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 $C_d \propto d^r$.

On the sample complexity of estimation in logistic regression

The logistic regression model is one of the most popular data generation model in noisy binary classification problems. In this work, we study the sample complexity of estimating the parameters of the logistic regression model up to a given $\ell_2$ error, in terms of the dimension and the inverse temperature, with standard normal covariates. The inverse temperature controls the signal-to-noise ratio of the data generation process. While both generalization bounds and asymptotic performance of the maximum-likelihood estimator for logistic regression are well-studied, the non-asymptotic sample complexity that shows the dependence on error and the inverse temperature for parameter estimation is absent from previous analyses. We show that the sample complexity curve has two change-points (or critical points) in terms of the inverse temperature, clearly separating the low, moderate, and high temperature regimes.

Representational Strengths and Limitations of Transformers

Attention layers, as commonly used in transformers, form the backbone of modern deep learning, yet there is no mathematical description of their benefits and deficiencies as compared with other architectures. In this work we establish both positive and negative results on the representation power of attention layers, with a focus on intrinsic complexity parameters such as width, depth, and embedding dimension. On the positive side, we present a sparse averaging task, where recurrent networks and feedforward networks all have complexity scaling polynomially in the input size, whereas transformers scale merely logarithmically in the input size; furthermore, we use the same construction to show the necessity and role of a large embedding dimension in a transformer. On the negative side, we present a triple detection task, where attention layers in turn have complexity scaling linearly in the input size; as this scenario seems rare in practice, we also present natural variants that can be efficiently solved by attention layers. The proof techniques emphasize the value of communication complexity in the analysis of transformers and related models, and the role of sparse averaging as a prototypical attention task, which even finds use in the analysis of triple detection.

Group conditional validity via multi-group learning

We consider the problem of distribution-free conformal prediction and the criterion of group conditional validity. This criterion is motivated by many practical scenarios including hidden stratification and group fairness. Existing methods achieve such guarantees under either restrictive grouping structure or distributional assumptions, or they are overly-conservative under heteroskedastic noise. We propose a simple reduction to the problem of achieving validity guarantees for individual populations by leveraging algorithms for a problem called multi-group learning. This allows us to port theoretical guarantees from multi-group learning to obtain obtain sample complexity guarantees for conformal prediction. We also provide a new algorithm for multi-group learning for groups with hierarchical structure. Using this algorithm in our reduction leads to improved sample complexity guarantees with a simpler predictor structure.

Intrinsic dimensionality and generalization properties of the $\mathcal{R}$-norm inductive bias

We study the structural and statistical properties of $\mathcal{R}$-norm minimizing interpolants of datasets labeled by specific target functions. The $\mathcal{R}$-norm is the basis of an inductive bias for two-layer neural networks, recently introduced to capture the functional effect of controlling the size of network weights, independently of the network width. We find that these interpolants are intrinsically multivariate functions, even when there are ridge functions that fit the data, and also that the $\mathcal{R}$-norm inductive bias is not sufficient for achieving statistically optimal generalization for certain learning problems. Altogether, these results shed new light on an inductive bias that is connected to practical neural network training.

Statistical-Computational Trade-offs in Tensor PCA and Related Problems via Communication Complexity

Tensor PCA is a stylized statistical inference problem introduced by Montanari and Richard to study the computational difficulty of estimating an unknown parameter from higher-order moment tensors. Unlike its matrix counterpart, Tensor PCA exhibits a statistical-computational gap, i.e., a sample size regime where the problem is information-theoretically solvable but conjectured to be computationally hard. This paper derives computational lower bounds on the run-time of memory bounded algorithms for Tensor PCA using communication complexity. These lower bounds specify a trade-off among the number of passes through the data sample, the sample size, and the memory required by any algorithm that successfully solves Tensor PCA. While the lower bounds do not rule out polynomial-time algorithms, they do imply that many commonly-used algorithms, such as gradient descent and power method, must have a higher iteration count when the sample size is not large enough. Similar lower bounds are obtained for Non-Gaussian Component Analysis, a family of statistical estimation problems in which low-order moment tensors carry no information about the unknown parameter. Finally, stronger lower bounds are obtained for an asymmetric variant of Tensor PCA and related statistical estimation problems. These results explain why many estimators for these problems use a memory state that is significantly larger than the effective dimensionality of the parameter of interest.