Author: Denis Avetisyan
New research reveals that different optimization algorithms can produce functionally different financial models with surprisingly similar predictive power.
The study demonstrates that optimizer selection impacts downstream decision-making, such as portfolio turnover, suggesting that evaluation metrics should extend beyond simple loss functions.
Despite achieving comparable out-of-sample error, neural network models applied to financial time series can exhibit surprisingly divergent behavior. This is the central finding of ‘Same Error, Different Function: The Optimizer as an Implicit Prior in Financial Time Series’, which demonstrates that differing optimizers yield qualitatively distinct learned functions even when predictive accuracy remains equivalent. Using large-scale volatility forecasting for S&P 500 stocks, the authors reveal that optimizer choice fundamentally reshapes non-linear response profiles and temporal dependencies, significantly impacting downstream decision-making-specifically, portfolio turnover as measured by a Sharpe-turnover frontier. Does this imply that model evaluation in underspecified financial settings must extend beyond scalar loss functions to encompass functional characteristics and real-world implications?
The Illusion of Prediction: Why Numbers Lie in Volatility Forecasting
The pervasive use of Normalized Mean Squared Error (NMSE) and similar metrics in volatility forecasting often cultivates an unwarranted confidence in model accuracy. While appearing to quantify predictive power, these measures are surprisingly susceptible to manipulation and can be misleadingly low even with largely random forecasts, particularly when evaluating models on limited datasets. This is because NMSE essentially assesses how closely a model’s predicted variance matches the realized variance, without necessarily validating the model’s ability to predict future price movements. A model can consistently underestimate volatility and still achieve a seemingly respectable NMSE score, leading practitioners to believe in its efficacy when, in reality, it’s merely reflecting a systematic bias rather than genuine forecasting skill. Consequently, reliance on NMSE alone can mask a model’s true limitations and contribute to the illusion of accurate financial prediction.
Financial markets are characterized by a persistent level of noise, stemming from the sheer volume of interacting agents and unpredictable external events. This inherent complexity means that apparent patterns in historical price data may often be the result of random fluctuations rather than genuine predictive signals. Distinguishing between true forecasting ability and mere statistical chance is therefore a substantial challenge, as models can easily appear successful during backtesting simply by capitalizing on random noise. Consequently, observed performance may not generalize to future, unseen data, leading to overconfidence in strategies that lack real predictive power and potentially significant financial risk. The difficulty lies not in a lack of sophisticated tools, but in the fundamental limitations imposed by the chaotic nature of financial time series.
Analyses of financial forecasting models are frequently distorted by survivorship bias, a systematic exclusion of data pertaining to strategies that have failed. This creates a deceptively optimistic picture of predictive power, as only successful strategies remain within the dataset used for evaluation. Consequently, reported performance metrics overestimate the true likelihood of success, implying a level of skill that may not exist. The omission of failed attempts leads to an artificially inflated average return and reduced perceived risk, potentially misleading investors and researchers alike. Addressing this bias requires a concerted effort to include data from all strategies, regardless of their outcome, to achieve a more realistic assessment of forecasting accuracy and avoid the illusion of consistent profitability.
Beyond the Score: Uncovering What Models Actually Do
Predictive equivalence, wherein multiple models achieve comparable performance metrics on a given task, does not necessitate similarity in their internal representations or decision-making processes. This phenomenon arises because numerous mathematical functions can approximate the same input-output relationship. Consequently, models may learn drastically different mappings – utilizing distinct feature weights, activation patterns, or algorithmic strategies – while still producing statistically similar predictions. This indicates that optimization for a single performance metric, such as accuracy or loss, does not inherently enforce a unique or interpretable solution, and can mask substantial differences in how models process and generalize from data.
A low loss value, while indicating a model’s ability to minimize errors on a training dataset, does not inherently confirm that the model has developed a genuinely meaningful or robust representation of the underlying data. Models can achieve comparable loss scores through fundamentally different learned functions, exhibiting predictive equivalence despite disparate internal mechanisms. This means a model with low loss may rely on spurious correlations, superficial features, or brittle patterns, leading to poor generalization performance when presented with novel or out-of-distribution data. Consequently, evaluating model understanding requires analysis beyond loss minimization, focusing on how a model arrives at its predictions, not simply that it produces accurate outputs on the training set.
Impulse Response Analysis (IRA) and SHAP (SHapley Additive exPlanations) values offer complementary methods for characterizing divergent model behaviors. IRA assesses a model’s sensitivity to small changes in input features, revealing how the model’s output reacts to localized perturbations; this highlights which features drive immediate responses. SHAP values, rooted in game theory, quantify each feature’s contribution to a specific prediction, providing a per-sample explanation of feature importance. By calculating SHAP values across a dataset, one obtains a distribution of feature impacts, revealing systematic differences in how models utilize information. Combining IRA’s dynamic response analysis with SHAP’s attribution of feature importance allows researchers to move beyond assessing predictive accuracy and understand how models arrive at their conclusions, identifying potentially spurious correlations or unintended biases.
