--- title: Projection Alignment Method (ZO-Align) url: https://www.emergentmind.com/topics/projection-alignment-method-zo-align type: topic --- # Projection Alignment Method (ZO-Align) 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 [2509.15552]. ## 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 [2509.15552]. 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 [2509.15552]. 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 [2509.15552]. ## 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 \[ x_{t+1}=x_t-Uy. \] Using \(L\)-smoothness, the paper writes \[ f(x-Uy)\le f(x)-\nabla f(x)^T(Uy)+\frac{L}{2}\|Uy\|^2. \] ZO-Align is obtained by minimizing this local quadratic upper bound over \(y\): \[ y^*=\arg\min_y \left[-\nabla f(x)^T(Uy)+\frac{L}{2}\|Uy\|^2\right]. \] The optimizer is \[ y^*=\frac{1}{L}(U^TU)^{-1}U^T\nabla f(x), \] which yields the subspace update \[ Uy^*= \frac{1}{L}U(U^TU)^{-1}U^T\nabla f(x). \] 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 [2509.15552]. 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: \[ \hat g_{\text{ALG}(x)=U(U^TU)^{-1} \begin{bmatrix} \frac{f(x+\mu u_1)-f(x)}{\mu}\ \vdots\ \frac{f(x+\mu u_q)-f(x)}{\mu} \end{bmatrix}. \] If \[ d(x)= \begin{bmatrix} \frac{f(x+\mu u_1)-f(x)}{\mu}\ \vdots\ \frac{f(x+\mu u_q)-f(x)}{\mu} \end{bmatrix}, \] then equivalently \[ \hat g_{\text{ALG}(x)=U(U^TU)^{-1}d(x). \] The optimization update is \[ x_{t+1}=x_t-\eta_t \hat g_{\text{ALG}(x_t)}. \] The key geometric object is \[ P_U = U(U^TU)^{-1}U^T, \] the orthogonal projector onto \(\mathrm{col}(U)\). In the idealized \(\mu\to 0\) regime, \[ \hat g_{\text{ALG}(x)\approx P_U \nabla f(x). \] Accordingly, ZO-Align is the orthogonal projection of the true gradient onto the sampled subspace. The paper also states \[ \mathbb{E}[\hat g_{\text{ALG}(x)] = \frac{q}{d}\nabla f(x). \] Unlike ZO-Avg, ZO-Align is therefore biased, but it is geometrically consistent within the sampled subspace [2509.15552]. 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 \(U(U^TU)^{-1}\), 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 [2509.15552]. In the **strongly convex** case, assuming \(f\) is \(L\)-smooth and \(\mu\)-strongly convex, and using step size \[ \eta_t=\frac{1}{L}, \] the paper proves \[ \mathbb{E}[f(x_T)-f(x^*)] \le \left(\prod_{t=0}^{T-1}\left(1-\frac{\mu q_t}{Ld}\right)\right)(f(x_0)-f(x^*)). \] Under a fixed budget \(K\), the optimal allocation is \[ q_t=d,\qquad T=\frac{K}{d}, \] yielding \[ \mathbb{E}[f(x_{K/d})-f(x^*)] \le \left(1-\frac{\mu}{L}\right)^{K/d}(f(x_0)-f(x^*)). \] The contraction factor improves linearly with \(q_t\), so more queries per step are better. In the **convex** case, assuming \(f\) is \(L\)-smooth and convex, with step size \[ \eta_t=\frac{1}{L}, \] the paper proves \[ \mathbb{E}[f(x_T)-f(x_*)] \le \frac{d}{2\sum_{t=0}^{T-1} q_t} \Big(L\|x_0-x_*\|^2+2(f(x_0)-f(x_*))\Big). \] Under a fixed total query budget \(K\), any allocation satisfying \[ \sum_{t=0}^{T-1} q_t = K \] gives \[ \mathbb{E}[f(x_T)-f(x_*)] \le \frac{d}{2K} \Big(L\|x_0-x_*\|^2+2(f(x_0)-f(x_*))\Big). \] Here the bound depends only on total queries \(K\), not on their distribution across iterations. In the **non-convex** case, assuming only \(L\)-smoothness and using \[ \eta_t=\frac{1}{L}, \] the paper proves \[ \frac{1}{T}\sum_{t=0}^{T-1}\mathbb{E}\big[\|\nabla f(x_t)\|^2\big] \le \frac{2Ld(f(x_0)-f^*)}{\sum_{t=0}^{T-1} q_t}. \] Under a fixed budget \(K\), \[ \frac{1}{T}\sum_{t=0}^{T-1}\mathbb{E}\big[\|\nabla f(x_t)\|^2\big] \le \frac{2Ld(f(x_0)-f^*)}{K}. \] Again, the bound is driven by total queries. In the **stochastic convex** case, assuming \(f(x)=\mathbb{E}_\xi[F(x,\xi)]\) is convex and \(L\)-smooth, with \[ \mathbb{E}_\xi[\|\nabla F(x^*,\xi)\|^2]\le \sigma^2, \] the paper uses the lemma \[ \mathbb{E}_\xi[\|\nabla F(x,\xi)\|^2]\le 4L(f(x)-f(x^*)) + 2\sigma^2. \] With diminishing step size \[ \eta_t=\frac{\eta_0}{\sqrt{t+1}},\qquad \eta_t\le \frac{1}{4L}, \] it gives a bound of the form \[ \mathbb{E}[f(\bar x_T)-f(x^*)] \le \frac{ d\|x_0-x^*\|^2 + 2\sigma^2 \sum_{t=0}^{T-1}\eta_t^2 q_t }{ \sum_{t=0}^{T-1}\eta_t q_t }. \] The stated optimal allocation under a fixed total budget is \[ q_t=d,\qquad T=\frac{K}{d}. \] 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 \(q_t=d\) 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 \[ \hat g_{\text{AVG}(x)=\frac{1}{q}\sum_{i=1}^q \frac{f(x+\mu u_i)-f(x)}{\mu}u_i. \] In the idealized \(\mu\to 0\) form, \[ \hat g_{\text{AVG}(x)\approx \frac{1}{q}UU^T\nabla f(x). \] The paper states that ZO-Avg is unbiased, \[ \mathbb{E}[\hat g_{\text{AVG}(x)] = \nabla f(x), \] with mean-squared error \[ \text{MSE}(\hat g_{\text{AVG})=\frac{d+1}{q}\|\nabla f(x)\|^2. \] For ZO-Align, the stated moments are \[ \mathbb{E}[\hat g_{\text{ALG}(x)] = \frac{q}{d}\nabla f(x), \] and \[ \text{MSE}(\hat g_{\text{ALG})=\frac{d-q}{d}\|\nabla f(x)\|^2. \] 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 \(1/q\), 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 \(q\) increases [2509.15552]. The resulting dichotomy is explicit. For ZO-Avg, the optimal allocation under budget \(K\) is \[ q_t=1\quad \text{for all }t. \] For ZO-Align, the optimal allocation is \[ q_t=d. \] 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 \[ P_U = U(U^TU)^{-1}U^T, \] if \(q=d\) and \(U\) is full rank, then \(\mathrm{col}(U)=\mathbb{R}^d\), so \[ P_U=I. \] In the idealized limit, ZO-Align then becomes \[ \hat g_{\text{ALG}(x)=\nabla f(x), \] which the paper interprets as recovery of the full gradient through a classical **full-subspace finite-difference method** [2509.15552]. The experimental study is conducted in dimension \(d=1000\) 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 \(q\) under two budgets, \(K=20{,}000\) and \(K=500\) [2509.15552]. The reported empirical findings mirror the theory. In the strongly convex case, ZO-Align with large \(q\) is dramatically superior, while ZO-Avg worsens as \(q\) 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 \(q\) again outperforms competitors. The paper further states that even when \(K