AS-BOX: Additional Sampling Method for Weighted Sum Problems with Box Constraints

TL;DR

The paper introduces AS-BOX, a new stochastic optimization method for weighted sum problems with box constraints, combining adaptive sampling, projected gradients, and line search to improve efficiency and convergence.

Contribution

It presents a novel adaptive sampling algorithm for constrained stochastic optimization that guarantees convergence and demonstrates practical effectiveness.

Findings

AS-BOX outperforms existing algorithms in numerical tests.

The method adapts sample size dynamically based on progress.

Theoretical convergence and complexity bounds are established.

Abstract

A class of optimization problems characterized by a weighted finite-sum objective function subject to box constraints is considered. We propose a novel stochastic optimization method, named AS-BOX (\text{A}ddi\-ti\-onal \text{S}ampling for \text{BOX} constraints), that combines projected gradient directions with adaptive variable sample size strategies and nonmonotone line search. The method dynamically adjusts the batch size based on progress with respect to the additional sampling function and on structural consistency of the projected direction, enabling practical adaptivity of AS-BOX, while ensuring theoretical support. We establish almost sure convergence under standard assumptions and provide complexity bounds. Numerical experiments demonstrate the efficiency and competitiveness of the proposed method compared to state-of-the-art algorithms.

Equations

$$\underset{x\in S}{\min }f(x):=\sum _{i=1}^N{w}_i{f}_i(x),S=\{x\in {\mathbb{R}}^n|l_i\le x_i\le u_i,i=1,...,n\},$$

$$l_i\in \mathbb{R}\cup \{-\infty \},u_i\in \mathbb{R}\cup \{\infty \},i=1,2,...,n,$$

$$\sum _{i=1}^N{w}_i=1,w_i\ge 0,i=1,..,N.$$

$$d(x):={\pi }_S(x-\nabla f(x))-x$$

$$f_{{\mathcal{N}}_k}(x):=\frac{1}{N_k}\sum _{i\in {\mathcal{N}}_k}{f}_i(x),$$

$$P(i_j^k=s)=w_s,s=1,...,N,j=1,...,N_k.$$

$$\mathbb{E}(f_{{\mathcal{N}}_k}(x)|x)=\mathbb{E}(\frac{1}{N_k}\sum _{j=1}^{N_k}{f}_{i_j^k}(x)|x)=\frac{1}{N_k}\sum _{j=1}^{N_k}\mathbb{E}(f_{i_j^k}(x)|x)=\frac{1}{N_k}\sum _{j=1}^{N_k}f(x)=f(x),$$

$$f_{{\mathcal{N}}_k}(x_k+t_k{p}_k)\le f_{{\mathcal{N}}_k}(x_k)+c_1{t}_k(\nabla f_{{\mathcal{N}}_k}(x_k){)}^T{p}_k+{\epsilon }_k,$$

$$\sum _{k=0}^{\infty }{\epsilon }_k\le \bar{\epsilon }<\infty .$$

$$f_{{\mathcal{D}}_k}(x):=\frac{1}{D_k}\sum _{i\in {\mathcal{D}}_k}{f}_i(x),$$

$$P(l_j^k=s)=w_s,s=1,...,N,j=1,...,D_k.$$

$$\underset{x\ge 0}{\min }f(x),$$

$$p_k={\pi }_S(x_k-\nabla f_{{\mathcal{N}}_k}(x_k))-x_k,$$

$$[p_k{]}_i=-[x_k{]}_i\text{mbox}if[x_k{]}_i<[\nabla f_{{\mathcal{N}}_k}(x_k){]}_i$$

$$[p_k{]}_i=-[\nabla f_{{\mathcal{N}}_k}(x_k){]}_i\text{mbox}if[x_k{]}_i\ge [\nabla f_{{\mathcal{N}}_k}(x_k){]}_i.$$

$$[I_{{\mathcal{N}}_k}{]}_i=\{\begin{array}{ll}1,& [x_k{]}_i<[\nabla f_{{\mathcal{N}}_k}(x_k){]}_i\\ 0,& [x_k{]}_i\ge [\nabla f_{{\mathcal{N}}_k}(x_k){]}_i.\end{array}$$

$$r_{{\mathcal{D}}_k}:=\Vert I_{{\mathcal{N}}_k}-I_{{\mathcal{D}}_k}\Vert .$$

$$f_{{\mathcal{N}}_k}(x_k+t_k{p}_k)\le f_{{\mathcal{N}}_k}(x_k)+c_1{t}_k(\nabla f_{{\mathcal{N}}_k}(x_k){)}^T{p}_k+{\epsilon }_k.$$

$$s_k={\pi }_S(x_k-\nabla f_{{\mathcal{D}}_k}(x_k))-x_k$$

$$r_{{\mathcal{D}}_k}=0\text{mbox}and{f}_{{\mathcal{D}}_k}({\bar{x}}_k)\le f_{{\mathcal{D}}_k}(x_k)-c\Vert s_k{\Vert }^2+C{\epsilon }_k,$$

$$f_{{\mathcal{D}}_k}({\bar{x}}_k)\le f_{{\mathcal{D}}_k}(x_k)-c\Vert s_k{\Vert }^2+C{\epsilon }_k$$

$${\mathcal{D}}_k^{+}=\{{\mathcal{D}}_k\subset \mathcal{N}|r_{{\mathcal{D}}_k}=0,f_{{\mathcal{D}}_k}({\bar{x}}_k)\le f_{{\mathcal{D}}_k}(x_k)-c\Vert s_k{\Vert }^2+C{\epsilon }_k\}.$$

$${\mathcal{D}}_k^{-}=\{{\mathcal{D}}_k\subset \mathcal{N}|r_{{\mathcal{D}}_k}>0\text{mbox}or{f}_{{\mathcal{D}}_k}({\bar{x}}_k)>f_{{\mathcal{D}}_k}(x_k)-c\Vert s_k{\Vert }^2+C{\epsilon }_k\}.$$

$$p_k^T\nabla f_{{\mathcal{N}}_k}(x_k)\le -\Vert p_k{\Vert }^2.$$

$$t_k\ge t_{min}:=\min \left\{1,\frac{2\beta (1-c_1)}{L}\right\}.$$

$$f(x_{k+1})\le f(x_k)-\bar{c}\Vert d(x_k){\Vert }^2+\bar{C}{\epsilon }_k,$$

$$f(x_{k+1})\le f(x_k)+c_1{t}_k(\nabla f(x_k){)}^{T}d(x_k)+{\epsilon }_k\le f(x_k)-c_1{t}_k\Vert d(x_k){\Vert }^2+{\epsilon }_k,$$

$$f(x_{k+1})\le f(x_k)-c_1{t}_{min}\Vert d(x_k){\Vert }^2+{\epsilon }_k.$$

$$f_j({\bar{x}}_k)\le f_j(x_k)-c\Vert s_k^j{\Vert }^2+C{\epsilon }_k,$$

$$f(x_{k+1})\le f(x_k)-c\sum _{j=1}^N{w}_j\Vert s_k^j{\Vert }^2+C{\epsilon }_k.$$

Full text

Key words: Projected Gradient Methods, Sample Average Approximation, Adaptive Variable Sample Size Strategies, Non-monotone Line Search, Additional Sampling, Almost Sure Convergence.

