Learning Gaussian Multi-Index Models with Gradient Flow: Time Complexity and Directional Convergence

This work focuses on the gradient flow dynamics of a neural network model that uses correlation loss to approximate a multi-index function on high-dimensional standard Gaussian data. Specifically, the multi-index function we consider is a sum of neurons $f^*(x) \!=\! \sum_{j=1}^k \! \sigma^*(v_j^T x)$ where $v_1, \dots, v_k$ are unit vectors, and $\sigma^*$ lacks the first and second Hermite polynomials in its Hermite expansion. It is known that, for the single-index case ($k\!=\!1$), overcoming the search phase requires polynomial time complexity. We first generalize this result to multi-index functions characterized by vectors in arbitrary directions. After the search phase, it is not clear whether the network neurons converge to the index vectors, or get stuck at a sub-optimal solution. When the index vectors are orthogonal, we give a complete characterization of the fixed points and prove that neurons converge to the nearest index vectors. Therefore, using $n \! \asymp \! k \log k$ neurons ensures finding the full set of index vectors with gradient flow with high probability over random initialization. When $ v_i^T v_j \!=\! \beta \! \geq \! 0$ for all $i \neq j$, we prove the existence of a sharp threshold $\beta_c \!=\! c/(c+k)$ at which the fixed point that computes the average of the index vectors transitions from a saddle point to a minimum. Numerical simulations show that using a correlation loss and a mild overparameterization suffices to learn all of the index vectors when they are nearly orthogonal, however, the correlation loss fails when the dot product between the index vectors exceeds a certain threshold.

Interactive Machine Teaching by Labeling Rules and Instances

Weakly supervised learning aims to reduce the cost of labeling data by using expert-designed labeling rules. However, existing methods require experts to design effective rules in a single shot, which is difficult in the absence of proper guidance and tooling. Therefore, it is still an open question whether experts should spend their limited time writing rules or instead providing instance labels via active learning. In this paper, we investigate how to exploit an expert's limited time to create effective supervision. First, to develop practical guidelines for rule creation, we conduct an exploratory analysis of diverse collections of existing expert-designed rules and find that rule precision is more important than coverage across datasets. Second, we compare rule creation to individual instance labeling via active learning and demonstrate the importance of both across 6 datasets. Third, we propose an interactive learning framework, INTERVAL, that achieves efficiency by automatically extracting candidate rules based on rich patterns (e.g., by prompting a language model), and effectiveness by soliciting expert feedback on both candidate rules and individual instances. Across 6 datasets, INTERVAL outperforms state-of-the-art weakly supervised approaches by 7% in F1. Furthermore, it requires as few as 10 queries for expert feedback to reach F1 values that existing active learning methods cannot match even with 100 queries.

One-layer transformers fail to solve the induction heads task

A simple communication complexity argument proves that no one-layer transformer can solve the induction heads task unless its size is exponentially larger than the size sufficient for a two-layer transformer.

Transformers Provably Learn Sparse Token Selection While Fully-Connected Nets Cannot

The transformer architecture has prevailed in various deep learning settings due to its exceptional capabilities to select and compose structural information. Motivated by these capabilities, Sanford et al. proposed the sparse token selection task, in which transformers excel while fully-connected networks (FCNs) fail in the worst case. Building upon that, we strengthen the FCN lower bound to an average-case setting and establish an algorithmic separation of transformers over FCNs. Specifically, a one-layer transformer trained with gradient descent provably learns the sparse token selection task and, surprisingly, exhibits strong out-of-distribution length generalization. We provide empirical simulations to justify our theoretical findings.

Group-wise oracle-efficient algorithms for online multi-group learning

We study the problem of online multi-group learning, a learning model in which an online learner must simultaneously achieve small prediction regret on a large collection of (possibly overlapping) subsequences corresponding to a family of groups. Groups are subsets of the context space, and in fairness applications, they may correspond to subpopulations defined by expressive functions of demographic attributes. In contrast to previous work on this learning model, we consider scenarios in which the family of groups is too large to explicitly enumerate, and hence we seek algorithms that only access groups via an optimization oracle. In this paper, we design such oracle-efficient algorithms with sublinear regret under a variety of settings, including: (i) the i.i.d. setting, (ii) the adversarial setting with smoothed context distributions, and (iii) the adversarial transductive setting.

