With the rising emergence of decentralized and opportunistic approaches to machine learning, end devices are increasingly tasked with training deep learning models on-devices using crowd-sourced data that they collect themselves. These approaches are desirable from a resource consumption perspective and also from a privacy preservation perspective. When the devices benefit directly from the trained models, the incentives are implicit - contributing devices' resources are incentivized by the availability of the higher-accuracy model that results from collaboration. However, explicit incentive mechanisms must be provided when end-user devices are asked to contribute their resources (e.g., computation, communication, and data) to a task performed primarily for the benefit of others, e.g., training a model for a task that a neighbor device needs but the device owner is uninterested in. In this project, we propose a novel blockchain-based incentive mechanism for completely decentralized and opportunistic learning architectures. We leverage a smart contract not only for providing explicit incentives to end devices to participate in decentralized learning but also to create a fully decentralized mechanism to inspect and reflect on the behavior of the learning architecture.
IoT devices are sorely underutilized in the medical field, especially within machine learning for medicine, yet they offer unrivaled benefits. IoT devices are low-cost, energy-efficient, small and intelligent devices. In this paper, we propose a distributed federated learning framework for IoT devices, more specifically for IoMT (Internet of Medical Things), using blockchain to allow for a decentralized scheme improving privacy and efficiency over a centralized system; this allows us to move from the cloud-based architectures, that are prevalent, to the edge. The system is designed for three paradigms: 1) Training neural networks on IoT devices to allow for collaborative training of a shared model whilst decoupling the learning from the dataset to ensure privacy. Training is performed in an online manner simultaneously amongst all participants, allowing for the training of actual data that may not have been present in a dataset collected in the traditional way and dynamically adapt the system whilst it is being trained. 2) Training of an IoMT system in a fully private manner such as to mitigate the issue with confidentiality of medical data and to build robust, and potentially bespoke, models where not much, if any, data exists. 3) Distribution of the actual network training, something federated learning itself does not do, to allow hospitals, for example, to utilize their spare computing resources to train network models.
Introduction: The paper addresses the challenging problem of predicting the short-term realized volatility of the Bitcoin price using order flow information. The inherent stochastic nature and anti-persistence of price pose difficulties in accurate prediction. Methods: To address this, we propose a method that transforms order flow data over a fixed time interval (snapshots) into images. The order flow includes trade sizes, trade directions, and limit order book, and is mapped into image colour channels. These images are then used to train both a simple 3-layer Convolutional Neural Network (CNN) and more advanced ResNet-18 and ConvMixer, with additionally supplementing them with hand-crafted features. The models are evaluated against classical GARCH, Multilayer Perceptron trained on raw data, and a naive guess method that considers current volatility as a prediction. Results: The experiments are conducted using price data from January 2021 and evaluate model performance in terms of root mean square error (RMSPE). The results show that our order flow representation with a CNN as a predictive model achieves the best performance, with an RMSPE of 0.85+/-1.1 for the model with aggregated features and 1.0+/-1.4 for the model without feature supplementation. ConvMixer with feature supplementation follows closely. In comparison, the RMSPE for the naive guess method was 1.4+/-3.0.
Device monitoring services have increased in popularity with the evolution of recent technology and the continuously increased number of Internet of Things (IoT) devices. Among the popular services are the ones that use device location information. However, these services run into privacy issues due to the nature of data collection and transmission. In this work, we introduce a platform incorporating Federated Kalman Filter (FKF) with a federated learning approach and private blockchain technology for privacy preservation. We analyze the accuracy of the proposed design against a standard Kalman Filter (KF) implementation of localization based on the Received Signal Strength Indicator (RSSI). The experimental results reveal significant potential for improved data estimation for RSSI-based localization in device monitoring.
