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Gradient ascent seeks a maximum by stepping along the gradient of a differentiable objective. It is gradient descent applied to $-J$. Taylor expansion gives $J(x+\eta \nabla J)=J(x)+\eta \Vert \nabla J{\Vert }^2+O({\eta }^2)$, so sufficiently small positive steps improve the objective when the gradient is nonzero. Line search, projection onto admissible controls, and noise-aware gradient estimates are common practical modifications; a local stationary result need not be globally optimal.

In adaptive experimental quantum control, parametrize a reproducible pulse by a finite real vector $c$, for example $f_c(t)={\sum }_{j=1}^p{c}_j{B}_j(t)$ in a calibrated waveform basis. Prepare the same initial state for each trial, apply that pulse without feedback during the run, and estimate the objective $J(c)$ from repeated measurements. No accurate dynamical model is required if the laboratory can evaluate the objective.

One concrete algorithm is experimental finite-difference gradient ascent. For every parameter, run trials at $c\pm \delta e_j$ and estimateUpdate $c$ to an admissible projection of $c+\eta g$, test the new pulse, and reduce the step size if performance deteriorates. Averaging many shots controls statistical noise; choose $\delta$ large enough for the difference to exceed the measurement uncertainty but small enough to approximate a derivative. Repeat until no statistically significant improvement remains. Multiple starting pulses help explore distinct basins, but this does not guarantee a global optimum. Resource penalties and hardware restrictions belong in the objective or the admissible parameter set.

The feedback is across experimental runs: a measured objective chooses the next pulse. Each individual trial is open-loop control. This distinction matters because the algorithm does not require an instantaneous nondestructive state measurement on the system being controlled.

Here a dense $100000$-by-$100000$ matrix has $10^{10}$ entries, about $80$ GB even with eight-byte storage per entry. Use limited-memory BFGS on the negative of the objective, or equivalently its positive-definite approximate-metric form of gradient ascent. It stores only a fixed small number $\ell$ of vector pairs, costing $O(\ell d)$ memory rather than $O(d^2)$.

Let $G(x)={\sum }_{i=1}^{100}{e}^{e^{-\Vert x-a_i{\Vert }^2}}$ and put $c_i(x)=e^{e^{-\Vert x-a_i{\Vert }^2}}{e}^{-\Vert x-a_i{\Vert }^2}$. Direct use of the chain rule givesIf $H_n$ denotes the positive-definite inverse-Hessian matrix approximation for $-G$, the iteration iswith a line search increasing $G$ and the usual safeguarded limited-memory updates. No exact Hessian matrix is required. With fixed memory and the usual bounded-metric and line search hypotheses, expect local linear convergence near a nondegenerate strict maximum; fixed-memory limited-memory BFGS does not generally inherit the superlinear theorem for full-memory BFGS.

Unlike the preceding objective, $G$ is not generally a concave function. Thus this is a method for finding a local maximum; comparison of different converged candidates or a global search is needed to certify a global maximum for arbitrary centres. The symbolic problem supplies no assumption that would justify an unconditional global-convergence claim.^[ref]

Take two visible bipolar units, no hidden units, zero thresholds, $\beta =1$, and $E_J(x)=-J{x}_1{x}_2$. As original training data, assign probabilities $3/8$ each to $(+,+)$ and $(-,-)$ and $1/8$ each to $(+,-)$ and $(-,+)$. The data pair product averages to $1/2$. Enumeration of all four states gives two weights $e^J$ and two $e^{-J}$, hence the partition function $Z(J)=4\cosh J$ andStarting at $J=0$, the positive-minus-negative difference is $1/2-0$. A gradient ascent step of size $1/2$ gives $J=1/4$. Each aligned state now has probability $0.311230$ and each opposed state $0.188770$, bringing the model toward the data. The log-likelihood increases from $-1.386294$ to $-1.292224$. Solving ${\ell }^{\prime }(J)=0$ gives $J_{\ast }=(\log 3)/2$, which reproduces the data exactly. Since ${\ell }^{\prime \prime }(J)=-1/{\cosh }^{2}J<0$, this is the unique maximum. These are exact enumerations and their decimal evaluations, with no sampling or laboratory measurements.