---
title: Sub-Sampled Trust-Region Methods
url: https://www.emergentmind.com/topics/sub-sampled-trust-region-methods
type: topic
---

# Sub-Sampled Trust-Region Methods

Sub-sampled trust-region methods are trust-region algorithms for large-scale smooth optimization in which the objective value, gradient, Hessian, or trust-region subproblem itself is approximated using only a subset of the available information. In the finite-sum setting,
\[
\min_{x\in\mathbb{R}^n} f(x)=\frac1d\sum_{i=1}^d f_i(x),
\]
the central objective is to reduce per-iteration cost without losing the globalization, stationarity, and worst-case complexity properties associated with classical trust-region methods. Across recent formulations, the common pattern is a local quadratic or higher-order model, a radius-constrained step computation, and an acceptance mechanism based on actual-versus-predicted reduction, but with derivative or function information replaced by noisy, inexact, or subsampled estimates [1802.06925] [2112.06176] [2507.17556].

## 1. Core trust-region structure

At the algorithmic level, the finite-sum trust-region model typically has the form
\[
m_k(d)=f(x_k)+\langle g_k,d\rangle+\frac12\langle B_k d,d\rangle,\qquad \|d\|\le \Delta_k,
\]
where \(g_k\) and \(B_k\) are exact or approximate derivatives. In first-order sub-sampled variants, only the gradient is subsampled, while \(B_k\) is any symmetric matrix satisfying a uniform bound; in second-order variants, both the gradient and Hessian are subsampled [2507.17556]. In large-scale non-convex optimization, a closely related inexact trust-region formulation uses approximate gradient \(g_t\), approximate Hessian \(H_t\), and an approximate subproblem solution \(s_t\), terminating when
\[
\|g_t\|\le \epsilon_g,\qquad \lambda_{\min}(H_t)\ge -\epsilon_H,
\]
so that the returned point is approximately second-order stationary [1802.06925].

The globalization mechanism remains recognizably classical. The standard trust-region ratio is
\[
\rho_k=\frac{f(x_k)-f(x_k+d_k)}{m_k(0)-m_k(d_k)},
\]
and successful iterations accept the trial step and enlarge the radius, while unsuccessful iterations reject the step and shrink it [2507.17556]. In the inexact Newton-type formulation, the same ratio is written as
\[
\rho_t=\frac{F(x_t)-F(x_t+s_t)}{-m_t(s_t)},
\]
with acceptance threshold \(\eta\) and multiplicative radius updates by a factor \(\gamma>1\) [1802.06925].

Sub-sampled trust-region methods differ from purely first-order stochastic schemes because they preserve this model-based structure. The step is not an arbitrary stochastic descent direction; it is the outcome of an approximate constrained minimization, typically required to satisfy a Cauchy-type decrease condition and, in second-order settings, also to exploit negative curvature when present [1802.06925] [2507.17556].

## 2. Inexactness, noise, and complexity theory

A major theoretical development is the extension of trust-region complexity analysis to noisy and probabilistic models. The TR\(\!q\)NE framework studies smooth unconstrained optimization with noisy objective values and noisy derivatives up to arbitrary order \(q\ge 1\). It defines the approximate optimality measure
\[
\phi_{f,j}^{\delta}(x)=\max_{\|d\|\le \delta}\Delta t_{f,j}(x,d),
\]
and calls \(x\) a \(q\)-th order \((\epsilon,\delta)\)-approximate minimizer when
\[
\phi_{f,j}^{\delta}(x)\le \epsilon_j \frac{\delta^j}{j!},\qquad j=1,\dots,q.
\]
Under suitable probabilistic assumptions on model accuracy and function-value accuracy, the expected number of inexact evaluations required to obtain such a point satisfies
\[
E[N_\epsilon]=\mathcal{O}(\epsilon^{-(q+1)}),
\]
which matches the sharp order known for deterministic trust-region methods with exact derivatives [2112.06176].

For second-order finite-sum optimization, inexact trust-region methods with approximate gradients, approximate Hessians, and approximate subproblem solves retain the optimal worst-case iteration complexity
\[
O\!\left(\max\{\epsilon_g^{-2}\epsilon_H^{-1},\ \epsilon_H^{-3}\}\right),
\]
provided the approximation errors satisfy
\[
\|g_t-\nabla F(x_t)\|\le \delta_g,\qquad \|H_t-\nabla^2F(x_t)\|\le \delta_H,
\]
with mild conditions linking \(\delta_g\) and \(\delta_H\) to \(\epsilon_g\) and \(\epsilon_H\) [1802.06925]. A deterministic adaptive subsampling framework later established the same asymptotic second-order trust-region complexity for finite-sum problems,
\[
\mathcal O\!\left(\max\{\varepsilon_g^{-2}\varepsilon_H^{-1},\varepsilon_H^{-3}\}\right),
\]
while also giving the first-order bound \(\mathcal O(\varepsilon_g^{-2})\) [2507.17556].

