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Why the Fast Inverse Square Root Matters

The Fast Inverse Square Root algorithm is among the most famous algorithms in computer science history. In this post, I want to go over what it is and why its so important in words anyone can understand.

Permutation Decision Trees

Decision Tree is a well understood Machine Learning model that is based on minimizing impurities in the internal nodes. The most common impurity measures are Shannon entropy and Gini impurity. These impurity measures are insensitive to the order of training data and hence the final tree obtained is invariant to any permutation of the data. This is a limitation in terms of modeling when there are temporal order dependencies between data instances. In this research, we propose the adoption of Effort-To-Compress (ETC) - a complexity measure, for the first time, as an alternative impurity measure. Unlike Shannon entropy and Gini impurity, structural impurity based on ETC is able to capture order dependencies in the data, thus obtaining potentially different decision trees for different permutations of the same data instances, a concept we term as Permutation Decision Trees (PDT). We then introduce the notion of Permutation Bagging achieved using permutation decision trees without the need for random feature selection and sub-sampling. We conduct a performance comparison between Permutation Decision Trees and classical decision trees across various real-world datasets, including Appendicitis, Breast Cancer Wisconsin, Diabetes Pima Indian, Ionosphere, Iris, Sonar, and Wine. Our findings reveal that PDT demonstrates comparable performance to classical decision trees across most datasets. Remarkably, in certain instances, PDT even slightly surpasses the performance of classical decision trees. In comparing Permutation Bagging with Random Forest, we attain comparable performance to Random Forest models consisting of 50 to 1000 trees, using merely 21 trees. This highlights the efficiency and effectiveness of Permutation Bagging in achieving comparable performance outcomes with significantly fewer trees.

Online Prompt Pricing based on Combinatorial Multi-Armed Bandit and Hierarchical Stackelberg Game

Generation models have shown promising performance in various tasks, making trading around machine learning models possible. In this paper, we aim at a novel prompt trading scenario, prompt bundle trading (PBT) system, and propose an online pricing mechanism. Based on the combinatorial multi-armed bandit (CMAB) and three-stage hierarchical Stackelburg (HS) game, our pricing mechanism considers the profits of the consumer, platform, and seller, simultaneously achieving the profit satisfaction of these three participants. We break down the pricing issue into two steps, namely unknown category selection and incentive strategy optimization. The former step is to select a set of categories with the highest qualities, and the latter is to derive the optimal strategy for each participant based on the chosen categories. Unlike the existing fixed pricing mode, the PBT pricing mechanism we propose is more flexible and diverse, which is more in accord with the transaction needs of real-world scenarios. We test our method on a simulated text-to-image dataset. The experimental results demonstrate the effectiveness of our algorithm, which provides a feasible price-setting standard for the prompt marketplaces.

Portfolio Optimization with Robust Covariance and Conditional Value-at-Risk Constraints

The measure of portfolio risk is an important input of the Markowitz framework. In this study, we explored various methods to obtain a robust covariance estimators that are less susceptible to financial data noise. We evaluated the performance of large-cap portfolio using various forms of Ledoit Shrinkage Covariance and Robust Gerber Covariance matrix during the period of 2012 to 2022. Out-of-sample performance indicates that robust covariance estimators can outperform the market capitalization-weighted benchmark portfolio, particularly during bull markets. The Gerber covariance with Mean-Absolute-Deviation (MAD) emerged as the top performer. However, robust estimators do not manage tail risk well under extreme market conditions, for example, Covid-19 period. When we aim to control for tail risk, we should add constraint on Conditional Value-at-Risk (CVaR) to make more conservative decision on risk exposure. Additionally, we incorporated unsupervised clustering algorithm K-means to the optimization algorithm (i.e. Nested Clustering Optimization, NCO). It not only helps mitigate numerical instability of the optimization algorithm, but also contributes to lower drawdown as well.

A multifractional option pricing formula

Fractional Brownian motion has become a standard tool to address long-range dependence in financial time series. However, a constant memory parameter is too restrictive to address different market conditions. Here we model the price fluctuations using a multifractional Brownian motion assuming that the Hurst exponent is a time-deterministic function. Through the multifractional Ito calculus, both the related transition density function and the analytical European Call option pricing formula are obtained. The empirical performance of the multifractional Black-Scholes model is tested by calibration of option market quotes for the SPX index and offers best fit than its counterparts based on standard and fractional Brownian motions.

Optimal market-neutral currency trading on the cryptocurrency platform

This research proposes a novel arbitrage approach in multivariate pair trading, termed the Optimal Trading Technique (OTT). We present a method for selectively forming a "bucket" of fiat currencies anchored to cryptocurrency for monitoring and exploiting trading opportunities simultaneously. To address quantitative conflicts from multiple trading signals, a novel bi-objective convex optimization formulation is designed to balance investor preferences between profitability and risk tolerance. We understand that cryptocurrencies carry significant financial risks. Therefore this process includes tunable parameters such as volatility penalties and action thresholds. In experiments conducted in the cryptocurrency market from 2020 to 2022, which encompassed a vigorous bull run followed by a bear run, the OTT achieved an annualized profit of 15.49%. Additionally, supplementary experiments detailed in the appendix extend the applicability of OTT to other major cryptocurrencies in the post-COVID period, validating the model's robustness and effectiveness in various market conditions. The arbitrage operation offers a new perspective on trading, without requiring external shorting or holding the intermediate during the arbitrage period. As a note of caution, this study acknowledges the high-risk nature of cryptocurrency investments, which can be subject to significant volatility and potential loss.