The Algorithm’s Fingerprint: How Optimization Shapes the Model
Common neural network architectures – Convolutional Neural Networks (CNNs), Long Short-Term Memory networks (LSTMs), Transformers, and Multilayer Perceptrons (MLPs) – are trained using a variety of optimization algorithms. Stochastic Gradient Descent (SGD) remains a foundational method, while Adam and its variants are widely used for their adaptive learning rates. Muon is a more recent optimization algorithm focused on second-order information. The selection of an optimizer is often determined empirically, as each algorithm exhibits differing performance characteristics based on the specific network architecture, dataset, and hyperparameter configuration. For example, Adam generally converges faster than SGD but may generalize less effectively in some cases, and Transformers frequently benefit from AdamW, a variant incorporating weight decay.
Optimization algorithms, such as Stochastic Gradient Descent (SGD), Adam, and Muon, directly impact the resulting weights and biases of a neural network, thereby defining the learned representation. While all aim to minimize a loss function, their differing update rules – considering factors like learning rate, momentum, and adaptive learning rates – lead to variations in the traversed loss landscape. This results in functionally divergent models, even when initialized identically and trained on the same dataset; a model optimized with Adam may prioritize different feature combinations or exhibit different generalization characteristics than one optimized with SGD. Consequently, the choice of optimizer isn’t solely a matter of convergence speed, but fundamentally alters the model’s internal representation and ultimately its performance on downstream tasks.
The predictive capability and inherent biases of a neural network are significantly determined by the interaction between its architecture and the optimization algorithm used during training. Different architectures, such as Convolutional Neural Networks (CNNs), Recurrent Neural Networks (LSTMs), and Transformers, possess varying inductive biases which predispose them to learn certain patterns in data. Simultaneously, optimization algorithms – including Stochastic Gradient Descent (SGD), Adam, and Muon – navigate the loss landscape in unique ways, potentially reinforcing or mitigating these architectural biases. For instance, an SGD optimizer might converge to a different local minimum compared to Adam, even with the same network architecture and dataset, resulting in variations in both predictive performance and the types of errors the model makes. Therefore, a comprehensive evaluation requires considering not only the architecture itself, but also the specific optimizer and its hyperparameters to fully understand the model’s behavior and potential biases.
Beyond Accuracy: The Real-World Impact on Portfolio Management
Accurate volatility estimation forms the bedrock of modern portfolio management, and the Garman-Klass estimator represents a frequently employed technique for this crucial task. This estimator, building upon historical price data – specifically, the open, high, low, and close prices – provides a robust, unbiased measure of volatility even with asynchronous trading data. Its widespread adoption stems from its relative simplicity and effectiveness in capturing the dynamic nature of financial markets, allowing investors to quantify the degree of price fluctuation inherent in an asset. Consequently, the Garman-Klass estimator directly influences risk assessment, option pricing, and the construction of efficient portfolios designed to balance risk and return, making it an indispensable tool for both institutional and individual investors.
Effective portfolio management hinges on the precision of volatility forecasts, as these directly influence two critical performance indicators: portfolio turnover and the Sharpe Ratio. Portfolio turnover, representing the rate at which assets are bought and sold, is heavily impacted by volatility estimations; higher perceived volatility often leads to more frequent trading as managers attempt to mitigate risk or capitalize on price swings. Simultaneously, the Sharpe Ratio, a measure of risk-adjusted return, is fundamentally dependent on an accurate assessment of volatility – the denominator in its calculation. A miscalculation of volatility can therefore distort the Sharpe Ratio, leading to an inaccurate evaluation of a portfolio’s true performance. Consequently, even seemingly minor improvements in volatility forecasting can translate into substantial changes in both trading activity and overall portfolio efficiency, underscoring the importance of robust volatility estimation techniques.
Ensemble learning offers a compelling strategy for navigating the challenges of functional divergence in volatility forecasting, potentially reducing risk by aggregating predictions from multiple, distinct models. However, simply combining models isn’t a guaranteed solution; careful attention must be paid to the inherent biases within each constituent model. While diversity amongst models is desirable, unaddressed biases can be amplified, leading to systematic errors in portfolio optimization. A successful ensemble approach necessitates a thorough understanding of each model’s strengths and weaknesses, coupled with a weighting or selection mechanism that actively mitigates the influence of problematic biases. Without this critical evaluation, the benefits of ensemble learning may be diminished, and the resulting portfolio could still be vulnerable to unexpected market behavior.
While predictive models may demonstrate statistically similar accuracy-as evidenced by a Normalized Mean Squared Error (NMSE) of 0.5730, a slight improvement over the 0.5751 achieved by ordinary least squares-their practical application within portfolio management can diverge considerably. Research indicates that different optimization algorithms, despite yielding comparable Sharpe Ratios, can produce drastically different trading strategies, leading to portfolio turnover rates that vary by a factor of three. This highlights a critical point: minimizing forecast error is not solely sufficient for optimal portfolio performance; the behavior of the model, specifically its trading frequency and sensitivity, plays a crucial role in realized returns and transaction costs, demanding careful consideration beyond simple statistical benchmarks.
