Online Portfolio Selection with ML Predictions
Authors: Ziliang Zhang
Tianming Zhao
Albert Zomaya
Venue: NeurIPS 2025 poster ()
Keywords: Universal Portfolio
Algorithms with Predictions
Learning-augmented Algorithms
Primary area: optimization
TL;DR: We propose an online portfolio selection algorithm that performs nearly optimal under accurate forecasts and remain provably robust.

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.

Online Portfolio Selection with ML Predictions
Ziliang Zhang
Tianming Zhao
School of Computer Science The University of Sydney Camperdown NSW 2050, Australia zzha0461 10ha0461@umi.... sydney. edu. au
School of Computer Science
The University of Sydney Camperdown NSW 2050, Australia tzha2101@uni sydney
.uyddeeeeee........eooooooo edu au
Albert Y. Zomaya
School of Computer Science The University of Sydney
Camperdown NSW 2050, Australia albert zomayassyddey edu.
Abstract
Online portfolio selection seeks determine a sequence of allocations maxi- mize capital growth. Classical universal strategies asymptotically match the best constant-rebalaneed portfolio but ignore potential forecasts. whereas heuristic methods often collapse when belief fails. We formalize this tension in a learning- augmented setting 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-gee -gadee 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.
Introduction
Online portfolio selection, the sequential allocation of capital across multiple assets (e.g., stocks), sits at the crossroads of mathematical finance, statistical learning, and online algorithmic design. In every trading period, an investor must commit to a nonnegative, unit-sum weight vector before observing the next price relatives] relying on two imperfect guides: extracted patterns from historical returns and predictive signals on future returns that may be noisy. The puzzle therefore two-fold: (i) distill from the past statistical regularities that remain informative, and (ii) fuse them with fallible forecasts in way that preserves worst-case guarantees, Resolving this tension is challenging because we can neither guarantee the persistence of past regularities trust the accuracy of future forecasts blindly.
Literature review. The growth-optimal portfolio paradigm traces back to Kelly's ① information- theoretic formulation of proportional betting, which showed that compounding wealth the exponen- tial rate is achievable when returns are i.i.d. Capital growth theory [2] later extended this principle multi-asset markets, providing a rigorous benchmark for ongghoriizon investment under more
Throughout, "price relative" and "return" are used interchangeably.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

realistic dynamics, Contemporary research has bifurcated into two main streams. The first is to find universal algorithms that provide provable guarantees against adversarial return sequences without committing to return model; the second relies on heuristic assumptions or data-driven forecasting powered by machine-learning models, trading theoretical certainty for empirical performance. These complementary approaches motivate our study, we therefore begin with a overview of each.
Universal algorithms, such as Universal Portfolio 3 and Exponential Gradient Update 4 make no market assumptions about price dynamics yet remain provably competitive with the best constant- rebalanced portfolio (BCRP) selected in hindsight. After n rounds, their cumulative wealth lags BCRP by at most polynomial factor, yielding sublinear regret in logarithmic growth. However, to hedge against adversarial sequences, universal algorithms deliberately diffuse capital across assets, forfeiting the windfalls that arise when strong predictive structure is present. A more frugal member of the provably safe camp the uniform buy-and-hold strategy [5]. Its final wealth is bounded at the arithmetic mean of single-asset returns, but the absence of rebalancing makes it even more conservative than the universal portfollo family, and empirical evidence shows it often lags behind whenever return dispersion is substantial. Further algorithmic families are reviewed comprehensively in the survey by Li and Hoi [6]. However, despite theoretical guarantees against offline benchmarks, these traditional algorithms share a cautious bias that may overlook significant predictive gains.
g to pattern matching approaches [9] rely on market assumptions to inform algorithmic decisions based on signaled patterns. Furthermore, there is growing interest in employing machine-learning (ML) models forecast future returns, heavily inclining the portfolio to the highest predicted asset in attempt to yield optimal return. Morris et al. [10 introduce an ensemble framework that combines data mining techniques with long short-term memory networks for forecasting stock and Bitcoin prices. Singh and Srivastava [I] deploy a deep neural network for stock price forecasting, demonstrating enhanced performance over recurrent neural networks. Zhang et al. (12) integrate a generative adversarial network with a long short-term memory module, achieving promising results in real-world closing-prit prediction. Feng et al. [33 propose an advanced machine learning approach to stock movement prediction, employing adversarial training to enhance the generalization capability of neural networks. These prediction- based strategies can exploit favorable market conditions and achieve superior returns if the data follows the underlying model. However, when the distribution of future returns deviates from the model, these approaches may make poor predictions and suffer unbounded losses compared to more conservative methods. In addition, ML models may suffer from overfitting if trained on insufficient or unrepresentative features. As wealth compounds exponentially, even a brief sequence of poor forecasts can rapidly erode capital, leading to catastrophic losses within a short time horizon.
On the other hand, heuristic methods ranging from mean reversion 7
Navigating preservation versus prediction-driven aggressiveness brings us to a fundamental question:
How to exploit forecasts yet preserve worst-case guarantees in portfolio selection?
Our answer is to weave ML forecasts into the fabric of classically robust schemes. The design is guided by the emerging paradigm of algorithms with predictions [14]. where an online procedure re- ceives a potentially faulty glimpse of the future and must convert that hint into improved average-case performance while keeping any inflation of worst-case regret provably minimal. In this framework, a baseline algorithm is augmented by some predictive signals that balances follow-the-predd predicton against hedge-for-adee adversits,,,, yielding an seror-sensitive competitive ratio that smoothly interpolates between perfect-forecass optimality and classical worst-case guarantees under adversarial prediction.
The dorming-augmeented paradigm has been instantiated across a wide spectrum of classic prob- lems. Lykouris and Vassilvitskii [15) demonstrate how single tunable parameter yields error sensitive competitive ratio for the paging problem, outperforming LRU [T6] whenever the predictor informative while retaining its worst-case bound. Kumar al. [1 adapt the same recipe both the ski-rental dilemma and non-clairvoyant job scheduling, Bai and Coester (18) present a sort- ing algorithm that harnesses potentially erroncous predictions to enhance computational efficiency. Learning- augmented techniques have also appeared in financal settings, including Pareto-optime threshold based algorithms for online conversion problems [9] and tight consistency-robusse bounds for Ane-Max-Searde [20]. For more comprehensive overview, we refer to the survey by Mitzenmacher and Vassilvitskii [1 and the online repository at [14. Online portfolio selection shares the same sequential-decision procedure: each round demands an irrevocable allocation before the next return vector revealed. By importing the augmmntte toolkit from algorithms with predictions into this framework, we aim to fuse the upside of ML forecasts with the downside pro-
2

---

tection of conservative strategies, yielding wealth trajectories that improves smoothly with accurate predication while retaining a measurable offline benchmark when predictive quality degrades.
To date, very few studies have embedded modern MI forecasts directly into the update rules of classical online portfolio algorithms. The main precursor is the side-information variant of Cover's s universal portfolio [22]. At every round, the investor receives discrete signal that labels the current market regime (i.e., bull or bear) and maintains a separate universal strategy for each regime, with the goal of matching the state-dependent BCRP (BSCRP). Borodin et al. 7 instead assume mean-reversion pattern in next-period prices, but offer no error-sensitive guarantees linking performance to the quality of its forecasts. More recent work has refined the side-in infornfformttinn idea: Bhatt al. [23] extend universal portfolios continuous side information via probabilistic partitioning that achieves first-ordes asymptotic optimality against BSCRP, while Yang et al. [24] employ an
framework to construct state-dependent expert ensembles that achieve the same asymptotic benchmark.
However, these constructions inherits several limitations. (i) Both BCRP and BSCRP are modest benchmarks, which can lag the true offline optimum that reallocates into the single best-performing asset each day by an exponential factor. (ii) Universal algorithms converge BCRP and/or BSCRP only at a polynomial rate in wealth, so the lower bound becomes practically vacuous once the exponential gap to the global optimum is accounted for. (iii) The regret bound on growth rate with side information grows linearly with the number of states, so a richer information alphabet paradoxically weakens the guarantee. (iv) Focusing solely on asymptotic growth obscures performance over the finite horizon which matters most in practice.
(v) Most critically, existing results are prediction- agnostie. They provide no quantitative characterization of how performance scales with prediction error, leaving unanswered how the algorithm fares when the side information is perfectly informative, entirely misleading, or somwehere in between.
These limitations motivate our approach, which fuses ML forecasts with adversarial safeguards and provides error-sensitive guarantees that explicitly couple portfolio performance prediction quality. We now formalize the problem and introduce the necessary notation.
Problem statement. x
We study online portfolio selection over n trading periods and m assetts. Let
x(i) = (z) (f), x2 (i), Im (m)) E IRT
(1)
where (i) is the price relative of asset j from period - 1 to A portfolio strategy defines
b" = -bb(1, (())))
b(i) = (z) bm (i)) E m-
with (1) 2 0 and ΣØ= (i) = 1 (self- finenefinnced, no margin/shorting). The wealth after n periods is
Sb x") = (b x(0) = I1 (1)2z() ==1==1
==1
We set the initial capital to Sn = 1; thus, Sn equals both the terminal wealth and total growth factor. At the start of each period i, the investor receives ranking forecast from black-box ML oracle. Let [m] == {1....,,7 m). The oracle outputs a permutation (i) of [m], where aii), denotes the index of the asset ranked k-th (highest) by predicted one-step return. For analysis, define y(i by reindexing the realized returns according to the predicted ranking:
(1)
(1))
(ie(i) == (o) (i)
Thus yı (i) (2)
> (i) reflects the oracle's predicted order (highest to lowest). Impor- tantly, our algorithm and guarantees depend only on the ranking oi); we do not assume access to, require, predicted magnitudes of next mext-period returns. The vector y() is introduced solely as analysis notation to track the realization under the predicted order.
We make no assumptions about how this vector (ranking) is generated, and it may come from any MLL model, e.g.: a neural network 100 that predicts one-step-ahead returns and then ranks them; a gradient- boosted tree model (25 that outputs the ranking directly; or any heuristic ranking rules. For analysis, we also define the clairvoyant order statistics x [1) (i) 2 ... 2 ((m) (i) as the entries of x(i) sorted in
We fix the number of assets the outset and treat it constant.