MANET is a collection of mobile nodes that communicate through wireless networks as they move from one point to another. MANET is an infrastructure-less network with a changeable topology; as a result, it is very susceptible to attacks. MANET attack prevention represents a serious difficulty. Malicious network nodes are the source of network-based attacks. In a MANET, attacks can take various forms, and each one alters the network's operation in its unique way. In general, attacks can be separated into two categories: those that target the data traffic on a network and those that target the control traffic. This article explains the many sorts of assaults, their impact on MANET, and the MANET-based defence measures that are currently in place. The suggested SRA that employs blockchain technology (SRABC) protects MANET from attacks and authenticates nodes. The secure routing algorithm (SRA) proposed by blockchain technology safeguards control and data flow against threats. This is achieved by generating a Hash Function for every transaction. We will begin by discussing the security of the MANET. This article's second section explores the role of blockchain in MANET security. In the third section, the SRA is described in connection with blockchain. In the fourth phase, PDR and Throughput are utilised to conduct an SRA review using Blockchain employing PDR and Throughput. The results suggest that the proposed technique enhances MANET security while concurrently decreasing delay. The performance of the proposed technique is analysed and compared to the routing protocols Q-AODV and DSR.
As various forms of fraud proliferate on Ethereum, it is imperative to safeguard against these malicious activities to protect susceptible users from being victimized. While current studies solely rely on graph-based fraud detection approaches, it is argued that they may not be well-suited for dealing with highly repetitive, skew-distributed and heterogeneous Ethereum transactions. To address these challenges, we propose BERT4ETH, a universal pre-trained Transformer encoder that serves as an account representation extractor for detecting various fraud behaviors on Ethereum. BERT4ETH features the superior modeling capability of Transformer to capture the dynamic sequential patterns inherent in Ethereum transactions, and addresses the challenges of pre-training a BERT model for Ethereum with three practical and effective strategies, namely repetitiveness reduction, skew alleviation and heterogeneity modeling. Our empirical evaluation demonstrates that BERT4ETH outperforms state-of-the-art methods with significant enhancements in terms of the phishing account detection and de-anonymization tasks. The code for BERT4ETH is available at: https://github.com/git-disl/BERT4ETH.
Graph neural networks (GNNs) have been utilized to create multi-layer graph models for a number of cybersecurity applications from fraud detection to software vulnerability analysis. Unfortunately, like traditional neural networks, GNNs also suffer from a lack of transparency, that is, it is challenging to interpret the model predictions. Prior works focused on specific factor explanations for a GNN model. In this work, we have designed and implemented Illuminati, a comprehensive and accurate explanation framework for cybersecurity applications using GNN models. Given a graph and a pre-trained GNN model, Illuminati is able to identify the important nodes, edges, and attributes that are contributing to the prediction while requiring no prior knowledge of GNN models. We evaluate Illuminati in two cybersecurity applications, i.e., code vulnerability detection and smart contract vulnerability detection. The experiments show that Illuminati achieves more accurate explanation results than state-of-the-art methods, specifically, 87.6% of subgraphs identified by Illuminati are able to retain their original prediction, an improvement of 10.3% over others at 77.3%. Furthermore, the explanation of Illuminati can be easily understood by the domain experts, suggesting the significant usefulness for the development of cybersecurity applications.
Rasoul Amirzadeh, Asef Nazari, Dhananjay Thiruvady, Mong Shan Ee
This study identifies the key factors influencing the price movements of major cryptocurrencies, Bitcoin, Binance Coin, Ethereum, Litecoin, Ripple, and Tether, using Bayesian networks (BNs). This study addresses two key challenges: modelling price movements in highly volatile cryptocurrency markets and enhancing predictive performance through discretisation-aware Bayesian Networks. It analyses both macro-financial indicators (gold, oil, MSCI, S and P 500, USDX) and social media signals (tweet volume) as potential price drivers. Moreover, since discretisation is a critical step in the effectiveness of BNs, we implement a structured procedure to build 54 BNs models by combining three discretisation methods (equal interval, equal quantile, and k-means) with several bin counts. These models are evaluated using four metrics, including balanced accuracy, F1 score, area under the ROC curve and a composite score. Results show that equal interval with two bins consistently yields the best predictive performance. We also provide deeper insights into each network's structure through inference, sensitivity, and influence strength analyses. These analyses reveal distinct price-driving patterns for each cryptocurrency, underscore the importance of coin-specific analysis, and demonstrate the value of BNs for interpretable causal modelling in volatile cryptocurrency markets.