MSC Classification: 90C15,90C26, 90C30, 65K05

1 Introduction

We consider a box-constrained optimization problem with the objective function in the form of a weighted finite sum, i.e.,

[TABLE]

where $S$ represents the feasible set defined by

[TABLE]

while $f_{i}:\mathbb{R}^{n}\to\mathbb{R},i=1,...,N$ are continuously-differentiable functions and $w_{1},...,w_{N}$ represent the weights such that

[TABLE]

This problem captures a wide class of practical problems. It generalizes classical unconstrained finite-sum formulations, where $w_{i}=1/N$ and $S=\mathbb{R}^{n}$ (e.g., empirical risk minimization with uniform weights) by allowing arbitrary positive weights and bound constraints. The formulation with unequal weights is motivated by the so-called local regression models (see e.g.,[13]) where the weights represent the importance (distance) of different data points in the training set with respect to a new data point. However if uniform sampling is more convenient from practical (implementation) point of view, the weights can be integrated to form the local cost functions $\tilde{f}_{i}(x):=w_{i}f_{i}(x)$ and equivalent problem $\min_{x\in S}\frac{1}{N}\sum_{i=1}^{N}\tilde{f}_{i}(x)$ can be solved. Although applying the proposed algorithm would yield different iterations due to randomness and different subsampling distributions (see (2.3) ahead), theoretical guarantees remain the same as for the weighted sum case that we consider. Further, special cases of the considered problem include nonnegative constraints $x\geq 0$, i.e., $x_{i}\geq 0,i=1,...,n$. Such box-constrained problems naturally arise in machine learning, signal processing, portfolio optimization, and computational statistics [29].

In large-scale optimization problems of form (1.1), evaluating the full gradient $\nabla f(x)=\sum_{i=1}^{N}w_{i}\nabla f_{i}(x)$ can be computationally expensive, especially when $N$ is large. Stochastic methods based on subsampling are widely used to reduce this cost. One possibility to reduce the cost while maintaining reasonably good approximations of the gradients is to use adaptive sampling strategies. These strategies dynamically refine gradient approximations during the optimization process, mainly by progressive increase of the sample size based on variance estimates or structural indicators such as descent quality or direction stability. This way one is able to maintain a balance between computational efficiency and convergence reliability. Adaptive sampling methods have been extensively studied in recent years [2, 3, 4, 6, 15, 16, 19, 23].

One effective subclass of adaptive sampling is the so-called additional sampling, which typically increases the sample size when a prescribed criterion fails [9, 17, 21, 22, 23, 24]. The criterion of progress is defined by an additional sampling, i.e., a new independent subsample (of modest size) is generated after each iteration and is used to accept/reject the iteration. As the additional sample is mainly of modest size, this approach avoids excessive computational cost while still ensures convergence in a stochastic sense. For example, in the IPAS method [22], the additional sampling is combined with projected gradient steps for problems with linear equality constraints. Sampling growth is governed by a descent-based condition that assess whether the current sample is sufficient to ensure meaningful progress. This mechanism allows the method to operate efficiently in early iterations with small batches and to increase precision only when needed. Furthermore, the ASPEN method [21] extends this idea to nonlinear equality-constrained problems by incorporating a quadratic penalty term. In this method, additional sampling is applied adaptively based on indicators such as gradient norm and descent quality. The method attempts to make progress using the current sample and increases the sample size only when necessary, making it particularly effective near critical points where variance in gradient estimates becomes more pronounced.

Another key component of the method proposed here is nonmonotone line search (NMLS). Classical Armijo-type condition requires a sufficient decrease in each iteration, which can be restrictive and lead to overly conservative steps in noisy settings. Nonmonotone line searches, on the other hand, allow temporary increases in the objective function, enabling better exploration of the landscape and improving practical performance. A number of NMLS is present in the literature, [11, 30, 25] and they are successfully applied in numerous deterministic and stochastic frameworks (e.g., [5, 14, 17, 18, 20, 24]). In stochastic settings, the effect of noise and variance by relaxing strict descent conditions are particularly important. The NMLS method we rely on is originally defined in [25].

Interior-point methods (IPMs) represent another popular class of algorithms for constrained optimization, known for their strong theoretical properties and practical efficiency. Numerous works have extended IPMs to accommodate large-scale and structured problems, including both deterministic and stochastic settings. For instance, classical interior-point frameworks tailored for convex programming and barrier methods are well-established (e.g., [26, 28]), while more recent advances incorporate stochastic elements or specific constraint structures (e.g., [12, 27]). In the stochastic optimization literature, interior-point methods have also been adapted to settings where exact gradients are either expensive or impossible to compute. These adaptations often involve inexact or sampled gradient approximations and the use of approximate barrier subproblems to preserve feasibility and convergence properties under uncertainty. A recent contribution in this direction is [8], where a stochastic gradient-based interior point method (SIPM) to solve box-constrained optimization problems is proposed. The SIPM algorithm extends the classical interior-point framework to the stochastic setting by augmenting the objective with a logarithmic barrier that enforces box constraints and by employing a prescribed decreasing sequence of barrier parameters rather than adaptive updates. Unlike standard interior-point methods, SIPM maintains iterates within progressively shrinking inner neighborhoods of the feasible box and avoids fraction-to-the-boundary rules or line searches, which are challenging to implement in stochastic regimes.

As the baseline algorithm, we use the Projected Stochastic Gradient Method (PSGM), which originates from the classical framework of projected gradient methods and is here adapted into a stochastic version following the implementation in [8]. PSGM is a projection-based method that iteratively computes stochastic gradient steps on the original objective and projects them back onto the feasible region, i.e., updates are of the form $x_{k+1}=\pi_{[l,u]}(x_{k}-\alpha_{k}g_{k})$, where $\pi_{[l,u]}$ denotes the projection onto the box constraints. While SIPM leverages barrier smoothing to handle boundaries implicitly, PSGM enforces feasibility explicitly through projection. These two methods are used for numerical comparison in this paper.

The method we propose here, AS-BOX is a novel stochastic optimization algorithm for weighted finite-sum problems with box constraints. Our key contributions include the following. A new stochastic method for solving the box constrained problems is proposed and analysed, both theoretically and numerically. The method relays on the well-established non-monotone line search along the projected subsampled gradient direction. The key innovation is additional sampling of modest size performed in each iteration, which yields two advantages. First, it resolved the theoretical issue of mutual dependence of the direction and stepsize and hence allowed us to prove a.s. convergence of the method. Second, the additional sampling results in a natural subsampling schedule, that is problem dependent (not predefined). The worst-case complexity is also analyzed, providing an expected number of iterations to reach the vicinity of the stationary points of the considered problem. Numerical results are presented on real-world data with logistic regression and Neural Network problems as test cases.