---

non-increasing order, and write (i) == ((m)).).))...(0....))) ((m) (4)) and x (1) = (()))),,(((()))))
Note the distinction: z. (i) is the return of asset (original index), whereas z(2 (t) is the j-th largest return at time (ranked value). Ties, when present, are broken by a fixed deterministic rule; following
analysis are invariant to this choice. We apply the same shorthand to the current weights by bi)i):
bd (i) (b))(0), (1), (1)
Our goal is to design online portfolio algorithm with prediction that translates the oracle's s noisy forecasts y" into higher wealth whenever they are informative, and preserves a provable worst-case guarantee matching the best prediction-agnoste baseline when the forecasts are arbitrarily inaccurate. We adopt the geometric mean of all returns referring the value line index, as such a baseline. Formally,
1/m
77
7
SGM 72
(1)
(4)
The best-known competitive (26] prediction-agnostcc strategy the Universal Portfolio B. which only guarantees wealth no less than 5 G Prop. 5J. Matching this floor therefore secures state-of- the-art worst-case performance while leaving headroom for upside when the oracle proves informative.
Our contribution. We propose a reamning-uugment portfolio with updating rules guided by ML forecasts, the Rebalanced Arithmetic Mean (RAM). For the prediction errot 7h E (0, 1] induced by any predictions y" = ( ............ y(n)), the resulting wealth of RAM after n periods satisfies:
3
1/m
GRAM
II b'(1 y(i) 2 max
SOM
(1)
Th II
b
(i) (i)
with weight updating rules b+ (i) followed by
strategs Dusing predicted permutations:
bi 1)ys (i
by (1) = (b()) ... (mm)(()) (i) = ΣØ= bú - 1)88 1) be(1) = m Vi,j, k (5)
When the predictions are maximally informative 7/, = and (t) (1), , Vi, RAM achieves wealth least a constant fraction of the hindsight-optimal "all-in-best-assed" benchmark:
SRAM
SOPT n
max xi)
Conversely, when forecasts are actively hostile (adversarial), we prove that RAM dominates the value line SGM, The results are independent market assumptions other than positive return factors.
We substantiate our theory with real-world equity experiments in progressively richer settings:
1. Prediction-free benchmark: We show that randomized, prediction-free variant of RAM outperforms the Universal Portfolio in expectation.
2. Controlled-noise study: Using synthesized predictions under controlled accuracy thresholds, RAM degrades gracefully under heavy noise and raises smoothly with improved predictions.
3. Real-world deployment: We deploy RAM with real-world ML model on recent market data, achieving significant gains over existing baselines, underscoring its practical utility.
2 Portfolio with predictions
Blindly following the forecasts. Consider two assets 21 and 22, where either doubles halves while the other does the opposite. Let the forecast be wrong with independent probability E (0, 1). An investor who completely follows the forecast accumulates wealth S, = 2(1-))n 2(1-e)m(0.5)0" = 2(1-2e)n When e < 0.5, the investor still gains, but at an exponential gap of 2 against the optimum 5 = 2 Conversely, when 0.5, the wealth decays exponentially zero. Therefore, even a small error rate produces unbounded competitive loss and expose the strategy to total ruin.
3 Formally defined Section 2.1 (Eq.
'Detailed in Section 2.1 Algorithm for pseudocode implementation.

---

Partially trusting the forecast. A natural repair is to invest fraction X [0, in the predicted winner and keep the remaining in a safe baseline such the uniform portfolio (0.5, 0.5). Lemma A. (Appendixx proves that, when the forecast is wrong with independent probability E (0. 1), the expected wealth [[S.] = 1. 1.25% (1 + 0.6A(1 - 2c)) Consequently, the expected per-period log- growth is (*,, = log 1 25 + (1 <) log + 0.6A) + log (1 0.6A). For every fixed X 2 1 1/3. there always exists a a (A) < 0.5 forcing (.(A = O. am adversary pushes the error rate even slightly above this threshold, eX, < 0 and wealth decays exponentially to zero. Meanwhile the competitive ratio against the optimal hindsight 2" still diverges exponentially whenever À > 0 and 0. Thus, fractional betting ameliorates but does not eliminate worst-case fragility.
Our intuition is to decouple magnitudes from directional signs through a two-stage architecture:
1. Base generator. At each period, prediction-agnoste strategy updates a weight vector using only realized return history, preserving the base algorithm's 's worst-case guarantees.
2. Permutation layer. The ML. predicted ranking (the ordering of upcoming returns) then re-labels these weights, concentrating larger masses on assets it considers promising.
This design inherits the robustness of the base generator yet exploits predictive insights whenever it is present. The crux pair a provably safe generator with a permutation rule that minimally degrades its guarantee while delivering gains under accurate predictions. The remainder of the paper therefore concentrates on this permutation -based portfolio class.
2.1 Rebalanced arithmetic mean
We propose the Rebalanced Arithmetic Mean (RAM) portfolio which overlays an oracle-supplied ranking of next-period gross returns onto the buy-and-hold baseline. Let the baseline assign weights (i)) period according to equation[5 where the pipeline follows by matching higher weights to higher predicted returns according to their ranks. Concurrently, an oracle outputs a permutation (t) intended to rank the forthcoming returns x(i): applying it produces the oracle-ordered vector y(i) as defined in equation 2 Conceptually, RAM is still a buy-and-ha policy, yet executed in a dynamically permuted coordinate system determined by permutations {o(1))
RAM operates by greedy matching: it pairs the largest weight ((y(()) with the asset which the oracle
predicts to deliver the highest return, the second-larged weight with the second-highest predicted return, and so on. The portfolio held at period therefore earns the inner product b (i) yi)).
Importantly, we highlight the key relation of bee x(i) b (i) (i) as Algorithm I visual
example of RAM's update process is shown in Figure !
Figure 1: RAM's 's rank matchint architecture: larger weights paired with higher predicted ranks.
Permutation Layer
Base Generator
Permutation Layer
True return
Prediction
Realized return
New weights
Ordered weights
Original weights
(nx())
x,00=
Tankk 2
Y:(/=1.06
6,(6 + 1)
0.491
bey(0-0.55
b,0)-0.3
bi(i) y(i)
(gp(())))
x>00=0.82
rank 3,'
y200=1.20
hg(i 1)
0.337
bgy/0000.3
bg(0m0.22
b-16) y(i
(Opu())))))
x 1.05
v360)-082
hg(B 1)
0.172
boy00=0.2
by())-055
b-18) -(()
To quantify the oracle's s accuracy, we compare its ordering with the clairvoyant optimal ordering of the upcoming returns x- (1). We define the per-period error rate (7))) and total error rate mhi
(1) (1) 71) = bl()) (1) be
E bo) THE
Th =
(6)
(7
'The daily error rate (1) is defined so that it decreases as predictions improve and increases as they degrade. The aggregated quantity behaves oppositely; although we phrase it as an overall error rate for brevity, it should be interpreted performance metric, where larger values indicate better performance.
5

