Title: A Learning-Augmented Approach to Online Allocation Problems
Venue: NeurIPS 2025
Accepted: yes
Keywords: online algorithm

learning-augmented algorithm

virtual circuit routing
Abstract: In online allocation problems, an algorithm must choose from a set of options at each step, where each option incurs a set of costs/rewards associated with a set of $d$ agents. The goal is to minimize/maximize a function of the accumulated costs/rewards assigned to the agents over the course of the entire allocation process. Such problems are common in combinatorial optimization, including minimization problems such as machine scheduling and network routing, as well as maximization problems such as fair allocation for welfare maximization.

In this paper, we develop a general learning-augmented algorithmic framework for online allocation problems that produces a nearly optimal solution using only a single $d$-dimensional vector of learned weights. Using this general framework, we derive learning-augmented online algorithms for a broad range of application problems in routing, scheduling, and fair allocation. Our main tool is convex programming duality, which may also have further implications for learning-augmented algorithms in the future.

A Learning-Augmented Approach to Online Allocation Problems

Abstract

In online allocation problems, an algorithm must choose from a set of options at each step, where each option incurs a set of costs/rewards associated with a set of d agents. The goal is to minimize/maximize a function of the accumulated costs/rewards assigned to the agents over the course of the entire allocation process. Such problems are common in combinatorial optimization, including minimization problems such as machine scheduling and network routing, as well as maximization problems such as fair allocation for welfare maximization.

In this paper, we develop a general learning-augmented algorithmic framework for online allocation problems that produces a nearly optimal solution using only a single d -dimensional vector of learned weights. Using this general framework, we derive learning-augmented online algorithms for a broad range of application problems in routing, scheduling, and fair allocation. Our main tool is convex programming duality, which may also have further implications for learning-augmented algorithms in the future.

1 Introduction

In the last few years, tremendous advances in machine learning have triggered much interest and rapid progress in a new domain called learning-augmented online algorithms. In many practical scenarios, one needs to solve optimization problems with dynamically evolving requirements (called online algorithms), where uncertainty about the future significantly affects the quality of the solution. The basic premise of the learning-augmented paradigm is to use (possibly noisy) machine-learned predictions as a proxy for the future, and design online algorithms that can take advantage of good predictions while not falling prey to bad ones. (The resulting algorithms are sometimes also referred to as 'online algorithms with predictions'.) The explosive growth of this new area over the last few years has spanned a large variety of domains such as caching, scheduling, matching, clustering, network design, and many others (see related work), and generated much excitement at the intersection of algorithm design and machine learning. While many novel and clever algorithms have been designed in the learning-augmented setting, these are typically tailored to the requirements of a specific problem at hand, and do not generalize to broader problem classes. Indeed, there is a surprising absence of unifying algorithmic models and general-purpose tools for online algorithms with predictions, particularly when contrasted with other models of computation where algorithmic progress has largely relied on broad, unifying models and algorithmic techniques.

In this paper, we aim to partially rectify this situation by studying a broad class of problems called online allocation in the learning-augmented setting. The setup is the following: in each online step, the algorithm is presented a set of options and has to choose one of them. Each option is represented by a vector, and the final objective is a (linear or non-linear) function of the (coordinate-wise) sum

Debmalya Panigrahi

of all the chosen vectors. As we will soon see, this captures a broad range of problems in network routing, scheduling, fair allocation, etc. In this paper, we develop a general algorithmic strategy for this entire class of problems in the learning-augmented setting that obtains a nearly optimal (fractional) solution. This is in sharp contrast to strong (super-constant) lower bounds without ML advice, for almost all problems captured by this framework.

The online allocation problem that we described above has many applications. Among the most important is that of online routing. In this problem, we are given a network and in each online step, the algorithm has to choose a route from a given source to a destination node. Eventually, the goal is to minimize the maximum congestion of any network link, which is defined as the number of chosen routes using that link. This is a classical and well-studied problem in online algorithms, and it is well-known that the best competitive ratio 1 achievable is logarithmic in the size of the network [3] (see also [4, 6, 7]). In sharp contrast, we obtain a (1 + ϵ ) -approximation for any ϵ > 0 using learning augmentation, by modeling the problem as online allocation where each coordinate in the vector represents a network link and the goal is to minimize the ℓ ∞ -norm of the congestion vector.

Asecond application domain for online allocation is that of online scheduling problems. Consider the problem of processing jobs online where each job comes with a slate of options given by vectors that represent the processing time on different machines. Overall, the goal is to minimize the makespan, i.e., the maximum load on any machine. This problem is also described via online allocation, where the coordinates of the vectors represent machines and the objective is again the ℓ ∞ -norm of machine loads given by the sum of the selected vectors. Our online allocation algorithm gives a (1 + ϵ ) -competitive learning-augmented solution to this problem, which again sharply contrasts with lower bounds that are logarithmic in the number of machines in the absence of learning augmentation [10].

We note that the flexibility of the online allocation framework also allows us to extend these problem formulations immediately to a much broader set of objectives, such as all ℓ p -norm objectives [5] rather than just the ℓ ∞ -norm. Another interesting direction is that of maximization problems. In contrast to minimization problems where the vector coordinates can be thought of as resources whose use needs to be minimized, we now imagine the coordinates to represent value added to agents when a particular option is selected. This allows us to model fair allocation problems, where a set of items have to be assigned in a way that maximizes the minimum value along all agents (or another function such as Nash social welfare, p -means, etc.) [13, 15] Again, we observe the sharp contrast between the traditional online setting without learning augmentation, where strong polynomial lower bounds exist for these problems [34, 18] and the learning-augmented setting where our online allocation algorithm immediately yields a (1 -ϵ ) -approximation for any ϵ > 0 .

We end this section by commenting on the machine-learned parameters that we use in our learningaugmented algorithm for the online allocation problem. As described above, we show that there exists a set of learned parameters that allows us to approximate the optimal objective within a (1 ± ϵ ) factor of the optimal value. Importantly, we show that these parameters are bounded as a function of ϵ and the number of coordinates (rather than the number of steps or options) and, moreover, that our learning-augmented algorithm is resilient to small errors in the values of the learned parameters. This allows us to derive PAC-learning results for the learned parameters and to use them in the learning-augmented algorithm to obtain a (1 ± O ( ϵ )) -approximation.

Organization. The formal model and statement of results appear in Section 2. Section 3 presents related work. The main result, which provides the learning-augmented algorithm, appears in Section 4. Applications of this result to routing and fair allocation problems are discussed in Section 5. In Appendix C, we derive sample complexity bounds for the learned parameters in the PAC model. All omitted details of proofs are in the appendix.

2 Our Contributions

First, we define the online allocation framework:

Online Allocation Problem. In this problem, we receive an initial input of a function f over a bounded domain in R d + . Subsequently, we proceed in steps. At every time step t = 1 , . . . , T , we receive a set A t ⊆ R d + of d -dimensional vectors. We must select one vector v † t ∈ A t before

proceeding to time step t +1 , using only information available up to time t . Denote the result vector v † tot := ∑ T t =1 v † t . The goal is to maximize or minimize f ( v † tot ) for some objective function f ( · ) .

For ease of notation, we fix an arbitrary ordering of the vectors in A t , and denote by v t,k ∈ R d + the k th vector in A t , i.e., A t = ⟨ v t, 1 , . . . , v t, | A t | ⟩ . Accordingly, we define K ( t ) = { 1 , . . . , | A t |} as the set of indices of these vectors, and for each k ∈ K ( t ) , we let v i,t,k denote the i th coordinate of vector v t,k in A t . Our results apply to well-behaved functions f ( · ) that we define as follows:

Well-Behaved Functions. Let f : R d + → R + be the objective function defined on the result vector. Then, f is well-behaved if it satisfies the following properties:

Monotonicity: f is said to be monotone if for any ℓ, ℓ ′ ∈ R d + such that ℓ i ≥ ℓ ′ i for all i ∈ [ d ] , we have f ( ℓ ) ≥ f ( ℓ ′ ) .