Paper organization. The paper is organized as follows. Section 2 provides the necessary preliminaries. In Section 3, we present the AS-NC method designed for problems with non-negativity constraints, including the algorithmic framework and convergence analysis. Although non-negativity constraints are a special case of the general box constraints we start with this case for clarity of exposition. Then, in Section 4 we generalize to AS-BOX method, our main contribution, for solving box-constrained problems. Numerical experiments are reported in Section 5, while Section 6 concludes the paper.

Notation. Throughout the paper, we use the following notation: $\mathbb{R}_{+}$ denotes the set of non-negative real numbers. The symbol $\|\cdot\|$ represents the standard Euclidean norm. The expectation operator is denoted by $\mathbb{E}(\cdot)$, and $\mathbb{E}(\cdot\mid\mathcal{F})$ stands for the conditional expectation given a $\sigma$-algebra $\mathcal{F}$. We use “a.s.” to abbreviate “almost sure”. For a finite set $A,$ $|A|$ denotes its cardinality.

2 Preliminaries

Let us denote by $\pi_{S}(y)$ the orthogonal projection of a point $y$ on the set $S$. One can show that the projected gradient direction of the form

[TABLE]

is a descent direction for function $f$ at point $x\in S$ unless $x$ is a stationary point of problem (1.1). More precisely, the following result is known.

Theorem 2.1**.**

[5]** Assume that $f\in C^{1}(S_{k})$ and $x\in S$. Then the projected gradient direction (2.1) satisfies:

• a)

$d^{T}(x)\nabla f(x)\leq-\|d(x)\|^{2}.$**

• b)

$d(x)=0$* if and only if $x$ is a stationary point of problem (1.1).*

We will be dealing with approximate evaluations of the objective function and its gradients. More precisely, we use the following sample-based estimate of the objective function at iteration $k$ in general [22]

[TABLE]

where $N_{k}:=|\mathcal{N}_{k}|$, $\mathcal{N}_{k}=\{i^{k}_{1},...,i^{k}_{N_{k}}\}$, and each $i^{k}_{j}\in\mathcal{N}_{k}$ takes the value $s\in\mathcal{N}:=\{1,...,N\}$ with probability $w_{s}$, i.e.,

[TABLE]

This way we have an unbiased estimate of $f$, i.e.,

[TABLE]

where $\mathbb{E}(\cdot|x)$ denotes conditional probability given the point $x$. However, this is not crucial for the analysis, and the convergence results hold for an arbitrary sampling of $\mathcal{N}_{k}$ as well. Moreover, since the method that will be proposed in the sequel may reach the full sample size, we will assume that when $N_{k}=N$ we simply take the whole sample, i.e., $\mathcal{N}_{k}=\mathcal{N}$. The approximate gradient will be taken as the gradient of the approximate function $\nabla f_{\mathcal{N}_{k}}$.

Since we work with approximate functions in general, non-monotone Armijo-type line search will be employed [25] to determine the step size $t_{k}$ given a direction $p_{k}=\pi_{S}(x_{k}-\nabla f_{\mathcal{N}_{k}}(x_{k}))-x_{k}$

[TABLE]

where $\varepsilon_{k}>0,k\in\mathbb{N}$ represents a predetermined sequence which satisfies the following condition

[TABLE]

Notice that the search direction $p_{k}$ is a descent direction for the function $f_{\mathcal{N}_{k}}$ at point $x_{k}$. Moreover, $x_{k}+p_{k}$ is feasible provided that $x_{k}$ is feasible as well, and due to the convexity of $S$, backtracking line search will ensure that $x_{k}+t_{k}p_{k}$ remains in the feasible set. Thus, starting from $x_{0}\in S$, the proposed algorithm will ensure the feasibility of all the iterates.

We apply an additional sampling technique to guide the sample size increase. Additional sampling is used to overcome bias that comes from the dependency of the candidate iterate $\bar{x}_{k}=x_{k}+t_{k}p_{k}$ on the sample $\mathcal{N}_{k}$. Moreover, it can be viewed as a check on the similarity of the local cost functions - if they are heterogeneous, then it is probably beneficial to increase the sample size since the mini-batch estimate is not good enough representative of the objective function. For more details one can see [22] and the references therein. We form an additional sampling function similarly to $f_{\mathcal{N}_{k}}$, but with a much smaller sample in general. Namely, we have

[TABLE]

where $D_{k}:=|\mathcal{D}_{k}|$, $\mathcal{D}_{k}=\{l^{k}_{1},...,l^{k}_{D_{k}}\}$, and each $l^{k}_{j}\in\mathcal{D}_{k}$ takes the value $s\in\mathcal{N}:=\{1,...,N\}$ with probability $w_{s}$, i.e.,

[TABLE]

Although $D_{k}$ may be arbitrary, it is assumed that it is significantly smaller than $N_{k}$, and the common choice is $D_{k}=1$ for all $k$. The additional sampling rule within this paper is adapted to box constraints. The additional sampling rule is also used to guide the acceptance of the candidate point. We will elaborate this in more detail in the next section. Finally, we emphasize that the additional sampling rule is constructed to determine if the sample size increase is needed, but allows an arbitrary increase.

For simplicity, we start our analysis by observing non-negativity constraints, and later on we extend it to general box constraints by introducing some simple modifications within the algorithm and the convergence analysis.

3 Nonnegativity Constraints: AS-NC method

Within this section we consider an important special case of problem (1.1) given by

[TABLE]

where the function $f$ is as in problem (1.1) and inequalities $x\geq 0$ are component-wise. Compared to the general box-constrained problem, this setting simplifies the structure of the feasible set, and we have $[\pi_{S}(y)]_{i}=\max\left\{y_{i},\,0\right\},i=1,..,n.$ Since our direction will be of the form

[TABLE]

we will distinguish two cases for each component $i\in\{1,...,n\}$:

[TABLE]

and

[TABLE]

Let us denote by $I_{\mathcal{N}_{k}}$ an indicator vector of the event $x_{k}<\nabla f_{\mathcal{N}_{k}}(x_{k}),$ with inequality defined by components. More precisely, for $i=1,\ldots,n$ we have

[TABLE]

Analogously, we define an indicator vector $I_{\mathcal{D}_{k}}$ of the event $x_{k}<\nabla f_{\mathcal{D}_{k}}(x_{k})$ and

[TABLE]

Given that $\mathcal{N}_{k},\mathcal{D}_{k}$ and $x_{k}$ are random, the values $r_{\mathcal{D}_{k}}$ are also random and will be used to check the similarity of local cost functions in terms of the structure of the search direction $p_{k}$. Namely, notice that if $r_{\mathcal{D}_{k}}=0$ then the structure of zero entries in $\pi_{S}(x_{k}-\nabla f_{\mathcal{N}_{k}}(x_{k}))$ is the same as for $\pi_{S}(x_{k}-\nabla f_{\mathcal{D}_{k}}(x_{k}))$.

