrence tensor.

Vatsal Sharan, Sham Kakade, Percy Liang et al.

We consider the problem of predicting the next observation given a sequence of past observations, and consider the extent to which accurate prediction requires complex algorithms that explicitly leverage long-range dependencies. Perhaps surprisingly, our positive results show that for a broad class of sequences, there is an algorithm that predicts well on average, and bases its predictions only on the most recent few observation together with a set of simple summary statistics of the past observations. Specifically, we show that for any distribution over observations, if the mutual information between past observations and future observations is upper bounded by $I$, then a simple Markov model over the most recent $I/ε$ observations obtains expected KL error $ε$---and hence $\ell_1$ error $\sqrtε$---with respect to the optimal predictor that has access to the entire past and knows the data generating distribution. For a Hidden Markov Model with $n$ hidden states, $I$ is bounded by $\log n$, a quantity that does not depend on the mixing time, and we show that the trivial prediction algorithm based on the empirical frequencies of length $O(\log n/ε)$ windows of observations achieves this error, provided the length of the sequence is $d^{Ω(\log n/ε)}$, where $d$ is the size of the observation alphabet. We also establish that this result cannot be improved upon, even for the class of HMMs, in the following two senses: First, for HMMs with $n$ hidden states, a window length of $\log n/ε$ is information-theoretically necessary to achieve expected $\ell_1$ error $\sqrtε$. Second, the $d^{Θ(\log n/ε)}$ samples required to estimate the Markov model for an observation alphabet of size $d$ is necessary for any computationally tractable learning algorithm, assuming the hardness of strongly refuting a certain class of CSPs.