Beyond Expectations: Learning with Stochastic Dominance Made Practical
Authors: Shicong Cen
Jincheng Mei
Hanjun Dai
Dale Schuurmans
Yuejie Chi
Bo Dai
Venue: Accepted by TMLR ()

Abstract: Stochastic dominance serves as a general framework for modeling a broad spectrum of decision preferences under uncertainty, with risk aversion as one notable example, as it naturally captures the intrinsic structure of the underlying uncertainty, in contrast to simply resorting to the expectations. Despite theoretically appealing, the application of stochastic dominance in machine learning has been scarce, due to the following challenges: i), the original concept of stochastic dominance only provides a partial order, therefore, is not amenable to serve as a general optimality criterion; and ii), an efficient computational recipe remains lacking due to the continuum nature of evaluating stochastic dominance. 

In this work, we make the first attempt towards establishing a general framework of learning with stochastic dominance. We first generalize the stochastic dominance concept to enable feasible comparisons between any arbitrary pair of random variables. We next develop a simple and computationally efficient approach for finding the optimal solution in terms of stochastic dominance, which can be seamlessly plugged into many learning tasks. Numerical experiments demonstrate that the proposed method achieves comparable performance as standard risk-neutral strategies and obtains better trade-offs against risk across a variety of applications including supervised learning, reinforcement learning, and portfolio optimization.

Published in Transactions on Machine Learning Research (1/2026)
Beyond Expectations:
Learning with Stochastic Dominance Made Practical
Shicong Cen Carnegie Mellon University
shioongcSonndre..cmn.dl
Jincheng Mei Google Research
jemeiiigoogle.co com com
Hanjun Dai Google Research
hadoi@google.. com
Dale Schuurmans
Cinaurmans@goooll...
Google Research & University of Alberta
Yuejie Chi
guejie..his@yale.com
Yale University
Bo Dai Google Research & Georgia Tech
bodai kodei@google.com
Reviewed on OpenReview:
Abstract
Stochastic dominance serves general framework for modeling broad spectrum of deci- sion preferences under uncertainty, with risk aversion captures the intrinsic structure of the underlying uncertainty, in contrast to simply resorting to the expectations. Despite theoretical appeal, the application of stochastic dominance in machine learning has been scarce, due to the following challenges: i), the original concept of stochastic dominance only provides a partial order, and therefore, not amenable to serve as general optimality criterion; and ii), an efficient computational recipe remains lacking due to the continuum nature of evaluating stochastic dominance.
notable example, it naturally
In this work, we make the first attempt towards establishing a general framework of learn- ing with stochastic dominance. We first generalize the stochastic dominance concept to enable feasible comparisons between any arbitrary pair of random variables. We next de- velop simple and computationally efficient approach for finding the optimal solution in terms of stochastic dominance, which can be seamlessly plugged into many learning tasks. Numerical experiments demonstrate that the proposed method achieves comparable perfor- mance as standard risk-neutral strategics and obtains better trade-offs against risk across a variety applications including supervised learning, reinforcement learning, and portfolio optimization.

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Published in Transactions on Machine Learning Research (1/2026)
1 Introduction
In machine learning and operations research, the prevalent paradigm of decision making in the presence of uncertain and stochastic outcomes is to maximize (resp. minimize) the expected utility (resp. loss) with respect to the decision variables. However, the expectation of the decision-dependent utility function alone often depicts an overly simplified snapshot its distribution, ignoring the intrinsic structure of the underlying uncertainly. As such, fails to distinguish decisions with the same expected utilities but drastically different outcomes or model behaviors, especially when taking risk into consideration.
There shortage of risk-sensitive applications where taming the risk is at least as important max- imizing the utility, examples including financial planning, medical examinations, robotics and autonomous systems, to mention few. In these high-stake applications, the principle of expectation may lead to inferior decisions due to its incertainty-agnostic nature. To motivate our discussions, we showcase three distinct applications where risk-averse solutions are of particular interest, which will run throughout this paper.
Risk-senssttin supervised learning. In standard supervised learning, one aims to find an optimally parameterized model such that the expected loss, which measures the difference between the model output and the target output given an input feature, is minimized. However, excessive prediction errors, even with extreme low probability, can pose significant risk to the system operation which may be undesirable.
Risk-sensitiv reinforcement learning. Reinforcement learning (RL) formulates sequential decision making problems as Markov decision processes (MDPs). The goal is to design an action selection rule, i.e., policy, which maximizes the expected cumulative reward collected over the trajectories by executing the policy. Nonetheless, risky actions, if not discouraged properly, might still be deployed when they are compensated by high reward in the long horizon.
Portfolio optimization. A lcading example finance planning is the selection between mutually exclusive investment opportunities or portfolios with uncertain returns. The allocation of the assets yields random variable representing the total return. Maximizing the expected total return alone can lead to large volatility and increases the probability of suffering significant loss.
1.1 Learning with Stochastic Dominance
One needs to go beyond expectations to handle risk, a topic that has been extensively researched in many disciplines. Broadly, existing approaches can be categorized into two paradigms. The first relies on a scalar metric that induces a total order over outcomes, thereby allowing every pair of random variables be com- pared. The mean risk (Markowitz & Todd, 2000) framework is a notable example: it quantifies the problem with two metries: mean that measures the expected outcome, and a risk that measures the variability of outcomes. Popular choices of risk measures include variance (Markowitz & Todd, 2000), semideviation (Ogryezak & Ruszczyúski, 1999), entropic risk (Rudloff et al., 2008), and so on. The mcan-risk approach models risk-averse preferences by penalizing the mean with the risk measure and allows simple trade-offs and efficient learning algorithms (Maurer & Pontil, 2009; Duchi & Namkoong, 2019). However, the design choices of the risk measure and corresponding trade-offs are usually ad-hoc, lacking rigorous justifications. In contrast, the second paradigm leverages partial orders to capture more fine-grained preference structures, among which stochastic dominance (Mann & Whitney, 1947) is the most prominent.
Stochastic dominance. Stochastic dominance (SD) (Mann & Whitney, 1947; Lehmann, 1955) provides principled scheme of comparing real-valued random variables by considering the full spectrum of their kth-order cumulative distribution functions, instcad of condensing into a single scalar metric. In fact, efficient solutions found by the mean-risk approach can be stochastically dominated by other feasible solutions (Ogryczak & Ruszczyíski, 2002), suggesting SD offers stronger guidance and finer granularity in modeling the risk-averse preference. In addition, the deployment of SD for comparing random variables does not need additional assumptions on the distribution (c.g., the mcan- variance approach requires normality (Levy, 2015)).
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Another nice justification of SD comes from expected utility theory (Boutilier et al., 2006; Armbruster & Delage, 2015). Specifically, if one solution stochastically dominates the other, it yields higher expected utility for any utility in a wide class of functions (egg, non-decreasing functions for first-order dominance). Since SD implics higher expectation in the risk-neutral sense by setting the utility function as the identity function, SD more selective in model selection as risk-averse criterion, without the need to specify utility functions.
Challenges. Despite the appealing theoretical properties of SD, applications of SD in machine learning remain scarce. truth, practical algorithms for finding a desirable solution under the criterion of SD remain lacking due to the following critical challenges:
SD, in its existing form, only defines a partial order all distributions of real-valued random variables, which is unsuitable to be optimized directly.
Evaluating SD involves comparisons along continuum of cumulative distribution functions and thus necessitates computationally efficient algorithms.
Existing literature on the computational aspects of stochastic dominance circumvents the first challenge by studying stochastic dominance constraint optimization (Dai et al., 2023), with the goal of maximizing the objective function - typically simply set to the expectation the utility - over a feasible set that consists of all dominating solutions versus a predefined reference solution. However, the fact that the reference solution being fixed compromises the optimality guarantee in the first place, as the approach fails to distinguish two feasible solutions with the same expected utility, even if one dominates the other. On the other end, while it is possible for standard optimization formulations over global scalar metric to yield an optimal SD solution (c.g., optimizing nondecrcasing and strictly concave utility function guarantees non-dominance in the second order), the induced total order introduces extra preference beyond SD that can be undesirable. This motivates the question:
Can we design practical algoritiom that finds an optimal solution in terms of stochastic dominance?
