Title: Online Portfolio Selection with ML Predictions
Venue: NeurIPS 2025
Accepted: yes
Keywords: Universal Portfolio

Algorithms with Predictions

Learning-augmented Algorithms
Abstract: Online portfolio selection seeks to determine a sequence of allocations to maximize capital growth. Classical universal strategies asymptotically match the best constant-rebalanced portfolio but ignore potential forecasts, whereas heuristic methods often collapse when belief fails. We formalize this tension in a learning-augmented setting in which an investor observes (possibly erroneous) predictions prior to each decision moment, and we introduce the Rebalanced Arithmetic Mean portfolio with predictions (RAM). Under arbitrary return sequences, we prove that RAM captures at least a constant fraction of the hindsight-optimal wealth when forecasts are perfect while still exceeding the geometric mean of the sequence even when the predictions are adversarial. Comprehensive experiments on large-scale equity data strengthen our theory, spanning both synthetic prediction streams and production-grade machine-learning models. RAM advantages over universal-portfolio variants equipped with side information across various regimes. These results demonstrate that modest predictive power can be reliably converted into tangible gains without sacrificing worst-case guarantees.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the online portfolio optimization problem in a learning-augmented setting. The setting involves a trader who must manage, in an online manner, a portfolio of $m$ assets over a period of $n$ trading days; on each day, the current price of each asset is reveled. In the standard version of the problem, the trader has no access to any information about the future prices of assets. In the ML-augmented setting, each day a prediction is provided via an ordering of the anticipated profitability of the assets. This information may naturally be erroneous, and the objective is to obtain trading algorithms  that have good robustness (competitiveness against worst-case, adversarial predictions), consistency (performance assuming foolproof predictions) and smoothness (quantifiable degradation as a function of the prediction error). \n\nThe main theoretical result is an algorithm with the following guarantees: It has consistency at least $1/m$ times the hindsight optimal revenue; has robustness at least that of the best-known prediction-agnostic algorithms (informally, a profit equal to the geometric mean of all revealed prices) and quantifiable smoothness that interpolates between these two extremes. The upper bound is the only theoretical result, and no lower bounds (impossibility results) are given.\n\nThe work also gives experimental results using both synthetic data, in which predictions are generated according to a specified rule, as well as on more realistic setting in which the predictions come from a gradiant-boosting tree is trained on real data. The performance of the algorithm is then compared to a number of benchmark algorithms.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* The problem is interesting and pertinent to the ML community. Using ML techniques to improve portfolio optimization is a problem with everyday applications.\n*There are both theoretical and experimental results. The experimental analysis incorporates realistic prediction models. \n\nWeaknesses:\n* The main weakness is the claim that the consistency is \"a constant factor from optimal\". This is technically incorrect, unless the number of assets is considered constant. The statement in line 230-233 seems to suggest it is not. Regardless, the absence of a lower bound is critical. One is left with the overall impression that the analysis only scratches the surface of the problem, especially considering that the upper bound is technically not difficult to obtain.\n* There are many related, albeit simpler problems such as one-max search and one-way trading that have been studied in several works under learning-augmented settings. I was looking forward to a discussion and comparison, especially considering that all these works present various types of impossibility results. On a similar vein, there are a lot of works on regret-based analysis of portfolio optimization under regret, but no discussion is given. I think the paper should also cite influential works and surveys by Li and Hoi, and Hazan.\n* While the paper is well written overall, in several places, the discussion is very unclear due to the choice of language and terminology that obfuscates instead of clarifying, particularly in the introduction. \n* As far as I can see, all algorithm benchmarks against which the obtain algorithm is compared come from the 90s. I find this extremely surprising given that portfolio optimization has been studied in many settings. The benchmark \"ML\" is very simple.