---

Algorithm RAM: rebalanced arithmetic mean with predictions
Require: return matrix [= C E oracle permutation o(i) <- FORECASTPERM(((()
1: initialize weights b(1) -
H.), wealth (0) <- 1
2: for 4 to do
3: Receive ranking prediction oi) from oracle.
4: b+ - (b))))
5: baew 0
6:
for j 4- to m do
b reverse pairing
7:
boce < bj
8:
end for
9: Receive return x(i).
10: R(i) bow x(i)
11: S(i) < S(i 1) x R(i)
12:
b(i+ - (b"ew o x(i))
13:
end for
14:
return S(n)
The cumulative penalty Th captures the aggregate loss in wealth relative to the clairvoyant optimal matcher. The final aggregated wealth RAM can therefore be expressed in exchangeable forms:
12
II b² (1)
(1) (1))
1
m
SRAM
II
(b((i)
y(1)
The
II (()(0)=() ==1j=1
(7)
This multiplicative decomposition cleanly separates the contribution of the underlying buy-and-hold strategy from the oracle-induced error factor. When the oracle's ranking is accurate, 1h1 1 and RAM achieves the clairvoyant optimum. Even under maximally adversarial rankings, 17, E (0, 1], RAM's wealth cannot drop below the value line index (the lower envelope of the arithmetic mean).
The reason RAM works. RAM is built on two simple intuitions. The first design goal is the
immediate optimality under perfect forecast. Universal portfolios with side information [22] only promise to approach the optimum in the limit, which means that they can lag for thousands of rounds and still be theoretically "optimal". RAM avoids that delay: perfect oracle yields the exact clairvoyant growth for each and every period. The second goal is to prove disciplined loss guarantee under adversarial regime. As inspired by the arithmetic-geomme -eeommereeeeeeriicc mean (AM-GM) inequality, we note key invariance: uniform buy-and-hold portfolio is unaffected by any fixed permutation of asset coordinates. Once the ordering is frozen, the AM-GM lower envelope applies the entire return sequence, independent of the chosen coordinate system. Consequently, re-indexing the next-period returns is essentially cost-freee, and the resulting wealth trajectory remains above the value line. Theorem 2.1 formaaizzs
claim with tight lower bounds under different levels of prediction quality. Theorem 2.1. For arbitrary market sequence x" as specified in equation I and predictions y" (2),),(22) (n)) where each follows by equation 2
we have claims for Algorithm 7
1. For arbitrary predictions (worst-case guarantee):
1/m
M
CRAM
(1)
Vnn (0, 1]
(8)
2. Under perfect predictions (best-case guarantee):
RAM
SPPT
1
II max (()), for Th
(9)
Proof. We first prove When predictions are perfect, y(1) =
x (1) Vi and nn =1:
GRAM
b b+(()
x (i)
bτ) (1)

---

By the monotonicity of (i) and x² (i):
b(1)(0) ... (i) and (()) 2 .... (mm) (1) ((n(0)=()))00)) 2 ... 2
Therefore, the index carrying larger weight at period must also carry larger wealth share after i:
(()())
(1)(0)=((
by + 1) دド= (()m((
= (i +1
Σド= b (i)
Apply (n) + 1) and expand the right-hand side recursively down to period 1 yields:
I TG) Th Eド=
bn) + 1)
by) (n + +1 =
T
bin
(1)
1 Σー T(
CRAM
ΣI
(t)
1 (1)
1 SOPT m
T 2j=1 66) (1)
m
j=1
proving 9 Next, prove
8 Recall under adversarial predictions SRAM
II b² and:
(1) (m)()))))))))
by (1) (i)
ba +
ba +))
Σド= (() (m-
ΣØ (1) (i)
where we have re-indexed the assets relative to the weight vector such that o(i)s denotes adversarially permuted pairing on weight (i), so the algebra is carried out in moving coordinate system attached to the weights, not in the fixed asset labeling. The specific indices (i)) are irrelevant to the wealth analysis, since the only property need is that every return value Iz(i), (i) is paired with exactly one weight coordinate. Apply by (n + 1) and expand the right-hand side yields:
1/m
I (o)
m R T (t)
m
by (n + 1) = II= b+ (i)
SRAM
ΣII
Z
IIII
k=1i=1
1=11=1
where the right-hand side follows by the arithmetic-geometre mean inequality.
Relative to existing universal strategies, RAM occupies distinctive midpoint. Under perfect oracle, RAM extracts a constant 1/m fraction of the hindsight optimum, whereas the side-information universal portfolio family converges only the modest best state-constant rebalanced portfolio and does so asymptotically. At the opposite end of the information spectrum, both RAM and universal portfolios enjoy the same distribution-free safe net, the value line index, ensuring no catastrophic under-performann
The gap between these extremes is where RAM's permutation layer becomes pivotal. Because weights are reassigned wholesale according to the forecast ordering. RAM reacts sharply on informative ranking, but with higher variance when the signal is noisy. Recent advances in machine learning have begun to yield predictors with demonstrably non-trivial forecasting power. RAM can harness these models immediately while its AM-GM floor caps downside risk. In addition, the algorithm runs in (m log m) per round, making readily deployable on large-scale portfolios where the exponential costs of richer universal mixtures are prohibitive.
Table I: Stock combinations on NYSE.
NYSE(O)
NYSE(N)
1: Iro & Kin
7: Cok & Ahp & Gm & Ge & Sc
2: Com & Kin
8: Dow & Mmm & Ame & Mer & Jnj
3: Com & Mei
9: Ame & Kim Alc & Ahp & Mor & Cok & Pan & Gm
4: Luk & Kin & Are
10: Mmm Dow & Mer Ge Ihm & Ing & Hp & Inj
5: Shr & Ge & Kin
11: Ge & Mer & Gm & Ahp & Inj & Pan & Ame & Ing & For & Dup & Dow
6: Gul & Esp & Pil
12: Alc & Cok & Hp & Ibm & Kim & Mmm & Mor & Se & Gm & Ahp & Pan
7

---

3 Empirical study
We begin by assessing RAM the canonical New York Stock Exchange (NYSE) benchmark, using i..dd random rankings to model fully oblivious and uninformative oracle. Our primary dataset is the original NYSE(() collection (271, which contains 36 stocks spanning 22 years (1962-1984)
over 5,651 trading days. To capture a broader range of market volatility and ensure more recent coverage, we also consider the extended NYSE(N) dataset [28], encompassing 21 assets from 1962 to 2006 (11,188 trading days). To avoid duplication, we preprocess NYSE(N) by retaining only the period from 1985 to 2006 (5,526 trading days), thus removing any overlap with NYSE(O). Each data point represents the ratio of an asset's 's price on a given trading day to its price on the previous day. Table details the various combinations of stocks: Combinations 1-3 have been widely adopted in seminal works such as B 4 Combinations 4-6 follow those introduced by Yang et al. [29]. and the remaining combinations were chosen for additional diversity and expanded market dimension.
To study RAM's s behavior under controllable signal quality, we draw rank- predddrdiittions from a one- parameter family (Djope which interpolates linearly between adversarial and optimal orderings:
(1-2p) 2p) d0t + Up
0 S p S 2
Dp p) + 1)
(10)
p <
where o¹ and of are the clairvoyant ascending and descending orderings of the upcoming returns, U is the uniform distribution over all m! permutations, and 6, denotes Dirac mass at permutation Consequently, p = 0 produces the worst-case forecast (deterministically ascending). p = $ gives full randomness, and p 1 provides the best-case forecast (deterministically descending). Moving p away from the middle point linearly transfers probability mass from the uniform component to the appropriate Dirac measure, facilitating transparent and smoothly tunable notion of forecast quality. For each fixed p we draw one permutation per period: with probability equal to the Dirac weight we output the deterministic order; otherwise we apply Fisher- Yates shuffle. The identically sampled permutation is fed to every applicable strategy under comparison, ensuring that performance differences arise solely from algorithmic design rather than additional stochasticity.
For each accuracy threshold p. we run 1,000 independent Monte Carlo simulations to obtain stable performance estimates. Table 2 summarizes the mean, standard deviation, and median, thereby reducing sensitivity to distributional skewness. For combinations I to 6, we ablate our method against the Universal Portfolio (UP) 3 and its side-information variant 22 owning to their strong theoretical guarantees. Because the exact UP incurs exponential computational cost in the number of assets, we also benchmark the Exponential Gradient (EG) 4 universal portfolio and its side- information extension on combination 7 to 12, which scale more favorably to large-scale portfolios. To ensure a fair comparison with iide-information portfolios, we recast each forecast as a categorical signal: the asset assigned rank defines the current state, yielding mn possible states. Under perfect forecasts, the resulting best state-constant rebalanced portfolio (BSCRP) matches the hindsight optimum. Conversely, the ranking that is adversarial to RAM still offers S1-portfolios exploitable structure, since every state inevitably identifies least one low-return asset. Constructing a ranking that is simultaneously worst-case for both methods is impractical, we therefore adopt this mapping, recognizing that is conservative for RAM. All experiments compute under 6h on one standard CPU. Source code is available at https: thub. com/ wwtmmm//opm/p// OPML
Our controlled Monte Carlo study reveals a clear, mono- tone relationship between forecast quality and portfolio performance. When the oracle is completely uninfor- mative (RAM-R), our algorithm still outperforms the vanilla UP in expectation. albeit with higher variance; its median wealth remains at least twice that of the geometrie-mean benchmark. On the other hand, UPSI exhibits lower dispersion and reliably tracks UP, but
Figure 2: Comb. I: RAM vs. EGSI over E [0.5.0.6] Similar trends hold for other comb. and p E (0.6, 1].
10t2
RAM
10le
EGSI
Final Final wealth
10a
its expected return trails RAM-R. Introducing even a modest informational edge at 53% forecast accuracy dramatically shifts the landscape: RAM now dominates both EGSI and standard HG in both mean and median wealth. The advantage widens with portfolio dimension: the regret for EGSI grows exponentially in the number
10a
10a
10?
0.50
0.52
0.54
0.56
0.58
0.60
Prediction accuracy
8