Ervin Moore, Ahmed Imteaj, Shabnam Rezapour, M. Hadi Amini
Federated Learning (FL) has gained widespread popularity in recent years due to the fast booming of advanced machine learning and artificial intelligence along with emerging security and privacy threats. FL enables efficient model generation from local data storage of the edge devices without revealing the sensitive data to any entities. While this paradigm partly mitigates the privacy issues of users' sensitive data, the performance of the FL process can be threatened and reached a bottleneck due to the growing cyber threats and privacy violation techniques. To expedite the proliferation of FL process, the integration of blockchain for FL environments has drawn prolific attention from the people of academia and industry. Blockchain has the potential to prevent security and privacy threats with its decentralization, immutability, consensus, and transparency characteristic. However, if the blockchain mechanism requires costly computational resources, then the resource-constrained FL clients cannot be involved in the training. Considering that, this survey focuses on reviewing the challenges, solutions, and future directions for the successful deployment of blockchain in resource-constrained FL environments. We comprehensively review variant blockchain mechanisms that are suitable for FL process and discuss their trade-offs for a limited resource budget. Further, we extensively analyze the cyber threats that could be observed in a resource-constrained FL environment, and how blockchain can play a key role to block those cyber attacks. To this end, we highlight some potential solutions towards the coupling of blockchain and federated learning that can offer high levels of reliability, data privacy, and distributed computing performance.
Blockchains are significantly easing trade finance, with billions of dollars worth of assets being transacted daily. However, analyzing these networks remains challenging due to the sheer volume and complexity of the data. We introduce a method named InnerCore that detects market manipulators within blockchain-based networks and offers a sentiment indicator for these networks. This is achieved through data depth-based core decomposition and centered motif discovery, ensuring scalability. InnerCore is a computationally efficient, unsupervised approach suitable for analyzing large temporal graphs. We demonstrate its effectiveness by analyzing and detecting three recent real-world incidents from our datasets: the catastrophic collapse of LunaTerra, the Proof-of-Stake switch of Ethereum, and the temporary peg loss of USDC - while also verifying our results against external ground truth. Our experiments show that InnerCore can match the qualified analysis accurately without human involvement, automating blockchain analysis in a scalable manner, while being more effective and efficient than baselines and state-of-the-art attributed change detection approach in dynamic graphs.
The metaverse, which is at the stage of innovation and exploration, faces the dilemma of data collection and the problem of private data leakage in the process of development. This can seriously hinder the widespread deployment of the metaverse. Fortunately, federated learning (FL) is a solution to the above problems. FL is a distributed machine learning paradigm with privacy-preserving features designed for a large number of edge devices. Federated learning for metaverse (FL4M) will be a powerful tool. Because FL allows edge devices to participate in training tasks locally using their own data, computational power, and model-building capabilities. Applying FL to the metaverse not only protects the data privacy of participants but also reduces the need for high computing power and high memory on servers. Until now, there have been many studies about FL and the metaverse, respectively. In this paper, we review some of the early advances of FL4M, which will be a research direction with unlimited development potential. We first introduce the concepts of metaverse and FL, respectively. Besides, we discuss the convergence of key metaverse technologies and FL in detail, such as big data, communication technology, the Internet of Things, edge computing, blockchain, and extended reality. Finally, we discuss some key challenges and promising directions of FL4M in detail. In summary, we hope that our up-to-date brief survey can help people better understand FL4M and build a fair, open, and secure metaverse.
This research study focuses primarily on Block-Chain-based voting systems, which facilitate participation in and administration of voting for voters, candidates, and officials. Because we used Block-Chain in the backend, which enables everyone to trace vote fraud, our system is incredibly safe. This paper approach any unique identification the Aadhar Card number or an OTP will be generated then user can utilise the voting system to cast his/her vote. A proposal for Bit-coin, a virtual currency system that is decided by a central authority for producing money, transferring ownership, and validating transactions, included the peer-to-peer network in a Block-Chain system, the ledger is duplicated across several, identical databases which is hosted and updated by a different process and all other nodes are updated concurrently if changes made to one node and a transaction occurs, the records of the values and assets are permanently exchanged, Only the user and the system need to be verified no other authentication required. If any transaction carried out on a block chain-based system would be settled in a matter of seconds while still being safe, verifiable, and transparent. Although block-chain technology is the foundation for Bitcoin and other digital currencies but also it may be applied widely to greatly reduce difficulties in many other sectors, Voting is the sector that is battling from a lack of security, centralized-authority, management-issues, and many more despite the fact that transactions are kept in a distributed and safe fashion.