1.2 Our contribution
In this work, we aim to establish a general framework of learning with stochastic dominance, by tackling the two challenges mentioned above. We handle the first challenge by quantifying the degree of stochastic dominance as functional and formulating SD optimality as a fixed point of the corresponding optimization process. This molivates the design of an iterative optimization procedure with non-stationary objective functions that can be solved efficiently. We summarize our contributions follows.
We first generalize the original stochastic dominance concept to enable feasible comparisons between any arbitrary pair of random variables, paving the way to a general machine learning framework that optimizes stochastic dominance.
We propose Learning with Stochastic Dominance (LSD), a novel first-order method for finding approximate optimal solutions in terms of stochastic dominance in the hypothesis space. The op- limality under LSD is equivalent to non-dominance under SD and hence allows direct combination with any risk measure or preference functional that is compatible with SD.
We establish convergence guarantees under mild technical assumptions despite the non-stationary nature of the optimization process. It is shown that LSD finds an e-approximate optimal solution within O iterations, which introduces minimal computational overhead compared with standard mini-batch stochastic gradient method.
We draw connections between SD and distributionally robust optimization (DRO), allowing us interpret the proposed method as optimizing surrogate distributionally robust loss.
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To the best of our knowledge, this work presents the first attempt towards a computationally tractable approach for learning stochastic dominance optimal solutions, both practically and theoretically. Numerical experiments are demonstrated to illustrate the effectiveness of our framework for finding risk-averse yet performant solutions a variety of learning tasks such as supervised learning, reinforcement learning, and portfolio optimization.
The rest of this paper is organized as follows. Section develops a general learning framework using stochas- tic dominance. Section 3 presents computationally efficient algorithm and its theoretical computational complexity. Numerical results presented in Section 4. Finally, we conclude the paper in Section 5, and leave the proofs and a connection to DRO in Appendix A.
Notation. We denote real-valued random variables by upper case letters, e.g., X, and the corresponding observed values by lower case letters, c.g., x. The set of probability mcasures over set A is denoted by A(A). (x)+ a shorthand notation of max(0, æ). Two random variables and are equal in distribution if they have the same distribution, denoted as X _ Y.
1.3 Related works
Stochastic-dominncce constrained optimization. In the literature, SD is often used to characterize the feasible set. of an optimization problem as a constrriint w.r.t. given competitor. In contrast, our SD learning framework, we seek the optimal solution in the stochastic dominance sense within the whole hypothesis space, instead of against fixed competitor. Many previous related works tackle the SD con- strained optimization by casting the comparison of kth-order cumulative distribution functions (Dentcheva & Ruszezynski, 2003; Dentcheva. & Ruszczyíski, 2004; Noyan et al., 2006) or its equivalent reformulations (Luedtke, 2008; Post, 2003; Armbruster & Delage, 2015) as linear programming and mixed-integer program- ming problems, which typically incurs a quadratic iteration complexity/ /mmorry consumption and is hence not applicable to large-scale practical problems. Dai et al. (2023) presents the latest effort towards efficiently solving SD constrained optimization and achieves (near) linear computation and memory cost. The kcy ingredient lies in the elficient solver for the inner oplimization in the Lagrangian, which is integral to our learning algorithm development as well.
Connections to other risk-sensitive approaches. Quantile statistics such as Value at Risk (VaR,, (Markowitz, 1952; Roy, 1952) and Conditional Value at Risk (CVaR,,) (Artzner et al., 1999) represent another popular choice of risk mcasure beyond variance. For a risk level a E (0, VaR, is given by the (1 a)- quantile of the loss, whereas CVaR,, takes a step further by focusing on the conditional expectation of loss beyond VaRa. Remarkably, second-order stochastic dominance (SSD) can be interpreted as a continuum of CVaR, comparison over the entire risk level set (0, 1) (Martin et al., 2020), which again justifies the superior theoretical properties of SD.
Beyond the uncertainty that stems from known data distribution, distributionally robust optimization (DRO) seeks to optimize the model against the uncertainty in the knowledge of the distribution itself, by focusing on the worst-case expectation under some distribution shift. Duchi et al. (2021) demonstrated that for uncertainty set induced from f-divergence balls, the DRO formulation is asymptotically equivalent to a mcan-risk treatment, with the risk measure given by the square root of variance.
Other generalization of stochastic dominance. Notable prior efforts of generalizing stochastic domi- nance and mitigating incomparability include almost stochastic dominance (Leshno & Levy, 2002), statistical preference (De Schuymer et al., 2003), gruded slochastic dominancc (Pérez-Feruándee et al., 2021), general- ized stochastic dominance for classifier comparison (Jansen et al., 2023) with further developments in Müller et al. (2025); Luo et al. (2017); De Schuymer et al. (2005); Jansen et al. (2024), among others. However, these contributions remain largely theoretical and do not yield a practical machine learning framework that is compatible with stochastic dominance, which we seek to address in this work.
Distributional RL. Distributional RL (Bellemare et al., 2023) provides a systematic approach towards learning the distribution of the cumulative rewards induced by executing a policy in RL. While this allows
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the decision maker to resort to mean-risk approaches for risk-averse policies, policy improvement scheme that fully utilizes the learned distributions in terms of stochastic dominance remains lacking. Martin et al. (2020) investigated the use of SD for action selection using a particle-based algorithm, which involved extra computation and thus fell short of providing an explicit policy.
2 Stochastic Dominance Learning
In this section, we first introduce the concept of stochastic dominance, and reveal the difficulty defining optimality in terms of stochastic dominance. We then resolve this difficulty and establish the stochastic dominance learning framework.
Stochastic dominance. Let X denote real-valued random variable. The k-th distribution function F is defined recursively as
F (n) Px < n))
(1)
F (n) = [" (a) (I: - 1)! Ex
(1 x
k
(2)
where F (1) is simply the standard cumulative distribution function (CDF). Then, X dominates Y in the k-th-order if (Mann& & Whilney, 1947; Dentcheva & Ruszczynski, 2003; Lehmann, 1955)
F (n) < FF E
(3)
denoted as X Z Y. By definition, the k-th-order dominance implies the (k: + 1)th-order dominance. In practice, the popular choices are k = 1 or & = 2. First-order stochastic dominance (FSD), by definition, pursues consistently a lower probability the random variable falling below a threshold, which equivalently asserts the existence of X 2 X and Y _ Y such that X Y a.s. Second-order stochastic dominance (SSD), due to the aforementioned intrinsic relationship CVaR, allows more fine-grained comparisons among random outcomes with the same expectations. Figure 1 illustrates two normal distributions centered at 0 with different variance, and their corresponding F2 function. SSD favours the one with smaller variance it yields a consistently lower 2 function.
5
0.4
((
Probability Densiny
3
0.2
2-
0.1
0
7j.
Figure 1: Probability density and second-order CDF of (o(0;) and (0;
Learning with SD. In many applications, we can model the quantity of interest by Xo. associated with some parameterized model / E O, which we would like to maximize in general. As examples, we can rethink a fow typical machine learning and decision making problems through the lens of random variable selection, to which we hope to leverage SD reason their preferences.
Supervised learning. Let feature vector and corresponding label y be sampled from data distri- bution D. The performance of parameterized model so is measured by the sample loss function
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3)). Typical choices of & include mean squared error for regression tasks and cross entropy loss for classification tasks. Empirical "risk" minimization seeks to optimize the expected sample loss (ixx))=l(ff((x)x))) by minimizing the empirical estimate * Ei= (x.f(x;).))) where the training datasct. We negate the sample loss in consistency with the definition of stochastic dominance, i.e., X3 -x(so(x),)) (x, ~D. D. One can compare different models via examining SD relations among Xg's over the hypothesis space of 0.
is
Reinforcement learning. Consider discounted MDP with state space S, action space A, reward function r 8 x A + R and transition kernel P(-|s, a) that defines the distribution of the next state upon choosing action a at state s. A policy S + (AA) defines a random action selection rule for each observed state, which induces random trajectory т = (8o, do, ro, 81, where ae ~ I( (8e), 8141 ~ P(-|st, az), Tt = r(s,, ae) from some initial state 80 E S. The cumulative reward of trajectory T is given by r(r) = Ei=orTT where ? E (0,1) the discount factor. Standard learning paradigm in RL is maximize the expected cumulative return E, [r()]], Le., value function, with respect paramcterized policy Ty. Stochastic dominance for comparison between policies be therefore deployed by setting X r(T), P n..