---

Table 2: Synthetic-forecas ablation on the NYSE dataset. Subscripts denote the forecast-accuraz parameter p. Reported statisties comprise the mean (μ)), standard deviation (σ), and median (p).. Results average over 1,000 Monte Carlo trials, each drawing predictions according to equation 10
Comb.
GM
UP
RAM-R P =
UPSI P 50%
/4
6.06
39.97
57.78
215.75
13.28
52.33
36.81
40.57
14.65
80.53
136.39
551.33
32.15
97.32
64.84
79.14
34.52
74.07
101.03
185.44
53.84
84.88
22.54
77.26
6.69
32.18
45,66
107.83
17.81
40.61
10.68
37.13
5
5.96
24,40
35.49
96.26
15.14
30.74
8.58
27.95
19.35
47.11
56.97
73.23
36.53
55.94
9.84
53.23
Comb.
GM
EG
RAM
EGSI
53%
53%
/
A
12.88
21.74
1.5E+03
1.9E+03
9.3E+02
22.94
0.43
22.89
8
22.85
33.37
1.0E+03
9.3E+02
7.2E+02
34.80
0.44
34.77
17.61
30.18
4.5E+03
6.3E+03
2.6E+03
31.18
0.29
31.15
18.43
31.50
4.6E+03
6.5E+03
2.6E+03
32.82
0.36
32.78
16.59
27.81
6,1E+03
8,2E+03
3,5E+03
28.61
0.17
28.60
12
14.67
27.86
1,5E+04
2.3E+04
7.3E+03
28.82
0.22
28.81
Comb,
RAM
EGSI
RAM
EGSI
100%
p=100%
60%
60%
1.1++1
4.9E+07
6.3E+08
8.6E+08
3.4E+08
36.25
2.35
36.19
1.5E+35
6.4E+05
5,8E+07
7.0E+07
3.8E+07
47.76
47.70
1.3E+49
6.7E+04
2.8E+10
4.6E+10
1.3E+10
39.59
1.15
39.52
10
3,4E+48
1,4++05
2.0E+10
3,4E+10
1.0E+10
42.47
1.43
42.34
11
2.5E+53
2.9E+03
1.5E+11
2.3E+11
7.3E+10
33.18
0.59
33.14
2.3E+59
1,8E+04
2.5E+12
5.0E+12
1.1E+12
35.10
0.83
35.08
of states, whereas RAM's guarantees are dimensionninneeeeddeee yielding superior performance in high-asset markets. At 60% accuracy, the gap becomes exponential. Under perfect forecasts where BSCRP coincides with the hindsight optimum: EGSI still converges only asymptotically. whereas RAM immediately captures the optimal growth. These results show that RAM exploits accurate predictive signals more effectively than universal-portfole baselines; Figure 2 the resulting monotone gap. We verified the sub-50% accuracy regime but omit the plots for brevity: the trajectories closely match our theoretical bound, with RAM's wealth smoothly converging to the value line without ever crossing below yet retaining domination in expectation. Collectively, these results demonstrate that RAM amplifies even modest predictive edges while provably guarding against catastrophic drawdowns.
illustrates
We further evaluate RAM on a live, production- grade setting that couples real market data with ceaninngto--rag engine. We use the nightly-refreshee S&P 500 historical
an calusstral-strenngee panel (30 available on Kaggle, containing 501 constituents from 2010 2024. The index liquid, broad-based, and widely adopted in ampirical-finane: research, ensuring reproducibility and practical relevance. stress-test robustness, we analyze two disjoint windows: (i) COVID-19 crash from July 2019 to May 2020, capturing extreme volatility: (ii) Recent market from Dec 2022 to Dec 2024. reflecting contemporary trading conditions. We construct three baskets as in Table 3 (i) a- Popular: nighhcapptallzztion names favored by retail brokers, mirroring real-world deployability; (ii) 3-Diversified: Two tickers from each of five GICS sectors for cross-sector hedging; - y-Large-scale: A broad slice of the index to probe scalability. Full stock names are in Table 5
We employ LightGBM LambdaMART 25 to forecast the ranks of next-day returns. The model is retrained each trading day using 250-day sliding window, featuring contemporaneous and three lagged returns per asset. A decaying factor with 0.995 prioritizes recent observations while discarding stale information. A 60-day hold-out slice inside the same window provides early-stopping signals, eliminating look-ahead bias. This protocol guarantees strict online deployment where only data available at decision time enter either training validation. Hyper- parameers and code are

---

Table 3: Basket specifications for the S&P 500 experiments. Regime A spans July 2019 to May 2020;
regime B covers December 2022 to December 2024.
Market Regime
Comb.
Comb. 3
Comb. y
Popularity-weighte B: Recent window stocks
A: High volatility
Diversified sector
Random draw 30 stocks
10 stocks
supplied in the supplementary material while we keep the exposition concise to focus on RAM's prediction-integraeed yet risk-free contribution.
Table 4: Empirical results on S&P 500 with production-grade machine-learning forecasts. Best denotes the clairvoyant benchmark of the single top-performing stock. ML allocates all capital each round to the asset with the highest predicted return (rank- 1) from the model.
Market Regime
Comp.
Best
GM
RAM
EGSI
ML
1.733
1.206
1.239
1,230
1.305
A (Covid-crash)
1,358
0.953
1.003
0.982
1.678
Y
1.733
0.885
0.915
0.924
0.532
O 8,293
2.470
2.512
2.629
1.676
B (Resent-window)
3.153
1.300
1.377
1.373
0.572
8.293
1.401
1,493
1,501
1.391
Across both market regimes, RAM consistently outperforms EGSI whenever the forecast surpass the value-line benchmark and remains resilient even when prediction collapses. In the turbulent COVID-19 window (regime A): RAM systematically exploits informative forecasts, capturing an additional 10.9% and 2.6% of excessive predictive edge over EGSI in baskets a and 33 respectively; when forecasts turn catastrophic in basket 7, RAM stays within 1% of EGSI while still eclipsing the value line by 3.3%. In the acceccy-ocused window (regime B), the gap widens: although every forecast underperforms the value line, RAM trails EGSI by only 4.5% in basket and 0.6% in basket y. Strikingly, under the worst-case basket 3 where the machine-leariing signal lags the value line by 56%, RAM nevertheless beats EGSI by 2.9% and outstrips the value line by 5.9%. In aggregate. these findings provide compelling empirical evidence that RAM converts even modest predictive cues into tangible gains while preserving a universal-style safety net when those cues deteriorate. Notably, this robustness is achieved with an off-the-shelf rank predictor trained solely on historical returns within narrow asset universe, which suggests that more sophisticated, finance-driven models would amplify RAM's advantage even further, as corroborated by our synthetic simulations.
4 Conclusion
This work advances learning-augmenter portfolio selection by introducing the Rebalanced Arithmetic Mean portfolio with predictions (RAM), a principled framework that overlays machine-learning forecasts on a classical rebalancing rule. We prove that RAM captures a constant fraction of the hindsight-optimal wealth when forecasts are perfect and dominates the market's geometric mean baseline even under maximally adversarial signals. Extensive experiments on real-world equity data complements the theory, spanning both synthetic forecast simulations and real-world MI. models: RAM harnesses informative predictive signs more effectively than side-information universal portfolios, delivering superior risk-adjusted returns across turbulent and benign regimes. These findings evidence the practical use of forecast without sacrificing worst-case safety
An important practical dimension left unexplored is transaction cost. Because RAM rebalances every round through permutation layer, incurs trading frictions analogous to those faced by universal portfolios. The attendant drag on portfolio wealth merits systematic investigation, for instance, through transaction-cost frameworks exemplified by the model in (3I.. Future work is also encouraged to establish the Pareto Pareto-optimal consistency-robuutness trade-off in online portfolio
selection. This work represents one operating point via RAM; whether the frontier can be improved (and whether RAM is Pareto-optimal) remains open.
10