3.1 The Algorithm

We state the algorithm for solving (3.1) as follows.

Algorithm 1: AS-NC (Additional Sampling - Nonnegativity Constraints)

• S0

Initialization. Input: $x_{0}\geq 0,N_{0}\in\mathbb{N},\beta,c,c_{1}\in(0,1),C>0$, $\{\varepsilon_{k}\}$ satisfying (2.4). Set $k:=0$.

• S1

Subsampling. If $N_{k}<N$, choose $\mathcal{N}_{k}$ such that (2.3) holds. Else, set $f_{\mathcal{N}_{k}}=f$.

• S2

Search direction. Compute $p_{k}=\pi_{S}(x_{k}-\nabla f_{\mathcal{N}_{k}}(x_{k}))-x_{k}.$

• S3

Step size. Find the smallest $j\in\mathbb{N}_{0}$ such that $t_{k}=\beta^{j}$ satisfies

[TABLE]

Set $\bar{x}_{k}=x_{k}+t_{k}p_{k}$.

• S4

*Additional sampling.

*If $N_{k}=N$, set $x_{k+1}=\bar{x}_{k}$, $k=k+1$ and go to step S1.

Else choose $\mathcal{D}_{k}$ via (2.5) and compute

[TABLE]

and $r_{\mathcal{D}_{k}}=\|I_{\mathcal{N}_{k}}-I_{\mathcal{D}_{k}}\|.$

• S5

*Sample size update.

*If

[TABLE]

$N_{k+1}=N_{k}$.

Else choose $N_{k+1}\in\{N_{k}+1,...,N\}$.

• S6

Iterate update.

If

[TABLE]

holds set $x_{k+1}=\bar{x}_{k}$. Else $x_{k+1}=x_{k}$.

• S7

Counter update. Set $k=k+1$ and go to Step S1.

Notice that the algorithm can yield two types of scenarios: the Mini-batch (MB) scenario, where $N_{k}<N$ for all $k\in\mathbb{N}$, and the Full sample (FS) scenario, where the full sample is eventually reached. Moreover, we say that AS-NC is in the MB phase at iteration $k$ if $N_{k}<N$. Otherwise, the full sample size is reached, i.e., if $N_{k}=N$, for some $k$ then all further iterations have the same property and we say that we are in the FS phase. In that case, the algorithm behaves as a deterministic projected gradient method. However, the sequence of iterates is still random due to sampling in the initial (MB) phase of the algorithm.

In the MB phase, we have sampling at two steps of the algorithm: S1 and S4. Although we propose unbiased estimators (2.3) in step S1, the sampling used for $\mathcal{N}_{k}$ can in fact be arbitrary. This allows many strategies which can be very important from a practical point of view. Moreover, the choice of $\mathcal{D}_{k}$ may be modified as well, but it has to meet certain requirements - it needs to be chosen independently of $N_{k}$ and it must allow positive probabilities for choosing each of the local cost function. Unbiased estimator is not essential for the convergence analysis.

Notice that the sequence of iterates is feasible due to the construction of the algorithm. The search direction $p_{k}$ is a descent direction for $f_{\mathcal{N}_{k}}$ and feasible with respect to constraints, while backtracking line search retains feasibility. The same type of direction is calculated in step S4, but with respect to $f_{\mathcal{D}_{k}}$, which is independent of $f_{\mathcal{N}_{k}}$. However, the Armijo-like condition is checked without performing any line search - it simply checks if the candidate point $\bar{x}_{k}$ is good enough for $f_{\mathcal{D}_{k}}$, which is an independent estimate of the objective function. Notice that in this check the constants $c$ and $C$ can be arbitrary small and large, respectively. If the value of $f_{\mathcal{D}_{k}}$ is good enough the candidate point is accepted at step S6. Otherwise, the step is rejected and the sample size $N_{k}$ is increased within step S5. The increase is arbitrary, as mentioned in Preliminaries. However, the sample size $N_{k}$ can be increased also due to the different structure of the projection considering two approximate gradients $\nabla f_{\mathcal{N}_{k}}$ and $\nabla f_{\mathcal{D}_{k}}$, which discloses through $r_{\mathcal{D}_{k}}>0.$ Overall, the condition (3.7) serves as the check of similarity of local cost functions and governs the sample size. Notice that calculating $r_{\mathcal{D}_{k}}$ does not yield additional costs since the structure observed in (3.3) is needed for forming the projections as well.

3.2 Convergence analysis

Within this section, we prove almost sure convergence of the proposed algorithm and analyze the complexity. We start the analysis by stating the following standard assumption.

Assumption A 1**.**

Each function $f_{i},i=1,...,N$ is continuously differentiable with $L$-Lispchitz continuous gradient and bounded from below by a constant $f_{low}$.

As usual for additional sampling framework analysis, we proceed by dividing the set of all possible outcomes at iteration $k$ into two complementary subsets. Namely, let us denote by $\mathcal{D}_{k}^{+}$ the subset of all possible outcomes of $\mathcal{D}_{k}$ at iteration $k$ for which the condition (3.7) is satisfied, i.e.,

[TABLE]

We denote the complementary subset of outcomes at iteration $k$ by

[TABLE]

We begin our analysis with the following lemma, which basically describes the situation in which the full sample is not reached, based on choices of $\mathcal{D}_{k}$ that violate (3.7). This lemma is conceptually aligned with Lemma 4.3 in [22], and the proof is the same as in [22], so it is omitted here.

Lemma 3.1**.**

[[22, Lemma 4.3]] Suppose that Assumption A1 holds. If $N_{k}<N$ for all $k\in\mathbb{N}$, then a.s. there exists $k_{1}\in\mathbb{N}$ such that $\mathcal{D}_{k}^{-}=\emptyset$ for all $k\geq k_{1}$.

The following lemma states the well-known result for backtracking line search under the stated assumptions since, according to Theorem 2.1 a), there holds

[TABLE]

Lemma 3.2**.**

Suppose that Assumption A1 holds. Then the step size $t_{k}$ obtained from step S3 satisfies

[TABLE]

Next, we prove the key result for the convergence analysis of AS-NC. Notice that (3.8) is related to the original objective function and $d$ defined as in (2.1), $d(x):=\pi_{S}(x-\nabla f(x))-x$, regardless of the scenario (MB or FS).

Theorem 3.3**.**

Suppose that Assumption A1 holds. Then a.s. there exists a finite, random iteration $\tilde{k}$ such that for all $k\geq\tilde{k}$ there holds

[TABLE]

where $\bar{c}=\min\{c,c_{1},2c_{1}(1-c_{1})\beta/L\}$ and $\bar{C}=\max\{1,C\}$.

Proof.