Porlfolio oplimization. Let random variable R& denole the return of stock i which can be heavy-lailed and correlated. Portfolio optimization seeks to get an ideal allocation Wa E Ax of assets among K stocks to achieve tradc-off between the expected return E [ R.] and some risk mcasure. Stochastic dominance naturally applies by focusing on the total return Xo Σ [wo]:
Generalized stochastic dominance for optimality. One might be tempted to search for a model 0* that dominates all 0 O, i.e., the greatest element under stochastic dominance rule in the aforementioned learning scenarios, which would imply (c.f. (3))
min (n) Fx,
(n) F (n) >0. YB 6.
(4)
However, such 0* is not guaranteed to exist due to the fact that SD only defines partial order among distributions. In other words, there exist two random variables such that their order cannot be distinguished in the sonse of SD. Therefore, it is impossible to define a general "optimality" in the sense of (4). This gap hinders the development of a learning framework under SD from both theoretical justification and optimization- based algorithm design, which motivates more general definition of SD. Consider the following two statements:
1. Xp- is a marimal element under the partial order of stochastic dominance, i.e., not dominated by other feasible solutions.
2. We have
max Fx. (1) Fk. (1) (m))=²) 00, 40 E,
where the equality achieved only when Xe D Xg..
In view of the equivalence of the above two statements, we propose the following Generalized Stochastic Dominance functional:
2x(X = max ( - 1 (n))
(6)
mE[a.]]
which quantifies the degree of stochastic dominance between X and Y over the interval [a, 6) in an unilateral way. Here, we follow the convention of enforcing stochastic dominance on an interval [a, b] to overcome numerical tractability issues and technical subtleties (Leshno & Levy, 2002; Dentcheva & Ruszczynski, 2003).
The proposition states the existence of such non-dominated solutions under mild conditions, with the proof deferred in Appendix C.
With slight abuse of notation, we drop the dependency on (a, bJ in the notation of Q for conciseness.

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Proposition 1 It is guaranteed that non-dominated solution l* crists long O is compoct and that Fx (1) is continuous with regard to 0 for every 1j E IR.
In view of the generalized SD in (5), it is now natural define general learning problem through an optimization lens, by seeking an approximate optimal solution &r such that for any 0 θ it holds that
Ω4 (x xx.) -6.
(7)
In other words, 8- is guaranteed to not be dominated by any other solution 0 by margin of over the interval [a, b].
3 LSD: First-order Optimization for Learning with SD
In this section, we design efficient first-order algorithm to solve (7), resolving the computational difficulty discussed in Section 1.
3.1
Stochastic Gradient for SD Learning
The optimality condition (5) can be written as
0* = arg min Q4 (Xo. Xo-).
which motivates us to interpret 0 fixed point of the following iterative process:
0111 arg min (Xo, Xo,)
(8)
Here, O, denote the choice of parameter 0 the t-th iteration. Rather than pursuing an exact optimal solution to (8), we show that it suffices to find d.+ such that
D (Xa Xo,) -e;
(9)
since failing find such 0.+1 implies the approximate optimality of 0g by definition (7). Since 94 (Xp, Xe.) 0, this amounts to making progress of the sub-problem
min Ω4 2x Xo.),
(10)
which makes gradient-basse based mothods an idcal candidate.
Subgradient calculation. remains unclear the optimization properties of (10) as well as how esti- mate gradients. To proceed, shall resort to the utility reformulation of Ω. Note that O (X, Y) can be equivalently written in variational form:
2 (X,Y)= max pEA((a))) /"
[F) (n))))))))
(11)
where the maximum laken over probability measures over (a, b). For every choice of μ, changing the order of integral by Fubini's theorem, we have
(F) d(())
1 (k 1)1
v) fx(=))d= (n (v(o)dy du((n)
(1) - fx(xx)d
(n -1 "fx(x)dp(m)dz 00 1 - (r))d(())d)
(A: 1)1

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f (x))f((())a)) L u(y)fr (d)dy
= Ex (xx) + By (x()].
(12)
where the utility function u is defined as
(k: 1)1 [(" (1) x) dd((n). =dd))))()).
u(x)
(13)
Therefore, we can write (x, Y) as
9x(XYY) = max
Ey Ex + [y())\)
{
wElli
Here, Us = (u : u(x) = - (1 dd(())) E A((a,b)) collects all utility functions that can be expressed in the form of (13).
Note that when k > 2, E Us is non-decreasing and concave, which guarantees u(zo) to be concave with regard to 0, as long as Ty is concave (Boyd & Vandenberghe, 2004). When the sampling probability of Xo is independent of 0, such as in supervised learning and portforlio optimization, (Xp, Y) takes maximum over a set of convex functions and is therefore convex as well. The subgradients of f2g (Xo, Y) (Bertsekas, 1971) is given by
Og(Ox (Xo, Y)] = conv - 8 (u(a)]] MEEE Ui)
= = - Ex. (xxx)oou(Xo)) u u Ui).
where Ut arg max L(X,Y,an u), and conv is the convex hull. The expectation formulation of the above equation allows estimation of the subgradient using sampling, i.e., the sample average, and subgradient chain rule (see e.g., Clason & Valkonen (2020, Theorem 4.19)), given by
Σ 0gu(z;)) N
1
1 Σ Oz, ((z)))001
(14)
where { N data points sampled from Xg. Here, omit the dependency with 0 in for notational simplicity. This allows interpreting our proposed method as stochastic gradient methods with each sample xi dynamically weighted by oz, u(x)). For learning tasks with model-dependent sampling probability (e.g.. RL), one can instead apply log-derivative trick (Williams, 1992) for gradient estimation (see Appendix B for more details).
Final algorithm. We summarize the algorithm procodure in Algorithm 1. Simply put, the algorithm follows a nested-loop design, where the inner loop focuses solving (10) by first obtaining ū* that maxinizes the sample estimate of I and then derive the stochastic subgradient with (14). We terminate the inner loop and update 0, when the progress condition (9) is approximately met. If (9) is not met within a certain mumber of iterations, we conclude that the current 6g is approximately optimal and return the solution.
3.2 Theoretical Analysis
Two questions arise naturally with regard to the theoretical guarantee of the proposed method: i), whether it is guaranteed to converge, and ii), whether it induces a significantly higher iteration complexity compared with standard minibatch SGD methods. The concern stems from the fact that the dynamics of Algorithm cannot be interpreted an optimization process targeting a fixed objective function, and that one round of inner loop alone can take (E)-?) iterations to end.
The following theorem addresses the concerns mentioned above by ensuring convergence within Ō(c=?) total iteration complexity.
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Algorithm 1: Learning with Stochastic dominance (LSD)
Input: Initialization 0o- for 1=0 0, ... Taax -1 do
Set 00.0 = 0r.
for / 0,... T moz 1 do
Sample data {
x and (ar,) X
Compute 8² = argmax T x(xo,, Xo., u), where
6
well.
N
N
:( Xo, u) =
Σ u(F,,,)) +
Σ u(zz,).
N
N im1
Update 0,,411 00,, - 1191.1 where
N
9t,7 E
1
Σ
Oz
N
if Og : Xo.) cc/2 then
7
Set Oc+1 = 0 01,111-
Break.
if 0 not updated then
10
Return i,.
Theorem For second-order stochastic dominance (k = 2), assume that za is concave with regand to 0, and bounded subgradient 119,,112 G2 and bounded k-th order CDF Fx(Xo. 7)) & C, An E b). Let n 1/Vi, and sample size N Ó(c=)), Taaz= T4C/c + 11, T (()-)). For any initialization 6, with probability 6, Algorithm 1 finds Og such that for any 0,
24 (Xo, > -
within Ō iterations.
Several remarks are in order.