With the development of blockchain technology, more and more attention has been paid to the intersection of blockchain and education, and various educational evaluation systems and E-learning systems are developed based on blockchain technology. Among them, Ethereum smart contract is favored by developers for its ``event-triggered" mechanism for building education intelligent trading systems and intelligent learning platforms. However, due to the immutability of blockchain, published smart contracts cannot be modified, so problematic contracts cannot be fixed by modifying the code in the educational blockchain. In recent years, security incidents due to smart contract vulnerabilities have caused huge property losses, so the detection of smart contract vulnerabilities in educational blockchain has become a great challenge. To solve this problem, this paper proposes a graph neural network (GNN) based vulnerability detection for smart contracts in educational blockchains. Firstly, the bytecodes are decompiled to get the opcode. Secondly, the basic blocks are divided, and the edges between the basic blocks according to the opcode execution logic are added. Then, the control flow graphs (CFG) are built. Finally, we designed a GNN-based model for vulnerability detection. The experimental results show that the proposed method is effective for the vulnerability detection of smart contracts. Compared with the traditional approaches, it can get good results with fewer layers of the GCN model, which shows that the contract bytecode and GCN model are efficient in vulnerability detection.
Detecting illicit nodes on blockchain networks is a valuable task for strengthening future regulation. Recent machine learning-based methods proposed to tackle the tasks are using some blockchain transaction datasets with a small portion of samples labeled positive and the rest unlabelled (PU). Albeit the assumption that a random sample of unlabeled nodes are normal nodes is used in some works, we discuss that the label mechanism assumption for the hidden positive labels and its effect on the evaluation metrics is worth considering. We further explore that PU classifiers dealing with potential hidden positive labels can have improved performance compared to regular machine learning models. We test the PU classifiers with a list of graph representation learning methods for obtaining different feature distributions for the same data to have more reliable results.
Stock market and cryptocurrency forecasting is very important to investors as they aspire to achieve even the slightest improvement to their buy or hold strategies so that they may increase profitability. However, obtaining accurate and reliable predictions is challenging, noting that accuracy does not equate to reliability, especially when financial time-series forecasting is applied owing to its complex and chaotic tendencies. To mitigate this complexity, this study provides a comprehensive method for forecasting financial time series based on tactical input output feature mapping techniques using machine learning (ML) models. During the prediction process, selecting the relevant indicators is vital to obtaining the desired results. In the financial field, limited attention has been paid to this problem with ML solutions. We investigate the use of feature selection with annealing (FSA) for the first time in this field, and we apply the least absolute shrinkage and selection operator (Lasso) method to select the features from more than 1,000 candidates obtained from 26 technical classifiers with different periods and lags. Boruta (BOR) feature selection, a wrapper method, is used as a baseline for comparison. Logistic regression (LR), extreme gradient boosting (XGBoost), and long short-term memory (LSTM) are then applied to the selected features for forecasting purposes using 10 different financial datasets containing cryptocurrencies and stocks. The dependent variables consisted of daily logarithmic returns and trends. The mean-squared error for regression, area under the receiver operating characteristic curve, and classification accuracy were used to evaluate model performance, and the statistical significance of the forecasting results was tested using paired t-tests. Experiments indicate that the FSA algorithm increased the performance of ML models, regardless of problem type.
We consider a project (model) owner that would like to train a model by utilizing the local private data and compute power of interested data owners, i.e., trainers. Our goal is to design a data marketplace for such decentralized collaborative/federated learning applications that simultaneously provides i) proof-of-contribution based reward allocation so that the trainers are compensated based on their contributions to the trained model; ii) privacy-preserving decentralized model training by avoiding any data movement from data owners; iii) robustness against malicious parties (e.g., trainers aiming to poison the model); iv) verifiability in the sense that the integrity, i.e., correctness, of all computations in the data market protocol including contribution assessment and outlier detection are verifiable through zero-knowledge proofs; and v) efficient and universal design. We propose a blockchain-based marketplace design to achieve all five objectives mentioned above. In our design, we utilize a distributed storage infrastructure and an aggregator aside from the project owner and the trainers. The aggregator is a processing node that performs certain computations, including assessing trainer contributions, removing outliers, and updating hyper-parameters. We execute the proposed data market through a blockchain smart contract. The deployed smart contract ensures that the project owner cannot evade payment, and honest trainers are rewarded based on their contributions at the end of training. Finally, we implement the building blocks of the proposed data market and demonstrate their applicability in practical scenarios through extensive experiments.