Let us consider the FS scenario first. Then there exists a finite $\tilde{k}_{1}$ such that for all $k\geq\tilde{k}_{1}$ we operate with the true objective function $f$ and according to (3.5) there holds

[TABLE]

where the last inequality comes from Theorem 2.1 a). Moreover, Lemma 3.2 implies that $t_{k}\geq t_{min}$ and thus we obtain

[TABLE]

Now, let us observe the MB scenario. According to Lemma 3.1 a.s. there exists some finite, random iteration $k_{1}$ such that $\mathcal{D}_{k}^{-}=\emptyset$ for all $k\geq k_{1}$. This means that the condition (3.7) holds for all the local cost functions222Otherwise the set $\mathcal{D}_{k}^{-}$ would not be empty since one could form at least one possible $\mathcal{D}_{k}$ that violates (3.7), e.g., $\mathcal{D}_{k}=\{j_{v},...,j_{v}\}$ where $j_{v}$ represents a local cost function that violates (3.7). . Therefore, for all $k\geq k_{1}$ and all $j\in\mathcal{N}$ there holds

[TABLE]

where $s^{j}_{k}:=\pi_{S}(x_{k}-\nabla f_{j}(x_{k}))-x_{k}.$ Using the fact that in the considered scenario the candidate point would be accepted, i.e., $x_{k+1}=\bar{x}_{k}$, multiplying both sides with $w_{j}$ and summing up, we obtain

[TABLE]

Let us consider the first condition of (3.7). Denote by $\mathcal{A}_{\mathcal{N}_{k}}$ the set of indices (components) $i\in\{1,...,n\}$ such that $[I_{\mathcal{N}_{k}}]_{i}=1$, i.e.,

[TABLE]

Furthermore, using the similar arguments as for the second condition of (3.7), we conclude that $\mathcal{D}_{k}^{-}=\emptyset$ for all $k\geq k_{1}$ implies that $r_{\mathcal{D}_{k}}=0$ for all the singleton choices $\mathcal{D}_{k}=\{1\},...,\mathcal{D}_{k}=\{N\}$ for all $k\geq k_{1}$. Having in mind the definition of $r_{\mathcal{D}_{k}}$ we conclude that for all $k\geq k_{1}$

[TABLE]

where $\mathcal{A}^{j}_{k}:=\{i\in\{1,...,n\}\;|\;[x_{k}]_{i}<[\nabla f_{j}(x_{k})]_{i}\},j=1,...,N.$ This further implies that all $k\geq k_{1}$, for all $i\in\mathcal{A}_{\mathcal{N}_{k}}$, for all $j\in\mathcal{N}$ there holds $[x_{k}]_{i}<[\nabla f_{j}(x_{k})]_{i}$ and thus

[TABLE]

Similarly, we obtain $[x_{k}]_{i}\geq[\nabla f(x_{k})]_{i}$ for all $i\notin\mathcal{A}_{\mathcal{N}_{k}}$ and due to (2.1) we conclude that the following holds for all $k\geq k_{1}$

[TABLE]

Now, let us estimate the norm of $d(x_{k})$ for $k\geq k_{1}$ as follows

[TABLE]

According to (3.12) we have for all $k\geq k_{1}$ and all $j\in\mathcal{N}$

[TABLE]

and thus for all $i\in\mathcal{A}_{\mathcal{N}_{k}}$

[TABLE]

which further implies

[TABLE]

Similarly, we conclude that for all $k\geq k_{1}$ and all $j\in\mathcal{N}$ there holds

[TABLE]

and we conclude that for all $i\notin\mathcal{A}_{\mathcal{N}_{k}}$

[TABLE]

which further implies

[TABLE]

Combining (3.2), (3.16) and (3.18) we obtain

[TABLE]

Combining this with (3.10) we obtain for all $k\geq k_{1}$

[TABLE]

Taking into account both scenarios (FS and MB), i.e., (3.20) and (3.9), we conclude the proof with $\tilde{k}=k_{1}$ in FS and $\tilde{k}=\tilde{k}_{1}$ in MB case. ∎

In order to obtain a.s. convergence, we impose the following assumption [24].

Assumption A 2**.**

There exists a constant $C_{b}$ such that $\mathbb{E}(|f(x_{\tilde{k}})|)\leq C_{b}$, where $\tilde{k}$ is specified in Theorem 3.3.

The above assumption is clearly fulfilled if the sequence $\{f(x_{k})\}$ is bounded. Moreover, in the case of bounded iterates (e.g., compact feasible set as a special case in Section 4) the assumption holds for many objective functions. But it also holds in more general situations as it allows the case when $f_{\mathcal{N}_{\tilde{k}}}(x_{\tilde{k}})$ is unbounded in general (for some sample paths), but the expectation over all possible sample paths is still bounded. Let us denote by $\mathbb{E}_{FS}(\cdot):=\mathbb{E}(\cdot\mid FS)$ the conditional expectation concerning all the sample paths falling into the FS scenario. Analogously, we define $\mathbb{E}_{MB}(\cdot):=\mathbb{E}(\cdot\mid MB)$. It can be shown (see [24] e.g.) that Assumption A2 implies

[TABLE]

for some constants $C^{FS}_{b},C^{MB}_{b}$ where $\tilde{k}_{1}$ and $k_{1}$ are as in the proof of Theorem 3.3.

Next, we state the main convergence result for AS-NC.

Theorem 3.4**.**

Suppose that Assumptions A1 and A2 hold. Then a.s. every accumulation point of sequence $\{x_{k}\}_{k\in\mathbb{N}}$ generated by AS-NC is a stationary point of the problem (3.1).

Proof.

According to (3.8) we have that a.s.

[TABLE]

for any $l\in\mathbb{N}$. Applying the expectation and using Assumption A2 together with sumability of $\varepsilon_{k}$ given in (2.4), by letting $l\to\infty$ we obtain

[TABLE]

Now, the extended Markov’s inequality and the Borel-Cantelli lemma (see e.g. [24] for details), we conclude that

[TABLE]

Let $x^{*}$ be an arbitrary accumulation point of the sequence $\{x_{k}\}$, and let $K_{0}\subset\mathbb{N}$ be a subsequence such that

[TABLE]

Due to continuity of the gradient and the projection operator, from (3.22) we conclude that a.s.

[TABLE]

and by Theorem 2.1 b) and the feasibility of the iterates, we conclude that $x^{*}$ is a.s. a stationary point of problem (3.1), which completes the proof. ∎

Next, we analyze the complexity of the proposed method. The analysis combines techniques of [3], [10], and [24]. We impose the assumption used in [24]. It states that the local cost functions are not homogeneous in the following sense.

Assumption A 3**.**

For each $k$ there exists at least one function $f_{i}$ such that the condition (3.7)

[TABLE]

is violated.

This assumption is likely to be satisfied in the case of data fitting if the data is heterogeneous or if the local cost functions $f_{i}$ are of a different type. In fact, considering, for instance, linear least squares problems, it can easily happen that a descent direction of one function is an ascent direction of another one.

Theorem 3.5**.**

Suppose that Assumptions A1, A2 and A3 hold. Then the expected number of iterations to reach $\|d(x_{k})\|<\nu$ is upper bounded by

[TABLE]

where $\bar{c}$ is as in Theorem 3.3, $C_{b}^{FS}$ as in (3.21) and $q=\min\{w_{1},...,w_{N}\}^{N-1}$.

