The \emph{Cheap Gradient Principle}~\citep{Griewank:2008:EDP:1455489} --- the computational cost of computing a $d$-dimensional vector of partial derivatives of a scalar function is nearly the same (often within a factor of $5$) as that of simply computing the scalar function itself --- is of central importance in optimization; it allows us to quickly obtain (high-dimensional) gradients of scalar loss functions which are subsequently used in black box gradient-based optimization procedures. The current state of affairs is markedly different with regards to computing sub-derivatives: widely used ML libraries, including TensorFlow and PyTorch, do \emph{not} correctly compute (generalized) sub-derivatives even on simple differentiable examples. This work considers the question: is there a \emph{Cheap Sub-gradient Principle}? Our main result shows that, under certain restrictions on our library of non-smooth functions (standard in non-linear programming), provably correct generalized sub-derivatives can be computed at a computational cost that is within a (dimension-free) factor of $6$ of the cost of computing the scalar function itself.

Finding tight linear bounds for activation functions in neural networks is an essential part of several state of the art neural network robustness certification tools.

An activation function is an arbitrary, nonlinear,scalar function $f: \mathbb{R}^d \rightarrow \mathbb{R}$. In the existing work on robustness certification, such bounds have been computed using human ingenuity for a handful of the most popular activation functions. While a number of heuristics have been proposed for bounding arbitrary functions,no analysis of the tightness optimality for general scalar functions has been offered yet, to the best of our knowledge. We fill this gap by formulating a concise optimality criterion for tightness of the approximation which allows us tobuild optimal bounds for any function convex in the region of interest $R$. Fora more general class of functions Lipshitz-continuous in $R$ we propose a sampling-based approach (SOL) which, given an instance of the bounding problem, efficiently computes the tightest linear bounds within a given $\varepsilon > 0$ threshold. We leverage an adaptive sampling technique to iteratively build a setof sample points suitable for representing the target activation function.

The \emph{Cheap Gradient Principle}~\citep{Griewank:2008:EDP:1455489} --- the computational cost of computing a $d$-dimensional vector of partial derivatives of a scalar function is nearly the same (often within a factor of $5$) as that of simply computing the scalar function itself --- is of central importance in optimization; it allows us to quickly obtain (high-dimensional) gradients of scalar loss functions which are subsequently used in black box gradient-based optimization procedures. The current state of affairs is markedly different with regards to computing sub-derivatives: widely used ML libraries, including TensorFlow and PyTorch, do \emph{not} correctly compute (generalized) sub-derivatives even on simple differentiable examples. This work considers the question: is there a \emph{Cheap Sub-gradient Principle}? Our main result shows that, under certain restrictions on our library of non-smooth functions (standard in non-linear programming), provably correct generalized sub-derivatives can be computed at a computational cost that is within a (dimension-free) factor of $6$ of the cost of computing the scalar function itself.

Among techniques for high-dimensional linear regression, Sorted L-One Penalized Estimation (SLOPE) generalizes the LASSO via an adaptive $l_1$ regularization that applies heavier penalties to larger coefficients in the model. To achieve such adaptivity, SLOPE requires the specification of a complex hierarchy of penalties, i.e., a monotone penalty sequence in $R^p$, in contrast to a single penalty scalar for LASSO. Tuning this sequence when $p$ is large poses a challenge, as brute force search over a grid of values is computationally prohibitive. In this work, we study the 2-level SLOPE, an important subclass of SLOPE, with only three hyperparameters. We demonstrate both empirically and analytically that 2-level SLOPE not only preserves the advantages of general SLOPE -- such as improved mean squared error and overcoming the Donoho-Tanner power limit -- but also exhibits computational benefits by reducing the penalty hyperparameter space. In particular, we prove that 2-level SLOPE admits a sharp, theoretically tight characterization of the trade-off between true positive proportion (TPP) and false discovery proportion (FDP), contrasting with general SLOPE where only upper and lower bounds are known. Empirical evaluations further underscore the effectiveness of 2-level SLOPE in settings where predictors exhibit high correlation, when the noise is large, or when the underlying signal is not sparse. Our results suggest that 2-level SLOPE offers a robust, scalable alternative to both LASSO and general SLOPE, making it particularly suited for practical high-dimensional data analysis.

Rotational symmetry plays a central role in physics, providing an elegant framework to describe how the properties of 3D objects -- from atoms to the macroscopic scale -- transform under the action of rigid rotations. Equivariant models of 3D point clouds are able to approximate structure-property relations in a way that is fully consistent with the structure of the rotation group, by combining intermediate representations that are themselves spherical tensors. The symmetry constraints however make this approach computationally demanding and cumbersome to implement, which motivates increasingly popular unconstrained architectures that learn approximate symmetries as part of the training process. In this work, we explore a third route to tackle this learning problem, where equivariant functions are expressed as the product of a scalar function of the point cloud coordinates and a small basis of tensors with the appropriate symmetry. We also propose approximations of the general expressions that, while lacking universal approximation properties, are fast, simple to implement, and accurate in practical settings.

The \emph{Cheap Gradient Principle} \citep{Griewank:2008:EDP:1455489} --- the computational cost of computing a d -dimensional vector of partial derivatives of a scalar function is nearly the same (often within a factor of 5) as that of simply computing the scalar function itself --- is of central importance in optimization; it allows us to quickly obtain (high-dimensional) gradients of scalar loss functions which are subsequently used in black box gradient-based optimization procedures. The current state of affairs is markedly different with regards to computing sub-derivatives: widely used ML libraries, including TensorFlow and PyTorch, do \emph{not} correctly compute (generalized) sub-derivatives even on simple differentiable examples. This work considers the question: is there a \emph{Cheap Sub-gradient Principle}? Our main result shows that, under certain restrictions on our library of non-smooth functions (standard in non-linear programming), provably correct generalized sub-derivatives can be computed at a computational cost that is within a (dimension-free) factor of 6 of the cost of computing the scalar function itself.

We propose a new topological tool for computer vision - Scalar Function Topology Divergence (SFTD), which measures the dissimilarity of multi-scale topology between sublevel sets of two functions having a common domain. Functions can be defined on an undirected graph or Euclidean space of any dimensionality. Most of the existing methods for comparing topology are based on Wasserstein distance between persistence barcodes and they don't take into account the localization of topological features. On the other hand, the minimization of SFTD ensures that the corresponding topological features of scalar functions are located in the same places. The proposed tool provides useful visualizations depicting areas where functions have topological dissimilarities. We provide applications of the proposed method to 3D computer vision. In particular, experiments demonstrate that SFTD improves the reconstruction of cellular 3D shapes from 2D fluorescence microscopy images, and helps to identify topological errors in 3D segmentation.

The ability to learn and predict simple functions is a key aspect of human intelligence. Recent works have started to explore this ability using transformer architectures, however it remains unclear whether this is sufficient to recapitulate the extrapolation abilities of people in this domain. Here, we propose to address this gap by augmenting the transformer architecture with two simple inductive learning biases, that are directly adapted from recent models of abstract reasoning in cognitive science. The results we report demonstrate that these biases are helpful in the context of large neural network models, as well as shed light on the types of inductive learning biases that may contribute to human abilities in extrapolation.

An easy to follow tutorial with beginners concepts of optimization, constrained optimization, feasible region and its importance in machine learning.