The approximation crror e incorporates the statistical error duc sampling that scale with 1/1 which necessitates choice of N ((e similar to the case for empirical risk minimization.
The iteration complexity of Ō is on par with that of subgradient methods for optimizing non-smooth convex functions, which sets LSD as appealing alternative to standard risk-neutral approaches in practice for risk-averse applications. The iteration complexity can be further improved by incorporating regularization terms of μe, e.g., entropy regularization, in the variational form (11) to ensure the uniqueness of the maximizer μ", which leads to differentiability of Ω by Danskin's theorem.
While state the theorem for k = for simplicity, the analysis can be easily generalized to & by adopting the Rademacher complexity of U4 and upper bound of Fg accordingly.
Algorithm can be used plug-and-play for achieving non-dominance in conjunction with standard empirical risk minimization, CVaR optimization, and related procedures, as long as the underlying risk or preference measure consistent with stochastic dominance.

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Algorithm 2: Utility solver for k =
Input: samples {= and {8r J Sort both sequences of samples in increasing order, and merge them into {re J for i= -,,,.., 2N do
(m)) = ((--)) + neta.//N
F+ (m.) P/ (m-1) + mntee:/
F (m)) ()- + (1: 1T Fx (1%-1)
(G(m) - (m=1) + (m - (0)--/11 P. (11-1):
Get ā E A arg max
- (mi))
for i = 2N, ... do
01 (1(++)) - 85 (m,),
(7.) = (2(7++1) + (7/+1 m.) (16+1))11.
Return U2-
A key ingredient of our analysis is to relate the sub-optimality gap ming Ωu (0, 0.) in the tth loop with the optimization progress in the subsequent rounds, despite the fact that they are associated with different objective functions. This is made possible by exploiting a triangular incquality with Ω which ensures that the inner loop generally lakes a smaller number of iterations than T naxx ensuring the final iteration complexity is still O(E 2). The proof is postponed to Appendix D.
3.3 Practical Implementation
When & & 3, the computation of 8 can be done in an efficient way that consumes O(N) memory and Õ(N) time (Dai et al., 2023). Below we demonstrate the case with k = 2. Recall that each candidate utility function u is associated with a probability measure μ by
u(x) = -Em-nl(11- - =)
(15)
For [(x, u) induced by samples {a:) and (w we still have Y, (P(0)
where (n) is an empirical estimate of (1) given by (r) - (n- Note Note that (()))F(mm) is wiece-wise linear and hence achieves its maximum among the sample values (xi) and {vi}. It is then straightforward to get î²* by assigning the weights to the maximizer(s) and obtaining ä* via (15). We summarize the procedure in Algorithm 2, where the terms involving with i = 0 and i = 2N + 1 are set to 0.
In general, (1) (r())) is picco-wisc polynomial of degree (k- (k-1), which allows closed-form solutions up k = 3. For k 2 4, which is less likely be considered in practice, one can resort to numerical approximations optimization treatment by parameterizing μ with special neural networks, discussed in Dai et al. (2023).
For further practical consideration, we remark that the extra cost of computing/ saaiiiin//////giggggggg from the reference 0. for the inner loop optimization can be alleviated by instead sampling from the set of samples from previous iterations, in similar way to experience replay techniques (Lin, 1992) in RL. This climinates the need of explicitly keeping and periodically updating the reference solution 0, and makes the algorithm more streamlined.
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4 Numerical Experiments
To demonstrate the versatility of our framework, we evaluate LSD on various tasks including supervised learning, reinforcement learning, and portfolio optimization.
4.1 Supervised Learning
We examined the performance of LSD on image classification lasks with MNIST and CIPAR-10 datasets. For MNIST, we train a simple 6-layer convolutional neural network for 10 epochs. For CIFAR-100 10, we use 20-layer Res.eet architecture and train for 200 epochs. In both experiments we set batch size to 128 and adopt stochastic gradient descent (SGD) method to optimize the models, with learning rate set to 0.1 and momentum set to 0.9. We repeat the training procedure on random seeds. The proposed method achieves comparable classification accuracy with SGD method, and more stable cross-entropy loss under -bounded distribution shift, characterized by the average absolute deviation from median (sce Theorem 3).
MNIST
CIFAR-10
Metric
LSD
SGD
LSD
SGD
Accuracy
99.16%
99.17%
91.2%
91.4%
CE loss
0.0283
0.0289
0.293
0.339
MAD
0.0286
0.0293
0.292
0.337
DRO loss (p = 0.1) derived from the above statistics
0.0312
0.0318
0.322
0.373
Table Test performance measures of LSD versus SGD on MNIST and CIFAR-10.
4.2
Reinforcement Learning
We then evaluate the ability of the proposed algorithm of learning risk-averse control policies in two rein- forcement learning tasks.
CliffWalking. We adopt a modified version of the ClifWalking environment from OpenAl Gym, as lustrated in Figure The action space the agent is given by {0, 1, 3}, representing moving by one step in four different directions. When an action is selected, with probability the agent will move in a random direction. The agent always start from so, and receives reward of whenever falls off the cliffs; reaching the goal instead assigns positive reward of +1. Under both circumstances the episode terminates immediately. Two strategies naturally arise: risky policy would take the shortest route to the goal and incur a higher chance of falling, while safe policy prefers taking detour to minimize the risk.
Safe Policy
Risky Policy
$g
-1
1
Figure 2: Illustration of the CliffWalking environment.
We tune the values e and & to ensure both strategies have similar expected cumulative rewards. We compare the policy learned by LSD (dctailed Appendix B) with the one Iearned by standard policy gradient method (REINFORCE) (Williams, 1992) under tabular parameterization. The two policies yield similar expected return (0.484 v.s. 0.479), yet approach achieves consistently lower value of F2,
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demonstrated in Figure 3 (left panel), which demonstrates the risk-averse nature of the learned policy; indeed this can be more evidently observed by examining the density of the return in Figure 3 (right panel), where REINFORCE leads to higher probability of falling (higher probability mass in negative returns).
0.30
REINFORCE
REINFORCE
14-
0.25
LSD
LSD
0.20
10
Density
0.15
0.05
0.00
-10
-0.5
0.0
-1.00
-0.75
-0.50
-0.25
0.25
0.50
0.75
return
Figure 3: The F2 (left panel) and density (right panel) of the cumulative return in the CliffWalking environ- ment, by executing the policy learned by REINFORCE and LSD-PG, respectively.
CartPole. We modify the CartPole-vl environment from OpenAl gym by perturbing the reward with zero-mean random noise when the cart enters into some certain regions. risk-aware policy is capable of avoiding such regions without compromising the cumulative return performance. The maximal episode length is set to 500, and the cart position is randomly initialized near the central point. We set the perturb region to {i < -0}, where is the cart position. The per-step reward is perturbed by 5 ~ Uniform((-&,:)) min 30][ as long as the cart stays in the region. The policy is parameterized by a multi-layer perceptron with two hidden layers of size (64, 64). We average all results over five random seeds. While all methods are able to achieve the optimal expected cumulative return 500, the policy learned by LSD is able to steer the cart away from the perturbed region, as demonstrated in the right panel of Figure 4. This leads to a more concentrated cumulative return distribution that stochastically dominates that induced by the policy learned by REINFORCE (sce the left two pancls of Figure 4). Noticcably, CVaR-PG method (Tamar et al., 2015) achieves comparable performance with LSD, but only with a carefully tuned choice of
1.75
REINFORCE
REINFORCE
LSD
150
1.50
CVall,pP
CN
0.6
CNaRaPGG
PG
1.25
CVVRg:PG CNARR
CV
CURBPP
Danesht
PG
1.00
0.70
0.04
Cumulative Reward
Locatin
Figure The F2 (left panel) and density (middle panel) of the cumulative return, as well as the density of the visited cart position (right panel) in the CartPole environment, by executing the policy learned by REINFORCE, LSD-PG and CV, respectively.