Proof.

Assumption A3 ensures that for every iteration $k$, there exists at least one function $f_{i}$ that violates the condition (3.7). Therefore, according to the distribution of $\mathcal{D}_{k}$ (2.5), we conclude that

[TABLE]

Further, let us denote by $S_{k}$ a random variable that counts the number of increments of the sample size until iteration $k$. Notice that $S_{k}$ can be represented as $S_{k}=I_{1}+I_{2}+\dots+I_{k},$ where $I_{k}$ is an indicator variable, i.e., $I_{k}=1$ if $N_{k}>N_{k-1}$ and $I_{k}=0$ otherwise. Furthermore, according to step S5 of AS-NC algorithm, the increase of the sample size happens if and only if $\mathcal{D}_{k}\in\mathcal{D}^{-}_{k}$ and thus

[TABLE]

which further implies

[TABLE]

Let $\tilde{N}$ represent the number of increments of the sample size needed to reach the full sample.333For instance, if we set $N_{k+1}=N_{k}+1$ at the end of step S5 of AS-NC, then $\tilde{N}=N-N_{0}$. See [24] and the text after Assumption 4 therein for further discussion on this topic. Requiring $\mathbb{E}(S_{\tilde{k}})=\tilde{N}$ and using (3.23), we conclude that the expected number of iterations to reach the full sample is bounded from above by $\lceil\tilde{N}/q\rceil$ which can further be upper bounded by

[TABLE]

Furthermore, let $\tilde{k}_{1}$ be the starting iteration of the FS phase. Then the decrease condition (3.9) holds and according to Assumptions A1, A2 and (2.4) we conclude that for any $j\in\mathbb{N}$ we have

[TABLE]

Obviously, for each FS scenario there holds $\lim_{k\to\infty}\|d(x_{k})\|=0$. Now, let us denote by $T$ the number of iterations (counting from $\tilde{k}_{1}$) needed to reach $\|d(x_{k})\|<\nu$. Then, we have

[TABLE]

and thus due to (3.25) we obtain

[TABLE]

Combining this with (3.24) we obtain the result. ∎

**Remark 1. ** Notice that the expected complexity bound $\hat{k}_{E}$ is very conservative. Since we observe the FS scenario in the previous theorem, instead of $\bar{c}$, we can use $c_{1}t_{min}$. Moreover, instead of $N-1$, we can take $\tilde{N}$, which reveals the influence of the dynamics of increase used in step S5 of AS-NC to the complexity bound. Finally, $q$ in fact depends on the size of the additional sample $D_{k}$. Thus, setting e.g. $D_{k}=1$ for each $k$ yields $q=\min\{w_{1},...,w_{N}\}$ which can be significantly larger than $q=\min\{w_{1},...,w_{N}\}^{N-1}$. The complexity bound is of order ${\cal O}(\nu^{-2})$ which corresponds to the deterministic case. If we take $N_{k}=N$ for all $k$, the bound becomes deterministic, i.e., the vicinity of a stationary point is reached after at most $\hat{k}=\left\lceil\frac{f(x_{0})-f_{low}+\bar{\varepsilon}}{\bar{c}\nu^{2}}\right\rceil,$ iterations. In this case, Assumptions A2 and A3 are not relevant and the method coincides with the standard projected gradient method well known from the literature, yielding the same complexity bounds as in [1].

We end this analysis by observing the strongly convex poblems. In that case, under Assumption A1, there exists a unique solution $x^{*}$ of problem (3.1).

Theorem 3.6**.**

Suppose that Assumption A1 holds and that the sequence $\{x_{k}\}_{k\in\mathbb{N}}$ generated by AS-NC is bounded. If the function $f$ is strongly convex, then $\lim_{k\to\infty}x_{k}=x^{*}$ a.s.

Proof.

Bounded iterates imply Assumption A2 and thus Theorem 3.4 implies that every accumulation point of the sequence $\{x_{k}\}_{k\in\mathbb{N}}$ is a stationary point of (3.1) a.s.. On the other hand, the strong convexity implies that $x^{*}$ is the unique stationary point of problem (3.1). Therefore, we conclude that all accumulation points of the sequence $\{x_{k}\}_{k\in\mathbb{N}}$ are equal to $x^{*}$ a.s., which further implies that the whole sequence converges to $x^{*}$ a.s. This completes the proof. ∎

4 Box Constraints: AS-BOX method

Within this section, we observe the general box-constrained problems (1.1). The analysis is essentially the same as for the non-negativity constraints case, and we focus on the differences needed to extend the results. The main difference is in the projection form, which further influences the changes in the definition of $r_{\mathcal{D}_{k}}$. These are, in fact, the only two modifications with respect to the AS-NC algorithm, as will be stated in the sequel. The convergence analysis is completely the same, except for the proof of Theorem 3.3, which needs to be adapted to the general case. We start by analyzing the projection operator and specifying the form of the search direction in this setting.

Considering the set $S=\{x\in\mathbb{R}^{n}\;|\;l_{i}\leq x_{i}\leq u_{i},\;i=1,...,n\},$ the search direction (3.2) is thus given by

[TABLE]

Analogously to (3.3) we define

[TABLE]

and the indicator vector $\tilde{I}_{\mathcal{D}_{k}}$ accordingly. Then, the sparsity similarity vector analogous to (3.4) is defined as

[TABLE]

4.1 The Algorithm

The algorithm differs from AS-NC only in steps S2 and S4. We state it for completeness.

Algorithm 2: AS-BOX (Additional Sampling - BOX constraints)

• S0

Initialization. Input: $x_{0}\in S,N_{0}\in\mathbb{N},\beta,c,c_{1}\in(0,1),C>0$, $\{\varepsilon_{k}\}$ satisfying (2.4). Set $k:=0$.

• S1

Subsampling. If $N_{k}<N$, choose $\mathcal{N}_{k}$ via (2.3). Else, set $f_{\mathcal{N}_{k}}=f$.

• S2

Search direction. Compute $p_{k}$ via (4.1).

• S3

Step size. Find the smallest $j\in\mathbb{N}_{0}$ such that $t_{k}=\beta^{j}$ satisfies

[TABLE]

Set $\bar{x}_{k}=x_{k}+t_{k}p_{k}$.

• S4

*Additional sampling.

*If $N_{k}=N$, set $x_{k+1}=\bar{x}_{k}$, $k=k+1$ and go to step S1.

Else choose $\mathcal{D}_{k}$ via (2.5) and compute

[TABLE]

and $\tilde{r}_{\mathcal{D}_{k}}$ defined by (4.2).

• S5

*Sample size update.

*If

[TABLE]

$N_{k+1}=N_{k}$.

Else choose $N_{k+1}\in\{N_{k}+1,...,N\}$.

• S6

Iterate update.

If

[TABLE]

holds set $x_{k+1}=\bar{x}_{k}$.

Else set $x_{k+1}=x_{k}$.

