Projection Alignment Method (ZO-Align)

• The paper introduces ZO-Align, a projection alignment method that combines multiple directional probes via local surrogate minimization for efficient zeroth-order optimization.
• ZO-Align is derived by constraining updates to the subspace spanned by queried directions, effectively leveraging full-subspace estimation to improve gradient accuracy.
• Empirical and theoretical results demonstrate that, unlike ZO-Avg, ZO-Align benefits from increased query counts per iteration to achieve faster convergence.

Projection Alignment Method, denoted ZO-Align, is a multi-query aggregation method for zeroth-order optimization in which multiple directional function evaluations are combined through a projection operator derived from local surrogate minimization. In the formulation reported in "The Multi-Query Paradox in Zeroth-Order Optimization," ZO-Align is contrasted with the de facto simple averaging baseline ZO-Avg and is analyzed under a fixed query budget, where the number of queries per iteration and the total number of iterations are inversely proportional. The central result is a dichotomy: for ZO-Avg, using more than one query per iteration is always query-inefficient, whereas for ZO-Align, increasing the number of queries per iteration is generally beneficial, with full-subspace estimation emerging as the optimal regime (Lin et al., 19 Sep 2025).

1. Problem setting and the multi-query paradox

Zeroth-order optimization considers minimization of an objective $f(x)$ when explicit gradients are unavailable and must be approximated using only function-value queries. A standard update has the form

$x_{t+1}=x_t-\eta_t \hat g(x_t).$

The basic single-query estimator is

$\hat g_u(x)=\frac{f(x+\mu u)-f(x)}{\mu}u,$

which, for small $\mu$, behaves like a directional gradient probe (Lin et al., 19 Sep 2025).

The multi-query setting uses $q$ query directions $u_1,\dots,u_q$ rather than a single direction. This can reduce estimator noise, but under a fixed query budget $K$, queries per iteration and the total number of optimization iterations are inversely proportional: $\sum_{t=0}^{T-1} q_t \le K.$ The associated trade-off is the paper’s multi-query paradox: larger $q_t$ may improve the gradient estimate at each iteration, but necessarily reduces the number of updates. The paper’s core conclusion is that this trade-off cannot be resolved independently of the aggregation rule. Instead, the optimal allocation depends entirely on how multiple directional probes are combined. For ZO-Avg, the optimal regime is the single-query regime; for ZO-Align, the optimal regime is the maximal-query regime (Lin et al., 19 Sep 2025).

A common misconception is that the fixed-budget setting should favor an intermediate query size balancing accuracy and iteration count. The reported theory rejects that view for the two aggregation rules studied: the choice is not between small, medium, and large query counts in any generic sense, but between two extremes induced by the aggregation mechanism itself (Lin et al., 19 Sep 2025).

2. Derivation from local surrogate minimization

ZO-Align is derived by constraining the update to the subspace spanned by the queried directions. Let

$U=[u_1,\dots,u_q]\in \mathbb{R}^{d\times q},$

with linearly independent columns, and consider updates of the form

$x_{t+1}=x_t-\eta_t \hat g(x_t).$0

Using $x_{t+1}=x_t-\eta_t \hat g(x_t).$1-smoothness, the paper writes

$x_{t+1}=x_t-\eta_t \hat g(x_t).$2

ZO-Align is obtained by minimizing this local quadratic upper bound over $x_{t+1}=x_t-\eta_t \hat g(x_t).$3: $x_{t+1}=x_t-\eta_t \hat g(x_t).$4 The optimizer is

$x_{t+1}=x_t-\eta_t \hat g(x_t).$5

which yields the subspace update

$x_{t+1}=x_t-\eta_t \hat g(x_t).$6

This derivation distinguishes ZO-Align from simple averaging. Rather than estimating the full gradient by averaging directional probes, ZO-Align constructs the best update within the sampled subspace according to a local surrogate model (Lin et al., 19 Sep 2025).

This suggests that ZO-Align should be interpreted geometrically rather than purely statistically. The method is not centered on variance reduction alone; it is centered on subspace-optimal descent induced by the local quadratic model. That distinction underlies the different query-allocation behavior later proved in the paper.

3. Aggregation rule and projection geometry

The ZO-Align estimator is formed by replacing directional derivatives in the subspace solution with finite differences: $x_{t+1}=x_t-\eta_t \hat g(x_t).$7 If

$x_{t+1}=x_t-\eta_t \hat g(x_t).$8

then equivalently

$x_{t+1}=x_t-\eta_t \hat g(x_t).$9

The optimization update is

$\hat g_u(x)=\frac{f(x+\mu u)-f(x)}{\mu}u,$0

The key geometric object is

$\hat g_u(x)=\frac{f(x+\mu u)-f(x)}{\mu}u,$1

the orthogonal projector onto $\hat g_u(x)=\frac{f(x+\mu u)-f(x)}{\mu}u,$2. In the idealized $\hat g_u(x)=\frac{f(x+\mu u)-f(x)}{\mu}u,$3 regime,

$\hat g_u(x)=\frac{f(x+\mu u)-f(x)}{\mu}u,$4

Accordingly, ZO-Align is the orthogonal projection of the true gradient onto the sampled subspace. The paper also states

$\hat g_u(x)=\frac{f(x+\mu u)-f(x)}{\mu}u,$5

Unlike ZO-Avg, ZO-Align is therefore biased, but it is geometrically consistent within the sampled subspace (Lin et al., 19 Sep 2025).

