How to allocate limited resources to projects that will yield the greatest long-term benefits is a problem that often arises in decision-making under uncertainty. For example, organizations may need to evaluate and select innovation projects with risky returns. Similarly, when allocating resources to research projects, funding agencies are tasked with identifying the most promising proposals based on idiosyncratic criteria. Finally, in participatory budgeting, a local community may need to select a subset of public projects to fund. Regardless of context, agents must estimate the uncertain values of a potentially large number of projects. Developing parsimonious methods to compare these projects, and aggregating agent evaluations so that the overall benefit is maximized, are critical in assembling the best project portfolio. Unlike in standard sorting algorithms, evaluating projects on the basis of uncertain long-term benefits introduces additional complexities. We propose comparison rules based on Quicksort and the Bradley--Terry model, which connects rankings to pairwise "win" probabilities. In our model, each agent determines win probabilities of a pair of projects based on his or her specific evaluation of the projects' long-term benefit. The win probabilities are then appropriately aggregated and used to rank projects. Several of the methods we propose perform better than the two most effective aggregation methods currently available. Additionally, our methods can be combined with sampling techniques to significantly reduce the number of pairwise comparisons. We also discuss how the Bradley--Terry portfolio selection approach can be implemented in practice.

We propose DeepAries , a novel deep reinforcement learning framework for dynamic portfolio management that jointly optimizes the timing and allocation of rebalancing decisions. Unlike prior reinforcement learning methods that employ fixed rebalancing intervals regardless of market conditions, DeepAries adaptively selects optimal rebalancing intervals along with portfolio weights to reduce unnecessary transaction costs and maximize risk-adjusted returns. Our framework integrates a Transformer-based state encoder, which effectively captures complex long-term market dependencies, with Proximal Policy Optimization (PPO) to generate simultaneous discrete (rebalancing intervals) and continuous (asset allocations) actions. Extensive experiments on multiple real-world financial markets demonstrate that DeepAries significantly outperforms traditional fixed-frequency and full-rebalancing strategies in terms of risk-adjusted returns, transaction costs, and drawdowns. Additionally, we provide a live demo of DeepAries at https://deep-aries.github.io/, along with the source code and dataset at https://github.com/dmis-lab/DeepAries, illustrating DeepAries' capability to produce interpretable rebalancing and allocation decisions aligned with shifting market regimes. Overall, DeepAries introduces an innovative paradigm for adaptive and practical portfolio management by integrating both timing and allocation into a unified decision-making process.

This paper introduces a new approach to generating sample paths of unknown stochastic differential equations (SDEs) using diffusion models, a class of generative AI models commonly employed in image and video applications. Unlike the traditional Monte Carlo methods for simulating SDEs, which require explicit specifications of the drift and diffusion coefficients, our method takes a model-free, data-driven approach. Given a finite set of sample paths from an SDE, we utilize conditional diffusion models to generate new, synthetic paths of the same SDE. To demonstrate the effectiveness of our approach, we conduct a simulation experiment to compare our method with alternative benchmark ones including neural SDEs. Furthermore, in an empirical study we leverage these synthetically generated sample paths to enhance the performance of reinforcement learning algorithms for continuous-time mean-variance portfolio selection, hinting promising applications of diffusion models in financial analysis and decision-making.

This algorithm incorporates a relative entropy projection step.

We revisit the classical problem of Bayesian ensembles and address the challenge of learning optimal combinations of Bayesian models in an online, continual learning setting. To this end, we reinterpret existing approaches such as Bayesian model averaging (BMA) and Bayesian stacking through a novel empirical Bayes lens, shedding new light on the limitations and pathologies of BMA. Further motivated by insights from online optimization, we propose Online Bayesian Stacking (OBS), a method that optimizes the log-score over predictive distributions to adaptively combine Bayesian models. A key contribution of our work is establishing a novel connection between OBS and portfolio selection, bridging Bayesian ensemble learning with a rich, well-studied theoretical framework that offers efficient algorithms and extensive regret analysis. We further clarify the relationship between OBS and online BMA, showing that they optimize related but distinct cost functions. Through theoretical analysis and empirical evaluation, we identify scenarios where OBS outperforms online BMA and provide principled guidance on when practitioners should prefer one approach over the other.

