### Limitations of Traditional Methods  
Traditional methods for portfolio optimization, such as mean-variance optimization (MVO) [4], risk parity, and Value at Risk (VaR), have long been central to the field of asset management. Originally developed within the framework of MPT, MVO aims to balance expected returns and risk and to construct an optimal portfolio by balancing expected returns and risks. The core objective of MVO is to minimize portfolio variance, given a target return, or equivalently, maximize return for a given level of risk, represented by variance or the standard deviation. The optimization problem can be expressed as  
\[\min_{w}w^{T}\Sigma w\quad\text{subject to}\quad w^{T}\mu\geq r_{min},\quad \sum w_{i}=1,\quad 0\leq w_{i}\leq 1 \tag{1}\]  
where \(w\) represents portfolio weights, \(\Sigma\) is the covariance matrix of asset returns, and \(\mu\) is the vector of expected returns [2]. Risk parity, on the other hand, aims to allocate assets such that each contributes equally to the overall portfolio risk. This method focuses on diversifying risk rather than returns, making it attractive for risk-averse investors. Value at Risk (VaR) is a widely used risk measure that estimates the potential loss in portfolio value over a specific time frame with a given confidence level. VaR is often used to determine the maximum expected loss under normal market conditions, though its reliance on historical data makes it vulnerable to extreme events [3]. Despite their widespread adoption, these traditional methods face significant limitations, particularly in the context of modern financial markets. One key issue is their lack of adaptability to rapidly changing market conditions. Methods like MVO rely on historical data and assume that future asset returns and risks will behave similarly, which is frequently invalid in volatile or non-stationary markets. This reliance on historical averages can result in overfitting, where the model becomes too tailored to past performance, leading to suboptimal decisions when market dynamics shift [5]. Moreover, MVO's assumption that risk can be fully captured by variance oversimplifies the complex nature of risk in financial markets, ignoring factors such as tail risk or extreme market movements. Another limitation of these traditional approaches is their reliance on strict assumptions about asset returns. MVO, for instance, assumes that asset returns follow a normal distribution and that correlations between assets are stable over time. In reality, financial returns tend to be skewed and exhibit fat tails, with extreme events occurring more frequently than predicted by normal distribution models [6]. Similarly, VaR assumes a linear relationship between risk factors and portfolio returns, which breaks down during periods of market stress, leading to an underestimation of risk. Furthermore, risk parity can be overly conservative in environments where certain assets exhibit low volatility but are exposed to systemic risks, thus potentially skewing allocations [7]. While this study focuses on RL methods for portfolio optimization, it is important to acknowledge the role and advances of traditional machine learning (ML) techniques in this domain. ML methods have been widely recognized for their ability to improve over classical approaches such as MVO by leveraging historical data and advanced predictive modeling. In [8], it was demonstrated that Long Short-Term Memory (LSTM) [9]-based portfolio optimization surpasses traditional MVO in cumulative returns and Sharpe ratios, while [10] showed the advantages of autoencoder-based models in annualized returns over MVO in specific scenarios. Similarly, advanced linear programming models [11] and robust optimization techniques [12] have been shown to achieve superior diversification, computational efficiency, and stability compared to traditional methods.