• S7

Counter update. Set $k=k+1$ and go to Step S1.

4.2 Convergence analysis

The convergence analysis is conducted under the same set of assumptions. Notice that the results of Lemma 3.1 and 3.2 also hold for AS-BOX. Now, we state the result analogous to Theorem 3.3. Notice that $\tilde{k}$ has the same role as in Theorem 3.3.

Theorem 4.1**.**

Suppose that Assumption A1 holds. Then a.s. there exists a finite, random iteration $\tilde{k}$ such that for all $k\geq\tilde{k}$ there holds

[TABLE]

where $\bar{c}=\min\{c,c_{1},2c_{1}(1-c_{1})\beta/L\}$ and $\bar{C}=\max\{1,C\}$.

Proof.

The first part of the proof is completely the same as the proof of Theorem 3.3. Consider first the FS scenario. Analogously as in the proof of Theorem 3.3 we derive the inequality

[TABLE]

In the MB case, proceeding as in Theorem 3.3 we conclude that for all $k\geq k_{1}$ there holds

[TABLE]

Notice that $k_{1}$ is again defined in Lemma 3.1. Now, let us define the following sets of indices

[TABLE]

Using the same arguments as in the proof of Theorem 3.3 we conclude that for all $k\geq k_{1}$ we have

[TABLE]

where for each $j=1,\dots,N$, we define

[TABLE]

This further implies that for all $k\geq k_{1}$, for all $i\in\mathcal{L}_{\mathcal{N}_{k}}$, for all $j\in\mathcal{N}$, there holds $[x_{k}]_{i}-[\nabla f_{j}(x_{k})]_{i}<l_{i}$, i.e., $[x_{k}]_{i}<[\nabla f_{j}(x_{k})]_{i}+l_{i},$ and thus

[TABLE]

Similarly, for all $i\in\mathcal{U}_{\mathcal{N}_{k}}$, we obtain $[x_{k}]_{i}-[\nabla f(x_{k})]_{i}>u_{i},$ and for all $i\in\mathcal{I}_{\mathcal{N}_{k}}$ $l_{i}\leq[x_{k}]_{i}-[\nabla f(x_{k})]_{i}\leq u_{i}.$ Therefore, for all $k\geq k_{1}$ we have

[TABLE]

Now, let us estimate the norm of $d(x_{k})$ for $k\geq k_{1}$ as follows

[TABLE]

Recalling the definition $s^{j}_{k}:=\pi_{S}(x_{k}-\nabla f_{j}(x_{k}))-x_{k}$, due to (4.8) we obtain

[TABLE]

for all $k\geq k_{1}$ and all $j\in\mathcal{N}$. Hence, for $i\in\mathcal{L}_{\mathcal{N}_{k}}$ and $k\geq k_{1}$ there holds

[TABLE]

and thus

[TABLE]

Similarly, for $i\in\mathcal{U}_{\mathcal{N}_{k}}$, we have

[TABLE]

and hence

[TABLE]

Now, for $i\in\mathcal{I}_{\mathcal{N}_{k}}$,

[TABLE]

which further implies

[TABLE]

Combining (4.2), (4.10), (4.11), and (4.12), we obtain

[TABLE]

Furthermore, combining this with (4.7) we obtain for all $k\geq k_{1}$

[TABLE]

Finally, Taking into account both scenarios (FS and MB), i.e., (4.14) and (4.6), we conclude the proof. ∎

The proofs of the following three main results for AS-BOX are the same as for AS-NC, so we only provide statements for completeness.

Theorem 4.2**.**

Suppose that Assumptions A1 and A2 hold. Then a.s. every accumulation point of sequence $\{x_{k}\}_{k\in\mathbb{N}}$ generated by AS-BOX is a stationary point of problem (1.1).

Theorem 4.3**.**

Suppose that Assumptions A1, A2 and A3 hold. Then the expected number of iterations to reach $\|d(x_{k})\|<\nu$ is upper bounded by

[TABLE]

where $\bar{c}$ is as in Theorem 3.3, $C_{b}^{FS}$ as in (3.21) and $q=\min\{w_{1},...,w_{N}\}^{N-1}$.

Theorem 4.4**.**

Suppose that Assumption A1 holds and that the sequence $\{x_{k}\}_{k\in\mathbb{N}}$ generated by AS-BOX is bounded. If the function $f$ is strongly convex, then $\lim_{k\to\infty}x_{k}=x^{*}$ a.s.

5 Numerical results

In this section, we present numerical experiments designed to evaluate the performance of the proposed Algorithm AS-BOX and to compare it with existing methods from the literature. In particular, we focus on a comparison with the stochastic gradient-based interior-point method (SIPM) developed in [8], which was designed for solving smooth optimization problems with box constraints. The second benchmark method we consider is PSGM (Projected Stochastic Gradient Method) as in [8]. In each iteration of the PSGM a stochastic gradient is computed and projected onto the feasible region defined by the box constraints. All the parameters used in our work that are related to SIPM and PSGM are the same as those provided in the numerical results section of [8].

The experiments comprise two binary classification models:

1. logistic regression (convex);
2. a single-hidden-layer neural network with cross-entropy loss (nonconvex). The experiments were conducted on datasets from the LIBSVM repository, namely *Mushrooms *($8124$ samples, $112$ features) and IJCNN1 ($49990$ samples, $22$ features). These datasets are widely used benchmarks due to their diversity in structure, which enables a comprehensive evaluation of algorithmic performance under varying problem structures. Labels were encoded in $\{-1,1\}$ format for binary classification.

We generated $x_{0}$ for each problem with elements drawn from a uniform distribution over $\left[-0.01,0.01\right]$. In both experiments, for the AS-BOX algorithm we use the following parameters: $D_{k}=1$, $C=1$, $\beta=0.1$, $\eta=10^{-4}$, $c=10^{-4}$, $\varepsilon_{k}=\frac{1}{k^{1.1}}$444We also tested several alternative schedules and observed qualitatively unchanged behavior, indicating robustness to the choice of $\varepsilon_{k}$.. The parameters given above were chosen following the standard settings used in prior work [21, 22, 24].

5.1 Logistic Regression Problem

The first benchmark problem is given by

[TABLE]

where $(a_{i},b_{i})$ are the training samples, $b_{i}\in\{-1,1\}$ are binary labels. This model is convex and commonly used as a baseline for large-scale classification tasks. Box constraints $[-1,1]^{n}$ were imposed to conform with the setup in [8] and to enable direct comparison with existing stochastic methods. We model the computational cost by $FEV_{k}$ – the number of scalar products required by the specified method to compute $x_{k}$, starting from the initial point $x_{0}$.

To evaluate the performance of the considered methods, we present: the distance between $x_{k}$ and the solution $x^{*}$ of the considered problem, i.e., $||x_{k}-x^{*}||$, against the computational cost measure $FEV_{k}$. The stopping criterion in all comparisons is a fixed budget of scalar products (FEV), so the methods are evaluated up to the same computational effort.

