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Adam, named for adaptive moment estimation, is an adaptive gradient method that scales a running gradient estimate by a running squared-gradient estimate. Diederik P. Kingma and Jimmy Lei Ba presented it as a first-order method for stochastic optimization. Its state makes the next optimizer update depend on gradient history as well as the current parameters.^[ref]

Starting from $m_0=v_0=0$, the basic minimization rule isHere $g_t$ is the supplied gradient, ${\eta }_t>0$ is the learning rate, $0\le {\beta }_1,{\beta }_2<1$, and $ϵ>0$. The Hadamard product, square root and division act coordinatewise. Adam bias correction supplies the two normalization factors. The accumulator $v_t$ concerns a raw second moment, rather than a centered variance.^[ref]

For an original scalar calculation, choose ${\theta }_0=0$, ${\beta }_1={\beta }_2=1/2$, ${\eta }_t=0.1$ and $ϵ=0.1$, and supply $g_1=2$, $g_2=-2$. The first step hasThe second hasEqual and opposite supplied gradients have not undone one another: the running histories give the second step a different magnitude. Plain gradient descent with the same constant learning rate would return to zero for this sequence.

The calculation specifies the recurrence, rather than a convergence guarantee for an unspecified objective function. An optimization checkpoint needs these histories and the step count to reproduce subsequent steps. AdamW adds decoupled weight decay; feeding a penalty gradient into Adam instead implements coupled L2 regularization and changes the histories.

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.

Decoupled weight decay applies a parameter-shrinkage contribution separately from the optimizer's processing of the data gradient. Write $u_t$ for the processed data-gradient direction, including any algorithm-specific optimizer state. Using the coefficient convention documented for PyTorch AdamW,The shrinkage term uses the pre-update parameter and is not inserted into the gradient moments.^[ref]

This differs from coupled L2 regularization, which processes $g_t+\lambda {\theta }_{t-1}$ as a single gradient input. Decoupled weight decay was introduced for adaptive optimization in the AdamW work; the original paper also writes decay coefficients with the learning-rate factor absorbed differently, so its coefficient must be translated before comparing equations.^[ref]

An original limiting case isolates the shrinkage. Suppose the processed direction is zero at every step and $\eta ,\lambda$ are constant. Repeated substitution givesWith $\eta =0.1$, $\lambda =0.2$ and ${\theta }_0=1$, one update gives 0.98 and 50 updates give approximately 0.36417. For varying learning rates, the corresponding factor isThus the same numerical decay coefficient can have a different accumulated effect under a different learning rate schedule. In this zero-direction model, $0<\eta \lambda <1$ gives sign-preserving shrinkage; other factors have different behaviour.

The update order is also part of the definition. For $\theta =1$, $u=2$, $\eta =0.1$ and $\lambda =0.2$, the stated formula gives $0.98-0.20=0.78$. Applying shrinkage after first taking the gradient step would instead give $(1-0.20)0.98=0.784$. Both operations may be described informally as decay plus a step, but they are different recurrences. An optimization checkpoint must retain the relevant state and settings to reproduce the chosen one.

Frank Hutter researches automated design and configuration of algorithms and machine learning systems. His work asks how experimental measurements of an algorithm's performance can guide the search for better settings, replacing some manual trial and error with an explicit computational procedure.^[ref]

Hutter and colleagues developed ParamILS, an algorithm configuration framework described in 2009. It searches settings of a target algorithm and limits unpromising evaluation runs to use the experimental budget more effectively.^[ref] A useful example is a solver with several search settings: choosing a setting changes the procedure used on the next problem instance, rather than changing that instance's answer. Performance measured on the tuning instances should therefore be checked on separate instances before claiming a general improvement.

With Ilya Loshchilov, Hutter also introduced the decoupled weight decay mechanism underlying AdamW. Their analysis showed why using coupled L2 regularization and directly shrinking parameters cease to be equivalent under general adaptive gradient scaling.^[ref] This contribution connects algorithm design to a precise update equation: an option called weight decay can describe different mathematical operations in different optimizers.

Ilya Loshchilov is a researcher in optimization and machine learning whose work with Frank Hutter includes SGDR and AdamW. His Freiburg institutional profile identifies the same author and lists the two papers; it is an alumni record, rather than evidence of a current appointment.^[ref]

Their 2017 SGDR paper introduced a learning rate schedule that decreases the rate along a cosine curve, then raises it at a warm restart while retaining the current parameters. In one cycle,Here $u$ is elapsed cycle progress and $T$ its length. With ${\eta }_{\min }=0$ and ${\eta }_{\max }=0.1$, the rates at $u=0,T/2,T$ are $0.1,0.05,0$. Raising the rate starts another exploration from the learned parameters; it does not reinitialize the model.^[ref]

Their 2019 AdamW work separates decoupled weight decay from the gradient-based update. For ordinary gradient descent, adding $\lambda \Vert \theta {\Vert }^2/2$ gives ${\theta }^{\prime }=(1-\eta \lambda )\theta -\eta g$. Coordinate-dependent scaling can act differently on the penalty contribution, so that equivalence does not extend unchanged to an adaptive gradient method.^[ref] Writing the equation makes the contribution clearer than comparing optimizer names alone.

Weight decay supplies a parameter-shrinkage contribution to an optimizer update, with a coefficient commonly denoted $\lambda$. Two implementations must be distinguished. Coupled L2 regularization adds the penalty $\lambda \Vert \theta {\Vert }^2/2$ to the objective function, replacing its data gradient $g$ by $g+\lambda \theta$. Decoupled weight decay applies parameter shrinkage separately from the optimizer's processing of the data gradient. These operations are equivalent for plain gradient descent with matched coefficients, but not in general for an adaptive gradient method.^[ref]

Use the PyTorch convention here: a decoupled update shrinks parameters by $1-\eta \lambda$, where $\eta$ is the learning rate. A source writing $1-\lambda$ instead has absorbed the step-size factor into its decay coefficient. For an unmodified gradient step,In AdamW, the decay contribution does not enter the stored first- or second-moment estimates of the gradient.^[ref]

An original fixed-scaling example exposes the difference. Let $\theta =(2,2{)}^T$, $g=(1,1{)}^T$, $\eta =0.1$, $\lambda =0.2$ and diagonal matrix $A=\mathrm{diag}(1,10)$. Applying the same scaling to the entire coupled gradient givesSeparating uniform parameter shrinkage givesThe second coordinate receives different shrinkage. This is an illustrative fixed matrix, not a full AdamW simulation. With history-dependent scaling, coupling the penalty into the gradient can also alter the optimizer state. State the update equation and coefficient convention when reporting a decay setting; the label alone does not specify either.