Mirror Polyak and a Primal-Dual Lifting

Abstract: First-order methods typically require a specific step-size that depends on the regularity conditions of the objective function, such as the smoothness, Lipschitz continuity, or strong convexity constants. The Polyak step-size is a classical alternative for subgradient descent on convex functions that only uses knowledge of the optimal value of the objective function and automatically adapts to the above-mentioned regimes. However, many optimization problems are better described by non-Euclidean geometries and are more amenable to mirror descent. Extending this adaptivity to mirror descent is subtle. Some existing generalizations of the Polyak step-size rely on norms instead of purely on relative geometry, excluding many of the use cases of mirror descent. In this work, we revisit a variant of the Polyak step-size based on Bregman projections due to Kiwiel (1997), which we call mirror Polyak. This method is known to converge asymptotically, but its convergence rate is not known. We show that mirror Polyak enjoys guarantees similar to its Euclidean counterpart, automatically adapting to relative notions of smoothness, Lipschitz continuity, or strong convexity. We then leverage mirror Polyak to avoid having to know the optimal value in some structured optimization problems such as regularized linear and logistic regression. We propose a lifted formulation based on convex duality with optimal value exactly zero and a natural mirror map given by the problem's structure. Mirror Polyak applied to the lifted problem enjoys the same worst-case guarantees as the Polyak step-size in the original problem if we knew the optimal value.

• The paper develops mirror Polyak, a Bregman-projection method that adapts Polyak step sizes to relative smoothness and achieves Euclidean-matching rates, including linear convergence for relatively smooth strongly convex objectives.
• The paper introduces a primal-dual lifting that sets the optimal value to zero, enabling parameter-free mirror Polyak for regularized least-squares and logistic regression at the cost of roughly twice as many gradient evaluations.
• Experiments show that mirror Polyak outperforms norm-based variants and requires less problem-specific tuning than level-estimation methods, while open issues include boundary optima, a logarithmic rate gap, and stochastic extensions.

The Polyak step-size is a classical device in first-order convex optimization: given the optimal value $f_\star$, gradient descent with step-size $\alpha_t = (f(x_t) - f_\star)/\|g(x_t)\|_2^2$ converges at rates that automatically adapt to smoothness, Lipschitz continuity, or strong convexity, without requiring knowledge of the corresponding constants (2608.17252). The paper "Mirror Polyak and a Primal-Dual Lifting" by Kunstner, D'Orazio, Portella, and Taylor makes two contributions. First, it provides a non-asymptotic analysis of a norm-free extension of the Polyak step-size to mirror descent, based on Bregman projections and originally due to Kiwiel, which the authors call mirror Polyak. Second, it introduces a primal-dual lifting construction that removes the requirement of knowing $f_\star$ for structured problems such as regularized linear and logistic regression, at the cost of doubling the per-iteration gradient evaluations.

The geometric view and the definition of mirror Polyak

The paper's starting observation is that the Euclidean Polyak step-size is best understood geometrically rather than through its closed-form expression. The update $x_{t+1} = x_t - \alpha_t g(x_t)$ is the projection of $x_t$ onto the hyperplane $$\{y : f(x_t) + \langle g(x_t), y - x_t\rangle = f_\star\}$$, which separates the iterate from the optimum by convexity. Equivalently, the step-size minimizes a convex upper bound on $\tfrac{1}{2}\|x_{t+1} - x_\star\|^2$ derived from convexity of $f$. This perspective suggests the natural generalization: replace the Euclidean projection with a Bregman projection induced by a Legendre-type reference function $h$. The resulting method is

$$x_{t+1} \in \arg\min_{y} D(y, x_t) \quad \text{s.t.} \quad f(x_t) + \langle g(x_t), y - x_t\rangle = f_\star,$$

where $\alpha_t = (f(x_t) - f_\star)/\|g(x_t)\|_2^2$0 is the Bregman divergence of $\alpha_t = (f(x_t) - f_\star)/\|g(x_t)\|_2^2$1. The update is a mirror descent step $\alpha_t = (f(x_t) - f_\star)/\|g(x_t)\|_2^2$2 with step-size implicitly defined by

$\alpha_t = (f(x_t) - f_\star)/\|g(x_t)\|_2^2$3

The authors contrast this with norm-based generalizations of the Polyak step-size, which substitute the Euclidean norm with a dual norm and require $\alpha_t = (f(x_t) - f_\star)/\|g(x_t)\|_2^2$4 to be strongly convex with respect to some norm. That assumption excludes important applications where the reference function is not strongly convex in any norm, such as the Burg entropy in Poisson regression or log-determinant problems, and it excludes objectives that are not smooth in any norm — a situation that arises precisely in the dual problem for logistic regression in the lifting construction. Mirror Polyak requires only relationships between $\alpha_t = (f(x_t) - f_\star)/\|g(x_t)\|_2^2$5 and $\alpha_t = (f(x_t) - f_\star)/\|g(x_t)\|_2^2$6 (relative regularity), making it, in the authors' view, the more faithful extension. Two supporting facts are established: the implicit step-size equation can be solved to floating-point precision by one-dimensional convex root-finding at negligible cost relative to a gradient evaluation, and whenever the norm-based variant is well-defined, mirror Polyak takes a step at least as large — a property the experiments confirm leads to better practical performance.