Portfolio optimization involves selecting asset weights to minimize a risk-reward objective, such as the portfolio variance in the classical minimum-variance framework. Sparse portfolio selection extends this by imposing a cardinality constraint: only $k$ assets from a universe of $p$ may be included. The standard approach models this problem as a mixed-integer quadratic program and relies on commercial solvers to find the optimal solution. However, the computational costs of such methods increase exponentially with $k$ and $p$, making them too slow for problems of even moderate size. We propose a fast and scalable gradient-based approach that transforms the combinatorial sparse selection problem into a constrained continuous optimization task via Boolean relaxation, while preserving equivalence with the original problem on the set of binary points. Our algorithm employs a tunable parameter that transmutes the auxiliary objective from a convex to a concave function. This allows a stable convex starting point, followed by a controlled path toward a sparse binary solution as the tuning parameter increases and the objective moves toward concavity. In practice, our method matches commercial solvers in asset selection for most instances and, in rare instances, the solution differs by a few assets whilst showing a negligible error in portfolio variance.

Portfolio management remains a crucial challenge in finance, with traditional methods often falling short in complex and volatile market environments. While deep reinforcement approaches have shown promise, they still face limitations in dynamic risk management, exploitation of temporal markets, and incorporation of complex trading strategies such as short-selling. These limitations can lead to suboptimal portfolio performance, increased vulnerability to market volatility, and missed opportunities in capturing potential returns from diverse market conditions. This paper introduces a Deep Reinforcement Learning Portfolio Management Framework with Time-Awareness and Short-Selling (MTS), offering a robust and adaptive strategy for sustainable investment performance. This framework utilizes a novel encoder-attention mechanism to address the limitations by incorporating temporal market characteristics, a parallel strategy for automated short-selling based on market trends, and risk management through innovative Incremental Conditional Value at Risk, enhancing adaptability and performance. Experimental validation on five diverse datasets from 2019 to 2023 demonstrates MTS's superiority over traditional algorithms and advanced machine learning techniques. MTS consistently achieves higher cumulative returns, Sharpe, Omega, and Sortino ratios, underscoring its effectiveness in balancing risk and return while adapting to market dynamics. MTS demonstrates an average relative increase of 30.67% in cumulative returns and 29.33% in Sharpe ratio compared to the next best-performing strategies across various datasets.

In this paper we propose and investigate a new class of Generalized Exponentiated Gradient (GEG) algorithms using Mirror Descent (MD) approaches, and applying as a regularization function the Bregman divergence with two-parameter deformation of logarithm as a link function. This link function (referred to as the Euler logarithm) is associated with a wide class of generalized entropies. In order to derive novel GEG/MD updates, we estimate generalized exponential function, which closely approximates the inverse of the Euler two-parameter logarithm. The characteristic/shape and properties of the Euler logarithm and its inverse -- deformed exponential functions are tuned by two or even more hyperparameters. By learning these hyperparameters, we can adapt to distribution of training data, and we can adjust them to achieve desired properties of gradient descent algorithms. The concept of generalized entropies and associated deformed logarithms provide deeper insight into novel gradient descent updates. In literature, there exist nowadays over fifty mathematically well-defined entropic functionals and associated deformed logarithms, so impossible to investigate all of them in one research paper. Therefore, we focus here on a wide-class of trace-form entropies and associated generalized logarithm. We applied the developed algorithms for Online Portfolio Selection (OPLS) in order to improve its performance and robustness.

We study continuous-time mean--variance portfolio selection in markets where stock prices are diffusion processes driven by observable factors that are also diffusion processes yet the coefficients of these processes are unknown. Based on the recently developed reinforcement learning (RL) theory for diffusion processes, we present a general data-driven RL algorithm that learns the pre-committed investment strategy directly without attempting to learn or estimate the market coefficients. For multi-stock Black--Scholes markets without factors, we further devise a baseline algorithm and prove its performance guarantee by deriving a sublinear regret bound in terms of Sharpe ratio. For performance enhancement and practical implementation, we modify the baseline algorithm into four variants, and carry out an extensive empirical study to compare their performance, in terms of a host of common metrics, with a large number of widely used portfolio allocation strategies on S\&P 500 constituents. The results demonstrate that the continuous-time RL strategies are consistently among the best especially in a volatile bear market, and decisively outperform the model-based continuous-time counterparts by significant margins.