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The optimal portfolio size for a venture capital (VC) fund is a topic often debated, but there is no consensus on the best strategy. This is because it is a function of many factors. It is not easy to find a general formula that can be applied to all situations, and it largely depends on the goal of the fund. In this report, we will go through the different factors step by step, studying how they affect fund returns and the optimal portfolio size, starting with some basic assumptions and then increasing the complexity of the model.

We introduce a framework for uncertainty estimation that both describes and extends many existing methods. We consider typical hyperparameters involved in classical training as random variables and marginalise them out to capture various sources of uncertainty in the parameter space. We investigate which forms and combinations of marginalisation are most useful from a practical point of view on standard benchmarking data sets. Moreover, we discuss how some marginalisations may produce reliable estimates of uncertainty without the need for extensive hyperparameter tuning and/or large-scale ensembling., Comment: Presented at the ICML 2021 Workshop on Uncertainty and Robustness in Deep Learning

We introduce a novel architecture for graph networks which is equivariant to any transformation in the coordinate embeddings that preserves the distance between neighbouring nodes. In particular, it is equivariant to the Euclidean and conformal orthogonal groups in $n$-dimensions. Thanks to its equivariance properties, the proposed model is extremely more data efficient with respect to classical graph architectures and also intrinsically equipped with a better inductive bias. We show that, learning on a minimal amount of data, the architecture we propose can perfectly generalise to unseen data in a synthetic problem, while much more training data are required from a standard model to reach comparable performance., Comment: Presented at the ICML 2021 Workshop on Subset Selection in Machine Learning: From Theory to Practice. arXiv admin note: text overlap with arXiv:2105.14058

Exploiting symmetries and invariance in data is a powerful, yet not fully exploited, way to achieve better generalisation with more efficiency. In this paper, we introduce two graph network architectures that are equivariant to several types of transformations affecting the node coordinates. First, we build equivariance to any transformation in the coordinate embeddings that preserves the distance between neighbouring nodes, allowing for equivariance to the Euclidean group. Then, we introduce angle attributes to build equivariance to any angle preserving transformation - thus, to the conformal group. Thanks to their equivariance properties, the proposed models can be vastly more data efficient with respect to classical graph architectures, intrinsically equipped with a better inductive bias and better at generalising. We demonstrate these capabilities on a synthetic dataset composed of $n$-dimensional geometric objects. Additionally, we provide examples of their limitations when (the right) symmetries are not present in the data.

In this draft paper, we introduce a novel architecture for graph networks which is equivariant to the Euclidean group in $n$-dimensions. The model is designed to work with graph networks in their general form and can be shown to include particular variants as special cases. Thanks to its equivariance properties, we expect the proposed model to be more data efficient with respect to classical graph architectures and also intrinsically equipped with a better inductive bias. We defer investigating this matter to future work.

The paper addresses large-scale, convex optimization problems that need to be solved in a distributed way by agents communicating according to a random time-varying graph. Specifically, the goal of the network is to minimize the sum of local costs, while satisfying local and coupling constraints. Agents communicate according to a time-varying model in which edges of an underlying connected graph are active at each iteration with certain non-uniform probabilities. By relying on a primal decomposition scheme applied to an equivalent problem reformulation, we propose a novel distributed algorithm in which agents negotiate a local allocation of the total resource only with neighbors with active communication links. The algorithm is studied as a subgradient method with block-wise updates, in which blocks correspond to the graph edges that are active at each iteration. Thanks to this analysis approach, we show almost sure convergence to the optimal cost of the original problem and almost sure asymptotic primal recovery without resorting to averaging mechanisms typically employed in dual decomposition schemes. Explicit sublinear convergence rates are provided under the assumption of diminishing and constant step-sizes. Finally, an extensive numerical study on a plug-in electric vehicle charging problem corroborates the theoretical results.

This paper deals with a network of computing agents aiming to solve an online optimization problem in a distributed fashion, i.e., by means of local computation and communication, without any central coordinator. We propose the gradient tracking with adaptive momentum estimation (GTAdam) distributed algorithm, which combines a gradient tracking mechanism with first and second order momentum estimates of the gradient. The algorithm is analyzed in the online setting for strongly convex cost functions with Lipschitz continuous gradients. We provide an upper bound for the dynamic regret given by a term related to the initial conditions and another term related to the temporal variations of the objective functions. Moreover, a linear convergence rate is guaranteed in the static setup. The algorithm is tested on a time-varying classification problem, on a (moving) target localization problem, and in a stochastic optimization setup from image classification. In these numerical experiments from multi-agent learning, GTAdam outperforms state-of-the-art distributed optimization methods.