Homogeneity: f is said to be homogeneous if for any ℓ, ℓ ′ ∈ R d + such that ℓ ′ i = α · ℓ i for all i ∈ [ d ] , we have f ( ℓ ′ ) = α · f ( ℓ ) .

These properties are satisfied by most objective functions studied in the literature including linear functions, p -norms, Nash Social Welfare (which is the geometric mean), p -means, etc.

Our algorithmic scheme will focus on developing a fractional selection rule, which can be interpreted as a probability distribution over the different choices in every step. We formally describe a fractional solution as follows:

Fractional Solution. At step t , the algorithm has to assign a fractional value to each option x t,k ∈ [0 , 1] for k ∈ K ( t ) such that ∑ | K ( t ) | k =1 x t,k = 1 . Accordingly, we have

Note that in some scenarios such as the allocation of divisible items or network flow routing, a fractional solution is already sufficient.

Learning-Based Online Scheme for Allocation Problems. Our main result is that for any d -dimensional instance of the online allocation problem with a well-behaved objective function, there exists a vector α ∈ R d such that an online algorithm guided by α achieves a competitive ratio of 1 + ϵ for minimization or 1 -ϵ for maximization, for any ϵ > 0 .

Theorem 2.1. Given an instance of the online allocation problem with a well-behaved objective and any ϵ > 0 , there exists a set of learned parameters α ∈ R d and an online algorithm that uses α such that the resulting fractional solution achieves a (1 ± O ( ϵ )) -approximation.

The online algorithm that establishes this result uses a simple exponential assignment rule , where given a learned parameter vector α ∈ R d , the fractional allocation x tk ( α ) of option k ∈ K ( t ) is:

We further show that the learned parameters can be bounded as a function of d and ϵ , and that the algorithm is robust to small perturbations in the parameter values. This structure allows the parameter space to be both bounded and discretized in terms of d, ϵ , independent of the time horizon. This property is crucial in applications where the number of steps (which may represent flow requests, jobs, or items) is significantly larger than the number of dimensions (which may represent network links, machines, or agents). Moreover, the discretization of the parameter space ensures that the parameters are learnable, and allows us to bound the number of samples required for learning.

Learnability of Parameters. Following the PAC framework of [26], we establish the learnability of parameter vectors for well-behaved objectives, under the additional assumption of superadditivity for maximization or subadditivity for minimization, as introduced in [18].

We consider a setting in which each online step (i.e., a set of vectors) is drawn independently (though not necessarily identically) from a distribution, and each step contributes only a small fraction of the total allocation. This latter property is formalized by the standard small items assumption (see, e.g., [1, 26, 24, 29]), which we state below:

Small Items Assumption: There exists a ζ = Θ ( log d ϵ 2 ) such that v itk ≤ L ζ for every i ∈ [ d ] , t ∈ [ T ] , and k ∈ K ( t ) .

Under this assumption, we derive a sample complexity bound for approximately learning a parameter vector α that ensures near-optimal performance. The following informal theorem summarizes our main learnability result; see Appendix C for details:

Theorem 2.2. Under standard PAC assumptions, the small items assumption, and mild regularity conditions on the objective (subadditivity for minimization and superadditivity for maximization), the parameter vector α ∈ R d can be learned from O ( d log d · log d ϵ ) i.i.d. samples.

Rounding the Fractional Solution. As mentioned earlier, a fractional solution is already sufficient for many applications. Furthermore, we show that under the small items assumption (see above), a fractional solution can be converted into a randomized online integral solution with a (1 ± ϵ ) loss in the objective value, where the parameter ϵ only depends on the parameters of the small items assumption.

Consider the following online randomized rounding algorithm: given an online fractional solution x tk for t ∈ [ T ] , the algorithm chooses option k with probability x t,k . Under the small items assumption, standard Chernoff bounds ensure that with high probability, the maximum coordinate in the rounded vector remains within a multiplicative factor of (1 ± O ( ϵ )) of its fractional counterpart. We get the following theorem:

Theorem 2.3. Let v f tot be a fractional solution, and let v † tot be the integer solution produced by randomized rounding of this fractional solution. Then, under the small items assumption, the following holds with high probability:

Now, combining the above theorem with Theorem 2.1, we get the following result for integer solutions under the small items assumption:

Corollary 2.4. Given an instance of the online allocation problem with a well-behaved objective and under the small-item assumption, for any ϵ > 0 , there exists a set of learned parameters α ∈ R d and an online algorithm that uses α such that the resulting integral solution achieves a (1 ± O ( ϵ )) -approximation.

Robustness-Consistency Tradeoff. Ideally, a learning-augmented algorithm should simultaneously exploit accurate predictions to achieve near-optimal performance (consistency), while also maintaining strong worst-case guarantees when predictions are unreliable (robustness). The (not learning-augmented) worst case lower bounds for allocation problems with minimization (resp., maximization) objectives is Ω(log d ) (resp., Ω( d ) ). Formally, an algorithm is said to be γ -consistent and δ -robust if it achieves a γ -approximation under accurate predictions (consistency), and a δ -approximation in the worst case when predictions are unreliable (robustness). In Appendix D, we give a slight modification of our scheme that preserves robustness while achieving consistency for both minimization and maximization objectives.

Theorem 2.5. For minimization objectives, there exists an algorithm that uses α ∈ R d , a predicted parameter vector, and achieves 1 -consistent and O (log d ) -robust approximation.

For maximization objectives, there exists an algorithm that uses α ∈ R d , a predicted parameter vector, and a parameter λ , which achieves a (1 -λ ) -consistent and λ · d -robust approximation.

Applications. In Section 5, we illustrate the utility of our framework in handling a broad class of objectives by demonstrating its use in two applications. The first is for the online routing problem, where we are given a network and in each time-step, there is request to route a given value of flow from a source to a destination vertex [3]. The flow needs to be routed in a way that minimizes maximum congestion on any edge cumulatively across all time-steps. We give the first learning-augmented algorithm for this problem as a simple corollary of our general online allocation framework. Next,

we consider a maximization problem, that of allocating items to agents so as to maximize the Nash Social Welfare [13]. Again, we give a learning-augmented algorithm for this problem as a simple corollary of our general framework, matching previous results in [18]. We note that our framework applies to many other application problems in domains such as scheduling (makespan and ℓ p -norm minimization) and fair allocation (Santa Claus and p -means maximization) that we do not state here for brevity.

3 Related Work

Previous work that is mostly closely related to ours is on learning-augmented online scheduling and assignment problems [23, 24, 26, 18]. These works focus on the assignment of items that arrive online to agents, where each choice affects only a single agent. In our setting, this corresponds to each choice being represented by a d -dimensional vector with a single nonzero dimension, where d corresponds to the number of agents.

Lattanzi et al. [23] focused on minimizing the makespan (maximum load) for restricted assignment. They showed that a suitable set of learned parameters enables a proportional allocation rule that obtains a nearly optimal result. In [24], it was further shown that these parameters are PAC-learnable.

Li and Xian [26] generalized this framework to handle unrelated scheduling. Their scheme introduces two learned parameters per machine. The first set of parameters transform the instance into a restrictedrelated setting, and they show that proportional assignment using the second set of parameters then applies to this restricted setting.

Cohen and Panigrahi [18] improved this result by using only a single set of parameters in an exponential allocation scheme, and also extended it to handle a large number of maximization and minimization objectives.

Our scheme generalizes these previous works to accommodate general vectors, which is crucial for applications such as online routing that cannot be solved using the previous techniques. Furthermore, we simplify the previous approaches as well by requiring only a single variable per dimension, thereby eliminating the dependence on an additional exponent base which further simplifies the analysis of the existence of such parameters. Our work introduces a new set of techniques based on perturbation/sensitivity analysis because the ideas previously used for bounding the learned parameters do not generalize to the case of arbitrary vector options.

