First-Order Optimization as Minimum-Time Control

Abstract: We formulate first-order optimization as a minimum-time control problem. The iterate is the state, the update, a combination of the gradients observed so far, is the control, and the points with gradient norm at most a prescribed tolerance form the target set. For a fixed objective and start, the minimum number of oracle queries needed to reach the target is a value function: it measures the complexity of the instance, not the worst case over a class. On a strongly convex quadratic, the conjugate gradient iterates emerge from the discrete Pontryagin conditions, and the value is a controllability index. Beyond the quadratic, a Hessian-generated reachable span replaces the controllability matrix, and feasibility becomes a reachability question: whether a critical point can be reached at all, and in how many steps, is read from the span. Curvature is a resource: a checkable condition certifies that an instance needs fewer steps than the controllability index of its quadratic model at the minimizer, and the gap can grow without bound with the dimension. The value is thus a benchmark for the intrinsic difficulty of each instance, against which any first-order method can be measured.

• The paper formulates per-instance first-order complexity as a minimum-time reachability problem, using gradient-tolerance sets and Pontryagin conditions without requiring Hessian evaluations.
• The paper shows that conjugate gradient is the causally optimal span method for strongly convex quadratics, reaching the solution in the number of distinct eigenvalues rather than a condition-number-dependent rate.
• The paper demonstrates that Hessian-driven curvature can reduce query counts below matched-quadratic controllability limits, including rank-based bounds for generalized linear models and a generic floor of order √(2d).

The minimum-time formulation

The paper recasts first-order optimization as a discrete-time minimum-time optimal control problem. For a continuously differentiable objective $f$ with minimizer $z^\star$ and an exact first-order oracle, the iterate is the state, and the control is the full coefficient array $\{c_{k,j}\}$ of the span-class iteration

$z_{k+1}=z_0-\sum_{j=0}^{k}c_{k,j}\,\nabla f(z_j).$

The target is the gradient-tolerance set $T_\varepsilon=\{z:\|\nabla f(z)\|\le\varepsilon\}$, chosen because it is computable from oracle outputs alone: an iterate certifies its own arrival without knowledge of $z^\star$ or $f^\star$. The value

$$V(z_0)=\min\{K:\ \exists\ \{c_{k,j}\}\ \text{with}\ z_K\in T_\varepsilon\}$$

is the per-instance oracle complexity: the fewest queries achievable over the entire span class with coefficients chosen with full knowledge of the instance. This contrasts with the classical worst-case program of Nemirovski–Yudin and Nesterov, and with performance-estimation and IQC-based synthesis, which produce a single method optimal for the hardest instance of a class. The value is explicitly a benchmark, not a runnable algorithm: its coefficients are clairvoyant, and the paper concedes that off the convex regime no transparent causal realization of $V$ is claimed.

The optimality system is derived twice. Dynamic programming on the lifted state $(z,G)$, where $z^\star$0 collects observed gradients, yields a Bellman recursion and a set-valued feedback law; the discrete Pontryagin minimum principle at each fixed horizon yields an open-loop schedule. A notable structural feature is the non-local costate recursion $z^\star$1, which arises because gradient reuse couples every stage—the plant carries the whole gradient history rather than a current state. The terminal condition $z^\star$2 with stationarity $z^\star$3 states that the final gradient is orthogonal to every previously observed gradient. The residuals are evaluated from observed gradients alone; no Hessian is formed, so the benchmark remains first-order.

Conjugate gradient as the minimum-time trajectory

On the strongly convex quadratic $z^\star$4, the paper's central identification is that the first-order span, the Krylov subspace, and the controllable subspace of the single-input pair $z^\star$5 coincide. The exact-reaching value is the grade of $z^\star$6 with respect to $z^\star$7, i.e. the controllability index:

$z^\star$8

bounded by $z^\star$9 via Cayley–Hamilton. The main theorem of this section shows that the discrete Pontryagin conditions hold along the conjugate gradient iterates alone: the terminal stationarity is exactly the Galerkin condition $\{c_{k,j}\}$0, which is necessary and sufficient for inner optimality over the affine Krylov slice, and the Hestenes–Stiefel residual orthogonality satisfies it at every horizon simultaneously. Conjugate gradient therefore emerges as the unique span-class trajectory satisfying the optimality system, and it attains the value causally, one evaluation per step. This is the paper's control-theoretic reading of the known instance-optimality of conjugate gradient on quadratics.

The quantitative gap to worst-case complexity is stark. The value on a quadratic depends only on the number $\{c_{k,j}\}$1 of distinct eigenvalues, independent of the condition number $\{c_{k,j}\}$2, while any method governed by a fixed contraction factor costs $\{c_{k,j}\}$3. On a $\{c_{k,j}\}$4, $\{c_{k,j}\}$5 instance with $\{c_{k,j}\}$6 and $\{c_{k,j}\}$7, conjugate gradient terminates at $\{c_{k,j}\}$8 queries while the triple-momentum method takes about $\{c_{k,j}\}$9. The paper is careful to note this does not contradict worst-case lower bounds, which are saturated by quadratics; it quantifies their looseness on structured instances.