Seasonality Patterns in 311-Reported Foodborne Illness Cases and Machine Learning-Identified Indications of Foodborne Illnesses from Yelp Reviews, New York City, 2022-2023

Restaurants are critical venues at which to investigate foodborne illness outbreaks due to shared sourcing, preparation, and distribution of foods. Formal channels to report illness after food consumption, such as 311, New York City's non-emergency municipal service platform, are underutilized. Given this, online social media platforms serve as abundant sources of user-generated content that provide critical insights into the needs of individuals and populations. We extracted restaurant reviews and metadata from Yelp to identify potential outbreaks of foodborne illness in connection with consuming food from restaurants. Because the prevalence of foodborne illnesses may increase in warmer months as higher temperatures breed more favorable conditions for bacterial growth, we aimed to identify seasonal patterns in foodborne illness reports from 311 and identify seasonal patterns of foodborne illness from Yelp reviews for New York City restaurants using a Hierarchical Sigmoid Attention Network (HSAN). We found no evidence of significant bivariate associations between any variables of interest. Given the inherent limitations of relying solely on user-generated data for public health insights, it is imperative to complement these sources with other data streams and insights from subject matter experts. Future investigations should involve conducting these analyses at more granular spatial and temporal scales to explore the presence of such differences or associations.

Transformers, parallel computation, and logarithmic depth

We show that a constant number of self-attention layers can efficiently simulate, and be simulated by, a constant number of communication rounds of Massively Parallel Computation. As a consequence, we show that logarithmic depth is sufficient for transformers to solve basic computational tasks that cannot be efficiently solved by several other neural sequence models and sub-quadratic transformer approximations. We thus establish parallelism as a key distinguishing property of transformers.

Multi-group Learning for Hierarchical Groups

The multi-group learning model formalizes the learning scenario in which a single predictor must generalize well on multiple, possibly overlapping subgroups of interest. We extend the study of multi-group learning to the natural case where the groups are hierarchically structured. We design an algorithm for this setting that outputs an interpretable and deterministic decision tree predictor with near-optimal sample complexity. We then conduct an empirical evaluation of our algorithm and find that it achieves attractive generalization properties on real datasets with hierarchical group structure.

Polynomial time auditing of statistical subgroup fairness for Gaussian data

We study the problem of auditing classifiers with the notion of statistical subgroup fairness. Kearns et al. (2018) has shown that the problem of auditing combinatorial subgroups fairness is as hard as agnostic learning. Essentially all work on remedying statistical measures of discrimination against subgroups assumes access to an oracle for this problem, despite the fact that no efficient algorithms are known for it. If we assume the data distribution is Gaussian, or even merely log-concave, then a recent line of work has discovered efficient agnostic learning algorithms for halfspaces. Unfortunately, the boosting-style reductions given by Kearns et al. required the agnostic learning algorithm to succeed on reweighted distributions that may not be log-concave, even if the original data distribution was. In this work, we give positive and negative results on auditing for the Gaussian distribution: On the positive side, we an alternative approach to leverage these advances in agnostic learning and thereby obtain the first polynomial-time approximation scheme (PTAS) for auditing nontrivial combinatorial subgroup fairness: we show how to audit statistical notions of fairness over homogeneous halfspace subgroups when the features are Gaussian. On the negative side, we find that under cryptographic assumptions, no polynomial-time algorithm can guarantee any nontrivial auditing, even under Gaussian feature distributions, for general halfspace subgroups.

Efficient Estimation of the Central Mean Subspace via Smoothed Gradient Outer Products

We consider the problem of sufficient dimension reduction (SDR) for multi-index models. The estimators of the central mean subspace in prior works either have slow (non-parametric) convergence rates, or rely on stringent distributional conditions (e.g., the covariate distribution $P_{\mathbf{X}}$ being elliptical symmetric). In this paper, we show that a fast parametric convergence rate of form $C_d \cdot n^{-1/2}$ is achievable via estimating the \emph{expected smoothed gradient outer product}, for a general class of distribution $P_{\mathbf{X}}$ admitting Gaussian or heavier distributions. When the link function is a polynomial with a degree of at most $r$ and $P_{\mathbf{X}}$ is the standard Gaussian, we show that the prefactor depends on the ambient dimension $d$ as $C_d \propto d^r$.