4.3 Portfolio Optimization
We evaluate the performance LSD on portfolio optimization with synthesized data and simulate the highly noisy return variables by deploying mixtures of Gaussians with random generated mean and covariance. We set the number of stocks 100 and the number of Gaussian mixtures to 20. To better reflect the heavy-tailed nature of the problem, multiply each Gaussian sample's distance to its center by random multiplier
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drawn from x3/3. Table 1 compares the constructed portfolio with those resulting from the mean-variance approach MVx (Markowitz & Todd, 2000) using different levels of variance penalty X, where Figure 5 further illustrates the density of the portfolio returns.
SGD
L50
20-
Metric
SGD
LSD
MV0.1
MVu.5
MV1.u
MiVaz
1.5
E[return]
0.501
0.195
0.501
0.207
0.128
ar [return]
7.496
0.112
7.467
0.134
0.059
Sharpe ratio
0.183
0.585
0.183
0.567
0.528
0.0
0.5
0.0
0.5
Table 2:
Comparison of LSD versus the mean-variance approach with different variance penalties for portfolio optimization.
Figure 5: Density of the portfolio re- turns achieved by different methods.
While none of the methods simultaneously achieves the highest expected return and the lowest variance, can evaluate whether the portfolio finds reasonable tradc-off through the Sharpe ratio (Sharpe, 1998), popular choice for measuring the risk-compensated performance, which distinguishes risky portfolios with catastrophically large variances (highlighted in red in Table 2). We conclude that LSD yields a reason- able construction portfolio and circumvents the ad-hoc regularization paramcter tuning in the mcan-risk approach.
5
Conclusion
This paper develops the first practical algorithm for finding an optimal solution in terms of (generalized) stochastic dominance for learning and decision making with uncertain outcomes. The method computation- ally efficient as it can be easily integrated with existing optimization methods with minimal computational overhead, and come with theoretical guarantees for finite-time convergence. Our work opens up opportunities to further explore the potential of stochastic dominance in risk-averse machine learning applications.
Acknowledgement
The work of S. Cen and Y. Chi is supported in part by the grants ONR N00014-19-122404, NSF DMS- 2134080, CCF-2106778 and CNS-2148212. S. Cen is also gratefully supported by Wei Shen and Xuehong Zhang Presidential Fellowship and JP Morgan AI Research Fellowship.
References
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A Connections with DRO
Before finishing up the paper, we demonstrate a connection between SD and DRO, which might be of independent interest. Given n samples { the distributionally robust formulation seek to maximize the return under adversarial distribution shifts, i.c.,
inf Ep[X], P P.B..)
where P., denote the empirical measure of the samples and B(P.) is uncertainty set centering around Pr.. The following theorem demonstrate that when the uncertainty set induced by loo norm, the objective function be written in mean-risk form. The proof is postponed to Appendix F.
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Theorem It holds that
inf Pe P..P...lP P. Epin
Ep(X] Bp [X] - PMAD P. (X],
where MAD [X] = - x4 - denoles the mean absolute deviation from sample median T.
It is possible to extend the above result to more general choices of uncertainty set, where the relationship holds asymptotically (similar to Duchi et al. (2021)), from which we refrain for simplicity. On the other hand, we have the following result characterizing the consistency between stochastic dominance and mean- semideviation models (Ogryezak & Ruszczyriski, 2001, Theorem 1).
Theorem 4 (Ogryczak & Ruszczyúski (2001)) Let k > 1 and X,Y E Lx. X I(k+1) Y, then E[X] [| and
8
[[ - z(h) -
Here, xx
denotes the kth central semideviation:
x = E [E[[]] xx 1x=xx]
k = 1,2,
In particular, the absolute semideviation at k = 1 can be written as
x[X
5001
[ x){(x)dr -= E[X]]
Note that it always holds that F - X] & B
[x]], where X the median, and μax is the mean. Tt. follows that when there exists such that Xg. ž Xo, vo E θ, then 0r can be interpreted as an approximate solution the robust optimization problem
[p || Px 2
sup inf 0 PEP.
for all E (0, 1/2), the sense that lr maximizes lower bound of the objective function. The approximation error is bounded by |ax - X).
B LSD for Policy Optimization
We detail the procedure of LSD applying to policy optimization in Algorithm
C Proof of Proposition 1
Let (0 be a chain under stochastic dominance rule, i.e., Xe, 2* Xo, when > j. The compactness of θ assures the existence of limit point Ø O, to which subsequence of (0t) converges. According the definition of stochastic dominance, for every 1) R the sequence { {Fx,, ( is non-decreasing. Since
Ft. (n) is continuous with regard to 0, we have
())( = lim F (n)-
By definition, Xy stochastically dominates X, for all 0 from the chain, establishing 0 as an upper bound of the chain (0 ( The existence of maximal element then guaranteed by Zorn't 's lemma.
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Algorithm 3: Stochastic Dominance Policy Optimization
Input: Initialization Po- for t= 0, ... mas do
Set 0..0 = l.
for 7=00 0, , maE 1 do
5
Sample trajectories [Treil with policy tH and (Trs)i with policy mee
Compute cumulative rewards R,,,, R(Teas), Re R(T.))
Compute ã- arg max L( Re, u), where
[( u,,
1
Σ
u
+
u (
N
N
==1
Update 0 with
N
0 0,441 = 0,, +
ū
(R
Vo log (T.r
N
if 1( Re) ee/2 then
10
Sct 8.+1 8,,711 Break.
11
if 0, is not updated then L Return i,
13
Proof of Theorem
We start by introducing the following lemma which bounds the statistical error due to sampling when
Lemma 5 Let (X,,Y max {- N ΣN Ei u(xi)+ N EN u(vi)}. where {a,, yi N are i. samples
from X and Y. It holds with probability 1 28' that
- 16([a + (8])
12g(2/8")
+ 6y
VN
N
For notational simplicity, we denote
N +
; = max weld
Σ
u(Io
u(20.s)) }
N
N
By setting = 5/ max ( max + 1)) in Lemma 5 and invoking the union bound, we have
[x .z Xo,)
1t Xo,) 5X
(16a)
7'
[Xo; Xo,) (x Xo.) _
(16b)
for all 0 & t < Taax: T < nax with probability 1-6 8, on which we shall condition in the remaining part of the proof. We remark that with the remaining ö probability the Algorithm may fail to find O, qualified for the output condition, or return sub-optimal solution that accidentally meet the condition. To proceed, we show that Or satisfies the following triangular inequality:
(X, 2)
max ((,(( (F) (v) (n) Fk (n). p (1)
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( (n) - Fy (1)) (z), (n)) + (FF (n) - FF (1)) (n)) }
max AE ((,,))))))
(FX (1) - -F (n),(()))
(FF (n) F (n), 1 (n))
max ((,,)(
max (E))((.))))
(X, Y) 2),
Let T be the total number of outer iterations. By (16a), for all 0 E
<6<
< T we have
12 (x & (x +
< -
4
4'
Denote 8 arg ming S2(Xo, Xx.) It follows that
T-1
(T t)
O2(Xo; Xo,) & Oe Xo.)
O2 ( Xo,) <
(17)
Step 1. We first show that with the choice of Tuax, Algorithm 1 is guaranteed to return an 8.. Otherwise, (17) holds for T Tax. On the other hand, we have g& Px (n) < C for all r E [a, bj. This gives
< 222(X Xoo) & Taae
or equivalently
40 Tnax
6
This contradicts with the choice of Taax-
Step 2. We then prove that the output Og satisfies
Q2 (Xot Xo.) 30 c.
According to the update rule, we have
10 10,,111
1, ||θ,, 1791,7 0,112
||0,r 8 2n (9.. Oet 0:) n
< |00,, 0 2n, (f2 Xx, Xo,) (2(XoozXa)) + 1l
The last step results from the convexity of п. Rearranging terms, have
217 12 1, Xo,) 22((Xo; Xo.)
|00,7 0: 1 |10,,2+1 - 03 + 11
+ 2n, [22 Xe,) - (Xo,r)
(22((N Xe.)] - 2n, [x2 Xo.) - 22( [o.))