In [14], the authors study the Online Nash Social Welfare problem with predictions. The goal is to compute an online divisible allocation of goods among d agents in a way that balances fairness and efficiency. In their setting, the prediction for each agent is their monopolist value -the utility the agent would obtain if all resources were allocated solely to them-and their algorithm achieves an O (log d ) -approximation under this assumption. In contrast, we show that by learning a different d -dimensional parameter vector, it is possible to achieve a nearly optimal allocation.

Various papers [12, 35, 19, 20] investigate the robustness-consistency tradeoff in the context of general packing and covering problems. These algorithms typically introduce an additional parameter λ ∈ [0 , 1] to encode the algorithm's confidence in the prediction. A primal-dual scheme is then employed to interpolate between a worst-case baseline and a prediction-driven strategy. Unlike our approach, these methods assume that the prediction consists of the full solution, which is often impractical in real-world settings.

More broadly, the study of learning-augmented online algorithms was initiated by Lykouris and Vassilvitskii [27] in the context of the caching problem and has since grown into a prominent research area. This framework enhances online algorithms by incorporating machine-learned predictions about the future, allowing them to surpass pessimistic worst-case competitive bounds. Over the past few years, numerous online allocation problems have been explored within this framework, including applications in scheduling [33, 8, 9, 11, 21, 30], online matching [2, 17, 22], and ad delivery [28, 25]. For a broader overview of learning-augmented online algorithms, we refer the reader to the surveys by Mitzenmacher and Vassilvitskii [31, 32].

4 Online Allocation for a Well-Behaved Objective via Learned Parameters

In this section, we show that for the general case of the online allocation problem, there exists a set of learned parameters that guarantees a near-optimal solution. Specifically, we show the following theorem, which is a more refined version of (and establishes) Theorem 2.1:

Theorem 4.1. Given an instance of the online allocation problem with a well-behaved objective and any ϵ > 0 , there exists a set of learned parameters α ∈ NET ( q, s ) and an online algorithm that uses α such that the resulting fractional solution achieves a (1 ± O ( ϵ )) -approximation.

Here, NET ( q, s ) = { i S | i ∈ [0 , q · s ] } d denotes a d -dimensional discrete net with parameters q, s that are bounded in poly( d, 1 /ϵ ) .

For the minimization of a well-behaved function, we first consider the MinMax objective, which is defined as the minimization of f , where f ( v ) = max i ∈ [ d ] v i . Similarly, for the maximization of a well-behaved function, we first consider the MaxMin objective, which is defined as the maximization of f , where f ( v ) = min i ∈ [ d ] v i .

In this section, we focus on the MinMax objective. In Appendix E, we address the MaxMin objective. In Appendix F, we explain how to extend these results to general well-behaved objectives.

4.1 Learning-Augmented Online Allocation for the MinMax Objective

We now state the main result for the MinMax objective:

Theorem 4.2. Given an instance of the online allocation problem with a MinMax objective and any ϵ > 0 , there exists a set of learned parameters α ∈ NET ( d 2 ϵ · ln ( d ϵ ) , d 3 ϵ 3 ) and an online algorithm that uses α such that the resulting fractional solution is a (1 + O ( ϵ )) -approximation.

To prove this result, we first apply a pre-processing step to convert arbitrary instances of the problem to structured instances that we say are balanced . Note that the conversion to balanced instances is an algorithmic technique and not a restriction on the input. We then leverage convex programming duality on a max-entropy style convex programming formulation to show the existence of learned parameters that can be used to obtain an near-optimal solution online. At this stage, the learned parameters can be of arbitrary precision and are not necessarily efficiently learnable. To ensure the latter, we need to obtain parameters that belong to NET ( d 2 ϵ · ln ( d ϵ ) , d 3 ϵ 3 ) . We do this in the last step by using tools from perturbation and sensitivity analysis in the convex programming literature. Together, these steps establish Theorem 4.2.

Preprocessing of MinMax Instances. Recall that the exponential assignment scheme computes online allocations based on the ratio v i,t,k m t ( v ) . This ratio can be highly sensitive and potentially unbounded under small perturbations to the input, thereby making it impossible to execute the last step of designing learned parameters of bounded precision. To overcome this difficulty, we introduce a pre-processing step that transforms the original instance into a balanced instance, incurring only a small loss in the objective value.

An instance ˜ I (˜ v, ˜ K ) is said to be balanced if it satisfies the following condition for all t ∈ [ T ] , i ∈ [ d ] , and k ∈ K ( t ) :

Note that in a balanced instance, we have v i,t,k m t ( v ) ≤ d 2 ϵ 2 . Moreover, by omitting redundant options, we may assume that the number of options per step is bounded, i.e., ln | K ( t ) | = O ( d · ln( d/ϵ )) . We construct a balanced instance I (˜ v, ˜ K ) as follows:

Given an instance I ( v, K ) ,

Define ˆ v such that

Define ˜ K by removing option k ′ from step t whenever there exists another option k with

Define ˜ v as follows: if ˆ v itk = 0 , then set ˜ v itk = 0 ; otherwise, set

By definition, I (˜ v, ˜ K ) is a balanced instance. In Appendix A, we show that transforming I ( v, K ) into I (˜ v, ˜ K ) incurs only a (1 + O ( ϵ )) loss in the objective value.

Lemma 4.3. Let I ( v, K ) be an instance of the allocation problem with the MinMax objective, and let ϵ > 0 . Then, any (1 + ϵ ) -approximate solution to the balanced instance I (˜ v, ˜ K ) constructed via the pre-processing algorithm yields a (1 + O ( ϵ )) -approximate solution to I ( v, K ) .

Learned Parameters for Balanced Instances. In light of Lemma 4.3, it suffices to only consider balanced instances in Theorem 4.2. We restate this goal as follows:

Lemma 4.4. Given a balanced instance of the online allocation problem with a MinMax objective and ϵ > 0 , there exists a set of parameters α ∈ NET ( d 2 ϵ · ln ( d ϵ ) , d 3 ϵ 3 ) such that the fractional solution defined by the exponential assignment scheme with parameters α is a (1 + O ( ϵ )) -approximation to the optimal objective.

The remainder of this section gives a proof of Lemma 4.4. We do this in two steps. First, we prove the existence of parameters α ( ϵ ) that are bounded as a function of d, ϵ using which the exponential assignment scheme produces an (1 + ϵ ) -approximate optimal solution. (For the special case of ϵ = 0 , the solution is precisely optimal but the parameters may be unbounded.) Then, we show that small errors (up to ϵ 3 /d 3 ) in the parameter values incur only a (1 + O ( ϵ )) loss in the objective value, enabling discretization of the learned parameters to NET ( d 2 ϵ · ln ( d ϵ ) , d 3 ϵ 3 ) .

We use L ∗ to denote the optimal MinMax value, and define the following convex program for ϵ > 0 :

Lemma 4.5. Given an instance of the online allocation problem with the MinMax objective and any ϵ ≥ 0 , there exists a vector α ( ϵ ) ∈ R d + such that the fractional solution defined by the exponential assignment scheme with parameters α ( ϵ ) is (1 + O ( ϵ )) -approximately optimal.