---

Acknowledgments
We thank the anonymous reviewers for their constructive feedback. Ziliang Zhang gratefully ac- knowledges the support of his wife, Wenyu Liu, and his parents, Chuanqing Zhang and Zhenying Huang.
References
[1] L. Kelly, A new interpretation of information rate. Bell System Technical Journal, 35(4):
917-926, 1956. doi: 10. 1002/// 1538-7700..95566.033800000
[2] N. H. Hakansson and W. T. Ziemba. Capital growth theory. In Handbooks in Operations Research and Management Science, Vol. Finance, chapter 3, pages 66-86. North-Hollane Amsterdam, 1995. doi: 10. 10.1016//0927-05077055538
[3] T. M. Cover. Universal portfolios. Mathematical Finance, 1(1):1-29, 1991. doi: 10.1111/j. 1467-9965. 1991. 11b0000.....
[4] D. P. Helmbold, R. E. Schapire, Y. Singer, and M. K. Warmuth. On-line portfolio selection using multiplicative updates. Mathematical Finance, 8(4):325-344 347, 1998. doi: 10.1111/1467-9965 00058.
[5] Burton G. Malkiel. A Random Walk Down Wall Street: The Time-Teste Tested Strategy for Successful Investing. W. W. Norton & Company, New York, 13 edition, 1973. ISBN 0-393-06245-77
[6] B. Li and S. C. H. Hoi. Online portfolio selection. ACM Computing Surveys, 46(3):1-36, 2014. doi: 10.1 145/2512962.
[7] A. Borodin, R. El-Yaniv, and V. Gogan. Can we learn to beat the best stock. Journal of Artificial Intelligence Research, 21:579-5555 2004. doi: 10. 10.1613/jair. 1336.
[8] B. Li, P. Zhao, S. C. H. Hoi, and Gopalkrishnan. Pamr: Passive aggressive mean reversion strategy for portfolio selection. Machine Learning, 87(2):221-255 2012. doi: 10.1007/ $10994-012-52814
[9] László Györfi, Gábor Lugosi, and Frederic Udina. Nonparametric kernel- based sequential investment strategies. Mathematical Finance, 16(2):337-357, 2006. doi: 10.11 146779496.9995. 2006.00274.x.
[10] K. J. Morris, S. E. Egan, J. L. Linsangan, C. K. Leung, A. Cuzzocrea, and X. Calvin. Token- based adaptive time-series prediction by ensembling linear and non-linear estimators: A machine learning approach for predictive analytics on big stock data. In Proceedings of the International Conference on Machine Learning and Applications (ICMLA), pages 1486-1491. IEEE, 2018. doi: 10.1 wtt.cmma..21..022...
[11] R. Singh and S. Srivastava. Stock prediction using deep learning. Multimedia Tools and Applications, 76(18):1859--1888 2016. doi: 10. 10.10077//11444-111444559
[12] K. Zhang, G. Zhong, J. Dong, S. Wang, and Y. Wang. Stock market prediction based on generative adversarial network. Procedia Computer Science, 147:400-40 406, 2019. doi: 10.1016/ pprocs.2019.01.
[13] Feng, H. Chen, X. He, J. Ding, M. Sun, and T.-S. Chua. Enhancing stock movement prediction with adversarial training. In Proceedings the Twenty-Eight International Joint Conference on Artificial Intelligence (IJCAI), pages 5843-5849, 2019, doi: 10.249663ijcai.201998
[14] ALPS. ALPS: Algorithms with Predictions. https://algorithme with with-prediccions github. i 2024. Accessed May 10, 2025.
[15] Lykouris and Vassilvitskii. Competitive caching with machine learned advice. In Pro- ceedings of the 35th International Conference on Machine Learning (ICML), volume 7, pages 5201-5213 Stockholm, Sweden, July 2018. International Machine Learning Society (IML.S.. ISBN 978-15 10866796-. Held from July 10-15, 2018, Code 141700.

---

[16] Daniel D. Sleator and Robert E. Tarjan. Amortized efficiency of list update and paging rules. Communications of the ACM, 28(2):202-208, 1985. doi: 10.1145/2786.7993.
[17] Ravi Kumar, Manish Purohit, and Zoya Svitkina. Improving online algorithms via ml predictions. In H. Larochelle, N. Cesa-Bianchi, H. Wallach, K. Grauman, S. Bengio, and R. Garnett, editors, Advances in Neural Information Processing Systems (NeurIPS), volume DDDeember, pages 96619966777 Montreal, Canada, December 2018. Neural Information Processing Systems Foundation. Held from December 2-8, 2018, Code 147412.
[18] Xingjian Bai and Christian Coester. Sorting with predictions. In Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS 2023), 2023, Full paper.
[19] Bo Sun, Russell Lee, Mohammad Hajiesmaili, Adam Wierman, and Danny H. K. Tsang. Pareto- optimal learing-auggmetedd
augmented algorithms for online conversion problems. In Advances in Neural Information Processing Systems, volume 34, pages 10339-10350, 2021.
[20] Ziyad Benomar, Lorenzo Croissant, Vianney Perchet, and Spyros Angelopoulos. Pareto- optimality, smoothness, and stochasticity in Bearning-augmenter one-max-search. In Pro- ceedings of the 42nd International Conference on Machine Learning (ICML), 2025. URL https:///peerreviex wwtp://oooooooddoooooooooo
ICML 2025 (poster).
[21] Michael Mitzenmacher and Sergei Vassilvitskii. Algorithms with predictions. Commun. ACM, 65(7):33-35, June 2022. ISSN 0001-0782. doi: 10. 145/3528087.
[22] T. M. Cover and E. Ordentlich. Universal portfolios with side information. IEEE Transactions on Information Theory, 42(2):348-363, 1996. doi: 10.1 0009//88888700
[23] Alankrita Bhatt, Jon Ryu, and Young- Han Kim. On universal portfolios with continuous side information. In Proceedings of the 26th International Conference on Artificial Intelligence and Statistics (AISTATS 2023), volume 206 of Proceedings of Machine Learning Reseurch, pages 4147-4163, Valencia, Spain, 2023. PMLR.
[24] Xingyu Yang, Xiaoteng Zheng, and Lina Zheng. Online portfolio selection strategy with side information based on learning with expert advice. Asia- ssia-Pacific Journal of Operational Research, 41(05):1-28, October 2024. doi: 10. N0l1142/S022155 75559233011999
[25] Guolin Ke, Qi Meng. Thomas Finley, Taifeng Wang, Wei Chen, Weidong Ma, Qiwei Ye, and Tie- Yan Liu. LightGBM: A highly efficient gradient boosting decision tree. In Advances in Neural Information Processing Systems 30 (NeurIPS 2017), pages 31 3146-3154, 2017.
[26] Erik Ordentlich and Thomas M. Cover. The cost of achieving the best portfolio in hindsight. Mathematics of Operations Research, 23(4):960-988) November 1998. doi: 10.1287/moor.23.4
960.
[27] L. Bin. Nyse (0) dataset.
edu, htt..a//aaaaat////aa///////////a NYSE_ TYSE___Datassee O_Dastset.....e html 2024. Accessed January 14, 2025.
[28] L. Bin. Nyse (n) dataset. //wp:://www.r mysmu. edu. sg/ ag//aaautt//cchhoo//s//aaaaaeee NYSE_N, 2_Daaseet....... html 2024. Accessed January 14, 2025.
[29] X. Yang, J. He, J. Xian, H. Lin, and Y. Zhang. Aggregating expert advice strategy for online portfolio selection with side information. Soft Computing, 2463):2077-2081, 2019. 2019. doi: 10.10077:305000019-4003333
[30] Kaggle contributor: Andrew MVD. S&P 500 Stocks (daily-updated) kaggle dataset. https //www. waggla.gee... http:///aaaaaaaaaaaa///////////0 May 2025.
2024. Dataset accessed 10
[31] A. Blum and A. Kalai. Universal portfolios with and without transaction costs. Machine Learning, 35(3):193-22 1999. ISSN 0885-6125. doi: 10.1023/A.1007:5377337444
12

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification:
The claims are rigorously grounded: Theorem 2.1 formalizes the complete
theoretical guarantee, and the experiments empirically corroborate these claims.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
is fine include aspirational goals as motivation long is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: The major limitation of this study is its omission of transaction-cost analysis. This choice aligns with much of the online-portfolio literature B 22 where the algorithmic framework is first developed and subsequent work addresses trading frictions separately. We therefore treat transaction-cost modeling distinct research track and confine our discussion of it to the future-work (Conclusion) section.
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed not penalize honesty concerning limitations.
3. Theory assumptions and proofs
13

---

Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
Our theoretical claim does not rely on ANY market assumptions beyond the mild requirement that each return factor is strictly positive (i.e., $1 stake cannot generate loss greater than $1).
Justification:
Guidelines:
The answer NA means that the paper does not include theoretical results. All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced the statement of any theorems.
The proof' can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Answer: [Yes]
Justification: All sources of randomness are explicitly documented, and the experiments can be reproduced by fixing the global random seed to 42. The paper (and codes) provides every implementation detail required to replicate the primary results in full.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is novel architecture, describing the architecture fully might suffice, or if the contribution a specific model and empirical evaluation, may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often good way to accomplish this, but reproducibility instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, other means that appropriate to the research performed.
also be provided via detailed
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) the contribution is primarily a new algorithm, the paper should make clear how to reproduce that algorithm.
(b) If the contribution is primarily new model architecture, the paper should describe the architecture clearly and fully.
(c) the contribution a new model (e.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
14