\n\nOverall evaluation, as summary of the above:\n\n- Quality: Technically sound work, but does not address the full potential of possible solutions.\n- Clarity: Overall the paper is well-written, but some parts use very obfuscating terminology, and misses important references.\n- Significance: Well motivated an interesting problem, with an approach that is lacking lower bounds or some solid theoretical evidence.\n- Originality: The obtained algorithm follows a simple idea with an analysis that is particularly involved, so the contribution is rather incremental.\n\nQUESTIONS:\n1. Does your algorithm have connections to multiplicative-update approaches used in regret-based analysis of similar online problems?\n\n2. Can you explain why all the algorithms you consider as benchmarks are quite old, are there no related recent algorithms in the literature that one could use? \n\n3. What is the conjectured lower bound on the consistency? Also, could one potentially improve the consistency at the expense of robustness (or vice versa).\n\nLIMITATIONS:\nI do not see any issues, other than explaining whether $m$ is assumed to be a constant or not.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new portfolio selection strategy that can incorporate any prediction of \"ranking\" of future returns, while maintaining the worst-case lower-bound guarantee.\nThe classical universal strategies represented by Cover's Universal Portfolio track the best constant strategy in hindsight, up to a polynomial factor that would diminish in the asymptotic regime. These classical approaches also enjoy the worst-case lower-bound guarantee.\nHowever, since there exists high-quality predictions, it is highly desirable to incorporate such a predictive power in the portfolio selection.\nThe challenge is to achieve the best of both worlds, by incorporating predictions and guaranteeing not to go bankrupt.\nThe proposed strategy receives a prediction of \"ranking\" of returns (i.e., price relatives) for the next day, sort the betting according to the ranking, readjust the betting based on the actual returns, and repeat. \nThe computational complexity is also light, only requiring to sort $m$ values at each step.\nThe theoretical guarantee is nice, showing that they are guaranteed to be lower bounded by the value line index same as Cover's UP, but also only away by a constant fraction from the optimal strategy when the ranking oracle is perfect.\nThe experimental results demonstrate the practical relevance of the algorithm.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\nThe problem of portfolio selection with prediction is very interesting in theory and also practically important problem.\nThe proposed solution not only is very simple while naturally incorporating any ranking predictions, but also enjoys nice theoretical guarantees. To my understanding, the solution is new, original, and refreshing.\n\n## Weaknesses\nThat said, I have a serious concern with the current form of writing, as the exposition can be much more improved.\nAfter careful reading, I figured out what the actual algorithm is, but the current writing and poor notation makes it really hard to understand.\nI think this paper has a VERY nice and important contribution, so it deserves better writing for its wider dissemination.\n\nI will be happy to increase the score on Quality and Clarity if the authors promise to substantially revise the methodological part.\nI listed a few suggestions below.\n\nQUESTIONS:\n## Suggestions on notation\nThere are too many inconsistent notation throughout. For example:\n- For permutation: \n  - Define $Q\\_m$.\n  - Given a permutation $\\sigma(i)$ at day $i$, it seems that its evaluation at index $k$ is denoted by both $\\sigma(i,k)$ and $\\sigma(i)_k$. The prior notation is extremely confusing, so please use the latter, if I am not mistaken.\n- After line 124, $x_{\\sigma(i,k)}(i)\\in \\mathbf{x}(i)$ is not well-defined.\n- It's super weird that daily error rate $\\eta(i)$ is defined as such, and the total error rate is the multiplication of the \"reciprocals\". To my view, the daily error rate should be defined as $\\eta(i) = \\text{(gain with RAM)} / \\text{(gain with perfect prediction)}$, which is the reciprocal of the current definition, so that the total error rate can be defined as the running multiplication of all daily error rates.\n- Choose one terminology from {return, price relative}, or explicitly mention that you use them interchangeably.