Convex Optimization

Semantic Faithfulness and Entropy Production Measures to Tame Your LLM Demons and Manage Hallucinations ArXiv ID: 2512.05156 “View on arXiv” Authors: Igor Halperin Abstract Evaluating faithfulness of Large Language Models (LLMs) to a given task is a complex challenge. We propose two new unsupervised metrics for faithfulness evaluation using insights from information theory and thermodynamics. Our approach treats an LLM as a bipartite information engine where hidden layers act as a Maxwell demon controlling transformations of context $C $ into answer $A$ via prompt $Q$. We model Question-Context-Answer (QCA) triplets as probability distributions over shared topics. Topic transformations from $C$ to $Q$ and $A$ are modeled as transition matrices ${"\bf Q"}$ and ${"\bf A"}$ encoding the query goal and actual result, respectively. Our semantic faithfulness (SF) metric quantifies faithfulness for any given QCA triplet by the Kullback-Leibler (KL) divergence between these matrices. Both matrices are inferred simultaneously via convex optimization of this KL divergence, and the final SF metric is obtained by mapping the minimal divergence onto the unit interval [“0,1”], where higher scores indicate greater faithfulness. Furthermore, we propose a thermodynamics-based semantic entropy production (SEP) metric in answer generation, and show that high faithfulness generally implies low entropy production. The SF and SEP metrics can be used jointly or separately for LLM evaluation and hallucination control. We demonstrate our framework on LLM summarization of corporate SEC 10-K filings. ...

Detecting AI Hallucinations in Finance: An Information-Theoretic Method Cuts Hallucination Rate by 92% ArXiv ID: 2512.03107 “View on arXiv” Authors: Mainak Singha Abstract Large language models (LLMs) produce fluent but unsupported answers - hallucinations - limiting safe deployment in high-stakes domains. We propose ECLIPSE, a framework that treats hallucination as a mismatch between a model’s semantic entropy and the capacity of available evidence. We combine entropy estimation via multi-sample clustering with a novel perplexity decomposition that measures how models use retrieved evidence. We prove that under mild conditions, the resulting entropy-capacity objective is strictly convex with a unique stable optimum. We evaluate on a controlled financial question answering dataset with GPT-3.5-turbo (n=200 balanced samples with synthetic hallucinations), where ECLIPSE achieves ROC AUC of 0.89 and average precision of 0.90, substantially outperforming a semantic entropy-only baseline (AUC 0.50). A controlled ablation with Claude-3-Haiku, which lacks token-level log probabilities, shows AUC dropping to 0.59 with coefficient magnitudes decreasing by 95% - demonstrating that ECLIPSE is a logprob-native mechanism whose effectiveness depends on calibrated token-level uncertainties. The perplexity decomposition features exhibit the largest learned coefficients, confirming that evidence utilization is central to hallucination detection. We position this work as a controlled mechanism study; broader validation across domains and naturally occurring hallucinations remains future work. ...

Entropy-Guided Multiplicative Updates: KL Projections for Multi-Factor Target Exposures ArXiv ID: 2510.24607 “View on arXiv” Authors: Yimeng Qiu Abstract We introduce Entropy-Guided Multiplicative Updates (EGMU), a convex optimization framework for constructing multi-factor target-exposure portfolios by minimizing Kullback-Leibler divergence from a benchmark under linear factor constraints. We establish feasibility and uniqueness of strictly positive solutions when the benchmark and targets satisfy convex-hull conditions. We derive the dual concave formulation with explicit gradient, Hessian, and sensitivity expressions, and provide two provably convergent solvers: a damped dual Newton method with global convergence and local quadratic rate, and a KL-projection scheme based on iterative proportional fitting and Bregman-Dykstra projections. We further generalize EGMU to handle elastic targets and robust target sets, and introduce a path-following ordinary differential equation for tracing solution trajectories. Stable and scalable implementations are provided using LogSumExp stabilization, covariance regularization, and half-space KL projections. Our focus is on theory and reproducible algorithms; empirical benchmarking is optional. ...

Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programs ArXiv ID: 2508.14995 “View on arXiv” Authors: Anastasis Kratsios, Ariel Neufeld, Philipp Schmocker Abstract Neural operators (NOs) are a class of deep learning models designed to simultaneously solve infinitely many related problems by casting them into an infinite-dimensional space, whereon these NOs operate. A significant gap remains between theory and practice: worst-case parameter bounds from universal approximation theorems suggest that NOs may require an unrealistically large number of parameters to solve most operator learning problems, which stands in direct opposition to a slew of experimental evidence. This paper closes that gap for a specific class of {“NOs”}, generative {“equilibrium operators”} (GEOs), using (realistic) finite-dimensional deep equilibrium layers, when solving families of convex optimization problems over a separable Hilbert space $X$. Here, the inputs are smooth, convex loss functions on $X$, and outputs are the associated (approximate) solutions to the optimization problem defined by each input loss. We show that when the input losses lie in suitable infinite-dimensional compact sets, our GEO can uniformly approximate the corresponding solutions to arbitrary precision, with rank, depth, and width growing only logarithmically in the reciprocal of the approximation error. We then validate both our theoretical results and the trainability of GEOs on three applications: (1) nonlinear PDEs, (2) stochastic optimal control problems, and (3) hedging problems in mathematical finance under liquidity constraints. ...

Measuring DEX Efficiency and The Effect of an Enhanced Routing Method on Both DEX Efficiency and Stakeholders’ Benefits ArXiv ID: 2508.03217 “View on arXiv” Authors: Yu Zhang, Claudio J. Tessone Abstract The efficiency of decentralized exchanges (DEXs) and the influence of token routing algorithms on market performance and stakeholder outcomes remain underexplored. This paper introduces the concept of Standardized Total Arbitrage Profit (STAP), computed via convex optimization, as a systematic measure of DEX efficiency. We prove that executing the trade order maximizing STAP and reintegrating the resulting transaction fees eliminates all arbitrage opportunities-both cyclic arbitrage within DEXs and between DEXs and centralized exchanges (CEXs). In a fully efficient DEX (i.e., STAP = 0), the monetary value of target tokens received must not exceed that of the source tokens, regardless of the routing algorithm. Any violation indicates arbitrage potential, making STAP a reliable metric for arbitrage detection. Using a token graph comprising 11 tokens and 18 liquidity pools based on Uniswap V2 data, we observe a decline in DEX efficiency between June 21 and November 8, 2024. Simulations comparing two routing algorithms-Yu Zhang et al.’s line-graph-based method and the depth-first search (DFS) algorithm-show that employing more profitable routing improves DEX efficiency and trader returns over time. Moreover, while total value locked (TVL) remains stable with the line-graph method, it increases under the DFS algorithm, indicating greater aggregate benefits for liquidity providers. ...

Transfer Learning Across Fixed-Income Product Classes ArXiv ID: 2505.07676 “View on arXiv” Authors: Nicolas Camenzind, Damir Filipovic Abstract We propose a framework for transfer learning of discount curves across different fixed-income product classes. Motivated by challenges in estimating discount curves from sparse or noisy data, we extend kernel ridge regression (KR) to a vector-valued setting, formulating a convex optimization problem in a vector-valued reproducing kernel Hilbert space (RKHS). Each component of the solution corresponds to the discount curve implied by a specific product class. We introduce an additional regularization term motivated by economic principles, promoting smoothness of spread curves between product classes, and show that it leads to a valid separable kernel structure. A main theoretical contribution is a decomposition of the vector-valued RKHS norm induced by separable kernels. We further provide a Gaussian process interpretation of vector-valued KR, enabling quantification of estimation uncertainty. Illustrative examples demonstrate that transfer learning significantly improves extrapolation performance and tightens confidence intervals compared to single-curve estimation. ...

Wasserstein Robust Market Making via Entropy Regularization ArXiv ID: 2503.04072 “View on arXiv” Authors: Unknown Abstract In this paper, we introduce a robust market making framework based on Wasserstein distance, utilizing a stochastic policy approach enhanced by entropy regularization. We demonstrate that, under mild assumptions, the robust market making problem can be reformulated as a convex optimization question. Additionally, we outline a methodology for selecting the optimal radius of the Wasserstein ball, further refining our framework’s effectiveness. ...