---

(d)
We recognize that reproducibility may be tricky in some cases, in which case authors are welcome describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, described in supplemental material?
Answer: [Yes]
Justification: Both the NYSE and S&P 500 datasets are publicly available; moreover, supply a one-click Colab notebook that fully replicates all reported experiments.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips.. cc/
for more details.
While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https
/ mnips.cc/public//
for more details.
The authors should provide instructions data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible in supplemental material (appended to the paper) recommended, but including URLs to data and code permitted.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: The experimentation section covers every key setup detail. Any peripheral settings omitted for space documented in the code.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, or as supplemental material.
7. Experiment statistical significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [Yes]
Justification: For every metric we report the mean, standard deviation, and median derived from 1,000 Monte Carlo trials, thereby conveying the same dispersion information that conventional error bars provide.
15

---

Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by enror bars, confi- dence intervals, or statistical significance tests, least for the experiments that support the main claims of the paper.
The factors of variability that the error bars capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall
with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
should be clear whether the eπror bar is the standard deviation or the standard error of the mean.
It is OK to report I-sigma error bars, but should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not show in tables figures symmetric error bars that would yield results that are out of range (c.g. negative error rates).
error bars are reported in tables or plots, The authors should explain the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
Question:
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
Answer: [Yes]
Justification: experiments compute under on one standard CPU. Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't 't make it into the paper).
9. Code of ethics
Question: Does the research conducted the paper conform, in every respect, with the NeurlPS Code of Ethics https: ftctpa:/nnerips..co http:///wwrrppp.cc/o//////ccccaa
Answer: [Yes]
Justification:
Guidelines:
The answer NA means that the authors have not reviewed the NeurlPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g.. there special consid- eration due to laws or regulations in their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive socictal impacts and negative societal impacts of the work performed?
Answer: [NA]
16

---

This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Justification:
Guidelines:
The answer NA means that there is no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied partieular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out, For example, it is legitimate to point that improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition attacks, mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Does the paper deseribe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: All data and model used publicly available.
Guidelines:
The answer NA means that the paper poses such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper. properly credited and are the license and terms of use explicitly mentioned and properly respected?
Answer: [Yes]
Justification:
All third-party assets are fully credited and their licenses respected.
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package dataset.
17

---

The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (e.g.. CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (e.g.. website). the copyright and terms of service of that souree should be provided.
assets are released, the license, copyright information, and terms use in package should be provided. For popular datasets, paperssi thcode com/ com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
If this information is not available online, the authors encouraged to reach out to the asset's ' creators.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer:
[NA]
Justification:
Guidelines:
The answer NA means that the paper does release new assets.
Researchers should communicate the details the datasct/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset used.
At submission time, remember to anonymize your assets (if applicable). You can either create anonymized URL or include an anonymized zip file.
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included the main paper.
According to the NeurIPS Code of Ethics, workers involved data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification:
Guidelines:
18

---

The answer NA means that the paper does not involve crowdsoureing nor research with
human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent)
may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and expect authors adhere the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
16. Declaration of LLM usage
Does the paper describe the usage of LLMs if it is important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
Question:
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.
Please refer to our LLM policy for what should or should not be described.
19

---

A Technical Appendices and Supplementary Material
Lemma A.1 (Fractional betting). Let the forecast be wrong with independent probability E (0, 1). For two assets whose return rectrrsactors in period i satisfy:
( (2, 2),
forecast is correct
(x1 (1), (1)
(22 2), forecast is wrong
For any constant X E [0, 1), let the portfolio be b(i) (eo(s) (1 - 2 2) where Co(1) puts weight on the predicted winner and O on the other. The following holds:
1. Expected wealth.
55 1
(
3 5
(SS.)
1+
A(1 -20)
1
2. Expected per-period log-growth.
en(A,)) [log [log S.] = log 4
5 + (1 - e) [1
3 52 + elog 1 (1-
3 51
3. Zero- -roo-ghowth error rate. For every fixed A 2 3. there exists a unique
log 5 1 + log (1 + (1
E 00. 1 2
(A) log (1 + (1 - log (1 {A
such that Wn = O. For e > E* (A). Wn < <0
4. Exponential competitive gap. Let S 2 be the hindsight-optimmm. Then
S%
8
3
1 +
A(1 - (2e)
E B[S..
5
which grows like eia) whenever X 0 and > 0.
Proof. (1) The per-period growth is R(i)= == b(i) x(i) = 5 + 2t. When predictions are correct
R+ (i) = 5 3 11, when predictions are wrong R (1)
1 +
1 Since R(i) are i..d,
n
E[Sa] =
R(()
E [R()) = (E((()))
=
2=1
5 3
5
3 A
)
5
3
3
11 c)
+
+
=
+
X
2 Xc
4 4
4
4
4 4
5
m
5 "
3
X(1 26)
1
X(1 26)
4
(2) Because R(i) are i.i.d.
E), Eloog Sml
1 71
1
Σ
Eloog Ri(i)]
Σ
n
E3 [log R(1) = Ellog R(1)]
n1
n
5 5
5 +elog
5 >
5 (1 c) log
[+ 3 {1
+elog ( 3
(1-31)
(1-e) log 1+
5
-log
4
4
4
(3) Observing W, (A, E) is strictly decreasing on e, AA E [0,
dW, (A. c)
log
1 - _\ - log
1 + _ > <0
de
5"
At = 0, W., = log 5 + log (1 + (2)
> At = }.
Wn (A, 1 = =loog $ 2 log (1 - ()))
which is negative if and only if 1 - (g))² 2
>>
J.. If 1.
W,, (A, is positive at 0.
20

---

negative at 2 and strictly decreasing on With the intermediate value theorem, there exists exactly one root e* (0, } such that Wi, (A,(ee"")) 0. Solving this yields:
log 4
+ log
1+ { > c² log (1+5))
3
log
1 1
5 )
3
log + log (1 + (
(0,
1
c*(a)= log (1 A) log (1 (1
A
3
Because (A <) is strictly decreasing in e and satisfies (A,,, c (A)) = 0, it immediately follows that Wn, (A, < 0, vc (A).
(4)
S
3 1 + A(1 - 26) 5
E[Sp]
3 1
1A11 2x) =
5
4
Since + -AA((11 2x) <
whenever > 0, 0, the ratio grows exponentially n.
Table 5: Ticker compositions of three baskets S&P 500.
Basket
Constituent tickers
a
T, MSFT, NVDA, AMZN,
B
CSCO, MSPT, WRB, RE, T, NELX, SBUX, BBY, ABT, BAX
7
CMCSA, AMP, HSIC, FSLR, FCX, VLO. BWA, ANSS, AMZN, C, EXPE, FDX, TJX, WST, EMN, PGR, FAST, PODD, HST, ADM, NVDA, PAYX, BRO, MO, ESS, DTE, WEC
21