\n- Choose a single notation for inner product. Currently both $\\intercal$ and dot product notation are used.\n\n## Typos/suggestions\n- In line 130, would it be better to say \"We adopt the geometric mean of buy-and-holds of each stock, which is referred to as the \\emph{value line index}, as such a baseline. Formally, ...\"?\n- I appreciate that the authors try to put their contribution early in the introduction in line 134, but the execution is poor. The prediction error $\\eta_n$ is neither defined nor referred to a future reference. The \"rank-matching strategy\" is also not defined at this point. Eq. (5) is not comprehensible. What is $j$ in $b_{(j)}$? This is a very bad notation.\n- Line 165: though -> through\n- In Eq. (7), := should be =.\n- The statement in Eq.(8) in Theorem 2.1 (and so the expression after line 136) is very weird, as the second argument in $\\max$ is another expression of $S\\_n^{\\rm RAM}$ as shown in Eq. (7). The first statement should be just the value line index as a lower bound, for the worst-case guarantee. If the authors want to keep it, maybe they can restate the alternative expression (in Eq. (7)) in Theorem 2.1 before the two items.\n- After line 212, after the $\\Longrightarrow$, consider having $1=\\sum_{j=1}^m b_{(j)}(n+1)=...$.\n- As alluded to earlier, the expressions after line 211 and line 214 are incomprehensible, whenever I see $b_k(...) \\propto b_{(j)}(...)$.\n- Table 2: \n  - Comp. should be Comb. (as a shorthand for Combination)?\n  - Consider using $p=...$ instead of the subscripts like 50% under RAM-R.\n- In the experiment, consider plotting the log-wealth performance of RAM varying $p\\in[1/2,1]$, while having some baseline performance indicated as horizontal lines. I believe that this will be very informative.\n\n- Most importantly, I think the paper can largely benefit from revising Algorithm 1, and also revising the rest of the manuscript accordingly. \nMy biggest confusion was whether RAM requires $\\mathbf{y}(i)$, which is the **true** price relative vector sorted by the prediction, or it just requires a ranking prediction, not the actual sorted relative vector. The confusion arises from the (too) early introduction of the quantity $\\mathbf{y}(i)$ both in Problem Statement and Algorithm 1, which is only required for the sake of analysis. Please do not define it early on, and make clear that the algorithm simply requires ranking prediction. For Algorithm 1, my suggestion is to \"time\" each line from the algorithm's perspective as follows:\n  - Remove line 3, as at this point the algorithm has no access to the price relative at day $i$.\n  - In place of line 3, \"Receive a ranking prediction $\\hat{\\sigma}(i)$ from oracle.\"\n  - Consider removing $(i)$ in $\\mathbf{b}^{\\rm new}(i)$.\n  - Line 7 should be $\\mathbf{b}\\_{\\hat{\\sigma}(i)\\_j}^{\\rm new} \\gets \\mathbf{b}\\_j^\\downarrow$\n  - After line 8, add \"Receive return / price relative $\\mathbf{x}(i)$.\n- And I believe that it is worth highlighting the key relation $(\\mathbf{b}^{\\rm new})^\\intercal \\mathbf{x}(i)= \\mathbf{b}^\\downarrow(i)^\\intercal \\mathbf{y}(i)$, maybe somewhere in the main text, but not in Algorithm 1.\n\nI believe incorporating these will greatly improve the readability of the paper.\n\nA final comment: I believe the proposed algorithm will undoubtedly stimulate further research within the mathematical finance community. I am also particularly excited to see whether this idea might influence recent developments in the application of universal portfolios to computational statistics (e.g., for anytime confidence intervals) and to online optimization (e.g., for parameter-free algorithms).\n\nLIMITATIONS:\nThe authors clearly addressed the limitations in the last section.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the fundamental challenge in online portfolio selection of how to effectively use potentially noisy machine-learning (ML) predictions without sacrificing worst-case performance guarantees. The authors introduce a new learning-augmented algorithm called the Rebalanced Arithmetic Mean (RAM) portfolio. The core idea of RAM is to decouple the weight-generation process from the asset-selection process. It uses a robust, prediction-agnostic base strategy to generate weight magnitudes and then applies a \"permutation layer,\" guided by an ML oracle's ranking of future returns, to assign these weights to specific assets.