Rongxin Xu, Shiva Raj Pokhrel, Qiujun Lan, Gang Li
Mobile Edge Computing (MEC) has been a promising paradigm for communicating and edge processing of data on the move. We aim to employ Federated Learning (FL) and prominent features of blockchain into MEC architecture such as connected autonomous vehicles to enable complete decentralization, immutability, and rewarding mechanisms simultaneously. FL is advantageous for mobile devices with constrained connectivity since it requires model updates to be delivered to a central point instead of substantial amounts of data communication. For instance, FL in autonomous, connected vehicles can increase data diversity and allow model customization, and predictions are possible even when the vehicles are not connected (by exploiting their local models) for short times. However, existing synchronous FL and Blockchain incur extremely high communication costs due to mobility-induced impairments and do not apply directly to MEC networks. We propose a fully asynchronous Blockchained Federated Learning (BFL) framework referred to as BFL-MEC, in which the mobile clients and their models evolve independently yet guarantee stability in the global learning process. More importantly, we employ post-quantum secure features over BFL-MEC to verify the client's identity and defend against malicious attacks. All of our design assumptions and results are evaluated with extensive simulations.
Biometric recognition as an efficient and hard-to-forge way of identification and verification has become an indispensable part of the current digital world. The fast evolution of this technology has been a strong incentive for integration into many applications. Meanwhile, blockchain, the decentralized ledger technology, has been widely received by both research and industry in the past few years, and it is being increasingly deployed today in many different applications, such as money transfer, IoT, healthcare, or logistics. Recently, researchers have started to speculate on the pros and cons and what the best applications would be when these two technologies cross paths. This paper provides a survey of the research literature on the combination of blockchain and biometrics and includes a first legal analysis of this integration based on GDPR to shed light on challenges and potentials. Although the integration of blockchain technology into the biometric sector is still in its infancy, with a growing body of literature discussing specific applications and advanced technological setups, this paper aims to provide a holistic understanding of blockchain applicability in biometrics. Based on published studies, this article discusses, among others, practical examples combining blockchain and biometrics for novel applications in PKI systems, distributed trusted services, and identity management. Challenges and limitations when combining blockchain and biometrics that motivate future work will also be discussed; e.g., blockchain networks at their current stage may not be efficient or economical for some real-time biometric applications. Finally, we also discuss key legal aspects of the EU General Data Protection Regulation (GDPR) related to this combination of technologies (blockchain and biometrics); for example, accountability, immutability, anonymity, and data protection elements.
Fazl Barez, Paul Bilokon, Arthur Gervais, Nikita Lisitsyn
This paper explores the novel deep learning Transformers architectures for high-frequency Bitcoin-USDT log-return forecasting and compares them to the traditional Long Short-Term Memory models. A hybrid Transformer model, called \textbf{HFformer}, is then introduced for time series forecasting which incorporates a Transformer encoder, linear decoder, spiking activations, and quantile loss function, and does not use position encoding. Furthermore, possible high-frequency trading strategies for use with the HFformer model are discussed, including trade sizing, trading signal aggregation, and minimal trading threshold. Ultimately, the performance of the HFformer and Long Short-Term Memory models are assessed and results indicate that the HFformer achieves a higher cumulative PnL than the LSTM when trading with multiple signals during backtesting.
Jiahua Xu, Yebo Feng, D. H. Cámpora Pérez, Benjamin Livshits
Decentralized finance (DeFi) is an integral component of the blockchain ecosystem, enabling a range of financial activities through smart-contract-based protocols. Traditional DeFi governance typically involves manual parameter adjustments by protocol teams or token holder votes, and is thus prone to human bias and financial risks, undermining the system's integrity and security. While existing efforts aim to establish more adaptive parameter adjustment schemes, there remains a need for a governance model that is both more efficient and resilient to significant market manipulations. In this paper, we introduce "Auto$.$gov", a learning-based governance framework that employs a deep Qnetwork (DQN) reinforcement learning (RL) strategy to perform semi-automated, data-driven parameter adjustments. We create a DeFi environment with an encoded action-state space akin to the Aave lending protocol for simulation and testing purposes, where Auto$.$gov has demonstrated the capability to retain funds that would have otherwise been lost to price oracle attacks. In tests with real-world data, Auto$.$gov outperforms the benchmark approaches by at least 14% and the static baseline model by tenfold, in terms of the preset performance metric--protocol profitability. Overall, the comprehensive evaluations confirm that Auto$.$gov is more efficient and effective than traditional governance methods, thereby enhancing the security, profitability, and ultimately, the sustainability of DeFi protocols.