10,7 0, 112 10,.+11 1| + G² + me,
where the last step results from (16a) and (16b). Summing over f, we obtain
T
T
T
2 m O2 12 No... Xo.) - 22 2x2 No.)] & ||0, 0
+
T/-
(18)
As Or is not updated in the f-th outer loop, step of Algorithm ensures
N2 Xo.) N2(Xo, Xo,) + c/4> -c/4
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for < 8 & T max- Combining with the above inequality, get
TEE
1
O2((X Xo,) &
10, 10112
G²
Σ 7=1
+
2 2
2===1 1=1
Note that T max = 2) is sufficient to get
T 2u G²
10:
0,11
T T== VE
4
[=1
which leads O2(Xo;; Xx.) -.
Finally, we bound the total number of iterations. Let Te be the number of inner iterations in t-th outer loop. By (18) we have
Step 3.
7.
T
T
2
Σ n, [2 O22(Xo; Xe:) & 11 0,11 G
77
(19)
1=1
[=1
1=1
Xo.) + c/4 Note that G2 o Xo,) > ;x inequality, we get
-e/4 for 1 & i < T. 1.
Combining with the above
T:-1
- O2(xo; Xo.) <
1
110
-0100 + G2
E
2
2
T.
1//
7=1
2
c 2
<
+
Recall from (17) that (Xo; Xo,) (T-r)e
We conchude that T, 5 - for T-3. Therefore,
the total number of iterations is bounded by
T Σ m=
T-3
T = 0
+ +37 nax =6
T
1)2
0m0
D.1 Proof of Lemma 5
Note that the empirical Rademacher complexity of U2 with N samples, denoted (22),, is the same that of ReLU functions, i.e.,
(l < 4(1a (B))
VN
Therefore, it holds with probability - 5 that (Mohri et al., 2018, Theorem 3.3)
N Σ
(og(2/0)
E
[uxx(X)
N (xs)))
< (2) + 33
N
1=1
Similarly,
1
log(2/8)
E 1u
< 2 + +3V
N
N
holds with probability 6. By union bound, holds with probability 28 that.
l2(XYY) V
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N
N
= max uEUl
{ - [x]] + B [u[()]]]
max welly
u
+
1
Σ u(vi)
N
N
1=1
N
max
- 18 [u()] (
ul (z:))
1
Σ
u(vi)}
N
1=1
16([a] + (6))
(2g(2/))
VN +61 +6V
N
E Proof of Theorem 3
The relationship can be established immediately by the following lemma, with being the distribution shift P- Pn.
Lemma 6 Let & represents the medion of {si, aiiiiii n), and
u { u R=11 u 0, |lulloo < <c/n].
We have
6 [; T
sup u wEll
T
Proof Note that U is convex polytope. Therefore u achieves maximum at one of the vertices of U. Note that the vertices of can be written as
c/n i
e//n iEA
M+1=1A-1- = 2
otherwise
When A+ collects the indices of |n/2] largest values in {x;} and A_ collecting the smallest values, To x achieves its maximum at
Σ
Σ
x
2
A
TEA
20


Reviews:
[{"review_id": "71tIkhGiw7", "reviewer": "Reviewer_dCTY", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors provide an algorithm to find an approximate non-dominated solution satisfying relation (7)", "claims_and_evidence_yn": "Yes", "claims_explanation": "The authors correctly calculate subgradients of $\\Omega$ and construct an algorithm", "audience_yn": "No", "audience_explanation": "Satisfying (7) is extremely weak. It simply means that $\\theta^*$ is non-dominated. \n\nAlmost every $\\theta^*$ that is optimal for an expected utility model will be non-dominated. By minimizing any coherent risk measure, we obtain non-dominated solutions. If we just maximize (or minimize in case of a loss) the expected value, we will obtain a non-dominated solution.\n\nUnless the authors show that their method selects particular non-dominated solutions enjoying some desirable properties, this is a non-result. In the now-standard approach, we require that the solution dominate a benchmark and optimize (under this condition) \nthe expected performance. It is more rational. We obtain a non-dominated solution that compares favorably to a benchmark.", "requested_changes": "Prove that your method finds particular nondominated solutions, for example, better in some sense than a benchmark, or some other idea.", "broader_impact_concerns": "I do not see any relevant issues here.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "mA4VdYMInF", "reviewer": "Reviewer_6wf5", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The article proposes establishing stochastic dominance as a risk-sensitive alternative to expected utility for comparing random variables. \nWhile this is already standard practice in some fields (e.g., finance, decision theory), such an approach is not widely used in machine learning.\nHowever, the authors show that important areas of machine learning, such as supervised learning or reinforcement learning, can also be \nseamlessly embedded in a stochastic dominance framework.\n\nThe article names two reasons that (in the authors' opinion) prevent the further spread of SD in machine learning: (1) SD leaves certain variables incomparable and thus \ndoes not correspond to the optimization mindset that prevails in ML, and (2) SD requires the comparison of variables across a continuum and is \ntherefore difficult to evaluate.\n\nThe article intends to address both of the challenges (1) and (2) mentioned above: For (1), a generalized stochastic dominance criterion is proposed, \nwhich should enable a comparison of any variables. For (2), a first-order algorithm is proposed, which makes it possible to efficiently calculate an \n(approximately) optimal solution according to the proposed generalized SD criterion. The article demonstrates the applicability of the proposed framework \nusing numerical examples from the fields of SL, RL, and portfolio optimization.\n\n\nStrenghts:\n\n- The paper addresses a relevant and interesting topic: Introducing SD-based Learning in ML seems to be a very good idea, as this allows for addressing \nseveral shortcomings of the traditional focus on the expected utility / expectation only. The concrete shortcoming mentioned here is its lack of risk-sensitivity, \nbut in my opinion the argument extends to nonstandard scaled data and other interesting areas.\n\n- The possibility of efficient algorithmic verification of SD seems very attractive to me. Progress in this direction has the potential to make SD methods, which \nappear very attractive in theory, relevant for practical work as well.\n\n- The possibility of treating seemingly different areas such as supervised learning, reinforcement learning, and portfolio optimization within the same formal \nframework strikes me as very elegant. It also reinforces the relevance for various fields of application.\n\n\nWeaknesses:\n\n- It is not really clear (to me) to what extent the concept of stochastic dominance is generalized here (as is claimed). As I understand it, the main proposal \nis to search for an undominated solution (which always exists under mild conditions, see Prop. 1) instead of a best solution in the sense of SD (which generally \ndoes not exist). To my knowledge, this approach is fairly standard in the context of SD (and non-total orders in general). However, this comment refers to the \nconceptual innovation and not to the proposed algorithm.\n\n- I find it difficult to identify the main contribution of the paper. While it is clear to me that an algorithm for more efficient discovery (compared to SoTA methods) of SD-undominated acts is proposed, the generality of the main theorem is unclear to me: Does this refer to SSD (as mentioned in line 1), or is an undominated act of order k actually found, as suggested by the last line of the theorem? I would appreciate some clarification on this point.\n\n- My biggest concerns relate to the interpretation of the “generalization” proposed here: While ERM solutions are unique (in many relevant cases), this is by no means true for undominated SD solutions. So if one algorithm (e.g., in an SL context) finds an undominated solution X and another finds an undominated solution X' (in our example, different forecasting algorithms), these will generally make very different predictions. The number of undetermined models is often very large, which means that the model found is, to a certain extent (except for certain worst-case guarantees), arbitrary. Shouldn't we rather try to sensibly strengthen the ordering power of SD methods instead of simply going with an underdetermined model?", "claims_and_evidence_yn": "Yes", "claims_explanation": "The main result, Theorem 2, is supported by a detailed mathematical proof.\n\nThe proposed algorithm is demonstrated along three different applications.", "audience_yn": "Yes", "audience_explanation": "I think incorporating SD in learning problems is a promising line of research that is often hindered by the complexity associated with checking for SD. In this sense, I think any step towards making SD more accessible for practitioners (i.e., more efficient to check) is very much needed and relevant.", "requested_changes": "- At several places it is claimed that SD is a partial order, i.e., antisymmetric. However, afaik, this is not the case: \nTwo variables may be equivalent with respect to SD, without being equal (just think of X,Y following the same distribution under FSD).\n\n- If I understand correctly, the “generalization” of stochastic dominance (compare Equation (6)) consists of checking whether (strict) \nstochastic dominance in the reverse direction does not exist. In other words, X dominates Y in the generalized sense if Y does not SD-dominate \nX strictly.  It is therefore not entirely clear to me to what extent this is actually a conceptual generalization.\n\n- In connection with this, I am also surprised by the wording in Proposition 1: “... THE non-dominated solution ...”. In general, there will be \nquite a few undominated variables. Thus, there will generally be no unique solution. Please reformulate this appropriately.\n\n- When discussing related work on generalizing stochastic dominance, the following recent attempts to do so seem to be missing. Both papers follow \nthe idea of strengthening the SD-order to increase comparability among variables, which seems a somewhat similar goal to what is pursued here.