Proof. Given such instance and for fixed ϵ consider the convex program of Figure 1. By our assumption, L ∗ is the optimal MinMax objective therefore there exists a feasible solution for the convex program for any ϵ ≥ 0 . Accordingly, define the Lagrangian L ( x, α, β ) as

From the KKT conditions for the optimal solution to the convex program as a function of ϵ x ( ϵ ) , α ( ϵ ) , β ( ϵ ) , the solution that allocates according to x ( ϵ ) is a (1 + ϵ ) -approximation to the optimal

objective L ∗ , and α ( ϵ ) i ≥ 0 for all i ∈ [ d ] . Furthermore,

For any two options k, r ∈ K ( t ) , we obtain:

Bounding the Learned Parameters. For ϵ = 0 , the learned parameters α ( ϵ ) in the previous lemma may be unbounded. However, for ϵ > 0 , we show that each α ( ϵ ) i can be bounded as a function of d and ϵ . Our main tool is perturbation and sensitivity analysis, following the framework of Boyd et al. [16]. (See Appendix B for details of perturbation and sensitivity analysis.)

Lemma 4.6. Let x ( ϵ ) , α ( ϵ ) , β ( ϵ ) be the optimal solution to the convex program in Figure 1, for some ϵ > 0 . Then, for all i ∈ [ d ] , it holds that α ( ϵ ) i ≤ d 2 ϵ · ln ( d ϵ ) .

Proof. We need the following claim:

Claim 4.7. For an n dimensional vector x ∈ [0 , 1] n such that ∑ n i =1 x i = 1 we have ∑ n k =1 x k ln x k ≥ -ln n .

Proof. By Jensen's inequality, we have

.

We define a perturbed convex program based on Figure 1, where u i corresponds to the constraint α i For each i ∈ [ d ] , setting u i = -ϵ · L ∗ and u i ′ = 0 for i ′ = i ensures that constraint i in the perturbed problem matches the original constraint, thereby guaranteeing a feasible solution. By Lemma B.1,

̸

where the last inequality follows from Claim 4.7.

where the first equality holds because x ∗ is a feasible solution, the first inequality follows since there must be at least one nonzero coordinate and by definition of m t ( v ) , and the second inequality follows from the linear program constraint for dimension i .

By our assumption on the instance, we have

Discretizing the Learned Parameters and Noise Resilience. In order to prove Lemma 4.4, we need to ensure that the learned parameters α ( ϵ ) belong to the discrete set NET ( d 2 ϵ · ln ( d ϵ ) , d 3 ϵ 3 ) . This may not be true of the Lagrangian multipliers derived above. We show that in discretizing the learned parameters, the competitive ratio only worsens by (1 + O ( ϵ )) . In particular, if we replace the Lagrangian vector α ( ϵ ) with a perturbed vector ˜ α , where -ϵ 3 d 3 ≤ ˜ α i -α ( ϵ ) i ≤ ϵ 3 d 3 for all i ∈ [ d ] , then the assignment fractions change by at most 1 ± 4 ϵ . Note that this lemma is also important from a noise resilience perspective: if the learned parameters have small error, this lemma shows that the resulting allocation is still approximately optimal.

Lemma 4.8. Let α ∗ ∈ R d be a vector, and let ˜ α ∈ R d be a perturbed vector such that | ˜ α i -α ∗ i | ≤ ϵ 3 d 3 for all i ∈ [ d ] . Then, the fractional assignment x tk (˜ α ) satisfies

Proof. We need the following claim:

Claim 4.9. For any ϵ > 0 , if we are given two sets of K weights a, a ′ ∈ R K + such that 1 -ϵ ≤ a k a ′ k ≤ 1 + ϵ , then for x, x ′ ∈ [0 , 1] K such that ∑ K k =1 x k = ∑ K k =1 x ′ k = 1 and x k ∝ a k and x ′ k ∝ a ′ k , we have 1 -4 ϵ ≤ x k x ′ k ≤ 1 + 4 ϵ .

Proof. Consider the assignment fractions x k . By definition, we have the following:

Define a tk ( α ) = exp ( -∑ i α i · v itk m t ( v ) ) . Then, we have

By the assumption that the instance is balanced, we have v itk m t ( v ) ≤ d 2 ϵ 2 , which implies

Thus, by Claim 4.9, the assignment fractions using ˜ α differ from those using α by at most (1 ± 4 ϵ ) .

Finally, we now put all the pieces together to establish Lemma 4.4:

Proof of Lemma 4.4. Fix a balanced instance I ( v, K ) and ϵ > 0 . By Lemma 4.6, there exists a parameter vector α ( ϵ ) such that α ( ϵ ) i ∈ [ 0 , d 2 ϵ · ln ( d ϵ ) ] for all i ∈ [ d ] . Therefore, there exists a vector ˜ α ∈ NET ( d 2 ϵ · ln ( d ϵ ) , d 3 ϵ 3 ) such that | ˜ α i -α ( ϵ ) i | ≤ ϵ 3 d 3 for all i ∈ [ d ] . By Lemma 4.8, the exponential assignment rule with ˜ α achieves a (1 + O ( ϵ )) -approximation.

5 Applications

Our framework applies broadly to a variety of online allocation problems. We briefly highlight two representative applications: online routing and Nash Social Welfare maximization.

Online Routing. In online routing (e.g., [3]), requests arrive over time and must be assigned to paths in a network to minimize congestion or delay. Each option (path) contributes load to

network edges, and the goal is to minimize the maximum edge utilization, which corresponds to the MinMax objective. Our approach uses a vector of learned edge-weight parameters to guide allocation decisions, combining strong worst-case guarantees with improved performance when input patterns are predictable.

Formally, we are given as input a directed graph G = ( V, E ) . At each time step t ∈ [ T ] , a flow request of amount r t between two vertices s t and t t , along with a set of candidate paths P t = { P t, 1 , P t, 2 , . . . } , is revealed. Here, each P t,k is a path from s t to t t in the graph. The algorithm must assign values q t,k to each path such that ∑ k q t,k = r t . The load on an edge e ∈ E is given by ℓ e ( q ) = ∑ t ∑ P t,k | e ∈ P t,k q t,k , and the objective is to minimize the maximum congestion on any edge, i.e., minmax e ∈ E { ℓ e ( q ) | ∑ k q t,k = r t for all t ∈ [ T ] } .

We note that online routing is a special case of the online allocation problem, where A t ⊆ { z t · f ⃗ | f ⃗ ∈ { 0 , 1 } d , z t ∈ R + } , and the objective is to minimize f ( v ) = max i ∈ [ d ] v i .

Our main result for online routing follows as a corollary of Theorem 4.1.

Corollary 5.1. Given an instance of online routing and any ϵ > 0 , there exists a set of parameters α ∈ NET ( d 2 ϵ · ln ( d ϵ ) , d 3 ϵ 3 ) such that the fractional solution x tk ∝ exp( -∑ i α i · f itk ) , for t ∈ [ T ] , approximates the optimal objective within a factor of (1 + O ( ϵ )) .

Online Nash Social Welfare. As mentioned, our scheme also applies to maximation problems where, at each step, a divisible resource has to be distributed among a set of d agents. The objective is to design an online algorithm that balances fairness and efficiency. At the start of each step t , the algorithm observes the value v i,t of each agent i for that resource, and then irrevocably determines the allocation without knowledge of future values. If agent i is allocated a fraction x i,t , their utility increases by x i,t v i,t . The total utility of agent i is then given by u i ( x ) = ∑ t v i,t x i,t .

The Nash Social Welfare (NSW) objective is known to provide a natural balance between fairness and efficiency. It is defined as the geometric mean of the agents' utilities: NSW ( x ) = ( ∏ i u i ( x )) 1 /d . By applying our method for maximizing well-behaved objectives, we obtain the following corollary:

Corollary 5.2. Given an instance of online NSW maximization and any ϵ > 0 , there exists a set of parameters α ∈ NET ( d 2 ϵ · ln ( d ϵ ) , d 3 ϵ 3 ) such that the fractional solution x t,k ∝ exp( ∑ i α i · v i,t,k ) , for t ∈ [ T ] , approximates the optimal objective within a factor of (1 -O ( ϵ )) .

Note that using our results, we may further generalize this to options with vector utilities, i.e., where an option can add (possibly different amounts of) value to multiple agents simultaneously.