Reachability beyond the quadratic

For twice-differentiable objectives, feasibility becomes a reachability question. The constant controllability matrix is replaced by a Hessian-generated reachable gradient span $z_{k+1}=z_0-\sum_{j=0}^{k}c_{k,j}\,\nabla f(z_j).$0, and the span-growth lemma shows the span grows by one dimension per step unless the current slice is $z_{k+1}=z_0-\sum_{j=0}^{k}c_{k,j}\,\nabla f(z_j).$1-invariant along it. The reachability theorem then states that $z_{k+1}=z_0-\sum_{j=0}^{k}c_{k,j}\,\nabla f(z_j).$2 is the first horizon at which the affine slice $z_{k+1}=z_0-\sum_{j=0}^{k}c_{k,j}\,\nabla f(z_j).$3 meets the critical set, that $z_{k+1}=z_0-\sum_{j=0}^{k}c_{k,j}\,\nabla f(z_j).$4 when no invariant slice disjoint from the critical set exists, and that infeasibility occurs exactly when every trajectory freezes at such a slice. For real-analytic $z_{k+1}=z_0-\sum_{j=0}^{k}c_{k,j}\,\nabla f(z_j).$5, genericity results show the reachable dimension is constant off a measure-zero set.

Two corollaries give the certificate its practical force. For generalized linear models $z_{k+1}=z_0-\sum_{j=0}^{k}c_{k,j}\,\nabla f(z_j).$6, the value is bounded by $z_{k+1}=z_0-\sum_{j=0}^{k}c_{k,j}\,\nabla f(z_j).$7, independent of the ambient dimension. The numerical demonstration is emphatic: on a rank-two logistic instance with $z_{k+1}=z_0-\sum_{j=0}^{k}c_{k,j}\,\nabla f(z_j).$8, $z_{k+1}=z_0-\sum_{j=0}^{k}c_{k,j}\,\nabla f(z_j).$9, and restricted condition number $T_\varepsilon=\{z:\|\nabla f(z)\|\le\varepsilon\}$0, the benchmark reaches the minimizer in $T_\varepsilon=\{z:\|\nabla f(z)\|\le\varepsilon\}$1 queries to tolerance $T_\varepsilon=\{z:\|\nabla f(z)\|\le\varepsilon\}$2, while triple momentum and gradient descent require roughly $T_\varepsilon=\{z:\|\nabla f(z)\|\le\varepsilon\}$3 and $T_\varepsilon=\{z:\|\nabla f(z)\|\le\varepsilon\}$4 queries on the same instance. The paper also notes the confinement is not a convexity phenomenon: $T_\varepsilon=\{z:\|\nabla f(z)\|\le\varepsilon\}$5 is nonconvex yet has $T_\varepsilon=\{z:\|\nabla f(z)\|\le\varepsilon\}$6 in every dimension.

Curvature as a resource

The most distinctive claim is that curvature can lower the per-instance count below the controllability index of the matched quadratic at the minimizer. A checkable coplanarity determinant $T_\varepsilon=\{z:\|\nabla f(z)\|\le\varepsilon\}$7, with a simple nonzero root at independent gradients, certifies $T_\varepsilon=\{z:\|\nabla f(z)\|\le\varepsilon\}$8 where the matched quadratic requires $T_\varepsilon=\{z:\|\nabla f(z)\|\le\varepsilon\}$9; the reduction persists on a neighborhood of the start. In dimension three, an explicit strongly convex real-analytic instance exhibits $z^\star$0, and the certifying first-step coefficient has sign opposite to both greedy first steps—so the minimum-time direction is one no descent method probes. In higher dimension the reduction compounds: the paper establishes a generic floor $z^\star$1, of order $z^\star$2, below the matched index $z^\star$3, with the gap $z^\star$4 growing without bound. This is a per-instance phenomenon invisible to worst-case analysis, since the classical lower bounds are saturated by the quadratics that carry the largest per-instance count. The floor is again a clairvoyant optimum; the gap a runnable method leaves to it is characterized as the price of causality.

Limitations and open questions

The paper is candid about scope. Off the quadratic, the Pontryagin conditions are necessary only: the inner slice problem is generally nonconvex, so the benchmark procedure returns an extremal and a certified upper bound, not a guaranteed optimum. The value function presumes full instance knowledge, and the paper claims no causal realization beyond the convex regime, where conjugate gradient attains it. The generic lower bound in the curvature theorem holds for a residual set of real-analytic objectives and almost every start, not universally, and no local bound read from the Hessian at $z^\star$5 can pin $z^\star$6 from below, since the shortcut can undercut the local grade. Whether the per-instance-versus-worst-case gap remains strict on smooth nonconvex classes, via the dimension-versus-rate mechanism seen on quadratics, is left open and tied to nonconvex worst-case lower bounds.

Conclusion

The paper establishes a minimum-time control formulation in which per-instance first-order complexity is a value function governed by reachability. On strongly convex quadratics, conjugate gradient emerges from the discrete Pontryagin conditions and the value is a controllability index; beyond, a Hessian-generated reachable span replaces the controllability matrix, the value is bounded by the dimension of the instance's reachable structure—rank of data for generalized linear models—and a checkable curvature condition drives the count below the matched index, compounding to a $z^\star$7 floor. The value serves as a clairvoyant benchmark against which any first-order method's per-instance query count can be measured.