A distinctive feature of the probabilistic analysis is its treatment of intrinsic noise. In the first-order specialization TR\(\!1\)NE, if the probabilistic assumptions fail for a particular realization, the failure itself implies a degraded stationarity statement: the realized iterate must satisfy a bound of the form
\[
\|\nabla_x f(x_k)\|<2\bar B.
\]
This does not recover the full theory, but it quantifies what can still be concluded when stochastic errors dominate near the minimizer [2112.06176].

## 3. Sampling rules and accuracy control

The literature separates into probabilistic batch-size control and deterministic sample-size control. In finite-sum inexact Newton-type trust-region methods, the gradient and Hessian are estimated by uniform random minibatches,
\[
g=\frac{1}{|\mathcal S_g|}\sum_{i\in \mathcal S_g}\nabla f_i(x), \qquad
H=\frac{1}{|\mathcal S_H|}\sum_{i\in \mathcal S_H}\nabla^2 f_i(x),
\]
and a sampling lemma states that if
\[
|\mathcal S_g|\ge \frac{16K_g^2}{\delta_g^2}\log\frac1\delta,\qquad
|\mathcal S_H|\ge \frac{16K_H^2}{\delta_H^2}\log\frac{2d}{\delta},
\]
then the desired derivative-accuracy bounds hold with probability at least \(1-\delta\) [1802.06925].

The probabilistic framework of TR\(\!q\)NE translates similar ideas to function values and derivatives through accurate-model and accurate-function events. In the finite-sum case, the function decrease estimate is formed from a batch difference,
\[
\bar\Delta f(x_k,s_k)=\frac1{|\mathfrak b_0(x_k)|}\sum_{i\in\mathfrak b_0(x_k)}\big(f_i(x_k)-f_i(x_k+s_k)\big),
\]
and concentration inequalities imply exponentially decaying tail bounds for the failure probabilities. The practical message stated in the paper is that larger batches are needed as one approaches a solution, because the relative noise must shrink enough for the trust-region acceptance and complexity theory to apply [2112.06176].

Deterministic adaptive subsampling replaces probability bounds by explicit radius-dependent rules. If a subsample omits only a fraction \(h\) of indices, then
\[
\|\nabla f(x)-\nabla f_{\mathcal G}(x)\|\le 2h\max_{i\in\mathcal N}\|\nabla f_i(x)\|,
\qquad
\|\nabla^2 f(x)-\nabla^2 f_{\mathcal H}(x)\|\le 2h\max_{i\in\mathcal N}\|\nabla^2 f_i(x)\|.
\]
The adaptive procedure chooses sample sizes through quantities such as \(h_k^j\), \(h_{k,g}^j\), and \(h_{k,H}^j\), all tied to the current trust-region radius \(\Delta_k\), so that smaller trust regions force more accurate derivative estimates [2507.17556].

A different deterministic mechanism appears in inexact restoration trust-region methods. There the sample size is treated as a feasibility variable, with infeasibility measure
\[
h(M)=\frac{N-M}{N},
\]
and a merit function
\[
\Psi(x,M,\theta)=\theta f_M(x)+(1-\theta)h(M).
\]
This schedules sample-size growth by balancing progress in the subsampled objective against progress toward full precision [1902.01710].

## 4. Principal algorithmic variants

The family now includes probabilistic, deterministic, non-monotone, and variance-reduced formulations.

| Variant | Distinctive mechanism | Stated guarantee |
|---|---|---|
| TR\(\!q\)NE [2112.06176] | Noisy derivatives, probabilistic model/function events, adaptive model order \(j_k\) | \(E[N_\epsilon]=\mathcal O(\epsilon^{-(q+1)})\) |
| Inexact TR [1802.06925] | Sub-sampled gradient and Hessian, approximate subproblem solve | \(O(\max\{\epsilon_g^{-2}\epsilon_H^{-1},\epsilon_H^{-3}\})\) |
| Deterministic STR [2507.17556] | Radius-dependent deterministic sample-size adjustment | \(\mathcal O(\varepsilon_g^{-2})\) and \(\mathcal O(\max\{\varepsilon_g^{-2}\varepsilon_H^{-1},\varepsilon_H^{-3}\})\) |
| ASNTR [2307.10038] | Non-monotone acceptance and independent additional sampling | \(\liminf_{k\to\infty}\|\nabla f(w_k)\|=0\) a.s. |
| TR-SVR [2412.00673] | SVRG-style control variate inside a trust-region/SQP model | \(O(1/(KS))\)-type average squared-gradient bound |

ASNTR is notable because it explicitly addresses the dependence problem created when the same minibatch is used both to build the step and to evaluate the trial point. It introduces an independent additional sample \(D_k\) and, in the stochastic phase, accepts a step only if both
\[
\rho_{N_k}\ge \eta \quad\text{and}\quad \rho_{D_k}\ge \nu.
\]
This permits a non-monotone trust-region method with noisy oracles and adaptive sample growth driven by either failure of the independent test or the relation \(\|g_k\|<\epsilon h(N_k)\) [2307.10038].