6 Closing Remarks

In this paper, we gave a general technique for designing online algorithms with predictions that applies to the entire spectrum of problems that can be modeled via the online covering framework. This has value in two respects. First, it gave the first learning-augmented results for important problems like online routing and scheduling that goes beyond simple assignment, and were beyond the scope of previous techniques. Second, and perhaps more importantly, it opens the door to the design of even more general-purpose methods for the design of online algorithms with predictions. For instance, can we give a technique for learning-augmented algorithms that applies to any covering LP where each online step reveals a new constraint in the LP? Techniques such as the online primal dual method that apply to this class of LPs have been hugely influential in the classical online algorithms literature (without predictions), which makes it a tempting proposition to explore similarly powerful tools in the learning-augmented setting.

Acknowledgments and Disclosure of Funding

Ilan Reuven Cohen's research was supported by the Israel Science Foundation grant No. 1737/21. Debmalya Panigrahi's research was supported in part by NSF grants CCF-1955703 and CCF-2329230.

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A Preprocessing for the MinMax Objective

We begin by introducing a preprocessing step for the MinMax objective. Given an instance I ( v, K ) , let MinMax ( v, K ) denote the optimal value of the corresponding MinMax objective. Specifically, we construct an instance I (˜ v, ˜ K ) that ϵ -approximates a given instance I ( v, K ) as follows:

Definition A.1. An instance I (˜ v, ˜ K ) is said to ϵ -approximate an instance I ( v, K ) if:

˜ K ( j ) ⊆ K ( j ) for all j ∈ N ,

MinMax (˜ v, ˜ K ) ≤ MinMax ( v, K ) · (1 + ϵ ) ,

For any feasible allocation ˜ x for I (˜ v, ˜ K ) , if the load in any dimension is at most L with respect to ˜ v , then the corresponding load with respect to v is at most L · (1 + ϵ ) .

(Note that ˜ x remains a feasible allocation for I ( v, K ) since ˜ K ( j ) ⊆ K ( j ) .)

Using this definition, we prove that in order to achieve a (1 + O ( ϵ )) to the instance I ( v, K ) it is sufficent to acheive (1 + ϵ ) approximation to I (˜ v, ˜ K ) .

Corollary A.2. Let I ( v, K ) be a given instance. Suppose the instance is transformed into an ϵ -approximate instance I (˜ v, ˜ K ) . If a (1 + ϵ ) -approximate allocation is computed for the transformed instance I (˜ v, ˜ K ) , then this allocation guarantees a (1+ O ( ϵ )) -approximation for the original instance I .

Proof. Let x be a feasible solution for I (˜ v, ˜ K ) such that

Using Definition A.1, we obtain:

We now show that each of the three steps defined in the transformation produces an ϵ -approximate instance with respect to the original instance.

Claim A.3 (Step (1) of the preprocessing) . Given an instance I ( v, K ) let ˆ v such that

the the instance I (ˆ v, K ) ϵ -approximate the instance I ( v, K ) .

Proof. By definition, MinMax (˜ v, ˜ K ) ≤ MinMax ( v, K ) . Next, given a solution x such that for all i ∈ [ d ] ,

For a fixed coordinate i ∈ [ d ] , define

we prove that

which represents the set of options for item t where the load on coordinate i is modified by the transformation. For an item t and an option k ∈ K ( t ) , let

denote the coordinate that experiences the maximum load for option k of item t .

Observe that if k ∈ R i ( t ) , then

Thus, we have:

where the first and final inequalities follow from the definition of x , and the second inequality follows from the transformation definition.

The

Claim A.4 (Step (2) of the Preprocessing) . Given an instance I ( v, K ) , let I (˜ v, K ′ ) be the instance obtained by removing option k ′ from item t whenever there exists another option k ∈ K ( t ) such that

where v max t,k = max i ∈ [ d ] v i,t,k .

Then I ( v, K ′ ) is an ϵ -approximation of the instance I ( v, K ) .

Proof. Clearly, an assignment with a load of at most L for all i ∈ [ d ] in I (˜ v, ˜ K ) results in a load of at most L in I ( v, K ) , since the retained options have identical loads. We now show that if an allocation x achieves a makespan of L in I ( v, K ) , then there exists a transformed allocation ˆ x that achieves a makespan of at most L · (1 + ϵ ) in I (˜ v, ˜ K ) .

We explicitly construct such an assignment ˆ x and show that it satisfies the required bound. Define

as the option with the smallest maximum load for item t , breaking ties by selecting the smallest index. Let

denote the coordinate where option k of item t imposes the maximum load.

Next, define the set

which identifies options with disproportionately high loads relative to the smallest maximum load. We then construct the transformed allocation ˆ x for each item t ∈ [ n ] as follows:

This transformation ensures that options in R ( t ) are reallocated to the option with the smallest maximum load, maintaining feasibility while ensuring that the makespan increases by at most a factor of (1 + ϵ ) . Specifically, we have:

where the first and final inequalities follow from the definition of x , and the second inequality follows from the fact that for k ′ ∈ R ( t ) ,

Claim A.5 (Step (3) of the Preprocessing) . Given an instance I ( v, K )

if v itk = 0 , then set ˜ v itk = 0 ; otherwise, set

Then I (˜ v, K ) is an ϵ -approximation of the instance I ( v, K ) .

Proof. By definition,

Therefore, I (˜ v, K ) is an ϵ -approximation of the instance I ( v, K ) .

By applying Claim A.3,Claim A.4,Claim A.5, and Corollary A.2 we proved the Lemma 4.3.

B Perturbation and Sensitivity Analysis

One of our main tools for bounding learned parameters is perturbation and sensitivity analysis, following the framework of Boyd et al. [16].

B.1 Perturbed Convex Programs

Consider a convex program represented by a tuple ( f, h ) , where f i : R d → R for i ∈ [0 , ˜ m ] and h i : R d → R for i ∈ [1 , ˜ n ] , formulated as:

We define the perturbed version of this problem as follows. Let u ∈ R ˜ m and v ∈ R ˜ n . Define perturbed constraints:

The perturbed problem becomes:

This coincides with the original problem when u = 0 and v = 0 . Positive u i relaxes the i th inequality; negative u i tightens it. The vector v perturbs the right-hand sides of the equality constraints.

Let p ∗ ( u, v ) denote the optimal value of the perturbed problem. If the problem is infeasible, we define p ∗ ( u, v ) = ∞ . When the original problem is convex, p ∗ ( u, v ) is convex in both u and v .

B.2 A Global Inequality via Duality

Assume strong duality holds and that the dual optimum is attained (which is the case under Slater's condition). Let ( λ ∗ , ν ∗ ) denote an optimal dual solution to the original problem. Then the following global bound holds.

Lemma B.1 (Perturbation Inequality, [16]) . For all u ∈ R ˜ m and v ∈ R ˜ n ,

B.3 Proof Sketch

To derive this inequality, consider any feasible solution x to the perturbed problem, i.e.,

By strong duality:

where the last inequality follows from f i ( x ) ≤ u i , h i ( x ) = v i , and λ ∗ ≥ 0 .

Rearranging, we conclude:

This inequality quantifies how much the objective value may change in response to perturbations of the constraints, depending linearly on the dual multipliers.

C Learnability of the Prediction Vectors

We consider the learning model introduced by [24], and show that under this model, the dual vector α can be learned efficiently from sampled instances. Specifically, we consider the following model: the t th step (i.e., the values of v t = ( v i,t,k : i ∈ [ d ] , k ∈ K ( t )) is independently sampled from a (discrete) distribution D j .

We set up the online allocation problem for the MinMax objective; the setup for the MaxMin objective is very similar and is omitted for brevity. Let L = E P ∼D [ MinMax ( P )] be the expected value of the MinMax objective in the optimal solution for an instance P drawn from D .