For constrained problems, the analysis focuses on the joint Bregman projection onto the intersection of the separating hyperplane and the constraint set. Under the assumption that $\alpha_t = (f(x_t) - f_\star)/\|g(x_t)\|_2^2$7, the authors derive three identities that underpin all subsequent rates: the KKT characterization of the update, the mirror Polyak condition with $\alpha_t = (f(x_t) - f_\star)/\|g(x_t)\|_2^2$8 on the right-hand side, and a monotone contraction of the divergence to the optimum, $\alpha_t = (f(x_t) - f_\star)/\|g(x_t)\|_2^2$9. The constraint qualification assumption is not merely technical: in an appendix the authors exhibit a setting (entropy mirror maps with an optimum on the boundary of the simplex) where the projection is realized only in the limit $f_\star$0, so the assumption guards against degenerate behavior.

Convergence rates under relative regularity

The main theoretical results establish that mirror Polyak inherits, up to minor factors, the adaptive guarantees of the Euclidean Polyak step-size, with all constants interpreted relative to $f_\star$1. The rates are summarized below.

Setting	Mirror Polyak rate	Euclidean Polyak rate
$f_\star$2-smooth, convex	$f_\star$3 (average)	$f_\star$4
$f_\star$5-smooth, $f_\star$6-strongly convex	$f_\star$7 (last iterate)	$f_\star$8
$f_\star$9-Lipschitz, convex	$x_{t+1} = x_t - \alpha_t g(x_t)$0 (average)	$x_{t+1} = x_t - \alpha_t g(x_t)$1
$x_{t+1} = x_t - \alpha_t g(x_t)$2-Lipschitz, $x_{t+1} = x_t - \alpha_t g(x_t)$3-strongly convex	$x_{t+1} = x_t - \alpha_t g(x_t)$4 (best iterate)	$x_{t+1} = x_t - \alpha_t g(x_t)$5

The relatively smooth and relatively Lipschitz cases follow proof strategies analogous to the Euclidean arguments, combining relative smoothness or Lipschitzness with the divergence contraction and Jensen's inequality. The relatively smooth and strongly convex case is the technically novel one. In the Euclidean setting, smoothness implies a constant lower bound on the step-sizes ($x_{t+1} = x_t - \alpha_t g(x_t)$6), which directly yields linear convergence; this bound fails in the relative setting, and the authors verify via performance estimation (PEPit) that the mirror Polyak step-size can be arbitrarily small even for relatively smooth and strongly convex objectives — the simple divergence-based and function-value-based one-step bounds are tight. The proof instead combines two Lyapunov analyses: if step-sizes are consistently large, the Bregman divergence contracts geometrically; if they are consistently small, the function gap contracts geometrically. An averaging argument over the two contraction products yields the rate $x_{t+1} = x_t - \alpha_t g(x_t)$7, matching the Euclidean result exactly. This is a strong claim: the method achieves the same linear rate as idealized Euclidean Polyak under purely relative assumptions, without any norm-based strong convexity of the mirror map.

For relatively Lipschitz and strongly convex functions, the paper obtains an $x_{t+1} = x_t - \alpha_t g(x_t)$8 rate for the best iterate, whereas the Euclidean Polyak step-size achieves $x_{t+1} = x_t - \alpha_t g(x_t)$9. The proof is by contradiction, using a lower bound on the step-sizes implied by relative Lipschitz continuity and the fact that strong convexity plus relative Lipschitzness bound the divergence to the optimum by $x_t$0. The authors are explicit that closing this logarithmic gap would require refined inequalities: the bounds they use are satisfiable by trajectories with gaps of order $x_t$1, so a matching $x_t$2 analysis, if possible, needs new techniques. This is the one place where mirror Polyak is provably (or at least provably with current tools) weaker than its Euclidean counterpart, and it is the only deviation from the Euclidean rates besides the log-factor noted in the abstract.

Removing the need to know $x_t$3 via primal-dual lifting

The second contribution addresses the practical obstacle that $x_t$4 is rarely known. Existing remedies — level estimation, target-value algorithms, restart schemes — either lose the convergence rate of the Polyak step-size or require fixing the iteration budget in advance. The paper's alternative is a lifted formulation. Given a primal $x_t$5 and a dual $x_t$6 with $x_t$7, one optimizes $x_t$8, which is convex, has optimal value exactly zero, and satisfies $x_t$9, so guarantees on the duality gap transfer to the primal suboptimality.