Reviews:
[{"review_id": "R0Qy8pZBLp", "reviewer": "Reviewer_rUoK", "summary": "This 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.", "questions": "Questions 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?", "limitations": "No limitations", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for recognizing the significance of our contribution and for your thoughtful evaluation. Your feedback is invaluable to the communities of learning-augmented algorithms, mathematical finance, and machine learning, and we appreciate the time and expertise you have devoted to this review. We address your concerns below.\n\n**Additional baselines**\n\nWe appreciate the suggestion and agree that adding more algorithms could be informative. However, our study focuses on a narrower yet fundamental question: How does algorithmic performance change as the quality of an external prediction signal varies? To address this, our experimental design requires baselines that (i) explicitly accept the same prediction signal as input and (ii) possess a principled mechanism for exploiting or defending against that signal. Universal portfolio variants equipped with side information (UPSI, EGSI) satisfy (almost) both criteria, making them the most appropriate comparators:\n\n- Scope and assumptions: PAMR and other mean-reversion or pattern-matching strategies rely on specific market assumptions and do not consume predictions. Their objective and information sets therefore differ materially from RAM’s.\n- Practical Fairness: In our experiments we feed an identical (almost) prediction signal to RAM and UPSI/EGSI. This choice is conservative for RAM because RAM must remain robust even when the signal is adversarial, whereas UPSI/EGSI can still partially exploit it. For algorithms that ignore predictions altogether (mean reversion, pattern matching, etc.), any attempt at a “fair” comparison would either give them a free pass on adverse signals or bias the evaluation against prediction-aware methods.\n\nFor these reasons, we limit the baseline set to UPSI/EGSI. We will add clarifying text to the manuscripts and an appendix discussion noting that incrementing non-prediction methods is non-trivial and left to future work.\n\n**Transaction cost**\n\nTransaction costs are crucial in practice. We originally experimented with incorporating them into RAM, but the resulting model proved sufficiently intricate that a full treatment would have overshadowed the main exposition. We therefore flagged the omission as a limitation in the conclusion. Below we summarize the insights we obtained and will incorporate as a brief discussion in our upcoming manuscript. Assume a fixed fee applies only when selling and let us analyze two rebalancing policies:\n\n- Full Liquidation: Every period the algorithm sells its entire holding and buys back according to the new weights. If the fee is $0.4%$%, wealth evolve as $S^{RAM,fee}_n=0.996^n\\cdot S^{RAM,free}_n$. Thus, the drag compounds exponentially, and the cost drives the portfolio below its value line in the worst case. On the other hand, whether this cost drives consistency below $\\frac{1}{m}OPT$ remains open.\n- Optimal partial rebalancer: Instead of liquidating everything, the algorithm sells only the excess of assets forecast to underperform. This greatly reduces turnover, but the net impact depends on the realized return sequence. While intuitively superior to full liquidation, it complicates the theory and can still deteriorate robustness. It remains open whether bounds analogous to the original ones survive up to a constant, logarithmic, or polynomial factor.\n\nFurthermore, extending RAM to cost-aware rebalancing requires a new adversarial model and proof technique, work we believe merits a separate study. Including it here would significantly enlarge the paper and blur the main contribution. We will add the above discussion to the revised manuscript and highlight a cost-aware extension of RAM as an important direction for future research.\n\n**Reproducibility**\n\nThank you for highlighting the reproducibility concern, and we apologize for the oversight. We summarize the key input features employed by our LightGBM model from line 297 to 302. To keep the main text centered on the algorithmic contribution, the complete hyper-parameter table together with the training and evaluation scripts are relocated to the supplementary material (line 302-304).\n\nTo enhance reproducibility in our upcoming manuscript, we will cite a GitHub link in the abstract containing the full training code and the exact LightGBM parameter grid (as well as all codes needed for replicating our experimental results). We hope these additions can address the reproducibility concern.\n\n**Tightness on consistency**\n\nThank you for raising this point. On real data, the bound becomes tight very quickly because wealth compounds exponentially. i) For a two-asset illustration, the inequality on line 213 telescopes to $S^{RAM}_n=(S^{OPT}_n+S^{ADV}_n)/2$, where ADV denotes the worst-performing asset each round. Over even moderate horizons, $S^{ADV}_n$ becomes negligible when compared to $S^{OPT}_n$, leaving $S^{RAM}_n\\approx\\frac{1}{2}S^{OPT}_n$. Hence the realized ratio approaches the theoretical floor in practice. ii) A similar argument extends to high-dimensional portfolios: the product of the period-wise \"minimum\" (any return except the highest one) decays so rapidly when compared to OPT that the empirical ratio is close to $\\frac{1}{m}$.\n\nWe will add a concise version of the above insights in our empirical section to the revised manuscript for clarity.\n\n**Relation to multiplicative update algorithms**\n\nThank you for highlighting the connection. RAM does share the multiplicative flavor of follow-the-winner algorithms such as EG, but its design philosophy diverges in three important ways:\n\n- Different objectives. EG balances two forces: it rewards assets that performed well in the most recent period and penalizes larger deviations using relative entropy. This built-in conservatism smooths the portfolio path. By contrast, RAM is prediction-centric: at every round, it re-weights fully toward the ordering supplied by the external predictor, with no explicit regularizer. If predictor says asset X will outperform, RAM immediately assigns it the highest weight, regardless of yesterday’s allocation.\n- Distinct risk profile. Because RAM can flip a large fraction of its wealth onto a single predicted winner, one bad prediction can sharply reduce its value. We accept this vulnerability intentionally; the algorithm is analyzed in two regimes: consistent with good predictions and robust with bad predictions. EG lacks this dual guarantee: it is more stable but sacrifices (the extent depends on the learning rate) the upside that RAM captures when the predictor is informative.\n- Theoretical guarantees. EG is competitive with BCRP (or BSCRP if side information is provided) in the asymptotic regime. RAM instead interpolates between the robustness and consistency extremes, because it is willing to follow the predictor without the inertia that EG enforces.\n\n**Significance on relative ordering**\n\nThank you for pointing this out. RAM transforms the entire predicted order into a graded weight vector, assigning higher-ranked assets with more weights while retaining positive shares for lower-ranked ones. Thus, even if the oracle misses the single best shot, any residual accuracy in the rest of the ranking directs capital toward several assets that ultimately do well (under the context of reasonably good relative ordering). The ML all-in strategy by contrast, has no such hedge. Theorem 2.1 bounds RAM’s loss when predictions are adversarial (above value line index). This guarantee is what protects the portfolio in the worst-case top-rank flip you highlight. However, while the whole order matters, the first rank is still the single largest weight (at least $\\frac{1}{m}$). A wrong call there reduces growth more than a low-rank error from any other “good” assets. To answer your question, a reasonably accurate ranking across the board does enhance RAM’s performance, but the size of the gain depends on the specific (and potentially arbitrary) return path. Because such paths can vary arbitrarily, it is difficult to provide a tighter, sequence-independent analysis beyond the universal bound (smoothness) in Theorem 2.1.\n\nThank you again for your valuable feedback. We hope the above response fully addresses your concerns. We will incorporate these clarifications into the revised manuscript and appreciate your continued support."}, {"review_id": "ztxGvaduHG", "reviewer": "Reviewer_jLHv", "summary": "This 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.", "questions": "## 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).", "limitations": "The authors clearly addressed the limitations in the last section.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "## 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.", "was_revised": true, "final_justification": "The authors' rebuttal sufficiently addressed my concerns on the clarity. I've updated my score from 3 to 5, increasing the score on Clarity from 1 to 3.", "rebuttal": "Thank you for recognizing our contribution and for the detailed suggestions on improving the methodology section. Your feedback is invaluable, and we greatly appreciate the time and effort you invested in your review.\n\nWe promise to substantially revising the methodological part to ensure greater clarity and accessibility, including: i) clarifying notations; ii) refining the algorithmic pseudocode; iii) improving descriptive text. As a first step, we would like to correct a typographical error in Theorem 2.1 (details below). We apologize for this oversight and will incorporate the correction, along all methodological improvements in the revised manuscript.\n\n**Typo (line 213)**\n\nThe expression following $\\geq$ should read\n$$\\frac{1}{m}\\prod_{i=1}^nx_{(1)}(i)$$\nnot $x_{(1)}(j)$. Below we describe the planned changes in detail to address your concern:\n\n**Notations**\n\n- For permutations: i) We will add a descriptive sentence defining that $Q_m$ is the set of all possible permutations from $m$ assets; ii) Throughout, a permutation evaluated at position $k$ on day $i$ will be written $\\sigma(i)_k$. We previously used $\\sigma(i,k)$ to avoid nesting subscripts, but recognize this was imprecise and will adopt the clearer $\\sigma(i)_k$ everywhere.\n- We will write $x_{(k)}(i)=x_{\\sigma(i)_k}(i)\\in\\mathbf{x}(i)$, meaning the k-th largest return at time $i$, which is the component of the vector $\\mathbf{x}(i)$ indexed by $\\sigma(i)_k$.\n- We apologize for the confusion on the definition of error rate. When defining the daily error factor $\\eta(i)$, we debated whether to use the ratio RAM/hindsight or its reciprocal. Intuitively, an error measure should decrease when performance improves and increase when it worsens. This interpretation favors hindsight/RAM, so that larger gains correspond to smaller error. However, this seems to complicate the expression and blur clarity. We will adopt your suggestions and use RAM/hindsight while explicitly explaining the meaning of “error” under this context.\n- On first use (Introduction, line 20) we will add the footnote: “Throughout, ‘return’ and ‘price relative’ are used interchangeably”.\n- We will use the dot product notation consistently in Algorithm 1 (line 9). The appearance of transpose will be replaced.\n\n**Typos/Suggestions**\n\n- We apologize for the oversight and will adopt your suggested correction on line 130.\n- i) We will add a forward reference for $\\eta(i)$ and $\\eta_n$ in the contribution section so readers can locate the definitions on them easily. ii) We will refine the description on “rank-matching strategy” and insert a forward reference where the mechanism is formally introduced. iii) We will state explicitly after equation (3) that $b_j$ denotes the $j$-th coordinate in a fixed basis, whereas $b_{(j)}$ is the $j$-th *largest* weight after the permutation applied by $\\sigma(i)_j$. iv) The intended meaning for equation (5) is that larger weights are paired with higher predicted returns (one-to-one by rank). We will correct the notation accordingly.