Morally, we would like to say that we can obtain the vector α that gives a nearly optimal solution (in expectation) using vector allocation (i.e., a MinMax objective of (1 + ϵ ) · L in expectation for some error parameter ϵ ) using a bounded (as a function of ϵ ) number of samples. Similar to [24], we need the following assumption:

Small Items Assumption: Conceptually, this assumption states that each individual item has a small utility compared to the overall utility of any agent in an optimal solution. Precisely, we need v itk ≤ L ζ

We show the following PAC learning theorem for the MinMax objective:

Theorem C.1. Fix an ϵ > 0 for which the small items assumption holds. Then, there is an (learning) algorithm that samples O ( d log d · log d ϵ ) independent instances from D and outputs (with high probability) a prediction vector α such that using α in the allocation scheme gives a MinMax objective of at least (1 + O ( ϵ )) · L in expectation over instances P ∼ D .

Proof Sketch. Recall that in PAC theory, the number of samples needed to learn a function from a family of N functions is about O (log N ) . Indeed, restricting α to be in the class NET ( K,S ) serves this role of limiting the hypothesis class to a finite, bounded set since | NET ( K,S ) | = ( K · S ) d where S = K = O ( poly ( d, ϵ )) . Using standard PAC theory, this implies that using about O ( d log K ) = O ( d · log d ϵ ) samples, we can learn the 'best' vector in NET ( K,S ) . Our main technical work is to show that this 'best' vector produces an approximately optimal solution when used.

C.1 Details of PAC Learning

Given an instance P , Let ℓ i ( P, α ) the value of the i th dimension of v f tot after applying our scheme on the instance P with the parameters α . Let K = d 2 ϵ and S = d 3 ϵ 3 .

Let us consider a combination of all instances in the support of the distribution D . For L processing matrices P (1) , P (2) , . . . , P ( L ) . We define P all = ⊕ L r =1 P ( r ) to be the instance defined by the n · L items. For every ℓ ∈ [ L ] and t ∈ [ T ] , we have a step t ( ℓ ) with values v ( ℓ ) t .

The following observation is immediate (subadditivity):

Using this observation, we can prove the following lemma, by considering the combination of all instances in D , scaled by their respective probabilities.

Lemma C.3. There exists α ∈ NET ( K,S ) , such that for every i ∈ [ d ] , we have

Proof. Consider the instance P = ⊕ Pr D [ P ] · P where Pr D [ P ] is the probability mass of P in D , and Pr D [ P ] · P is the matrix P multiplied by Pr D [ P ] . By Observation C.2, we have

We can apply Lemma 4.4 to the combined instance to show there exists α ∗ ∈ NET ( K,S ) such that for every i ∈ [ d ] , we have,

where t indexes over all steps in P . Notice that x t,k ( P , α ∗ ) depends on the utility vector for step t , which is part of the instance P ∈ D that t belongs to. Therefore, the left side of the above inequality is exactly

as required.

For any real numbers A,B,ϵ,C , we use A ≈ ϵ,C B to denote | A -B | ≤ ϵ · max( B,C ) . The next lemma appears in [26]:

Lemma C.4 (Lemma D.6 in in [26]) . For any α ∈ NET ( K,S ) , with high probability over P ∼ D , we have

The learning algorithm. We sample H = O ( d log d log d ϵ ) instances P 1 , P 2 , . . . , P H independently and randomly form D . We output ˜ α ∈ NET ( K,S ) that maximizes min i ∈ [ d ] 1 H ∑ H h =1 ℓ i ( P h , ˜ α ) .

The next lemma also appears in [26]:

Lemma C.5 (Lemma D.7 in [26]) . With probability at least 1 -1 ( K · S ) m , for every α ∈ NET ( K,S ) and for every i ∈ [ d ] , we have

Now assume the event in Lemma C.5 happens. Then by Lemma C.3, there exists some α ∈ NET ( K,S ) such that

In particular, since ˜ α maximizes min i ∈ [ m ] 1 H ∑ H h =1 ℓ i ( P h , ˜ α ) for ˜ α ∈ NET ( m,ϵ ) , we can conclude that

Applying Lemma C.5 again, we get

We now apply Lemma C.4 to ˜ α . We have that with high probability over P ∼ D , for every i ∈ [ m ] the following holds:

Therefore, MinMax ( P, ˜ α ) ≤ (1 + Ω( ϵ )) · L ∗ . This completes the proof of Theorem C.1.

D Robustness-Consistency Tradeoff

In this section, we show that our learning-augmented scheme can be modified to balance consistency and robustness, achieving near-optimal performance when predictions are accurate while retaining strong worst-case guarantees when they are not.

Recall that an algorithm is said to be γ -consistent and δ -robust if it achieves a γ -approximation under accurate predictions (consistency), and a δ -approximation in the worst case when predictions are unreliable (robustness).

For allocation problems with minimization objectives, the worst-case approximation ratio without any predictions is O (log d ) , and for maximization objectives it is O ( d ) , where d is the number of agents.

We show that our learning-augmented scheme can be modified to satisfy this robustness-consistency tradeoff.

Modified Algorithm for Minimization Objectives. Let α ∈ R d be the predicted parameter vector. The algorithm operates in two phases:

Prediction Phase: At each time step, use the exponential assignment scheme with parameters α .

Fallback Phase: Monitor the cumulative objective value. If it exceeds the optimal value by a factor larger than O (log d ) , the algorithm switches to a standard worst-case online algorithm.

Let η be the approximation factor achieved using α . Then, the final approximation ratio is min( η, O (log d )) , ensuring both consistency and robustness.

Modified Algorithm for Maximization Objectives. Let α ∈ R d be the predicted parameter vector, and let λ ∈ [0 , 1] be a confidence parameter reflecting trust in the prediction. The algorithm allocates each item as follows:

Allocate a fraction 1 -λ of the item using the exponential assignment scheme with parameters α .

Allocate the remaining λ fraction using a worst-case robust algorithm (e.g., uniform allocation or greedy).

This strategy guarantees:

Consistency: The portion allocated by the learned parameters achieves an approximation ratio of (1 -λ )(1 -ϵ ) , assuming the predicted parameters yield a (1 -ϵ ) -approximation.

Robustness: The worst-case portion contributes at most λ · d , matching the lower bound of known worst-case algorithms.

Hence, the algorithm achieves a (1 -λ )(1 -ϵ ) -consistent and λ · d -robust guarantee.

E Learning-Augmented Online Allocation for the MaxMin Objective

In this section, we prove our main result for the MaxMin objective.

Theorem E.1. Given an instance of the online allocation problem with a MaxMin objective and any ϵ > 0 , there exists a set of learned parameters

and an online algorithm that uses the exponential assignment scheme with -α , such that the resulting fractional solution is a (1 -O ( ϵ )) -approximation.

To prove this theorem, we proceed similarly to the MinMax case: we begin with a preprocessing step that transforms the instance into a balanced form. However, the details of the preprocessing differ in the MaxMin setting.

E.1 Preprocessing for the MaxMin Objective

We begin by describing a transformation that modifies the input instance into a balanced form suitable for our learning-augmented algorithm. The transformation has two steps:

Step 1: Remove agents with large monopolist values. For each step t ∈ [ T ] , define

Define the monopolist value of agent i as a i = ∑ t ∈ [ T ] v ∗ it , and let a min = min i ∈ [ d ] a i . For each agent i such that

we allocate an ϵ/d fraction of the resource at each step to their preferred option k ∗ it .

Step 2: Zero out negligible values. For the remaining instance (with the reduced set of agents), define a new instance ˆ v by zeroing out small entries:

Let J i ( t ) ⊂ K ( t ) be the set of options that were modified for agent i at step t , and define

Step 3: Quantize values to form a balanced instance. Define the final preprocessed values ˜ v as:

This rounding ensures that all nonzero values are restricted to a logarithmic grid defined by the base 1 + ϵ , thereby yielding a balanced instance.