This formulation also clarifies the method’s name. The alignment is a projection alignment: directional finite-difference measurements are aligned with the sampled subspace through $\hat g_u(x)=\frac{f(x+\mu u)-f(x)}{\mu}u,$6, rather than merely averaged componentwise. A plausible implication is that the method’s effectiveness depends on how accurately the queried subspace captures the useful gradient directions at each iterate.

4. Convergence results under fixed query budgets

The paper derives explicit convergence rates that expose the dependence on the number of queries per iteration in strongly convex, convex, non-convex, and stochastic settings (Lin et al., 19 Sep 2025).

In the strongly convex case, assuming $\hat g_u(x)=\frac{f(x+\mu u)-f(x)}{\mu}u,$7 is $\hat g_u(x)=\frac{f(x+\mu u)-f(x)}{\mu}u,$8-smooth and $\hat g_u(x)=\frac{f(x+\mu u)-f(x)}{\mu}u,$9-strongly convex, and using step size

$\mu$0

the paper proves

$\mu$1

Under a fixed budget $\mu$2, the optimal allocation is

$\mu$3

yielding

$\mu$4

The contraction factor improves linearly with $\mu$5, so more queries per step are better.

In the convex case, assuming $\mu$6 is $\mu$7-smooth and convex, with step size

$\mu$8

the paper proves

$\mu$9

Under a fixed total query budget $q$0, any allocation satisfying

$q$1

gives

$q$2

Here the bound depends only on total queries $q$3, not on their distribution across iterations.

In the non-convex case, assuming only $q$4-smoothness and using

$q$5

the paper proves

$q$6

Under a fixed budget $q$7,

$q$8

Again, the bound is driven by total queries.

In the stochastic convex case, assuming $q$9 is convex and $u_1,\dots,u_q$0-smooth, with

$u_1,\dots,u_q$1

the paper uses the lemma

$u_1,\dots,u_q$2

With diminishing step size

$u_1,\dots,u_q$3

it gives a bound of the form

$u_1,\dots,u_q$4

The stated optimal allocation under a fixed total budget is

$u_1,\dots,u_q$5

Across these settings, the theory implies that ZO-Align is not merely tolerant of larger multi-query blocks; it is typically improved by them, with the full-subspace choice $u_1,\dots,u_q$6 singled out as optimal in the strongly convex and stochastic convex analyses and as the natural extremal regime elsewhere.

5. Contrast with ZO-Avg

The comparison with ZO-Avg is the organizing contrast of the paper. The simple averaging estimator is

$u_1,\dots,u_q$7

In the idealized $u_1,\dots,u_q$8 form,

$u_1,\dots,u_q$9

The paper states that ZO-Avg is unbiased,

$K$0

with mean-squared error

$K$1

For ZO-Align, the stated moments are

$K$2

and

$K$3

The paper’s interpretation is geometric: ZO-Avg tries to estimate the full gradient by averaging noisy single-direction probes, so its variance reduction scales only as $K$4, making additional queries query-inefficient under fixed budget. ZO-Align instead builds the best update in the sampled subspace by solving a local surrogate problem, so its error decreases strongly as $K$5 increases (Lin et al., 19 Sep 2025).

The resulting dichotomy is explicit. For ZO-Avg, the optimal allocation under budget $K$6 is

$K$7

For ZO-Align, the optimal allocation is

$K$8

This is the principal controversy resolved by the paper: multi-query zeroth-order optimization does not have a universal answer. The answer is aggregation-specific.

6. Full-subspace estimation and empirical validation

The relationship between ZO-Align and full-subspace estimation is one of the paper’s main conceptual points. Because

$K$9

if $\sum_{t=0}^{T-1} q_t \le K.$0 and $\sum_{t=0}^{T-1} q_t \le K.$1 is full rank, then $\sum_{t=0}^{T-1} q_t \le K.$2, so

$\sum_{t=0}^{T-1} q_t \le K.$3

In the idealized limit, ZO-Align then becomes

$\sum_{t=0}^{T-1} q_t \le K.$4

which the paper interprets as recovery of the full gradient through a classical full-subspace finite-difference method (Lin et al., 19 Sep 2025).

The experimental study is conducted in dimension $\sum_{t=0}^{T-1} q_t \le K.$5 on four problem types: a strongly convex quadratic function, convex logistic regression on synthetic data, the non-convex Rosenbrock function, and stochastic logistic regression. Both ZO-Avg and ZO-Align are tested over various query sizes $\sum_{t=0}^{T-1} q_t \le K.$6 under two budgets, $\sum_{t=0}^{T-1} q_t \le K.$7 and $\sum_{t=0}^{T-1} q_t \le K.$8 (Lin et al., 19 Sep 2025).

The reported empirical findings mirror the theory. In the strongly convex case, ZO-Align with large $\sum_{t=0}^{T-1} q_t \le K.$9 is dramatically superior, while ZO-Avg worsens as $q_t$0 increases. In the convex and non-convex cases, the same qualitative pattern holds, though the advantage of ZO-Align is more modest than in the strongly convex regime. In the stochastic case, ZO-Align with large $q_t$1 again outperforms competitors. The paper further states that even when $q_t$2, so full-subspace estimation is not possible, the largest feasible $q_t$3 for ZO-Align is still best.

These findings position ZO-Align as a projection-based multi-query estimator whose natural operating regime is the full-subspace extreme rather than an intermediate query size. More broadly, the paper reframes query allocation in zeroth-order optimization as a choice between two classical algorithmic behaviors: sequential single-query estimation through simple averaging, and subspace projection through alignment-based aggregation (Lin et al., 19 Sep 2025).