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AdaGrad is a family of adaptive gradient methods that derives update geometry from accumulated gradient information. Its diagonal form uses a separate cumulative squared-gradient history for each coordinate. John Duchi, Elad Hazan and Yoram Singer's original paper develops diagonal and full-matrix versions, including constrained and regularized settings.^[ref]

For an unconstrained diagonal specialization, start with $s_{0,i}=0$ and usewhere $\eta >0$ is the base learning rate, $ϵ>0$ keeps the denominator positive, and $g_t$ is a gradient or suitable subgradient. The additive stabilizer is outside the square root in this declared variant. The accumulator belongs to the optimizer state and must be retained to reproduce later updates.

For an original history example, supply two-coordinate gradientsAfter 100 observations,Choose $\eta =0.1$ and $ϵ=1$. The effective coordinate learning rates are thenAlthough the recent supplied gradients agree, the early large first-coordinate value still strongly reduces its update scale.

Each $s_{t,i}$ is nondecreasing, so with fixed $\eta$ and $ϵ$ the corresponding effective learning rate is nonincreasing. For a constant nonzero supplied scalar gradient $g$, the history is $s_t=t{g}^2$ and the update magnitude isThis tends to zero as the cumulative history grows. Adam instead uses a decaying squared-gradient history, giving old observations a different influence. The choice between these histories is part of the algorithm, rather than merely a different initial learning rate.

An adaptive gradient method uses information from supplied gradients to adjust subsequent optimizer updates, often with a different scale for each coordinate. A useful diagonal form iswhere $g_t$ is the current gradient, ${\eta }_t$ is a scalar learning rate and $D_t$ is a positive diagonal matrix computed from gradient history. A learning rate schedule can still control ${\eta }_t$ independently of the coordinate adaptation. The broader family includes AdaGrad, Adam and AdamW.

The 2011 AdaGrad paper by John Duchi, Elad Hazan and Yoram Singer formalized adaptive geometry from accumulated gradient information, including diagonal and full-matrix constructions.^[ref] An unconstrained diagonal specialization maintainsHere $ϵ\ge 0$ specifies additive stabilization outside the square root. The accumulator is part of the optimizer state, rather than determined by the current parameters alone.

For an original example, start at ${\theta }_0=(0,0)$ and supply $g_1=(10,1)$, with $\eta =0.1$ and $ϵ=0$. Both accumulated entries are positive, so the denominators are defined. The first step givesPlain gradient descent with the same scalar learning rate would instead give $(-1,-0.1)$. If the next supplied gradient is again $(10,1)$, then $s_2=(200,2)$ and each coordinate's additional movement is $-0.1/\sqrt{2}$. These are explicit supplied-gradient calculations, rather than a convergence claim.

In this cumulative version, an accumulated squared-gradient entry never decreases, so its effective coordinate learning rate does not increase. Other adaptive gradient methods retain decaying averages instead, changing how strongly old gradients influence later updates. Adaptation therefore specifies both the state recurrence and how that state transforms the gradient; the word “adaptive” alone does not determine either.

Yoram Singer coauthored the 2011 AdaGrad paper with John Duchi and Elad Hazan. Their adaptive gradient methods use past subgradients to change the geometry of later optimizer updates.^[ref]

An unconstrained diagonal specialization uses a diagonal AdaGrad accumulatorwith learning rate $\eta >0$ and stabilization $\delta >0$ outside the square root. The accumulator is part of the optimizer state.^[ref]

For an original supplied-gradient example, let the current accumulator be $s_t=(100,1)$, the current gradient be $g_t=(1,1)$, and choose $\eta =0.1$, $\delta =1$. These accumulator values include the current gradient. The next displacement isThe identical current gradient coordinates produce different movements because their histories differ. Plain gradient descent with the same learning rate would move by $(-0.1,-0.1)$. This calculation exposes the mechanism without asserting that either update improves every possible objective function.

The example also explains why saving only the parameter vector is insufficient to reproduce a later adaptive step. Two runs at the same $x_t$, receiving the same $g_t$, can move differently if their stored accumulators differ.