Lemma E.2. Let I ( v, K ) be an instance of the allocation problem with the MaxMin objective, and let ϵ > 0 . Then, the transformed instance I (˜ v, K ) , obtained via the three-step preprocessing, satisfies the following: any (1 -ϵ ) -approximate solution for I (˜ v, K ) , when combined with the allocations reserved in Step 1, yields a (1 -O ( ϵ )) -approximate solution for the original instance I ( v, K ) .

Proof. We begin by analyzing the impact of Step 1. The utility of any agent i removed during this step is guaranteed to be at least

so these agents are fully satisfied by the reserved allocation. Moreover, the total amount of resource allocated to these agents is at most an ϵ -fraction of the total, ensuring that the remaining instance is affected by at most a (1 -ϵ ) loss in the objective value.

Now consider the remaining agents in the modified instance I (ˆ v, K ) . In Step 2, we zero out negligible values to reduce the dynamic range. Let ˜ v it = max k ∈ J i ( t ) v itk , where J i ( t ) is the set of coordinates zeroed out for agent i at step t . For each i ∈ [ d ] , we bound the total value removed as:

where the last inequality uses the fact that MaxMin ( v, K ) ≥ a min /d , as each agent can receive a 1 /d share of their monopolist option.

Hence, the approximation loss from zeroing out small values is bounded by ϵ · MaxMin ( v, K ) , and the resulting instance I (ˆ v, K ) differs from the original by at most an O ( ϵ ) factor.

Finally, Step 3 introduces a geometric rounding of values to the nearest power of (1 + ϵ ) . As shown in the MinMax case, this quantization step results in an additional loss of at most a (1 -ϵ ) factor.

Combining the effects of the three steps, the overall degradation in objective value is at most a (1 -O ( ϵ )) factor. Thus, a (1 -ϵ ) -approximate solution to the preprocessed instance yields a (1 -O ( ϵ )) -approximate solution for the original instance.

E.2 Existence of Discretized Parameters for Balanced Instances for MaxMin

In light of Lemma E.2, to prove Theorem E.1. it suffices to consider balanced instances, as stated in the following lemma:

Lemma E.3. Given a balanced instance of the online allocation problem with a MaxMin objective and ϵ > 0 , there exists a set of parameters

such that the fractional solution defined by the exponential assignment scheme with parameters -α is a (1 -O ( ϵ )) -approximation to the optimal objective.

As in the MinMax objective, we define a slightly perturbed convex program (see Figure 2). We use L ∗ to denote the optimal MinMax value, and define the following convex program for ϵ > 0 :

Lemma E.4. Given an instance of the online allocation problem with the MaxMin objective and any ϵ ≥ 0 , there exists a vector α ( ϵ ) ∈ R d + such that the fractional solution defined by the exponential assignment scheme with parameters -α ( ϵ ) is (1 -O ( ϵ )) -approximately optimal.

Proof. Given such instance and for fixed ϵ consider the convex program of Figure 2. By our assumption, L ∗ is the optimal MaxMin objective therefore there exists a feasible solution for the convex program for any ϵ ≥ 0 . Accordingly, define the Lagrangian L ( x, α, β ) as

From the KKT conditions for the optimal solution to the convex program as a function of ϵ x ( ϵ ) , α ( ϵ ) , β ( ϵ ) , the solution that allocates according to x ( ϵ ) is a (1 + ϵ ) -approximation to the optimal objective L ∗ , and α ( ϵ ) i ≥ 0 for all i ∈ [ d ] . Furthermore,

For any two options k, r ∈ K ( t ) , we obtain:

Bounding the Learned Parameters. As in the MinMax objective, we bound the learned parameters using perturbation and sensitivity analysis techniques.

Lemma E.5. Let x ( ϵ ) , α ( ϵ ) , β ( ϵ ) be the optimal solution to the convex program in Figure 1, for some ϵ > 0 . Then, for all i ∈ [ d ] , it holds that α ( ϵ ) i ≤ d 2 ϵ · log ( d ϵ ) .

Proof. We define a perturbed convex program based on Figure 2, where u i corresponds to the constraint α i .

̸

For each i ∈ [ d ] , setting u i = -ϵ · L ∗ and u i ′ = 0 for i ′ = i ensures that constraint i in the perturbed problem matches the original constraint, thereby guaranteeing a feasible solution. By Lemma B.1,

where the second inequality follows from Claim 4.7, and the third inequality by ∑ t m t ( v ) ≤ d · L ∗ and log | K ( t ) | ≤ log ( d ϵ ) .

Finally, we now put all the pieces together to establish Lemma E.3:

Proof of Lemma E.3. Fix a balanced instance I ( v, K ) and ϵ > 0 . By Lemma E.5, there exists a parameter vector α ( ϵ ) such that α ( ϵ ) i ∈ [ 0 , d 2 ϵ · log ( d ϵ ) ] for all i ∈ [ d ] . Therefore, there exists a vector ˜ α ∈ NET ( d 2 ϵ · log ( d ϵ ) , d 3 ϵ 3 ) such that | ˜ α i -α ( ϵ ) i | ≤ ϵ 3 d 3 for all i ∈ [ d ] . By Lemma 4.8, the exponential assignment rule with ˜ α achieves a (1 + O ( ϵ )) -approximation.

F Generalization to Well-Behaved function

We now complete the proof of Theorem 4.1

Proof of Theorem 4.1. Fix an objective function f and an instance I ( v, K ) . Let ℓ f i denote the load in the i th dimension in an optimal solution for objective function f . Also, let x i,t denote the fraction at step t assigned to option k in this optimal solution.

Now, consider the instance ˜ I (˜ v, K ) , where ˜ v itk = v itk ℓ f i . By the monotonicity property of f , the optimal objective value for ˜ I is 1. Therefore, by Lemma 4.4, there exists ˜ α such that using an allocation, we get ℓ ∗ ( ˜ I, α ) ≥ 1 -ϵ for maximization and ℓ ∗ ( ˜ I, α ) ≤ 1 + ϵ for minimization.