\n\n- https://proceedings.neurips.cc/paper_files/paper/2024/hash/b1f140eeee243db24e9e006481b91cf1-Abstract-Conference.html\n\n- https://jmlr.org/papers/v24/22-0902.html\n\n- In Equation (5), how do you choose the interval boundaries a and b in practice? Cutting off the tails of the distribution functions might \nimply that the undominated variables above the interval are not necessarily undominated on the entire real line. Thus, this restriction may \nresult in dominated variables being selected, correct? If so, the choice of a and b should be well justified. It would also be nice to learn \na little more about the “technical subtleties” mentioned (see paragraph under equation (5)).\n\n- I think there is a chnage in notation when comparing Equations (1),(2),(3) and Figure 1 and the paragraph above it: While in the equations the kth order\ndistribution is donoted by F^k, in the figure and its description it is denoted by F_k.\n\n- I find the explanations surrounding equation (6) somewhat confusing:  The equation itself specifies the condition that X_{\\theta^*} is not SD-dominated by X_{\\theta}. \nBelow this, it is stated that X_{\\theta^*} is not dominated by any other variable. It seems to me that an all-quantifier is missing before equation (6) (\\forall \\theta).\nOnly then does the reformulation at the beginning of Section 3.1 make sense.\n\n- The idea of the relaxation in Equation (7) seems to be close to the notion of almost SD, just replaced by almost non-SD. While I think the idea is nice in \nprinciple and can understand it from an optimization standpoint, I still wonder how strong the statement is in the end: Especially when combined with the truncated \ntails from equation (5), it seems to me that a lot of precision is lost (especially since Theorem 2 also only provides an almost undominated solution). \nA little discussion on this would certainly be enlightening.\n\n- The reformulation of condition (6) at the beginning of Section 3.1 seems a little sloppy to me. I understand that a maximum element $X_{\\theta^*}$ can be characterized equivalently by requiring that \n\n$\n\\min_{\\theta \\in \\Theta} \\, \\Omega(X_{\\theta}, X_{\\theta^*})  \\geq  0 .\n$\n\nHowever, no requirement is made for the $\\arg\\min$ specified here. Perhaps one could describe in more detail that this $\\arg\\min$ is a candidate for an undominated solution. If one starts with a dominated $X_{\\theta^*}$, one still finds an $\\arg\\min$, but this does not satisfy the condition.\n\n- I believe the symbol x_theta (lowercase x), which is first used in the bottom paragraph on page 7, was never formally defined.", "broader_impact_concerns": "No broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "1Yk7OwQ8ZZ", "reviewer": "Reviewer_Y1PH", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors present a comprehensive framework based on stochastic dominance to solve optimisation problems under uncertainty. They first motivate the method reasoning about how stochastic dominance provides a naturally risk-aware optimisation objective without the need for trade-offs or multiple objective formulations. They analyse the main caveat of using SD as an optimisation objective; the fact that SD only provides a partial ordering of variables, and does not guarantee a global optimum, and propose a solution by relaxing the criteria into a scalar formulation of a stochastic dominance gap. They provide an algorithmic approach to learning with SD, theoretical analysis of convergence properties and some empirical results demonstrating the intuition behind SD learning solutions obtained in standard classification, RL and decision making problems. Overall I find it a good paper with interesting insights. \n## Strengths\n- The paper introduces a seemingly practical method to learn with SD and provides both theoretical and empirical results.\n- The authors cover comprehensively the intuition behind optimizing the proposed generalised SD functional.\n- The paper is well structured and easy to follow.\n\n## Weaknesses\n- Some sections or concepts are missing detail, see below for questions.\n- The practical utility of learning with SD is unclear to me yet, but I trust that it could be a powerful approach in some problems.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Yes, the authors provide justification, results and evidence for the claims in the paper. There are no unsupported claims.", "audience_yn": "Yes", "audience_explanation": "Yes. The work proposes a general-purpose approach to risk-aware learning based in stochastic dominance, relevant to learning theory, robust optimization, risk-sensitive RL/decision making and to other field-specific (e.g. finance) mean–risk trade-off problems. The paper connects SD to utility theory and DRO, which will be useful for readers working on principled robustness and distributional optimization approaches. The empirical section shows the proposed method is practical across standard ML settings.", "requested_changes": "I have some questions and comments that I would appreciate the authors clarifying.\n- There is not really a discussion or analysis on the particular choice of the intervals $[a,b]$, particularly in how it influences the generalised SD functional or how the choice of bounds would affect learning dynamics. Some comment on this would be useful.\n- There is some discussion on how SD with $k=2$ is somehow equivalent to the CVaR, but there is no specific comparison in the empirical examples against CVaR optimisation methods (very prevalent in e.g. RL [1]). Could the authors comment on specific advantages of SD versus CVaR approaches? \n- The authors mention that existing risk-aware optimisation frameworks (mean-risk, CVaR) often require trade-off design or tuning of relative weights and these decisions are not usually formally justified. I agree, but because of this added design choice, these methods also provide a way to control the risk aversion induced in the optimal solutions obtained. Would there be an analogous way of tuning risk-aversion in favour of  e.g. higher cumulated probability mass in higher utility regions of the space?\n- There is plenty of discussion on using first and second order distribution functions, but it is not clear to me under what circumstances one would want to use higher order $k$, and what would that mean intuitively for the risk aversion of the solutions obtained.\n- I think the paper would benefit of having a general, abstracted subsection where Learning with SD is formally introduced in an objective agnostic way *before* the examples provided (i.e. the examples in pg 5 were very clear and helped ground the SD approach to specific optimisation problems but it would be clearer if an abstract, formal definition of the framework would be presented before, and then particularised to each example).\n\n\n[1] Chow, Yinlam, et al. \"Risk-sensitive and robust decision-making: a cvar optimization approach.\" Advances in neural information processing systems 28 (2015).", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "GnV7ESEyFR", "reviewer": "Reviewer_mLRE", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "## Summary Of Contributions\n- **General framework for risk-aware learning:** This paper tackles the limitation of conventional expectation-based learning by incorporating **stochastic dominance (SD)** as an optimality criterion. SD offers a principled way to compare random outcomes by considering their entire distribution (not just the mean), which is important in high-stakes applications like finance, healthcare, and robotics. However, SD defines only a partial order over distributions, making it non-trivial to use as a training objective. The authors address this by **quantifying the degree of stochastic dominance** as a continuous functional, thereby enabling comparisons between *any* two random variables. This conceptual generalization is a significant contribution that effectively transforms SD from a theoretical preference relation into a usable objective for machine learning models.\n\n- **Novel first-order optimization method (LSD):** Building on the above idea, the paper proposes **Learning with Stochastic Dominance (LSD)**, a new iterative algorithm to find approximately SD-optimal solutions. The method uses a nested-loop optimization: in each outer iteration, it finds a \"worst-case\" utility function under which the current model is most dominated, and then updates the model to reduce that dominance gap. Importantly, LSD is a **first-order, gradient-based procedure** that can be integrated with standard training (e.g. SGD) with minimal modifications. The authors prove a convergence guarantee: under mild assumptions, LSD will find an $\\epsilon$-approximate SD-optimal solution in at most $O(\\epsilon^{-2})$ iterations. This theoretical result indicates **minimal computational overhead**, comparable to standard stochastic gradient methods – a strong point in favor of LSD’s practicality.\n\n- **Theoretical guarantees and insights:** The paper provides rigorous analysis to support LSD. It formalizes the SD-optimal solution as a **fixed-point of an optimization process**, ensuring that no other solution stochastically dominates it by more than $\\epsilon$. Despite the algorithm’s objective changing at each iteration (non-stationary), the authors establish **finite-time convergence** of LSD to an SD-optimal point. They also draw a connection between their SD-based criterion and **distributionally robust optimization (DRO)**. In particular, they show that optimizing for second-order SD is equivalent to optimizing a surrogate DRO loss: for example, under an $\\ell_\\infty$ distribution shift, the worst-case expected loss equals the empirical mean loss plus a term proportional to the **mean absolute deviation**. This insight situates LSD within known risk-sensitive frameworks, lending additional intuition and credibility to their approach.