We present a distributed optimization algorithm for solving online personalized optimization problems over a network of computing and communicating nodes, each of which linked to a specific user. The local objective functions are assumed to have a composite structure and to consist of a known time-varying (engineering) part and an unknown (user-specific) part. Regarding the unknown part, it is assumed to have a known parametric (e.g., quadratic) structure a priori, whose parameters are to be learned along with the evolution of the algorithm. The algorithm is composed of two intertwined components: (i) a dynamic gradient tracking scheme for finding local solution estimates and (ii) a recursive least squares scheme for estimating the unknown parameters via user's noisy feedback on the local solution estimates. The algorithm is shown to exhibit a bounded regret under suitable assumptions. Finally, a numerical example corroborates the theoretical analysis.

The recently developed Distributed Block Proximal Method, for solving stochastic big-data convex optimization problems, is studied in this paper under the assumption of constant stepsizes and strongly convex (possibly non-smooth) local objective functions. This class of problems arises in many learning and classification problems in which, for example, strongly-convex regularizing functions are included in the objective function, the decision variable is extremely high dimensional, and large datasets are employed. The algorithm produces local estimates by means of block-wise updates and communication among the agents. The expected distance from the (global) optimum, in terms of cost value, is shown to decay linearly to a constant value which is proportional to the selected local stepsizes. A numerical example involving a classification problem corroborates the theoretical results., Comment: arXiv admin note: text overlap with arXiv:1905.04214

In this paper, we introduce the concept of collective learning (CL) which exploits the notion of collective intelligence in the field of distributed semi-supervised learning. The proposed framework draws inspiration from the learning behavior of human beings, who alternate phases involving collaboration, confrontation and exchange of views with other consisting of studying and learning on their own. On this regard, CL comprises two main phases: a self-training phase in which learning is performed on local private (labeled) data only and a collective training phase in which proxy-labels are assigned to shared (unlabeled) data by means of a consensus-based algorithm. In the considered framework, heterogeneous systems can be connected over the same network, each with different computational capabilities and resources and everyone in the network may take advantage of the cooperation and will eventually reach higher performance with respect to those it can reach on its own. An extensive experimental campaign on an image classification problem emphasizes the properties of CL by analyzing the performance achieved by the cooperating agents., Comment: update references

In this paper, the extension of the framework of Learning from Constraints (LfC) to a distributed setting where multiple parties, connected over the network, contribute to the learning process is studied. LfC relies on the generic notion of "constraint" to inject knowledge into the learning problem and, due to its generality, it deals with possibly nonconvex constraints, enforced either in a hard or soft way. Motivated by recent progresses in the field of distributed and constrained nonconvex optimization, we apply the (distributed) Asynchronous Method of Multipliers (ASYMM) to LfC. The study shows that such a method allows us to support scenarios where selected constraints (i.e., knowledge), data, and outcomes of the learning process can be locally stored in each computational node without being shared with the rest of the network, opening the road to further investigations into privacy-preserving LfC. Constraints act as a bridge between what is shared over the net and what is private to each node and no central authority is required. We demonstrate the applicability of these ideas in two distributed real-world settings in the context of digit recognition and document classification.

In this paper we introduce DISROPT, a Python package for distributed optimization over networks. We focus on cooperative set-ups in which an optimization problem must be solved by peer-to-peer processors (without central coordinators) that have access only to partial knowledge of the entire problem. To reflect this, agents in DISROPT are modeled as entities that are initialized with their local knowledge of the problem. Agents then run local routines and communicate with each other to solve the global optimization problem. A simple syntax has been designed to allow for an easy modeling of the problems. The package comes with many distributed optimization algorithms that are already embedded. Moreover, the package provides full-fledged functionalities for communication and local computation, which can be used to design and implement new algorithms. DISROPT is available at github.com/disropt/disropt under the GPL license, with a complete documentation and many examples.

