arXiv research

This work is an analytical and numerical study of the composition of several fractals into one and of the relation between the composite dimension and the dimensions of the component fractals. In the case of composition of standard IFS with segments of equal size, the composite dimension can be expressed as a function …

Study on deep neural networks using branching processes and Mehler's formula.

The stochastic gradient descent has been widely used for solving composite optimization problems in big data analyses. Many algorithms and convergence properties have been developed. The composite functions were convex primarily and gradually nonconvex composite functions have been adopted to obtain more desirable prop…

This paper explores optimising acquisition functions in Bayesian optimisation.

Deep networks learn sparse hierarchical features without CoD.

Stochastic compositional optimization arises in many important machine learning tasks such as value function evaluation in reinforcement learning and portfolio management. The objective function is the composition of two expectations of stochastic functions, and is more challenging to optimize than vanilla stochastic o…

Improved subgradient method tackles ill-conditioned composite optimization problems.

Gaussian processes struggle with compositional functions, but deep Gaussian processes can outperform.

Many machine learning, statistical inference, and portfolio optimization problems require minimization of a composition of expected value functions (CEVF). Of particular interest is the finite-sum versions of such compositional optimization problems (FS-CEVF). Compositional stochastic variance reduced gradient (C-SVRG)…

In the past few years, off-policy reinforcement learning methods have shown promising results in their application for robot control. Deep Q-learning, however, still suffers from poor data-efficiency and is susceptible to stochasticity in the environment or reward functions which is limiting with regard to real-world a…

In this paper, we specify what functions induce the bounded composition operators on a reproducing kernel Hilbert space (RKHS) associated with an analytic positive definite function defined on $\mathbf{R}^d$. We prove that only affine transforms can do so in a pretty large class of RKHS. Our result covers not only the …

In this paper, we consider the convex and non-convex composition problem with the structure $\frac{1}{n}\sum\nolimits_{i = 1}^n {{F_i}( {G( x )} )}$, where $G( x )=\frac{1}{n}\sum\nolimits_{j = 1}^n {{G_j}( x )}$ is the inner function, and $F_i(\cdot)$ is the outer function. We explore the variance reduction based met…

Deep neural networks can approximate complex functions through repeated compositions of a fixed-size ReLU network.

We consider composite loss functions for multiclass prediction comprising a proper (i.e., Fisher-consistent) loss over probability distributions and an inverse link function. We establish conditions for their (strong) convexity and explore the implications. We also show how the separation of concerns afforded by using …

We study losses for binary classification and class probability estimation and extend the understanding of them from margin losses to general composite losses which are the composition of a proper loss with a link function. We characterise when margin losses can be proper composite losses, explicitly show how to determ…

Paper proposes iLPA for solving DC composite optimization problems, with applications to matrix completion with outliers.

We consider the problem of minimizing the composition of a smooth (nonconvex) function and a smooth vector mapping, where the inner mapping is in the form of an expectation over some random variable or a finite sum. We propose a stochastic composite gradient method that employs an incremental variance-reduced estimator…

We consider the composition optimization with two expected-value functions in the form of $\frac{1}{n}\sum\nolimits_{i = 1}^n F_i(\frac{1}{m}\sum\nolimits_{j = 1}^m G_j(x))+R(x)$, { which formulates many important problems in statistical learning and machine learning such as solving Bellman equations in reinforcement l…

Deep learning can learn compositional functions more efficiently by breaking them into stages.

Sublinearly structured DNNs achieve feature learning consistency for compositional functions.

Defines 'nowhere coexpanding functions' and studies their fixed points.

Here we study non-convex composite optimization: first, a finite-sum of smooth but non-convex functions, and second, a general function that admits a simple proximal mapping. Most research on stochastic methods for composite optimization assumes convexity or strong convexity of each function. In this paper, we extend t…

Unified algorithm for minimizing composite functions with flexible design.

New geometric approach for analyzing compositional data like gut microbiomes.

We present a compositional embedding framework that infers not just a single class per input image, but a set of classes, in the setting of one-shot learning. Specifically, we propose and evaluate several novel models consisting of (1) an embedding function f trained jointly with a "composition" function g that compute…

Unified analysis of multi-task functional linear regression with manifold and composite penalties.

Model predicts composite structures assembly quality with input uncertainty.

In this paper, we present C-ADAM, the first adaptive solver for compositional problems involving a non-linear functional nesting of expected values. We proof that C-ADAM converges to a stationary point in $\mathcal{O}(δ^{-2.25})$ with $δ$ being a precision parameter. Moreover, we demonstrate the importance of our resul…

We consider optimization of composite objective functions, i.e., of the form $f(x)=g(h(x))$, where $h$ is a black-box derivative-free expensive-to-evaluate function with vector-valued outputs, and $g$ is a cheap-to-evaluate real-valued function. While these problems can be solved with standard Bayesian optimization, we…

Consider the stochastic composition optimization problem where the objective is a composition of two expected-value functions. We propose a new stochastic first-order method, namely the accelerated stochastic compositional proximal gradient (ASC-PG) method, which updates based on queries to the sampling oracle using tw…

BOIS optimizes complex systems by leveraging structural knowledge.

New method for causal inference with complex treatment compositions.

In this paper we determine a number of meaningful compositions of higher order of a set of functions, which is considered in Malesevic (1998), in implicit and explicit form. Results which are obtained are applied to the vector analysis in order to determine the number of meaningful differential operations of higher ord…

Curvature penalties improve interpretability of KANs without sacrificing accuracy.

We consider in this work a system of two stochastic differential equations named the perturbed compositional gradient flow. By introducing a separation of fast and slow scales of the two equations, we show that the limit of the slow motion is given by an averaged ordinary differential equation. We then demonstrate that…

Classical stochastic gradient methods are well suited for minimizing expected-value objective functions. However, they do not apply to the minimization of a nonlinear function involving expected values or a composition of two expected-value functions, i.e., problems of the form $\min_x \mathbf{E}_v [f_v\big(\mathbf{E}_…

This work theoretically investigates the performance of a composite neural network. A composite neural network is a rooted directed acyclic graph combining a set of pre-trained and non-instantiated neural network models, where a pre-trained neural network model is well-crafted for a specific task and targeted to approx…

New samplers improve compositional generation with diffusion models.

New sparse GP model learns compositional kernels efficiently.

This work investigates the framework and performance issues of the composite neural network, which is composed of a collection of pre-trained and non-instantiated neural network models connected as a rooted directed acyclic graph for solving complicated applications. A pre-trained neural network model is generally well…

In this contribution we describe an approach to evolve composite covariance functions for Gaussian processes using genetic programming. A critical aspect of Gaussian processes and similar kernel-based models such as SVM is, that the covariance function should be adapted to the modeled data. Frequently, the squared expo…

Paper analyzes stability and generalization of SCO algorithms.

Study dual representations for quasiconvex systemic risk measures.

Transformer architecture struggles with complex tasks due to limitations in function composition.

Lower bounds set for infinite-precision transformers.

This paper explores the non-convex composition optimization in the form including inner and outer finite-sum functions with a large number of component functions. This problem arises in some important applications such as nonlinear embedding and reinforcement learning. Although existing approaches such as stochastic gr…

Adding noise controls capacity of function compositions.

AutoBayes simplifies variational inference by composing models and optimizing them.