By the definition of the allocation, x ∗ t,k is proportional to

Thus, if we define α such that α i = ˜ α i ℓ f i , then the corresponding allocation gives a (1 -ϵ ) -approximate solution for maximization and a (1 + ϵ ) -approximate solution for minimization.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers the online resource allocation problem, where the objective is to maximize the cumulative reward of an allocation strategy, with each allocation yielding a corresponding reward. The focus is on learning-augmented algorithms, where machine learning is used to predict future inputs, and these predictions are incorporated into the algorithmic design. The paper analyzes the problem within a unified framework by considering the existence of learnable parameters and strategies that can achieve a $1 + O(\\epsilon)$ approximation. It also examines the sensitivity and robustness of the algorithms under small perturbations to the input or predictions. The paper highlights two specific applications—online routing and Nash social welfare—and presents corollaries of the main theoretical results in these contexts.\n\nSTRENGTHS AND WEAKNESSES:\nStrength \n\n- The technical results are new, especially the construction of the balanced instance,\n\n- The paper considers a relatively general class of online resource allocation problem, and shows a unified framework of learning-augmented online algorithms, by characterizing the approximation ratio. \n\n- The paper also conducts the sensitivity analysis by considering the perturbation. \n\nWeakness\n\n- The paper could benefit from another round of polishing and it current form at is less ready for publication, since the current structure is a bit hard to follow with respect to the logic connections. For example, the introduction part focuses on heavily on the applications, but the applications mentioned in Section 5 do not match these applications exactly. One can get easily lost in the last two paragraphs before the Organization in Introduction. These two paragraphs do not help introducing this paper and its main theme. Also, merging the problem formulation with the contributions is somewhat confusing. Having a separate section of the problem formulation would make things clearer. \n\n- The proof of Lemma 4.4 is missing, or I may miss something. It would be helpful if the authors could point to the place. \n\n- MaxMin and MinMax are not defined with mathematic rigor. \n\n- I assume that the balanced instance is $\\tilde{I}$ instead of $I$ in the statements of the theorems. \n\n- The paper seems to over claim the contributions in the introduction, since essentially the results hold for balanced instances, instead of all the instances in the online resource allocation problems. It would be helpful to clarify these, to avoid ambiguity. \n\n- the concept of agents do not seem to appear in the general problem formulation. This necessitates further clarifications.\n\nQUESTIONS:\nPlease see the above weakness section.\n\nLIMITATIONS:\nNA", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a new, general framework for learning-augmented online allocation problems. Online allocation problems involve an algorithm making choices from a set of options at each step, which are represented as vectors, with the goal of minimizing or maximizing accumulated costs or rewards for a set of agents. This is represented by the value of an objective function on the sum of the vectors chosen at each step by the algorithm. Some problems that can be formalized in this way include machine scheduling, network routing (minimization), and fair allocation (maximization). \n\n\nThe proposed framework uses a single d-dimensional vector $\\alpha$ of learned weights to produce a nearly optimal solution using an exponential assignment rule. The main result achieves a (1+$\\epsilon$)-approximation for minimization or (1−$\\epsilon$)-approximation for maximization for any $\\epsilon>0$.  To that end, It leverages convex programming duality and a preprocessing step to convert the instance to a so-called “balanced” one. The authors also provide a modified algorithm to achieve a trade-off between consistency (performance with accurate predictions) and robustness (worst-case guarantees with unreliable predictions). The paper finally demonstrates the applicability of this framework to various problems in routing, scheduling, and fair allocation including online routing and online Nash Social Welfare maximization. \n\n\nMinor comments: \nI would suggest adding the relevant references to the first paragraph (e.g lines 24-25) of the introduction for clarity.\n\nSTRENGTHS AND WEAKNESSES:\nThis paper's main strength is its development of a general learning-augmented algorithmic framework for online allocation problems, which can be widely applicable. The results leverage machine learned predictions to achieve almost optimal allocations with no prediction error while being robust to prediction error as well. In terms of techniques, the novel application of convex programming duality could also influence future research in the field. On the other hand, I would like to see more justification for the “small items assumption”, which is required both for learnability of the predictions and the rounding of the fractional solutions.\n\nQUESTIONS:\n1. Can you be more precise on the consistency robustness trade-off? Specifically, how is the confidence parameter $\\lambda$ chosen and what is the dependence on the error of the learned parameter vector?  Is there an adaptive mechanism to set the confidence parameter dynamically based on the perceived accuracy of predictions, rather than a fixed value?\n\n2. Could the convex programming formulation be adapted to handle different types of constraints or other settings beyond allocation problems?\n\n3. For problems requiring integral solutions, how significant is the \"small items assumption\" in practice, and what are the implications if it doesn't hold perfectly?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper develops a unified learning-augmented framework for online allocation problems. The key contribution shows that a single learned parameter vector α enables an exponential assignment rule achieving (1±ε)-approximations for well-behaved objectives (routing, scheduling, fair allocation). Using convex programming duality and perturbation analysis, the authors prove existence of bounded, learnable parameters.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\nUsing convex duality to derive the exponential assignment rule from KKT conditions provides good theoretical grounding. The perturbation analysis showing dual variables can be bounded and discretized to poly(d,1/ε) is also well-executed. \n\nWeakness: \nThe convex program requires knowing the entire instance, making the \"learning\" aspect somewhat artificial. The paper solves fractional problems, with integral solutions in the appendix. The (1±ε) guarantees are relative to fractional problems, not integral ones. There seems to be a mismatch of what the paper claims to contribute. The three-step balanced instance transformation also requires better motivation. It is essential to bound the exponential terms but deviate from reality it claims to model.\n\nQUESTIONS:\nCan you 1.provide concrete examples where fractional solutions are directly applicable without rounding? The paper's main guarantees are for fractional solutions, yet most applications (routing, scheduling) require integral decisions.\n\n2.The three-step preprocessing appears quite aggressive, potentially changing the instance structure significantly. For example, provide an example where the algorithm fails without preprocessing? Also, it's helpful to quantify how much the preprocessing changes typical instances (what fraction of options are removed in Step 2)?\n\n3.Provide the explicit algorithm for learning α from samples?\n\nLIMITATIONS:\nThe authors partially address limitations throughout the paper but would benefit from a dedicated \"Limitations\" section consolidating:\n\nKey restrictions: Fractional solutions only, required preprocessing that alters problem structure, i.i.d. assumption, and limitation to monotone/homogeneous objectives\n\nPractical applicability gap: The theoretical unification is elegant but preprocessing and fractional nature limit direct deployment. The \"learning\" aspect is somewhat misleading—it's really about proving good fixed parameters exist rather than adaptive learning.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies a general framework to solve online allocation problems via convex programming in the algorithms-with-predictions  model.  In the online algorithms with predictions framework, the goal is to leverage machine-learned predictions to make improved decisions when dealing with an online problem.  The goal is to achieve a better competitive ratio compared to a worst-case algorithm without any predictions.  The paper uses this model to provide an algorithm for a general online optimization problem such as online routing in which the algorithm is provided a prediction vector \\alpha (with one parameter per edge) and goal is the minimize a global objective such as minimizing the max congestion one each edge.   Without predictions,  the best known competitive ratio for algorithms covered in this framework is logarithmic in the size of the input and the proposed learning-augmented algorithm achieves a (1+/- \\eps) competitive ratio under prefect predictions.  To achieve robustness with respect to prediction errors, they propose a modified algorithm that essential reverts to the worst-case algorithm if the error accumulates.\n\nTheir framework, algorithm and results follow closely that of Cohen and Panigrahi [17] and generalize it to handle inputs that are vectors and use it solve problems that are not covered by [17] such as routing and fair allocation.\n\n\nMinor typos:\nLine 46 typo:  is logarithmic in the \"number of size\" of the the network [2]\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n-  The paper provides a way to solve general online convex optimization problems in the algorithms-with-predictions model.\n-  Applying their general framework to online nash social welfare with predictions improves upon the prior best-known competitive ratio from logarithmic to (1+/eps).\n-  They provide the first learning-augmented algorithm for the online routing problem which is a fundamental optimization problem \n-  The paper shows that the predictions are PAC learnable, as well as provide new techniques such as perturbation analysis which are of independent interest to the community.\n\nWeakness:\n - The main weakness of the paper is that it does adequately address the overlap with prior work Cohen and Panigrahi [17] in the current write up.  The framework and algorithm borrow significantly from [17] and the new ideas and insights are not clearly addressed.   See questions below.\n\nQUESTIONS:\nAll my questions are about my only concern:  identifying the overlap with past work: citation [17] in the paper.  The following questions are trying to get more details about this.\n\n1. What is the difference between the PAC learnability proof (Theorem 2.2) from the learnability bounds in [17]?\n2. What is the main technical difference between the problems that cannot be handled by the framework in [17] vs the current paper?  The description that [17] cannot handle general vector needs explanation:  one way to interpret the divisible weights p_{i, j} in [17] is to view it as a d-dimensional vector (one per agent) that is received each time an item j appears.  \n3.  On line 177, the paper says in comparison to [17] \"we simplify the previous approaches as well by requiring only a single variable per dimension\":   it appears that [17] also requires one learned-parameter per dimension d so this needs further clarification. \n4.  Is the fair allocation problem with Nash Social welfare objective already covered by the framework in [17]?  What does \"matching previous results in [17]\" mean on line 154 in the paper.\n\nI am willing to raise my score to accept if this overlap is sufficiently addressed by the authors clarifying the new insights/contributions provided by this paper over [17].\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-15"}