BPQP: A Differentiable Convex Optimization Framework for Efficient End-to-End Learning ArXiv ID: 2411.19285 “View on arXiv” Authors: Unknown Abstract Data-driven decision-making processes increasingly utilize end-to-end learnable deep neural networks to render final decisions. Sometimes, the output of the forward functions in certain layers is determined by the solutions to mathematical optimization problems, leading to the emergence of differentiable optimization layers that permit gradient back-propagation. However, real-world scenarios often involve large-scale datasets and numerous constraints, presenting significant challenges. Current methods for differentiating optimization problems typically rely on implicit differentiation, which necessitates costly computations on the Jacobian matrices, resulting in low efficiency. In this paper, we introduce BPQP, a differentiable convex optimization framework designed for efficient end-to-end learning. To enhance efficiency, we reformulate the backward pass as a simplified and decoupled quadratic programming problem by leveraging the structural properties of the KKT matrix. This reformulation enables the use of first-order optimization algorithms in calculating the backward pass gradients, allowing our framework to potentially utilize any state-of-the-art solver. As solver technologies evolve, BPQP can continuously adapt and improve its efficiency. Extensive experiments on both simulated and real-world datasets demonstrate that BPQP achieves a significant improvement in efficiency–typically an order of magnitude faster in overall execution time compared to other differentiable optimization layers. Our results not only highlight the efficiency gains of BPQP but also underscore its superiority over differentiable optimization layer baselines. ...

A Krasnoselskii-Mann Proximity Algorithm for Markowitz Portfolios with Adaptive Expected Return Level ArXiv ID: 2409.13608 “View on arXiv” Authors: Unknown Abstract Markowitz’s criterion aims to balance expected return and risk when optimizing the portfolio. The expected return level is usually fixed according to the risk appetite of an investor, then the risk is minimized at this fixed return level. However, the investor may not know which return level is suitable for her/him and the current financial circumstance. It motivates us to find a novel approach that adaptively optimizes this return level and the portfolio at the same time. It not only relieves the trouble of deciding the return level during an investment but also gets more adaptive to the ever-changing financial market than a subjective return level. In order to solve the new model, we propose an exact, convergent, and efficient Krasnoselskii-Mann Proximity Algorithm based on the proximity operator and Krasnoselskii-Mann momentum technique. Extensive experiments show that the proposed method achieves significant improvements over state-of-the-art methods in portfolio optimization. This finding may contribute a new perspective on the relationship between return and risk in portfolio optimization. ...

Profit Maximization In Arbitrage Loops ArXiv ID: 2406.16600 “View on arXiv” Authors: Unknown Abstract Cyclic arbitrage chances exist abundantly among decentralized exchanges (DEXs), like Uniswap V2. For an arbitrage cycle (loop), researchers or practitioners usually choose a specific token, such as Ether as input, and optimize their input amount to get the net maximal amount of the specific token as arbitrage profit. By considering the tokens’ prices from CEXs in this paper, the new arbitrage profit, called monetized arbitrage profit, will be quantified as the product of the net number of a specific token we got from the arbitrage loop and its corresponding price in CEXs. Based on this concept, we put forward three different strategies to maximize the monetized arbitrage profit for each arbitrage loop. The first strategy is called the MaxPrice strategy. Under this strategy, arbitrageurs start arbitrage only from the token with the highest CEX price. The second strategy is called the MaxMax strategy. Under this strategy, we calculate the monetized arbitrage profit for each token as input in turn in the arbitrage loop. Then, we pick up the most maximal monetized arbitrage profit among them as the monetized arbitrage profit of the MaxMax strategy. The third one is called the Convex Optimization strategy. By mapping the MaxMax strategy to a convex optimization problem, we proved that the Convex Optimization strategy could get more profit in theory than the MaxMax strategy, which is proved again in a given example. We also proved that if no arbitrage profit exists according to the MaxMax strategy, then the Convex Optimization strategy can not detect any arbitrage profit, either. However, the empirical data analysis denotes that the profitability of the Convex Optimization strategy is almost equal to that of the MaxMax strategy, and the MaxPrice strategy is not reliable in getting the maximal monetized arbitrage profit compared to the MaxMax strategy. ...