Towards Resilient Forecasting: A Glimpse into the Future
Current forecasting evaluation overwhelmingly relies on loss minimization, a quantitative approach that often fails to capture crucial qualitative differences between models. While a model with lower loss may appear superior, it doesn’t necessarily translate to greater robustness, generalization, or interpretability. Future research must therefore prioritize the development of novel metrics capable of assessing these nuanced characteristics. These metrics could explore a model’s sensitivity to input perturbations, the consistency of its internal representations, or its ability to extrapolate beyond the training data distribution. Such advancements would move beyond simply identifying the ‘best’ performing model in a narrow sense, and instead facilitate the selection of models that are truly resilient, reliable, and understandable – ultimately leading to more effective and trustworthy forecasting systems.
A model’s capacity to withstand unexpected market shifts isn’t solely determined by its predictive accuracy, but is deeply intertwined with how it internally represents the data. Research suggests that the structure of these internal representations – the patterns and relationships the model learns – directly impacts its resilience. A robust model doesn’t just memorize historical data; it develops an understanding of underlying principles and dependencies, allowing it to generalize effectively even when faced with novel conditions. Investigations are now focused on deciphering how specific features within a model’s architecture contribute to this stability, examining whether certain learned representations are inherently more resistant to perturbations or ‘shocks’ than others. Understanding this connection promises not just improved forecasting, but the ability to design models that are fundamentally less vulnerable to the inherent unpredictability of financial markets.
The concept of an ‘edge of stability’ – the point at which a complex system is most susceptible to change – offers a compelling framework for understanding forecasting model resilience. Research suggests that models operating near this edge can exhibit heightened sensitivity to even minor market fluctuations, leading to unpredictable functional divergence – where models initially aligned in their predictions rapidly drift apart. Investigating this phenomenon could unlock strategies for constructing more robust models, potentially through regularization techniques that constrain the model’s operating point away from instability, or by developing methods to actively monitor and adjust the model’s parameters in response to shifts in market dynamics. Ultimately, understanding how a model’s proximity to the edge of stability impacts its forecasting accuracy is vital for building systems that can reliably navigate unpredictable economic landscapes.
A recent analysis of training dynamics reveals a critical distinction between the Adam and Stochastic Gradient Descent (SGD) optimization algorithms in forecasting models. Researchers observed that models refined with Adam converge to solutions characterized by a significantly higher maximum Hessian eigenvalue – reaching 111.5 – compared to those trained with SGD, which plateau at 63.1. This \lambda_{max} value, a measure of the curvature of the loss landscape, suggests Adam-trained models settle into sharper minima. While potentially achieving lower initial loss, these sharper minima are often associated with increased sensitivity to perturbations and a greater propensity for volatile behavior when confronted with unseen data or market shifts, implying a trade-off between immediate performance and long-term robustness.
The pursuit of optimal forecasting, as detailed in this exploration of optimizer choice, feels less like science and more like arranging deck chairs on the Titanic. The paper meticulously illustrates how functionally divergent models can achieve predictive equivalence-a fancy way of saying different paths arrive at the same, often illusory, destination. Donald Davies observed, “The computer is a tool for extending man’s mental reach, not for replacing his mind.” This rings true; the optimizer, despite its complexity, merely implements a prior, obscuring the underlying assumptions. The Sharpe-Turnover frontier demonstrates this: performance metrics beyond simple loss are critical, yet still only capture a slice of the inherent chaos. It’s not about finding the optimal model, but understanding how the chosen optimizer shapes the resulting ‘book of pain’-the inevitable errors that accumulate in production. They don’t deploy-they let go.
The Road Ahead
The observation that optimizer choice sculpts functionally divergent models, despite predictive equivalence, feels less like a breakthrough and more like a re-statement of an old truth. The pursuit of minimizing a loss function-any loss function-will inevitably lead to solutions that prioritize different aspects of the underlying data generation process. That these differing emphases manifest as altered portfolio turnover, and are largely invisible to standard evaluation, suggests the Sharpe ratio, despite its ubiquity, offers a surprisingly narrow view of model quality. It’s a metric that rewards what is predicted, not how it is predicted.
Future work will likely focus on quantifying this ‘functional divergence’ beyond simple turnover metrics. One anticipates a proliferation of new ‘model health’ indicators, each attempting to capture some elusive aspect of model behavior. The history of machine learning suggests these indicators will prove transient; each will be gamed, bypassed, or rendered obsolete by the next architectural innovation. The real challenge isn’t finding a better metric, but accepting that a truly comprehensive assessment is likely impossible.
Ultimately, the field may circle back to the understanding that the choice of optimizer isn’t merely a technical detail, but an implicit prior-a statement about the desired characteristics of the solution. And, as with all priors, it should be chosen with careful consideration, not blind faith. It’s a lesson repeatedly learned, and repeatedly forgotten, in the relentless drive for incremental improvement.
Original article: https://arxiv.org/pdf/2603.02620.pdf
Contact the author: https://www.linkedin.com/in/avetisyan/
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2026-03-04 06:35