Computing Approximate $\ell_p$ Sensitivities

Recent works in dimensionality reduction for regression tasks have introduced the notion of sensitivity, an estimate of the importance of a specific datapoint in a dataset, offering provable guarantees on the quality of the approximation after removing low-sensitivity datapoints via subsampling. However, fast algorithms for approximating $\ell_p$ sensitivities, which we show is equivalent to approximate $\ell_p$ regression, are known for only the $\ell_2$ setting, in which they are termed leverage scores. In this work, we provide efficient algorithms for approximating $\ell_p$ sensitivities and related summary statistics of a given matrix. In particular, for a given $n \times d$ matrix, we compute $\alpha$-approximation to its $\ell_1$ sensitivities at the cost of $O(n/\alpha)$ sensitivity computations. For estimating the total $\ell_p$ sensitivity (i.e. the sum of $\ell_p$ sensitivities), we provide an algorithm based on importance sampling of $\ell_p$ Lewis weights, which computes a constant factor approximation to the total sensitivity at the cost of roughly $O(\sqrt{d})$ sensitivity computations. Furthermore, we estimate the maximum $\ell_1$ sensitivity, up to a $\sqrt{d}$ factor, using $O(d)$ sensitivity computations. We generalize all these results to $\ell_p$ norms for $p > 1$. Lastly, we experimentally show that for a wide class of matrices in real-world datasets, the total sensitivity can be quickly approximated and is significantly smaller than the theoretical prediction, demonstrating that real-world datasets have low intrinsic effective dimensionality.

Hardness of Low Rank Approximation of Entrywise Transformed Matrix Products

Inspired by fast algorithms in natural language processing, we study low rank approximation in the entrywise transformed setting where we want to find a good rank $k$ approximation to $f(U \cdot V)$, where $U, V^\top \in \mathbb{R}^{n \times r}$ are given, $r = O(\log(n))$, and $f(x)$ is a general scalar function. Previous work in sublinear low rank approximation has shown that if both (1) $U = V^\top$ and (2) $f(x)$ is a PSD kernel function, then there is an $O(nk^{\omega-1})$ time constant relative error approximation algorithm, where $\omega \approx 2.376$ is the exponent of matrix multiplication. We give the first conditional time hardness results for this problem, demonstrating that both conditions (1) and (2) are in fact necessary for getting better than $n^{2-o(1)}$ time for a relative error low rank approximation for a wide class of functions. We give novel reductions from the Strong Exponential Time Hypothesis (SETH) that rely on lower bounding the leverage scores of flat sparse vectors and hold even when the rank of the transformed matrix $f(UV)$ and the target rank are $n^{o(1)}$, and when $U = V^\top$. Furthermore, even when $f(x) = x^p$ is a simple polynomial, we give runtime lower bounds in the case when $U \neq V^\top$ of the form $\Omega(\min(n^{2-o(1)}, \Omega(2^p)))$. Lastly, we demonstrate that our lower bounds are tight by giving an $O(n \cdot \text{poly}(k, 2^p, 1/\epsilon))$ time relative error approximation algorithm and a fast $O(n \cdot \text{poly}(k, p, 1/\epsilon))$ additive error approximation using fast tensor-based sketching. Additionally, since our low rank algorithms rely on matrix-vector product subroutines, our lower bounds extend to show that computing $f(UV)W$, for even a small matrix $W$, requires $\Omega(n^{2-o(1)})$ time.

Getting aligned on representational alignment

Biological and artificial information processing systems form representations of the world that they can use to categorize, reason, plan, navigate, and make decisions. To what extent do the representations formed by these diverse systems agree? Can diverging representations still lead to the same behaviors? And how can systems modify their representations to better match those of another system? These questions pertaining to the study of \textbf{\emph{representational alignment}} are at the heart of some of the most active research areas in contemporary cognitive science, neuroscience, and machine learning. Unfortunately, there is limited knowledge-transfer between research communities interested in representational alignment, and much of the progress in one field ends up being rediscovered independently in another, when greater cross-field communication would be advantageous. To improve communication between fields, we propose a unifying framework that can serve as a common language between researchers studying representational alignment. We survey the literature from the fields of cognitive science, neuroscience, and machine learning, and demonstrate how prior work fits into this framework. Finally, we lay out open problems in representational alignment where progress can benefit all three fields. We hope that our work can catalyze cross-disciplinary collaboration and accelerate progress for all communities studying and developing information processing systems. We note that this is a working paper and encourage readers to reach out with their suggestions for future revisions.

Leveraging Contextual Counterfactuals Toward Belief Calibration

Beliefs and values are increasingly being incorporated into our AI systems through alignment processes, such as carefully curating data collection principles or regularizing the loss function used for training. However, the meta-alignment problem is that these human beliefs are diverse and not aligned across populations; furthermore, the implicit strength of each belief may not be well calibrated even among humans, especially when trying to generalize across contexts. Specifically, in high regret situations, we observe that contextual counterfactuals and recourse costs are particularly important in updating a decision maker's beliefs and the strengths to which such beliefs are held. Therefore, we argue that including counterfactuals is key to an accurate calibration of beliefs during alignment. To do this, we first segment belief diversity into two categories: subjectivity (across individuals within a population) and epistemic uncertainty (within an individual across different contexts). By leveraging our notion of epistemic uncertainty, we introduce `the belief calibration cycle' framework to more holistically calibrate this diversity of beliefs with context-driven counterfactual reasoning by using a multi-objective optimization. We empirically apply our framework for finding a Pareto frontier of clustered optimal belief strengths that generalize across different contexts, demonstrating its efficacy on a toy dataset for credit decisions.