Estimating the limbs pose in a wearable way may benefit multiple areas such as rehabilitation, teleoperation, human-robot interaction, gaming, and many more. Several solutions are commercially available, but they are usually expensive or not wearable/portable. We present a wearable pose estimation system (WePosE), based on inertial measurements units (IMUs), for motion analysis and body tracking. Differently from camera-based approaches, the proposed system does not suffer from occlusion problems and lighting conditions, it is cost effective and it can be used in indoor and outdoor environments. Moreover, since only accelerometers and gyroscopes are used to estimate the orientation, the system can be used also in the presence of iron and magnetic disturbances. An experimental validation using a high precision optical tracker has been performed. Results confirmed the effectiveness of the proposed approach., Comment: 8 pages, 8 figures

In this paper we deal with a network of computing agents with local processing and neighboring communication capabilities that aim at solving (without any central unit) a submodular optimization problem. The cost function is the sum of many local submodular functions and each agent in the network has access to one function in the sum only. In this \emph{distributed} set-up, in order to preserve their own privacy, agents communicate with neighbors but do not share their local cost functions. We propose a distributed algorithm in which agents resort to the Lov\`{a}sz extension of their local submodular functions and perform local updates and communications in terms of single blocks of the entire optimization variable. Updates are performed by means of a greedy algorithm which is run only until the selected block is computed, thus resulting in a reduced computational burden. The proposed algorithm is shown to converge in expected value to the optimal cost of the problem, and an approximate solution to the submodular problem is retrieved by a thresholding operation. As an application, we consider a distributed image segmentation problem in which each agent has access only to a portion of the entire image. While agents cannot segment the entire image on their own, they correctly complete the task by cooperating through the proposed distributed algorithm.

In this paper we introduce a class of novel distributed algorithms for solving stochastic big-data convex optimization problems over directed graphs. In the addressed set-up, the dimension of the decision variable can be extremely high and the objective function can be nonsmooth. The general algorithm consists of two main steps: a consensus step and an update on a single block of the optimization variable, which is then broadcast to neighbors. Three special instances of the proposed method, involving particular problem structures, are then presented. In the general case, the convergence of a dynamic consensus algorithm over random row stochastic matrices is shown. Then, the convergence of the proposed algorithm to the optimal cost is proven in expected value. Exact convergence is achieved when using diminishing (local) stepsizes, while approximate convergence is attained when constant stepsizes are employed. The convergence rate is shown to be sublinear and an explicit rate is provided in the case of constant stepsizes. Finally, the algorithm is tested on a distributed classification problem, first on synthetic data and, then, on a real, high-dimensional, text dataset.

This paper presents a fully asynchronous and distributed approach for tackling optimization problems in which both the objective function and the constraints may be nonconvex. In the considered network setting each node is active upon triggering of a local timer and has access only to a portion of the objective function and to a subset of the constraints. In the proposed technique, based on the method of multipliers, each node performs, when it wakes up, either a descent step on a local augmented Lagrangian or an ascent step on the local multiplier vector. Nodes realize when to switch from the descent step to the ascent one through an asynchronous distributed logic-AND, which detects when all the nodes have reached a predefined tolerance in the minimization of the augmented Lagrangian. It is shown that the resulting distributed algorithm is equivalent to a block coordinate descent for the minimization of the global augmented Lagrangian. This allows one to extend the properties of the centralized method of multipliers to the considered distributed framework. Two application examples are presented to validate the proposed approach: a distributed source localization problem and the parameter estimation of a neural network., Comment: arXiv admin note: substantial text overlap with arXiv:1803.06482

This work addresses the distributed estimation problem in a set membership framework. The agents of a network collect measurements which are affected by bounded errors, thus implying that the unknown parameters to be estimated belong to a suitable feasible set. Two distributed algorithms are considered, based on projections of the estimate of each agent onto its local feasible set. The main contribution of the paper is to show that such algorithms are asymptotic interpolatory estimators, i.e. they converge to an element of the global feasible set, under the assumption that the feasible set associated to each measurement is convex. The proposed techniques are demonstrated on a distributed linear regression estimation problem.

This paper addresses a class of constrained optimization problems over networks in which local cost functions and constraints can be nonconvex. We propose an asynchronous distributed optimization algorithm, relying on the centralized Method of Multipliers, in which each node wakes up in an uncoordinated fashion and performs either a descent step on a local Augmented Lagrangian or an ascent step on the local multiplier vector. These two phases are regulated by a distributed logic-AND, which allows nodes to understand when the descent on the (whole) Augmented Lagrangian is sufficiently small. We show that this distributed algorithm is equivalent to a block coordinate descent algorithm for the minimization of the Augmented Lagrangian followed by an update of the whole multiplier vector. Thus, the proposed algorithm inherits the convergence properties of the Method of Multipliers.