The subtlety is that the lifted problem may lose Euclidean regularity. The paper's resolution is to choose the reference function $$\{y : f(x_t) + \langle g(x_t), y - x_t\rangle = f_\star\}$$0 matched to the problem structure. For a class of regularized data-fitting problems $$\{y : f(x_t) + \langle g(x_t), y - x_t\rangle = f_\star\}$$1 with Fenchel dual $$\{y : f(x_t) + \langle g(x_t), y - x_t\rangle = f_\star\}$$2, strong duality holds under a standard relative-interior qualification. When $$\{y : f(x_t) + \langle g(x_t), y - x_t\rangle = f_\star\}$$3 and $$\{y : f(x_t) + \langle g(x_t), y - x_t\rangle = f_\star\}$$4 is $$\{y : f(x_t) + \langle g(x_t), y - x_t\rangle = f_\star\}$$5-smooth, both $$\{y : f(x_t) + \langle g(x_t), y - x_t\rangle = f_\star\}$$6 and the lifted $$\{y : f(x_t) + \langle g(x_t), y - x_t\rangle = f_\star\}$$7 are $$\{y : f(x_t) + \langle g(x_t), y - x_t\rangle = f_\star\}$$8-smooth and $$\{y : f(x_t) + \langle g(x_t), y - x_t\rangle = f_\star\}$$9-strongly convex relative to $\tfrac{1}{2}\|x_{t+1} - x_\star\|^2$0 with $\tfrac{1}{2}\|x_{t+1} - x_\star\|^2$1 — the same condition number as the primal. Consequently, mirror Polyak on the lifted problem achieves the same worst-case rate as the idealized Euclidean Polyak step-size with known $\tfrac{1}{2}\|x_{t+1} - x_\star\|^2$2, at twice the per-iteration cost.

Two instances illustrate the range of applicability. For regularized least-squares, the lifted problem is smooth and strongly convex even in the Euclidean norm. For regularized logistic regression, the dual involves $\tfrac{1}{2}\|x_{t+1} - x_\star\|^2$3, the negative entropy, whose gradient diverges as any dual variable approaches 0 or 1; the dual is neither smooth nor Lipschitz in any norm. Yet the lifted problem is $\tfrac{1}{2}\|x_{t+1} - x_\star\|^2$4-smooth and $\tfrac{1}{2}\|x_{t+1} - x_\star\|^2$5-strongly convex relative to the entropy-based mirror map. This is precisely the situation where the norm-based Polyak generalizations are undefined and the Bregman-projection formulation is essential — the lifting construction and the mirror Polyak analysis are complementary rather than independent results.

The approach has acknowledged costs. It doubles the dimensionality, requires evaluating gradients of both primal and dual (feasible here because the dual is the same one used by SDCA), and is not a black-box method: it requires structural knowledge of the problem beyond a gradient oracle for $\tfrac{1}{2}\|x_{t+1} - x_\star\|^2$6.

Experiments

The empirical section compares mirror Polyak on the lifted duality gap against level-estimation methods (with and without hyperparameter tuning) and TwinPolyak, using regularized linear regression on the abalone dataset and regularized logistic regression on the a1a dataset, with the idealized Polyak step-size knowing $\tfrac{1}{2}\|x_{t+1} - x_\star\|^2$7 as a reference. The findings support the theoretical claims: mirror Polyak on the gap is fully specified by the problem structure, whereas level-estimation methods perform well only when their parameters $\tfrac{1}{2}\|x_{t+1} - x_\star\|^2$8 and $\tfrac{1}{2}\|x_{t+1} - x_\star\|^2$9 are tuned to the initial optimality gap and distance to the optimum, with no single setting working across problems. On logistic regression, the norm-based variant underperforms mirror Polyak, consistent with the step-size domination result. The authors also observe that mirror Polyak on the lifted problem need not improve the primal gap monotonically — progress may occur only on the dual side.

Limitations and open questions

Several limitations are conceded explicitly. The convergence rates require the constraint qualification $f$0, and boundary optima with entropy-type mirror maps can make the projection attainable only in the limit. The $f$1 rate in the relatively Lipschitz and strongly convex case is weaker than the Euclidean $f$2, and the authors show their current inequalities cannot close the gap. The lifting requires an efficiently evaluable dual and problem structure, so it does not apply to black-box objectives. Most significantly, stochastic extensions remain open: under interpolation, mirror Polyak converges asymptotically via existing contraction arguments, but its non-asymptotic stochastic behavior is unknown, and whether the lifting can improve adaptive stochastic schemes is left as a question.

Conclusion

The paper establishes that Kiwiel's Bregman-projection variant of the Polyak step-size, rather than norm-based generalizations, is the appropriate extension of Polyak's method to mirror descent: it matches the Euclidean adaptive rates — exactly for relative smoothness, relative smoothness plus strong convexity, and relative Lipschitz continuity, and up to a $f$3 factor in the relatively Lipschitz strongly convex case — without assuming the mirror map is strongly convex in any norm. The primal-dual lifting then converts structural knowledge of regularized regression problems into the optimal value information the step-size requires, preserving worst-case guarantees. The combination yields a parameter-free method for regularized linear and logistic regression whose only costs are doubled per-iteration gradient evaluations and the need to identify a suitable dual and mirror map.