TR-SVR occupies a different position. It uses a two-loop structure with a full gradient at an outer reference point \(x_{k,0}\) and a variance-reduced estimator
\[
\bar g_{k,s}=\tilde g_{k,s}-(\tilde g_{k,0}-g_{k,0}),
\]
whose variance satisfies
\[
\mathbb{E}_{k,s}\!\left[\|\bar g_{k,s}-g(x_{k,s})\|^2\right]\le \frac{L^2}{b}\|x_{k,s}-x_{k,0}\|^2.
\]
The paper explicitly notes that it does not use a classical ratio-based acceptance test and that the radius update is less formal than in standard trust-region algorithms [2412.00673].

## 5. Trust-region subproblems and low-dimensional reductions

Sub-sampling concerns not only derivatives but also the inner trust-region subproblem. Inexact second-order trust-region methods rely on steps that are at least as good as the Cauchy and eigen points,
\[
m_t(s_t)\le m_t(s_t^C),\qquad m_t(s_t)\le m_t(s_t^E),
\]
so approximate subproblem solves are acceptable provided they preserve sufficient decrease along gradient and negative-curvature directions [1802.06925]. Deterministic adaptive STR uses a Cauchy-type decrease condition in its first-order form and a stronger condition exploiting either gradient reduction or negative curvature reduction in its second-order form; numerically, it uses Dogleg [2507.17556]. Inexact restoration variants use CG-Steihaug for quadratic models and the normalized gradient step for linear models [1902.01710].

Several related works reduce the dimensionality of the subproblem itself. Random projection methods solve a projected lower-dimensional surrogate obtained from a Gaussian matrix \(P\), replacing the original constrained trust-region subproblem by one in \(\mathbb R^d\) and then lifting the solution back with \(P^\top\). The approximation guarantees depend on Johnson–Lindenstrauss-type norm preservation, feasibility transfer, and the fullness of the feasible set [1706.02730]. A different randomized reduction, TLTR, computes a full-space trust-region step \(p_k^F\), then adds a correction from a randomly sketched subspace,
\[
p_k=p_k^F+\alpha_k S_k^T p_k^S,
\]
accepting the composite step through a trust-region ratio that combines the full-space model decrease with the actual decrease produced by the subspace correction [2409.05479].

For repeated trust-region subproblems with slowly varying or reused Hessians, the TREK method supplies a reusable inner solver. It builds an extended Krylov subspace from one factorization of \(A\), reduces the trust-region problem to a tiny projected problem, and then solves the projected problem cheaply by root-finding. The paper explicitly positions this as useful when a sampled or approximate Hessian is reused and the trust-region radius changes across outer iterations [2511.11135].

## 6. Reported performance, limitations, and recurring issues

The empirical results consistently frame sub-sampled trust-region methods as mechanisms for reducing derivative cost rather than changing the outer trust-region logic. In the inexact Newton-type study, inexact TR was reported as roughly **3–5× faster than SubH TR** and **5–10× faster than Full TR** in terms of propagation count on non-convex binary classification problems [1802.06925]. In deterministic adaptive STR, reported cost reductions relative to full TR were **4%** for \(d=100\), **39%** for \(d=500\), **33%** for \(d=1000\), and **50%** for \(d=3000\); for \(d=3000\), the sample size sometimes dropped as low as **273 components** [2507.17556].

For deep-learning-oriented stochastic trust-region methods, ASNTR was reported to achieve higher training and testing accuracy and lower loss than STORM for the same gradient-evaluation budget, while often requiring significantly fewer gradient evaluations [2307.10038]. TLTR reported that larger random-subspace dimensions reduce iteration count but increase cost per iteration, and that sketch sizes around **20–30% of \(n\)** were often a good compromise in the experiments [2409.05479].

Several limitations recur across the literature. First, derivative accuracy requirements typically tighten near stationarity; subsampling is therefore not a license for permanently small batches. The probabilistic analysis of noisy trust-region methods makes this explicit by showing that larger batches are needed near a solution [2112.06176]. Second, not every stochastic trust-region method preserves the classical trust-region acceptance framework: TR-SVR lacks an explicit actual-versus-predicted reduction ratio and provides an average-based first-order guarantee rather than a detailed oracle complexity statement [2412.00673]. Third, intrinsic noise may prevent probabilistic assumptions from holding forever; in that regime the strongest complexity results may fail, although degraded stationarity conclusions remain available in the first-order setting [2112.06176].

A plausible implication is that “sub-sampled trust-region methods” should be understood less as a single algorithm and more as a design space. The unifying principle is the preservation of trust-region globalization under cheaper information acquisition, while the main technical differences lie in how derivative accuracy is certified, how sample sizes are adapted, and how the trust-region subproblem is solved or reduced.

Source: https://www.emergentmind.com/topics/sub-sampled-trust-region-methods