\n\nThe authors provide a rigorous theoretical analysis of RAM, proving that under arbitrary, even adversarial, market sequences, its performance has a strong lower bound tied to the market's geometric mean. At the same time, if the ML predictions are perfect, RAM's wealth is guaranteed to be at least a constant fraction (1/m) of the hindsight-optimal \"all-in-best-asset\" strategy. This formally captures the desired properties of a learning-augmented algorithm: consistency (good performance with good predictions) and robustness (a strong safety net with bad predictions). The paper substantiates these theoretical claims with comprehensive experiments on historical NYSE and S&P 500 data, using both synthetically-generated forecasts and a production-grade ML model.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nElegant and Novel Algorithm Design: The core idea of RAM—decoupling weight generation from asset permutation—is both simple and powerful. It provides an intuitive and principled way to integrate external predictions into a classical, robust portfolio strategy. This two-stage architecture is a novel contribution to the learning-augmented algorithms literature.\nStrong Theoretical Guarantees: The paper's main theoretical result (Theorem 2.1) is impressive. It formalizes the algorithm's consistency and robustness, showing that RAM smoothly interpolates between near-optimal performance with perfect predictions and a solid worst-case guarantee that matches other universal strategies. Proving these bounds for arbitrary (adversarial) return sequences is a significant technical achievement.\nComprehensive Empirical Validation: The experimental section is thorough and convincing. The authors test RAM under multiple scenarios:\nA prediction-free baseline (RAM-R) to show the strength of the core rebalancing rule.\nA controlled-noise study with synthetic predictions to demonstrate the smooth trade-off between prediction quality and performance.\nA realistic deployment with a production-grade ML model (LightGBM) on recent, volatile market data (including the COVID-19 crash). This multi-faceted evaluation provides strong evidence for the practical utility of RAM.\nClarity and Motivation: The paper is very well-written. The introduction clearly frames the central tension between conservative universal portfolios and aggressive heuristic methods, motivating the need for a learning-augmented approach. The problem statement and algorithm are described clearly.\n\nWeaknesses and Areas for Improvement:\nComparison to a Broader Range of Baselines: While the comparison to Universal Portfolio (UP) and Exponential Gradient (EG) is appropriate, the paper could be strengthened by including other well-known online portfolio strategies, such as mean-reversion (e.g., PAMR ) or pattern-matching algorithms, in the empirical evaluation. This would provide a more complete picture of where RAM stands in the landscape of existing methods.\n\nImpact of Transaction Costs: The authors rightly mention in the conclusion that transaction costs are a key practical dimension that is not explored. While it's standard to omit this in initial theoretical work, the rebalancing nature of RAM suggests that transaction costs could be a significant factor. A brief discussion or even a simple simulation showing the impact of a fixed transaction cost would add significant practical depth.\n\nNature of the ML Oracle: The paper treats the ML model as a black-box ranker. While this is a valid and general approach, a little more detail on the features used for the LightGBM model in the S&P 500 experiments (beyond \"contemporaneous and three lagged returns\") could be beneficial for reproducibility and understanding.\n\nQUESTIONS:\nQuestions for the Authors:\n\n1. Your theoretical analysis guarantees that with perfect predictions, RAM achieves at least 1/m of the optimal wealth. In your experiments, especially on lower-dimensional portfolios, did you observe performance much closer to the optimum than this bound suggests? Is the 1/m factor a loose bound in practice?\n2. The update rule for the weights in RAM is b(i+1) = (b_new(i) ⊙ x(i)) / R(i). This resembles the \"follow the winner\" dynamic of Exponential Gradient. Could you elaborate on the connection between RAM's update rule and other multiplicative-update algorithms?\n\n3. In the S&P 500 experiments (Table 4), the standard ML approach of going all-in on the top-ranked asset sometimes performs very poorly (e.g., 0.532 in basket γ, regime A). In these cases, RAM still performs well. Does this suggest that the value of the ML oracle for RAM lies not just in identifying the single best asset, but in providing a reasonably good relative ordering across the board?\n\nLIMITATIONS:\nNo limitations", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}