In Figure 1, the comparison of the four algorithms (AS-BOX, SIPM, PSGM, and FULL SAMPLE (AS-BOX full sample, for all $k:N_{k}=N,$)) is demonstrated on the Mushrooms dataset. The graph shows the Euclidean distance $||x_{k}-x^{*}||$ as a function of the number of scalar products $FEV_{k}$, where AS-BOX achieves the fastest convergence toward the reference solution. SIPM shows a slower but steady decrease, while PSGM seems to be stagnating. On this dataset, the FULL SAMPLE ($N_{k}=N$) performs the worst overall, exhibiting the slowest convergence.

A similar behavior can be observed on the IJCNN1 dataset (Figure 2), where AS-BOX again achieves the best performance and reaches the smallest distance to the reference solution, attaining an accuracy level of $10^{-1}$ between $150{,}000$ and $200{,}000$ $FEV_{k}$, SIPM follows with moderate convergence, and PSGM remains the slowest method. On this dataset, the FULL SAMPLE ($N_{k}=N$) is slightly better than PSGM, but still clearly worse than AS-BOX and SIPM.

In Figure 3 we report the evolution of the subsample size $N_{k}$ as a function of the computational budget $\mathrm{FEV}_{k}$ for AS-BOX on two datasets. The growth of $N_{k}$ is monotone and staircase-like - $N_{k}$ increases only when the acceptance test fails, and remains flat otherwise. On Mushrooms ($m=8{,}124$), $N_{k}$ peaked at $168$ (about $2.1\%$ of the data); on IJCNN1 ($m=49{,}990$), it peaked at $504$ (about $1.0\%$). In neither case did the method reach the full sample size, indicating that AS-BOX makes steady progress without resorting to $N_{k}=N$, which underlies its computational efficiency at a fixed $\mathrm{FEV}_{k}$ budget.

5.2 Neural Network with Cross-Entropy Loss

The second problem considers training a single-hidden-layer neural network for binary classification. Let $\tanh(\cdot)$ be the activation function in the hidden layer, while the output layer uses a sigmoid activation. The network output is therefore given by

[TABLE]

where $W_{1},W_{2}$ are weight matrices and $b_{1},b_{2}$ are bias vectors. The training objective is the average cross-entropy loss

[TABLE]

where $x$ collects all parameters $(W_{1},W_{2},b_{1},b_{2})$ and $d$ denotes the total number of network parameters. As in the logistic regression case, the parameters are constrained to lie within $[-1,1]$.

This problem is inherently nonconvex and poses a stronger challenge to optimization methods. Its inclusion in the test suite allows us to assess the robustness of AS-BOX when applied to neural network training under box constraints. Since the solution of such problem is not unique in general, we plot the optimality measure $||d(x_{k})||$ against the computational cost measure $FEV_{k}$ to evaluate the performance of the considered methods.

Figure 4 part a) shows the cross-entropy loss trajectory on the Mushrooms dataset. AS-BOX again outperforms its competitors, with the loss dropping from approximately $2\times 10^{-1}$ to below $10^{-2}$ within $10^{5}$ evaluations. SIPM displays a slower descent, converging around $5\times 10^{-2}$, whereas PSGM initially outperforms AS-BOX but stagnates around $3\times 10^{-2}$. The results suggest that AS-BOX maintains its advantage across problems of different structure. Figure 4 part b) reports the stationarity measure for the Mushrooms dataset. Consistent with the loss plots, AS-BOX achieves the most significant reduction, descending from roughly $10^{-1}$ to about $10^{-2}$ and exhibiting a stable convergence pattern despite minor stochastic fluctuations. SIPM steadily decreases but remains above $3\times 10^{-2}$ by the end, while PSGM flattens out near $4\times 10^{-2}$ early in the run.

Next, Figure 5 part a) illustrates the evolution of the cross-entropy loss with respect to the number of function evaluations (FEV) for the IJCNN1 dataset. All algorithms start from a comparable initial loss of approximately $4.2\times 10^{-1}$. The proposed AS-BOX method demonstrates the fastest and most consistent decrease, reaching a loss below $3.2\times 10^{-1}$ after roughly $4\times 10^{5}$ evaluations. The part b) on Figure 5 presents the stationarity measure $||d(x_{k})||$ as a function of $FEV_{k}$ on the same dataset. This metric reflects how close the iterates are to satisfying the first-order optimality conditions under box constraints. AS-BOX exhibits the steepest decline, dropping below $2\times 10^{-2}$ by the end of the run, indicating near-stationarity. SIPM converges more slowly, stabilizing around $4\times 10^{-2}$, while PSGM decreases rapidly at first but stagnates near $4\times 10^{-2}$. This highlights AS-BOX’s ability to achieve higher stationarity accuracy compared to the other methods. These results confirm the superior long-term convergence behavior of AS-BOX.

When comparing the results obtained on the IJCNN1 and Mushrooms datasets, a consistent trend emerges: the AS-BOX algorithm achieves the biggest decrease in both cross-entropy loss and stationarity measure within the considered FEV budget, outperforming SIPM and PSGM across all experiments. However, the rate of convergence differs between the datasets due to their structural properties. The Mushrooms dataset, being smaller and with more separable data, allows all algorithms to achieve lower loss values more quickly, with AS-BOX reaching near-optimal performance within $10^{5}$ evaluations. In contrast, IJCNN1 is higher-dimensional and less separable, which slows down convergence for all methods; nevertheless, AS-BOX maintains a significant advantage over SIPM and PSGM, achieving roughly twice the reduction in stationarity by the end of the run.

6 Conclusions

A novel method (AS-BOX) for box-constrained weighted finite sum problems has been proposed. This method falls into the framework of stochastic projected gradient methods and uses non-monotone line search to adaptively determine the step size sequence, while retaining the feasibility of the iterates. The main novelty of AS-BOX lies in the adaptation of an additional sampling technique to box-constrained weighted finite-sum problems. Thus, the resulting method adaptively changes the sample size and conforms to different structures of the problems. AS-BOX also has a theoretical background - a.s. convergence is proved under a standard set of assumptions, without imposing the convexity. This makes it suitable for NN problems as well. Moreover, complexity analysis has been conducted as well, and a stronger convergence result is provided for strongly convex problems such as regularized logistic regression. Numerical study showed the efficiency of AS-BOX.

Future work naturally tends to additional sampling methods for finite-sum problems with general, nonlinear equality and inequality constraints.

**Funding. ** N. Krejić and N. Krklec Jerinkić are supported by the Science Fund of the Republic of Serbia, Grant no. 7359, Project LASCADO. T. Ostojić is partially supported by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia (Grants No. 451-03-136/2025-03/200156) and by the Faculty of Technical Sciences, University of Novi Sad through project “Scientific and Artistic Research Work of Researchers in Teaching and Associate Positions at the Faculty of Technical Sciences, University of Novi Sad 2025” (No. 01-50/295). N. Vučićević is partially supported by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia (Grant no. 451-03-137/2025-03/ 200122)