\n\n- **Broad empirical evaluation:** The contributions are demonstrated across **diverse learning tasks** – supervised learning, reinforcement learning (RL), and portfolio optimization. In image classification (MNIST and CIFAR-10), LSD achieves essentially the **same accuracy as standard SGD** while yielding a lower and more stable loss under distribution shifts. In RL (CliffWalking and CartPole), LSD learns policies with **similar expected return** to risk-neutral policy gradients, but with significantly reduced risk of catastrophic outcomes. In portfolio optimization, LSD automatically finds a balanced portfolio that avoids extreme variance, achieving a **higher Sharpe ratio** than portfolios from mean-variance trade-offs without needing ad-hoc penalty tuning. These experiments underscore LSD’s versatility and practical impact in **delivering risk-averse solutions without sacrificing performance**. A minor weakness is that comparisons to other established risk-sensitive methods (e.g. CVaR or distributional RL algorithms) are limited, but the provided results strongly support the paper’s main claims.", "claims_and_evidence_yn": "Yes", "claims_explanation": "- *Claim 1: “First general framework for learning with stochastic dominance.”* – Supported by a survey of prior work showing earlier SD methods were theoretical or impractical. The authors provide a concrete algorithm (LSD) with proofs of convergence and existence of SD-optimal solutions. This fills a genuine gap in the literature with a practical solution.\n\n- *Claim 2: “LSD has minimal overhead and converges to an $\\epsilon$-optimal solution.”* – Theoretical complexity analysis proves convergence in $O(\\epsilon^{-2})$ iterations, matching SGD. Algorithm description and proofs provide confidence that this claim is accurate.\n\n- *Claim 3: “LSD achieves comparable performance to risk-neutral methods while improving risk profiles.”* – Empirical results (MNIST, CIFAR-10, RL, portfolio optimization) confirm equal expected performance but reduced variance, improved tail outcomes, and higher Sharpe ratios. This directly backs the claim.\n\n- *Claim 4: “Broad applicability and versatility.”* – Demonstrated through experiments across classification, RL, and finance. Although more baselines would be useful, the results clearly support the claim of generality.", "audience_yn": "Yes", "audience_explanation": "Individuals in TMLR's audience would be interested in this work for the following reasons:\n\n- **Risk-sensitive learning is important:** The work tackles a key issue in ML – reliability under uncertainty – that spans multiple application areas.\n\n- **Novel methodology with practical impact:** LSD is both theoretically interesting and practically implementable in standard training pipelines.\n\n- **Cross-domain appeal:** Experiments across vision, RL, and finance ensure the paper is relevant to multiple sub-communities.\n\n- **Theoretical and conceptual interest:** Bridges stochastic dominance from economics/decision theory to modern ML and DRO, sparking potential new research directions.", "requested_changes": "## Requested Changes\n\n- **(Critical) Include comparisons with other risk-aware baselines:**  \n  While the experiments demonstrate LSD’s ability to maintain expected performance while improving risk profiles, the empirical evaluation would be much stronger with direct comparisons to **established risk-sensitive methods**. For example, in supervised learning, including a baseline such as CVaR (Conditional Value at Risk) optimization or a distributionally robust training objective (e.g., adversarial DRO or worst-case risk minimization) would show how LSD performs relative to methods already in practice. In reinforcement learning, comparing against risk-aware RL algorithms (such as CVaR Policy Gradient or distributional RL methods like C51 or QR-DQN) would contextualize LSD’s gains in reducing catastrophic outcomes. Without these baselines, it is difficult to assess whether LSD truly advances the state of the art or simply achieves what existing methods can already provide. Adding these comparisons is essential to establish LSD as a genuinely superior approach rather than just an alternative formulation.\n\n- **(Critical) Clarify the role of different dominance orders:**  \n  The paper introduces a general framework for $k$-th order stochastic dominance, but in practice, all experiments focus on second-order dominance (which aligns with risk-averse decision-making and variance sensitivity). It is unclear whether LSD is equally effective or even tractable for higher orders, such as third-order dominance (capturing skewness preferences) or beyond. The authors should explicitly state whether higher-order extensions are supported, whether the theory and algorithm remain valid, and if so, whether they have practical significance. If the focus is mainly on $k=2$, this limitation should be acknowledged clearly to avoid overselling the generality of the contribution. Alternatively, even a small-scale demonstration of $k=3$ on synthetic data would reassure readers that the framework is not limited to just second-order dominance.\n\n- **(Critical) Discuss limitations and edge cases:**  \n  Currently, the paper emphasizes the strengths of LSD but provides little discussion of its limitations. It would improve transparency and academic rigor to explicitly acknowledge edge cases. For instance, in hypothesis spaces with multiple **non-dominated solutions**, LSD may converge to one arbitrary solution without guarantees of uniqueness; readers should know whether this affects interpretability or robustness. Similarly, the assumptions underlying the theoretical results, such as compactness of the hypothesis space and continuity of loss distributions, may not hold in high-dimensional deep learning settings. The authors should comment on how violations of these assumptions might impact LSD’s performance. Finally, discussion of computational scaling in very large models would be useful, since even small overheads in the inner–outer loop structure could become significant. Highlighting these points will strengthen the credibility of the work by showing awareness of its boundaries.\n\n- **(Suggested) Add a Broader Impact discussion:**  \n  The submission would benefit from a dedicated section reflecting on the societal and ethical implications of risk-sensitive learning. On the positive side, LSD could reduce the frequency of catastrophic failures in applications like healthcare, finance, or autonomous systems, where tail risks are critical. On the other hand, excessive risk aversion could result in **overly conservative models**, potentially harming efficiency or fairness (e.g., consistently rejecting high-risk/high-reward loan applicants, or under-treating patients who could benefit from riskier but more effective therapies). The authors should also mention possible misuse, such as financial institutions employing LSD-based models in ways that unfairly disadvantage certain groups. Including this discussion would not only meet TMLR expectations but also demonstrate responsible research practice.\n\n- **(Suggested) Minor clarity and presentation improvements:**  \n  While the paper is generally well-written, several improvements could make it more accessible. First, an **illustrative toy example** early in the paper (e.g., comparing two distributions with simple means and variances) would help readers unfamiliar with stochastic dominance understand the intuition behind the dominance functional before diving into heavy formalism. Also, a careful proofreading pass to fix minor grammatical issues and streamline exposition would enhance readability. These are not critical changes, but they would improve the presentation.", "broader_impact_concerns": "## Broader Impact Concerns\n\n- The paper lacks an explicit broader impact section.  \n- Positive impacts: safer AI systems in finance, healthcare, or robotics by reducing catastrophic risk.  \n- Risks: overly conservative decisions might lower utility or unfairly disadvantage certain groups.  \n- Authors should reflect on both sides and encourage careful application of LSD in real-world deployments.", "additional_comments": "## Additional Comments\n\nThe paper is generally clear, but the intuition could be strengthened with a simple illustrative example of stochastic dominance before introducing the heavier formalism, which would help orient readers less familiar with the concept. The results convincingly demonstrate LSD’s ability to reduce risk without sacrificing expected performance, with the reinforcement learning experiments providing particularly strong evidence of this balance. A further practical advantage worth highlighting is that LSD eliminates the need for manual tuning of risk trade-off parameters, setting it apart from many mean-variance or DRO-based methods. Finally, future research directions could include extending LSD to fairness-sensitive machine learning, scaling it to large deep learning models, or tailoring it to specific high-stakes domains such as finance or healthcare, where risk considerations are central.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)