\n- We apologize for the oversight and will correct the typo on “though”.\n- In equation (7), you are correct: the equality should be \"$=$\" rather than \"$:=$\" as the error has been defined in equation (6). We will fix this.\n- We will incorporate your suggestion on equation (8): mention the alternative expression earlier and state explicitly that Claim 1 gives only the worst-case robustness lower bound.\n- We will add the transition proof after line 212 to improve logical flow.\n- The proofs after line 211 and 214 mark the switch from a fixed coordinate system to the permuted one. We will add explanatory text to clarify the change of basis.\n- For Tables: “Comp.” stands for composition (stock composition in Table 2; basket composition in Tables 3 and 4); we omit the parameter $p$ elsewhere to avoid redundancy but will reiterate this in the table.\n- Thank you for the suggestion on log-growth plots; we will include additional plots and discussion to illustrate smoothness more transparently.\n- Thank you for noting the ambiguity in the pseudocode. We will clarify that the algorithm only requires the predicted rank ordering; the numerical return estimates appear purely for theoretical exposition and can be removed from the implementation. In the revision we will: i) move the formal definition of the predictor from the problem statement to the analysis sections; and ii) update Algorithm 1 as suggested: line 3 will consume the ranking prediction, the portfolio update will use $\\mathbf{b}^{new}$, the subscript in line 7 will be corrected, and an explicit \"observe return\" step will be added after line 8.\n- We will insert a brief clarification in the main-mechanism paragraph explicitly stating that $(\\mathbf{b}^{new})^\\top\\mathbf{x}(i)=\\mathbf{b}^{\\downarrow}(i)^\\top\\mathbf{y}(i)$. Thank you for highlighting this point.\n\nWe are sincerely grateful for your thorough and constructive review. Your detailed comments not only strengthened the methodological and expository aspects of our submission but also affirmed the broader relevance of our work to learning-augmented algorithms, mathematical finance, and machine learning research. We commit to a thorough revision for clarity, incorporating all of your recommendations so that the final manuscript is fully polished and ready for dissemination. Finally, we appreciate the time and expertise you devoted to guiding these improvements."}, {"review_id": "4sFuhYBNxH", "reviewer": "Reviewer_Q1Qe", "summary": "This 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.", "questions": "1. 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).", "limitations": "I do not see any issues, other than explaining whether $m$ is assumed to be a constant or not.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "After the very helpful rebuttal by the authors, I am upgrading by overall score. I still believe that the paper is borderline for the following three  reasons:\n\n*  The experimental evaluation is against very old algorithms, and seems to miss developments and baselines from regret-based optimization.\n\n* The consistency is a fraction of $OPT$ only insofar as $m$ is treated as a constant, which may or mat not be a realistic assumption. \n\n* Related to the above, an upper bound (impossibility result) would be very useful, if not necessary, in par with the recent progress in learning-augmented online algorithms. This does not necessarily imply the identification of the entire Pareto front between  consistency and robustness, but rather a few significant points, e.g., what if we require that the algorithm has robustness close to the \"optimal\" one. This becomes even more significant considering that the lower bound (main algorithm) comes from a very simple analysis, and the theoretical part of the paper is thus somewhat limited. \n\nNotwithstanding the above observations, overall the work is interesting. I am not pushing for rejection, but I am not enthusiastic either.", "rebuttal": "Thank you for your evaluation. We regret the notation and description imprecision that may have caused confusion and appreciate the chance to clarify. Below we respond to your comments in detail.\n\n**Clarification**\n\nOur submission introduces RAM, a learning-augmented portfolio strategy that updates asset weights directly from externally supplied ranking predictions. RAM is designed to provide consistency, robustness, and smoothness:\n\n- *Consistency* (good predictions with good performance): Theorem 2.1 Claim 2 shows that with perfect predictions, RAM achieves a payoff at least $\\frac{1}{m}OPT$. RAM can in practice outperform this bound but can never fall below it under perfect forecasts.\n- *Robustness* (worst-case guarantee under adversarial predictions): Theorem 2.1 Claim 1 establishes that under adversarial predictions and arbitrary return sequences, RAM’s wealth is bounded below by the value line index (the geometric mean of each buy and holds). Because the arithmetic-geometric mean inequality is tight, this bound is essentially tight without introducing sequence-dependent variables.\n- *Smoothness* (error-sensitive degradation linked to prediction quality): Claim 1 also quantifies how RAM’s performance degrades as a function of prediction error, yielding a continuous transition between the robustness and consistency regimes.\n\nBoth the robustness and consistency results are lower bounds; aside from the obvious upper bound OPT, no upper bounds are asserted. Finally, our paper contains no conjecture, and all stated results are rigorously proved and supported by empirical evaluation.\n\n**Is $m$ a constant?**\n\nWe regret the earlier ambiguity. In our setting, the asset universe size $m$ is chosen once at the start of trading and remains fixed for all $n$ rounds. All theoretical results (including RAM's $\\frac{1}{m}OPT$-consistency guarantee) are therefore stated with $m$ treated as a constant parameter. We recognize (as you highlighted) that $m$ is a priori a variable. The claim should be read as: the consistency is a constant factor from optimum under fixed $m$. We will clarify this in the revised manuscript to address the concern of the technical ambiguity.\n\n**Discussion to other problems and regret-based analysis**\n\nThank you for pointing out the link between our portfolio-selection setting and the classical online problems of time series search (one-max search) and one-way trading. All three aim to maximize wealth against an adversary, yet they differ fundamentally in action space and reversibility:\n\n- Time series search: a single irreversible \"stop\" decision chooses one price; performance is gauged against the sequence maximum (or fluctuation ratio).\n- One-way trading: capital can flow from cash to an asset over multiple rounds, but never back; each partial conversion is irreversible.\n- Portfolio selection (our setting): at every round the investor freely rebalances across $m\\geq 2$ assets, moving weight both into and out of any asset. Trades are thus fully reversible, and the benchmarks include: i) the best constant-rebalanced portfolio, and ii) the global optimum of all-in highest stake at each round.\n\nConsequently, one-way trading is a strict special case of portfolio selection with $m=2$ and a one-direction constraint, while time series search is the even narrower all-in-once variant. Our analysis therefore requires different techniques (e.g., continuous weight dynamics, the compounding effect of multiplication, logarithmic growth regret, etc.) and yields guarantees that do not follow directly from the simpler problems. Therefore, while one-max search and one-way trading are instructive pedagogical benchmarks, Cover's portfolio selection framework captures the full multi-asset landscape in which dynamic rebalancing lies at the core of modern online portfolio theory.\n\nWe apologize for omitting this discussion. In the revision we will add a concise subsection contrasting the three settings and clarify why our focus is on the fully reversible, multi-asset scenario.\n\nLines 25-45 and lines 98-108 of the introduction survey existing regret-based approaches, which typically establish asymptotic convergence (to the modest best constant-rebalanced portfolio benchmark) in logarithmic growth. By contrast, our analysis supplies non-asymptotic (exact) guarantees on realized wealth, which is the metric of greatest practical relevance for investors.\n\n**Additional references**\n\nWe appreciate your pointer to the comprehensive survey “Online Portfolio Selection: A Survey”. We acknowledge its foundational importance and regret the omission. We will add an explicit citation in the introduction's literature review section to properly situate our contribution within existing methods.\n\n**Writing clarity**\n\nWe agree that the manuscript’s exposition can be sharpened and have already enumerated specific revisions detailed in our reply to Reviewer jLHv, which we respectfully direct you to consult, as these updates will be integrated into the next version. \n\nYou note that portions of the discussion (particularly in the introduction) are “very unclear”. To address this constructively, could you please identify the exact passages or arguments that require clarification? Concrete pointers will enable us to revise the text with the precision and transparency expected in the NeurIPS community.\n\n**Baselines**\n\nThank you for raising the concern that some of our baselines appear dated. To the best of our knowledge, we are unaware of any existing algorithm that simultaneously (i) incorporates explicit external predictions into portfolio updates and (ii) provides formal guarantees of consistency, robustness, and error-smoothness. Hence, Cover’s side-information variant and its immediate extensions remain the most relevant points of comparison. \n\nThank you for pointing that our ML component may appear overly simplistic. This choice is deliberate: to foreground the consistency-robustness trade-off we wish to study, we use a lightweight LightGBM forecaster that relies solely on lagged returns. The resulting signal is transparent, enabling us to sweep forecast quality without the confounding effects of a sophisticated model. This setup allows us to evaluate how RAM’s learning-augmented updates react across varying levels of predictive accuracy, rather than to demonstrate state-of-the-art forecasting performance.\n\n**Relationship between consistency and robustness**\n\nWe appreciate the question “Can consistency be improved at the expense of robustness, or the reverse?”. In principle, yes. For example, RAM guarantees a consistency floor of $\\frac{1}{m}OPT$ while remaining at least as robust as the value line. One could boost consistency by adopting an aggressive “all-in” strategy that invests solely in the asset with the highest predicted return each period. However, this approach suffers dramatic loss in robustness, rendering it of limited practical value. Determining the optimal consistency-robustness trade-off therefore remains an open problem. Theorem 2.1 quantifies a specific trade-off for RAM, but whether it is information-theoretically tight remains open. We will add a detailed discussion of these open directions in the conclusion of the revised manuscript.\n\nWe hope these clarifications fully address your concerns. Thank you once again for the time and care you devoted to reviewing our work; your feedback has been invaluable."}]

Author rebuttal:

Meta-review: The paper proposes an online portfolio selection algorithm which leverages ML predictions. The proposed algorithm guarantees a worst-case return when the ML predictions are completely wrong, while yielding higher wealth when the predictions are accurate. The authors also derive a prediction error-dependent lower bound on the resulting wealth.

Reviewers overall agree that the paper makes a significant contribution by proposing an algorithm that leverages ML predictions in online portfolio selection in a safe way, while guaranteeing high returns when the predictions are correct. One main concern is the clarity of writing. Authors are strongly encouraged to incorporate reviewers’ suggestions to improve the clarity.
Decision: Accept (poster)