Set Learning for Accurate and Calibrated Models

Model overconfidence and poor calibration are common in machine learning and difficult to account for when applying standard empirical risk minimization. In this work, we propose a novel method to alleviate these problems that we call odd-$k$-out learning (OKO), which minimizes the cross-entropy error for sets rather than for single examples. This naturally allows the model to capture correlations across data examples and achieves both better accuracy and calibration, especially in limited training data and class-imbalanced regimes. Perhaps surprisingly, OKO often yields better calibration even when training with hard labels and dropping any additional calibration parameter tuning, such as temperature scaling. We provide theoretical justification, establishing that OKO naturally yields better calibration, and provide extensive experimental analyses that corroborate our theoretical findings. We emphasize that OKO is a general framework that can be easily adapted to many settings and the trained model can be applied to single examples at inference time, without introducing significant run-time overhead or architecture changes.

Leveraging Initial Hints for Free in Stochastic Linear Bandits

We study the setting of optimizing with bandit feedback with additional prior knowledge provided to the learner in the form of an initial hint of the optimal action. We present a novel algorithm for stochastic linear bandits that uses this hint to improve its regret to $\tilde O(\sqrt{T})$ when the hint is accurate, while maintaining a minimax-optimal $\tilde O(d\sqrt{T})$ regret independent of the quality of the hint. Furthermore, we provide a Pareto frontier of tight tradeoffs between best-case and worst-case regret, with matching lower bounds. Perhaps surprisingly, our work shows that leveraging a hint shows provable gains without sacrificing worst-case performance, implying that our algorithm adapts to the quality of the hint for free. We also provide an extension of our algorithm to the case of $m$ initial hints, showing that we can achieve a $\tilde O(m^{2/3}\sqrt{T})$ regret.

Random Hypervolume Scalarizations for Provable Multi-Objective Black Box Optimization

Single-objective black box optimization (also known as zeroth-order optimization) is the process of minimizing a scalar objective $f(x)$, given evaluations at adaptively chosen inputs $x$. In this paper, we consider multi-objective optimization, where $f(x)$ outputs a vector of possibly competing objectives and the goal is to converge to the Pareto frontier. Quantitatively, we wish to maximize the standard hypervolume indicator metric, which measures the dominated hypervolume of the entire set of chosen inputs. In this paper, we introduce a novel scalarization function, which we term the hypervolume scalarization, and show that drawing random scalarizations from an appropriately chosen distribution can be used to efficiently approximate the hypervolume indicator metric. We utilize this connection to show that Bayesian optimization with our scalarization via common acquisition functions, such as Thompson Sampling or Upper Confidence Bound, provably converges to the whole Pareto frontier by deriving tight hypervolume regret bounds on the order of $\widetilde{O}(\sqrt{T})$. Furthermore, we highlight the general utility of our scalarization framework by showing that any provably convergent single-objective optimization process can be effortlessly converted to a multi-objective optimization process with provable convergence guarantees.

Regularized Weighted Low Rank Approximation

The classical low rank approximation problem is to find a rank $k$ matrix $UV$ (where $U$ has $k$ columns and $V$ has $k$ rows) that minimizes the Frobenius norm of $A - UV$. Although this problem can be solved efficiently, we study an NP-hard variant of this problem that involves weights and regularization. A previous paper of [Razenshteyn et al. '16] derived a polynomial time algorithm for weighted low rank approximation with constant rank. We derive provably sharper guarantees for the regularized version by obtaining parameterized complexity bounds in terms of the statistical dimension rather than the rank, allowing for a rank-independent runtime that can be significantly faster. Our improvement comes from applying sharper matrix concentration bounds, using a novel conditioning technique, and proving structural theorems for regularized low rank problems.

Gradientless Descent: High-Dimensional Zeroth-Order Optimization

Zeroth-order optimization is the process of minimizing an objective $f(x)$, given oracle access to evaluations at adaptively chosen inputs $x$. In this paper, we present two simple yet powerful GradientLess Descent (GLD) algorithms that do not rely on an underlying gradient estimate and are numerically stable. We analyze our algorithm from a novel geometric perspective and present a novel analysis that shows convergence within an $\epsilon$-ball of the optimum in $O(kQ\log(n)\log(R/\epsilon))$ evaluations, for {\it any monotone transform} of a smooth and strongly convex objective with latent dimension $k < n$, where the input dimension is $n$, $R$ is the diameter of the input space and $Q$ is the condition number. Our rates are the first of its kind to be both 1) poly-logarithmically dependent on dimensionality and 2) invariant under monotone transformations. We further leverage our geometric perspective to show that our analysis is optimal. Both monotone invariance and its ability to utilize a low latent dimensionality are key to the empirical success of our algorithms, as demonstrated on BBOB and MuJoCo benchmarks.