The cryptocurrency ecosystem has been the centre of discussion on many social media platforms, following its noted volatility and varied opinions. Twitter is rapidly being utilised as a news source and a medium for bitcoin discussion. Our algorithm seeks to use historical prices and sentiment of tweets to forecast the price of Bitcoin. In this study, we develop an end-to-end model that can forecast the sentiment of a set of tweets (using a Bidirectional Encoder Representations from Transformers - based Neural Network Model) and forecast the price of Bitcoin (using Gated Recurrent Unit) using the predicted sentiment and other metrics like historical cryptocurrency price data, tweet volume, a user's following, and whether or not a user is verified. The sentiment prediction gave a Mean Absolute Percentage Error of 9.45%, an average of real-time data, and test data. The mean absolute percent error for the price prediction was 3.6%.
Ali Nikhalat-Jahromi, Ali Mohammad Saghiri, Mohammad Reza Meybodi
Learning Automaton (LA) is an adaptive self-organized model that improves its action-selection through interaction with an unknown environment. LA with finite action set can be classified into two main categories: fixed and variable structure. Furthermore, variable action-set learning automaton (VASLA) is one of the main subsets of variable structure learning automaton. In this paper, we propose VDHLA, a novel hybrid learning automaton model, which is a combination of fixed structure and variable action set learning automaton. In the proposed model, variable action set learning automaton can increase, decrease, or leave unchanged the depth of fixed structure learning automaton during the action switching phase. In addition, the depth of the proposed model can change in a symmetric (SVDHLA) or asymmetric (AVDHLA) manner. To the best of our knowledge, it is the first hybrid model that intelligently changes the depth of fixed structure learning automaton. Several computer simulations are conducted to study the performance of the proposed model with respect to the total number of rewards and action switching in stationary and non-stationary environments. The proposed model is compared with FSLA and VSLA. In order to determine the performance of the proposed model in a practical application, the selfish mining attack which threatens the incentive-compatibility of a proof-of-work based blockchain environment is considered. The proposed model is applied to defend against the selfish mining attack in Bitcoin and compared with the tie-breaking mechanism, which is a well-known defense. Simulation results in all environments have shown the superiority of the proposed model.
Non-Fungible Tokens (NFTs) represent deeds of ownership, based on blockchain technologies and smart contracts, of unique crypto assets on digital art forms (e.g., artworks or collectibles). In the spotlight after skyrocketing in 2021, NFTs have attracted the attention of crypto enthusiasts and investors intent on placing promising investments in this profitable market. However, the NFT financial performance prediction has not been widely explored to date. In this work, we address the above problem based on the hypothesis that NFT images and their textual descriptions are essential proxies to predict the NFT selling prices. To this purpose, we propose MERLIN, a novel multimodal deep learning framework designed to train Transformer-based language and visual models, along with graph neural network models, on collections of NFTs' images and texts. A key aspect in MERLIN is its independence on financial features, as it exploits only the primary data a user interested in NFT trading would like to deal with, i.e., NFT images and textual descriptions. By learning dense representations of such data, a price-category classification task is performed by MERLIN models, which can also be tuned according to user preferences in the inference phase to mimic different risk-return investment profiles. Experimental evaluation on a publicly available dataset has shown that MERLIN models achieve significant performances according to several financial assessment criteria, fostering profitable investments, and also beating baseline machine-learning classifiers based on financial features.
Decentralized Exchanges (DEXs) are new types of marketplaces leveraging Blockchain technology. They allow users to trade assets with Automatic Market Makers (AMM), using funds provided by liquidity providers, removing the need for order books. One such DEX, Uniswap v3, allows liquidity providers to allocate funds more efficiently by specifying an active price interval for their funds. This introduces the problem of finding an optimal strategy for choosing price intervals. We formalize this problem as an online learning problem with non-stochastic rewards. We use regret-minimization methods to show a liquidity provision strategy that guarantees a lower bound on the reward. This is true even for non-stochastic changes to asset pricing, and we express this bound in terms of the trading volume.